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Investigating the dark sector of the universe using cosmological observables Forestell, Lindsay 2019

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Investigating the Dark Sector of the Universe using CosmologicalObservablesbyLindsay ForestellB.Sc., The University of Alberta, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)August 2019c© Lindsay Forestell, 2019The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled:Investigating the Dark Sector of the Universe using Cosmological Observablessubmitted by Lindsay Forestell in partial fulfillment of the requirements for the degree of Doctor ofPhilosophy in Physics.Examining Committee:Kris Sigurdson, PhysicsCo-SupervisorDavid Morrissey, PhysicsCo-SupervisorMark Van Raamsdonk, PhysicsUniversity ExaminerMark Thachuk, ChemistryUniversity ExaminerHeather Logan, PhysicsExternal ExaminerAdditional Supervisory Committee Members:Gary Hinshaw, PhysicsSupervisory Committee MemberJeremy Heyl, PhysicsSupervisory Committee MemberAlison Lister, PhysicsSupervisory Committee MemberiiAbstractAlthough the Standard Model of particle physics has been a phenomenal success in modelling knownparticles and predicting new, theoretically founded particles, it is known to be incomplete. And whilethe Standard Model of cosmology has been a phenomenal success in modelling the evolution of theUniverse, it too has open questions that remain unresolved. In this thesis, we aim to address propertiesof new physics models that are being developed that aim to answer these questions. In particular, wewish to focus on and examine in detail the connection between the dark sector of the Universe and thevisible sector. In examining this connection, we may use cosmological observables to place strict limitson new theories that go beyond the Standard Model.In the first part of this thesis we will address the flow of energy from the visible sector to the hiddenvia a phenomenon known as freeze-in. Here, we explore the effects that early-time, ultraviolet energytransfer may have on the infrared, late-time evolution of a dark matter candidate. We use a simplifiedhidden-sector model to highlight the notion that operators that are typically considered early may haverelevant late-time effects.Following this, we consider the reverse energy flow, and consider how dark-sector energy injectionvia decays of electromagnetic radiation may affect the products of Big Bang Nucleosynthesis. In thissection, we focus on arbitrary light particle (< 100 MeV) decays, and identify how direct and indirectalteration of the light element abundances can be constrained using the measured values today. Directalteration is caused by photodissociation, while indirect effects are felt through changes in the radiationenergy density.Finally, we consider a full and rich dark sector, consisting of a non-Abelian SU(3) gauge force.This new gauge field presents itself as glueballs after a confining transition. We study the effects of thisconfining transition, as well as the subsequent dynamic evolution of the spectrum of glueballs produced.In the final chapter, we examine how decays to Standard Model particles via higher-dimensional, non-renormalizable operators can place stringent limits on the parameter space of this gauge force.iiiLay SummaryWhile science has made incredible discoveries throughout the ages, the Universe is still a vast, incrediblemystery. A large portion of the Universe is considered to be ‘dark matter,’ a new type of matter thatdoes not have an explanation within our current particle models, but seems to be required by our mostup-to-date cosmological models. Beyond dark matter, we have puzzles in our particle models that hintat the need for new unknown particles to create consistent explanations within all our theories.This thesis aims to explore the relationship between cosmological and particle models, and how newphysics may connect the two. We examine methods for creating dark matter, how destroying it mightaffect the world around us, and how it might behave in different physical scenarios. In doing so, wehope to shed new light on some of the problems that face the physics community today.ivPrefaceParts II, III, and IV of this thesis are based on published (and peer-reviewed) work in collaboration withothers, while parts I and V are original to this thesis.Part II is based on L. Forestell and D. E. Morrissey, Infrared Effects of Ultraviolet Operators on DarkMatter Freeze-In, [arXiv:1811.08905] [1]. I participated in constructing the models to study, and wasresponsible for the numerical calculations carried out, as well as some of the analytic calculations. DavidMorrissey, the supervisory author on this work also provided analytic estimates, as well as providingself-interaction constraints. Both authors contributed to the composition of the manuscript.Part III is based on L. Forestell, D. E. Morrissey, and G. White, Limits from BBN on Light Elec-tromagnetic Decays, JHEP, 1901, (2018), 074, [arXiv:1809.01179] [2]. The supervisory author, DavidMorrissey, provided photon spectrums, while Graham White and I provided numerical and analytic es-timates of the effects on Big Bang Nucleosynthesis. I also provided the numerical and analytic inputsfor section 4.4.1, and provided the constraint estimates shown in Figures 4.11. An expanded section onNe f f effects is included as well. David Morrissey wrote the initial draft of the manuscripts, while allauthors contributed and edited.Part IV is based on two papers: L. Forestell, D. E. Morrissey, and K. Sigurdson, Non-Abelian DarkForces and the Relic Densities of Dark Glueballs, Phys. Rev. D, 95, (2016), 015032, [arXiv:1605.08048][3] and L. Forestell, D. E. Morrissey, and K. Sigurdson, Cosmological Bounds on Non-Abelian DarkForces, Phys. Rev. D, 97, (2018), 075029, [arXiv:1710.06447] [4] I was responsible for the major-ity of numerical results throughout both manuscripts. David Morrissey and Kris Sigurdson were thesupervisory authors. We jointly composed and edited these manuscripts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxI Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Gauge Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 The Physical Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Problems with the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Cosmology and Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 The ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Thermal Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vi2.3.2 Boltzmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 The Universe Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.1 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.2 Candidates and their Production Mechanisms . . . . . . . . . . . . . . . . . . 442.5.3 Observational Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6 Beyond the Standard Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49II From the Visible to the Dark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Infrared Effects of Ultraviolet Operators and Dark Matter Freeze-In . . . . . . . . . . 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Populating the Dark Sector through UV Freeze-In . . . . . . . . . . . . . . . . . . . . 563.2.1 Transfer without the Dark Vector . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Thermalization with the Dark Vector . . . . . . . . . . . . . . . . . . . . . . . 573.3 Freeze-Out and Late Transfer in the Dark Sector . . . . . . . . . . . . . . . . . . . . . 593.3.1 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.2 Analytic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.3 Numerical Results for Freeze-Out . . . . . . . . . . . . . . . . . . . . . . . . 633.4 Dark Matter Self-Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66III From the Dark to the Visible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 Limits from BBN on Light Decays and Annihilations . . . . . . . . . . . . . . . . . . . . 704.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Development of the Electromagnetic Cascade . . . . . . . . . . . . . . . . . . . . . . 734.2.1 Computing the Electromagnetic Cascade . . . . . . . . . . . . . . . . . . . . 744.2.2 Review of the Universal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 784.2.3 Results for Photon Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.4 Results for Electron Injection . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 Effects of Electromagnetic Injection on BBN . . . . . . . . . . . . . . . . . . . . . . 814.3.1 Photodissociation of Light Elements . . . . . . . . . . . . . . . . . . . . . . . 814.3.2 BBN Constraints on Photon Injection . . . . . . . . . . . . . . . . . . . . . . 834.3.3 BBN Constraints on Electron Injection . . . . . . . . . . . . . . . . . . . . . 844.4 Other Constraints on Low Energy Decays . . . . . . . . . . . . . . . . . . . . . . . . 854.4.1 Constraints from Ne f f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.4.2 Constraints from the CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90viiIV A Complete Dark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Non-Abelian Dark Forces and the Relic Densities of Dark Glueballs . . . . . . . . . . . 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Glueball Spectrum and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2.1 Glueball Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2.2 Glueball Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Freeze-out of the Lightest Glueball . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3.1 Single-State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.2 Dynamics of the Confining Transition . . . . . . . . . . . . . . . . . . . . . . 1025.4 Freeze-Out with Multiple Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4.1 Glueball Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4.2 Relic Densities of C-Even States . . . . . . . . . . . . . . . . . . . . . . . . . 1085.4.3 Relic Densities of C-Odd States . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5 Dark Matter Scenarios and Connections to the SM . . . . . . . . . . . . . . . . . . . . 1135.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 Cosmological Bounds on Non-Abelian Dark Forces . . . . . . . . . . . . . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Glueball Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2.1 Glueball Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3 Connections to the SM and Glueball Decays . . . . . . . . . . . . . . . . . . . . . . . 1186.3.1 Dimension-8 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3.2 Dimension-6 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.3 Decay Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Glueball Densities in the Early Universe . . . . . . . . . . . . . . . . . . . . . . . . . 1256.4.1 Glueball Formation and Freeze-Out without Connectors . . . . . . . . . . . . 1256.4.2 Glueball Freeze-Out with Connectors . . . . . . . . . . . . . . . . . . . . . . 1266.4.3 Comments on Theoretical Uncertainties . . . . . . . . . . . . . . . . . . . . . 1326.5 Cosmological Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5.1 Decay Constraints from BBN . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5.2 Decay Constraints from the CMB . . . . . . . . . . . . . . . . . . . . . . . . 1336.5.3 Decay Constraints from Gamma Rays . . . . . . . . . . . . . . . . . . . . . . 1346.5.4 Application to Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139V Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 Conclusions and Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A Calculation of Transfer Rates during Freeze-In . . . . . . . . . . . . . . . . . . . . . . . 165B Thermalization Rates for Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168ixList of TablesTable 1.1 Vector gauge bosons and their gauge charges. . . . . . . . . . . . . . . . . . . . . 6Table 1.2 Fermions of the SM and their gauge charges. The top group corresponds to quarks,while the bottom group corresponds to leptons. . . . . . . . . . . . . . . . . . . . . 7Table 1.3 Higgs Boson and its gauge charges. . . . . . . . . . . . . . . . . . . . . . . . . . . 7Table 4.1 Processes included in our calculation of photodissociation effects from electromag-netic injections, as well as their threshold energies and peak cross sections. . . . . . 82Table 5.1 Masses of known stable glueballs in SU(2) [5] and SU(3) [6]. . . . . . . . . . . . 96Table 5.2 List of stable glueball states and mass ratios for SU(3), from Ref. [6]. . . . . . . . . 105xList of FiguresFigure 1.1 Feynman diagrams contributing to the one-loop corrections to the Higgs mass inthe SM. From left to right, these include fermion loops (particularly the top), Higgsself-couplings, and massive gauge boson loops. All three diagrams are quadraticallydivergent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.1 Evolution of the relativistic degrees of freedom, g∗ and g∗s as a function of temper-ature in the SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.2 Brief timeline of the evolution of the Universe. Note that inflation could in principlehappen at much lower energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.3 Generic freeze-out of a WIMP, the most commonly considered production mech-anism for dark matter. Going to lower branches is equivalent to moving to largercross-sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.4 Three fundamental methods to detect dark matter. . . . . . . . . . . . . . . . . . . 48Figure 2.5 Visualization of the connections possible between the visible and dark sectors. . . 49Figure 3.1 Flow of information considered in this chapter. The visible sector transfers energyand number density to the dark sector, which may go through further self-processingeffects that interplay with the inflow of energy. In this chapter, the visible sectorprovides an interaction via the Higgs boson, while the dark sector consists of afermionic DM candidate and a massless vector boson. . . . . . . . . . . . . . . . 53Figure 3.2 General behaviour for the yield Y = nx/s of various production mechanism for darkmatter. The red curve shows the classical freeze-out behaviour (with the dottedline following equilibrium), while the blue and green curves show ultraviolet (UV)and infrared (IR) freeze-in, respectively. Freeze-out occurs near x ∼ 20, while IRfreeze-in is dominant around x ∼ 1 and UV freeze-in occurs almost completely atxmin ∼ xRH  1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.3 Feynman diagrams for a possible UV completion of the Higgs portal. The leftdiagram is the full UV model, mediated by a scalar particle. On the right, the scalarparticle has been integrated out, and its propagator and couplings combine to createa new coupling, with approximate strength 1M . . . . . . . . . . . . . . . . . . . . 55xiFigure 3.4 Interactions involved in the dark sector. The left hand S-channel (and the equiva-lent T and U channels) will contribute to the thermalization of the hidden sector.The right hand T-channel (as well as a corresponding U-channel) will contribute tothermalization as well as eventual ψ freeze-out. . . . . . . . . . . . . . . . . . . 58Figure 3.5 Minimum consistent values of ξ (TRH) in the M–mψ plane for αx = 10−1 (left),10−2 (middle), 10−3 (right). The black line indicates where ξ (TRH)→ 1 and ourassumption of non-thermalization with the SM breaks down. . . . . . . . . . . . . 59Figure 3.6 Evolution of the relevant rates in the upper panels and the ψ density in the lowerpanels for αx = 0.1, ξ = 0.1, mψ = 104 GeV, and M= 1012 GeV (left) and 1015 GeV (right).64Figure 3.7 Enhancement of the ψ relic density due to late transfer effects relative to the valuewithout this effect, Ωψ/Ωno−trψ for αx = 0.1 (left) and 0.01 (right) and ξ = ξmin. . 65Figure 3.8 Values of mψ that give the correct relic density of ψ dark matter as a function ofM for αx = 0.1 (left) and 0.01 (right) for various fixed values of ξ . Each solidline corresponds to the correct ψ relic density for the corresponding value of ξ .The red shaded upper region is excluded due to overproduction of ψ relic densityfor any consistent value of ξ . The lower blue shaded regions indicate exclusionsfrom the effects of ψ dark matter self-interactions from the observed ellipticity ofgalactic halos, with the dark blue indicating a conservative exclusion and the lightblue showing a more aggressive one. The dotted line indicates a DM self-scatteringtransfer cross-section per mass in dwarf halos of σT/mψ = 10cm2/g. . . . . . . . 66Figure 4.1 Flow of information considered in this chapter. The dark sector now transfers energyvia decays and annihilations to the visible sector, which will go through further self-processing effects that interplay with the inflow of energy. In this chapter, the visiblesector consists of relevant BBN particles, while the dark sector consists of a singlespecies X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.2 Most important reactions for the development of the electromagnetic cascade. Toprow: high energy photons scatter off background photons or nuclei. Bottom row:Compton scattering for either high energy photons or e± (left), as well as final stateradiation (right). The fastest processes tend to be 4P (top left) and IC (bottom left). 74Figure 4.3 Photon spectrum f¯γ(E) for single photon injection with energy EX = 1000 GeV (left)and 100, GeV (right), for temperatures T = 1, 10, 100 eV. Also shown are the pre-dictions of the universal spectrum (solid) and the parametrizations of Kawasaki andMoroi given in Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.4 Photon spectrum f¯γ(E) for photon injection with EX = 100 MeV (left), EX = 30 MeV (mid-dle), and EX = 10 MeV (right), with T = 1, 10, 100 eV. Also shown are the predic-tions of the universal spectrum (solid) and the low-energy prescription of Ref. [8]. 80xiiFigure 4.5 Photon spectrum f¯γ(E) for electron plus positron (e+e−) injection with energiesEX = 100 MeV (left), 30 MeV (middle), and 10 MeV (right), with T = 1, 10, 100 eV.The solid lines show the full spectrum, while the dashed lines show the result whenFSR is not taken into account. Also shown is the universal spectrum for the sametotal injected energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.6 Limits on EX YX from BBN on the monochromatic photon decay of species X asa function of the lifetime τX for photon injection energies EX = 10 MeV (left),30 MeV (middle), and 100 MeV (right). Bounds are given for the effects on thenuclear species D, 3He, and 4He. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.7 Combined limits on EX YX as a function of τX and EX for the decay of a species Xwith lifetime τX injecting a single photon with energy EX . . . . . . . . . . . . . . 84Figure 4.8 Limits on EX YX from BBN on the monochromatic e+e− decay of species X as afunction of the lifetime τX for individual electron injection energies EX = 10 MeV (left),30 MeV (middle), and 100 MeV (right). Bounds are given for the effects on the nu-clear species D, 3He, and 4He, and contributions to the electromagnetic cascadesfrom FSR are included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.9 Same as Fig. 4.8 but without FSR effects. . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.10 Combined limits on EX YX as a function of τX and EX for the decay of a species Xwith lifetime τX injecting an electron-positron pair each with energy EX , with FSReffects included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.11 Effects of low energy decays on Ne f f for decays to neutrinos (top, blue), and electro-magnetic species (bottom, red). Shown are effects for various particle lifetimes. Thecentral grey band corresponds to the conservative estimate given in Eq. (4.42), inagreement with Planck and BBN estimates[9, 10]. The enlarged green region showsthe extra phase space that could be allowed if a sterile neutrino with ∆Ne f f = 1 isincluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 4.12 Other bounds on electromagnetic decays in the early universe as a function of thelifetime τX and the total electromagnetic injection ∆E YX relative to limits derivedfrom BBN. In both panels, the red line shows ∆Ne f f = 0.31± 0.16, while thesolid (dotted) blue lines show the current and projected CMB frequency boundsfrom COBE/FIRAS (PIXIE). The left panel also indicates the limits derived fromBBN for photon injection with energy EX = 10, 30, 100 MeV with green dotted,dashed, and solid lines. The right panel shows the corresponding BBN bounds frommonochromatic e+e− injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 5.1 Flow of information considered in this chapter. We start our inspection of a non-Abelian dark force with a completely hidden sector, entirely isolated from the vis-ible. However, even though isolated, the dark sector will still have a complexset of interactions, with many stable glueball states that can be involved in self-interactions, annihilations, and so on. . . . . . . . . . . . . . . . . . . . . . . . . 94xiiiFigure 5.2 Feynman diagrams for the scalar 0++ glueball state, which include a typical self-interaction (left), as well as a number changing interaction (right). . . . . . . . . . 98Figure 5.3 Temperature evolution of the hidden sector while the 0++ is freezing-out. Shown arethe inverse of the dark and visible sector temperatures (xx(x) = mx/Tx(T )). Beforefreeze-out, the temperature drops (xx rises) much slower than the visible sector, asthe glueballs reheat themselves through the 3→ 2 process (dashed red line). Afterfreeze-out, the dark temperature scales as Tx ∝ a−2 (solid red line). The dashedblack line shows the comparison with how the temperatures would evolve if therewas no reheating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.4 Mass-weighted relic yields in the single-state simplified model discussed in the textwith N = 3 as a function of the mass Λx = mx and entropy ratio R. The solidwhite line indicates where the glueball density saturates the observed dark matterabundance Ωxh2 = 0.1186 [9]. The dark masked region at the lower right indicateswhere freeze-out occurs for x f ox < 5 and our freeze-out calculation is not applicabledue to the unknown dynamics of the confining phase transition. . . . . . . . . . . 102Figure 5.5 Mass-weighted relic yields in the single-state simplified model with the initial den-sity set by f = Yx(xcx)/Yx(xcx,µx = 0) at Tx = Λx/5 with R = 10−9 and N = 3. . . . 104Figure 5.6 Mass-weighted relic yields of the four lightest C-even glueballs in SU(3), JPC =0++, 2++, 0−+, 2−+, as a function of the dark glueball temperature variable xx =mx/Tx computed using the simplified reaction network discussed in the text. Thesolid lines show the yields derived from the reaction network while the dashed linesindicate the yields expected if the states were to continue following equilibrium withµi = 0. Top left: (Λx/GeV, R) = (1, 10−9). Top right: (Λx/GeV, R) = (105, 10−9).Bottom left: (Λx/GeV, R) = (1, 10−3). Bottom right: (Λx/GeV, R) = (105, 10−3). 109Figure 5.7 Mass-weighted relic yields of the 0−+ dark glueball in SU(3) as functions of Λx =mx and R, computed using the simplified C-even reaction network discussed in thetext. For reference, we also indicate the yield corresponding to the observed darkmatter density. Note that the yield of the 0++ state is much larger. . . . . . . . . . 110Figure 5.8 Mass-weighted relic yields of the 1+− dark glueball in SU(3) as a function ofΛx = mx and R, computed in the simplified two-state network discussed in the text.For reference, we also indicate the yield corresponding to the observed dark matterdensity. Note that the yield of the 0++ state is much larger. . . . . . . . . . . . . 112Figure 6.1 Flow of information considered in this chapter. We now have a fully realized darksector, with complex interactions. The dark sector may also transfer energy to thevisible sector via decays, while the visible sector may influence the production ofdark glueballs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116xivFigure 6.2 Diagrams that contribute to the effective Lagrangian of Eqs. (6.9) and (6.10). Effec-tive operators are created via integration of loops of the heavy mediator fermions[11].The left hand diagram will loosely correspond to decays of 0++ (Eq. (6.9)), whilethe right hand diagram will also contain terms that describe 1+− decays (Eq. (6.10)). 119Figure 6.3 Decay lifetimes τ = 1/Γ of the 0++ (left) and 1+− (right) glueball states due to thedimension-8 operators as a function of M and m0 for χi = χY = 1 and Gx = SU(3).The masked regions at the upper left show where m0 > M/10 and our treatment interms of effective operators breaks down, while the white dotted, solid, and dashedlines indicate reference lifetimes of τ = 0.1s, 5×1017 s, 1026 s. . . . . . . . . . . 122Figure 6.4 Diagram that contributes to the effective Lagrangian of Eq. (6.22). Effective opera-tors are created via integration of loops of the heavy mediator fermions[12]. . . . 122Figure 6.5 Decay lifetime τ = 1/Γ of the 0++ glueball due to the combined dimension-6 anddimension-8 operators as a function of M and m0 for χi = χY = 1, ye f f = 1, andGx = SU(3). The masked region at the upper left shows where m0 > M/10 and ourtreatment in terms of effective operators breaks down, while the dotted, solid, anddashed white lines indicate lifetimes of τ = 0.1s, 5×1017 s, 1026 s. . . . . . . . . 124Figure 6.6 Mass-weighted relic yields of the 0++ (left) and 1+− (right) glueballs in the m0–Rplane in the absence of connectors for Gx = SU(3). The solid white lines in eachpanel indicate where the relic density saturates the observed dark matter abundance.The dark masked region at the lower right of both panels shows where 0++ freeze-out occurs for x f ox < 5 and our freeze-out calculation is not applicable due to theunknown dynamics of the confining phase transition. . . . . . . . . . . . . . . . . 126Figure 6.7 Values of the minimal entropy ratio Rmin in the M–m0 plane for energy transfer viadimension-8 (left) and dimension-6 (right) operators for Gx = SU(3). The blackshaded region at the upper left indicates where our treatment in terms of effectiveoperators breaks down. The diagonal black dotted, solid, and dashed lines showreference values of Rmin = 10−3, 10−6, 10−9. In the cyan region in the right panel,thermalization between the visible and dark sectors is maintained at least until con-finement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 6.8 Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 1with dominant dimension-8 operators and broken Cx. The upper two panels haveR = Rmin, Rmax, while the lower three panels have fixed R = 10−9, 10−6, 10−3. Thegrey shaded region in each panel indicates where our theoretical assumptions fail,while R < Rmin to the left of the dashed line. . . . . . . . . . . . . . . . . . . . . 135Figure 6.9 Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 2with dominant dimension-8 operators and conserved Cx. The upper two panels haveR = Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. Thegrey shaded region indicates where our theoretical assumptions fail, while to the leftof the dashed line we find R < Rmin. . . . . . . . . . . . . . . . . . . . . . . . . . 136xvFigure 6.10 Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 3with dominant dimension-6 operators and broken Cx. The upper two panels haveR = Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. Theblack shaded region indicates where our theoretical assumptions fail, while to theleft of the dashed line we find R < Rmin. . . . . . . . . . . . . . . . . . . . . . . . 137Figure 6.11 Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 4with dominant dimension-8 operators and conserved Cx. The upper two panels haveR = Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. Theblack shaded region indicates where our theoretical assumptions fail, while to theleft of the dashed line we find R < Rmin. . . . . . . . . . . . . . . . . . . . . . . . 138xviGlossaryACT Atacama Cosmology TelescopeBBN Big Bang NucleosynthesisBSM Beyond the Standard ModelCMB Cosmic Microwave BackgroundCKM Cabibbo-Kobayashi-MaskawaCP charge-conjugation parityCS Compton scatteringDLA damped Lyman-αDM dark matterEM electromagneticEWSB electroweak symmetry breakingEW electroweakEFT effective field theoryFI freeze-inFIMP feebly-interacting massive particleFRW Friedmann-Robertson-WalkerFSR final state radiationGB glueballGUT Grand Unified TheoryIC inverse ComptonxviiIGM inter-galactic mediumIR infraredΛCDM Λ cold dark matterLHC Large Hadron ColliderLSND Liquid Scintillator Neutrino DetectorLSP lightest super-partnerNGB Nambu-Goldstone BosonPCN pair creation on nucleiPMNS Pontecorvo-Maki-Nakagawa-SakataPP photon photon scatteringQCD Quantum ChromodynamicsQED Quantum ElectrodynamicsQFT quantum field theoryRG renormalization groupSIMP self-interacting massive particleSM Standard ModelSPT South Pole TelescopeSSB spontaneous symmetry breakingSUSY supersymmetryUV ultravioletvev vacuum expectation valueWIMP weakly-interacting massive particle4P photon photon pair productionxviiiAcknowledgmentsThis thesis would not have been possible without the support of a large group of mentors, collaborators,friends, family, and financial supporters. Here I wish to acknowledge and thank those who made this allpossible.First I would like to thank the Natural Science and Engineering Research Council of Canada, aswell as the University of British Columbia and TRIUMF for the financial support, without which noneof this would have been possible.Next, I would like to thank everyone who I was able to collaborate with to write interesting papers,troubleshoot various technical malfunctions, and listen to and improve various presentations over theyears. In particular, the theory group at TRIUMF has been a great support. I am grateful to everyone whohas come and gone over the years, including Mirko Miorelli the ever patient office-mate and GrahamWhite for the helpful discussions.I am also grateful to my doctoral committee, Jeremy Heyl, Alison Lister, Gary Hinshaw, Kris Sig-urdson, and David Morrissey, for the insightful and thought-provoking questions and feedback over thepast few years.Most importantly, I need to thank my supervisors Kris Sigurdson and David Morrissey for guidingme over the last five years. To Kris, I thank you for stimulating discussions and always pushing thebounds on the vast world of theoretical physics. To David, I thank you for always having an open doorand supporting, encouraging words whenever I felt uncertain along this path.Finally, to my friends and family, especially my parents. I could not have done this without you.You have kept me rooted and sane when the Universe threatened to steal me away for good! I would notbe here without the incredible support, helping hands, listening ears, and encouraging words from all ofyou.Thank you, everyone.xixTo Emily, Bentley, Lachlan, Jack, and Everett. You guys have inspired me to see the world in an entirelynew, open and curious light. Thank you for being you, and for being a joy in my life.To Aron, thank you for being there every day, good and bad, always encouraging me in my drive tocomplete this goal. Your support and love have meant the world to me, and have made this entirelypossible.xxPart IIntroduction1Chapter 1The Standard Model1.1 IntroductionThe Standard Model (SM) of particle physics is one of the most successful physical theories of modernphysics, successfully explaining the strong, weak and electromagnetic forces. It has been tested, suc-cessfully predicting new particles such as the Higgs Boson and the massive electroweak gauge bosons[13–15]. Although we have successfully built, used and tested this powerful model, there are still somekey problems that we wish to address. Doing so inevitably require introducing new physics. However,well motivated physics models need to do more than just answer the question they were proposed tosolve. They must also pass tests in the rest of the physics realms as well. In particular, new models mustnot contradict currently known and well understood effects. In this thesis, we aim to study exactly hownew physics may interact with old, from a variety of angles. In order to do this, we must first understandhow our successful models work.In this chapter we will build up the Standard Model, focusing on three key pieces that build thefundamental Lagrangian:L =LGauge+LHiggs+LYukawa (1.1)where each piece of this Lagrangian will be explained in detail in this chapter. Following this, wewill address some problems that have arisen that the SM cannot solve, and give common solutions thatincorporate new physics. Reviews of quantum field theory (QFT) and the SM can be found in Refs.[16–20].Throughout this thesis, unless stated otherwise we will use natural units, where h¯ = c = kB = 1, andtypically use units of GeV as our natural energy unit.21.2 The Standard Model1.2.1 Gauge SymmetriesIn this section, we wish to explicitly defineLgauge, the portion of the Standard Model that describes theinteractions between fermions via gauge forces. This will rely heavily on group theory and Lie algebras,for which good references can be found at [21, 22], while gauge field theories can be found in Ref. [23].All of the known particles can be well defined by the SM, a theoretical description of the Strong,Weak, and Electromagnetic forces. These forces define the interactions that exist between differentparticles, and are explicitly identified in nature by a gauge symmetry of the form:SU(3)C×SU(2)L×U(1)Y (1.2)The first term, SU(3)C, corresponds to Quantum Chromodynamics (QCD), the fundamental descriptionof the strong force. This covers the interactions of quarks and gluons (or bound states of quarks andgluons, hadrons, at low energy). The next two terms, SU(2)L×U(1)Y , together form electroweak (EW)theory, which as we shall see can be broken down to the Weak force and the electromagnetic (EM) force(or in QFT terms, the EM force is often described by Quantum Electrodynamics (QED)) separately afterthe symmetry is explicitly broken. The EM force will independently preserve a new U(1)EM symmetry,with the group charge corresponding to the physical charge of particles that we see (eg. Q = -1 forelectrons). The Weak force remains responsible for events such as β decay of a nucleus, and is the onlyexplicit connection between neutrinos and other particles in the SM.In the modern view of particle physics, we take as our starting point that any particles that arecharged under these gauge groups must be invariant under local gauge transformations of the form:Ur = eiαatar (1.3)where αa = αa(x) has local spatial dependence, and tar correspond to the Hermitian generators of theLie algebra in question. Some examples of such generators include the Pauli matrices for SU(2), wherethey are typically normalized as tar =12σa:σ1 =(0 11 0)σ2 =(0 −ii 0)σ3 =(1 00 −1)(1.4)3or the Gell-Mann matrices for SU(3), with normalization tar =12λa:λ1 = 0 1 01 0 00 0 0 λ2 = 0 −i 0i 0 00 0 0 λ3 = 1 0 00 −1 00 0 0λ4 = 0 0 10 0 01 0 0 λ5 = 0 0 −i0 0 0i 0 0λ6 = 0 0 00 0 10 1 0 λ7 = 0 0 00 0 −i0 i 0 λ8 = 1√3 1 0 00 1 00 0 −2(1.5)Under these transformations, fermions will transform as ψ →Urψ and ψ¯ → ψ¯U†r . Here we use thestandard definitions, ψ¯ = iσ2ψ∗ and σ2 is the Pauli matrix. Note that here we refer to ψ (ψ¯) as left(right)-handed Weyl spinors [24, 25]. To get the more familiar 4-component Dirac spinors, we requiretwo 2-component Weyl spinors:ΨD =(ψχ¯), (1.6)while the Pauli matrices (including σ0 = I2) can be promoted to the more familiar gamma matricesusing:γµ =(0 σ µσ¯µ 0)(1.7)where σ¯ = (I2,−~σ) and for the special case of Majorana fermions, χ¯ = ψ¯ . Although we are typicallyonly interested in the 4-component Dirac notation for the spinors, it is instructive to use the chiral Weyl-spinors here to explicitly identify the left and right-handed dependencies within the SM. If we now wishto apply this transformation to a canonical kinetic term for a fermion, we will immediately run intoproblems:ψ†iσ¯µ∂µψ → ψ†iσ¯µ∂µψ+ψ†iσ¯ µU†r (∂µUr)ψ (1.8)which is not invariant under the transformation. However, we can remedy this by introducing a vectorfield, Aµ , and promote it to a matrix via the specific representation we are interested in: Arµ = Aaµtar . Ifwe allow Aµ to transform under the adjoint representation of the group:Arµ →UrArµU†r +1igUr(∂µU†r ) (1.9)4We can then show that:(igArµ +∂µ)ψ →Ur(igArµ +∂µ)ψ (1.10)So, if we promote our derivatives to covariant derivatives,Dµ = igAaµtar +∂µ (1.11)then the kinetic terms will remain invariant under the gauge transformation:ψ†iσ¯µDµψ → ψ†iσ¯µDµψ (1.12)and so we see that adding an interaction with a vector field, and simultaneously transforming bothfields will preserve the symmetry. Thus, gauge vector interactions with any particles charged under thatspecific gauge are necessary to preserve our gauge symmetries.Now that we have introduced the gauge vector field, we must also define its own kinetic term. To begauge invariant, this can be done using:Fµν = ∂µAν −∂νAµ −g f abcAbµAcν (1.13)where f abc are the structure constants for the group representation. Finally, we write the kinetic term as:L ⊃−14FaµνFa µν (1.14)Here it is important to note that for Abelian gauge groups, such as U(1), the structure constants are allzero, and so there will be no gauge boson self-interactions. However, non-Abelian gauge groups willhave non-zero structure constants, leading to the possibility of interesting self-interacting effects. Theseself-interactions for a particular non-Abelian extension to the SM are explored in much more detail inPart IV.In the Standard Model, we thus have to introduce three sets of gauge vector bosons, one for eachgauge symmetry. For SU(3)C, this will be the gluons, Gaµ , where a runs from 1 to 8.1 We also have3 Weak bosons, W aµ for the SU(2)L group, and finally Bµ for the U(1)Y hypercharge group. To beconcrete, we can explicitly express all the vector gauge bosons in terms of their gauge representations,as shown in Table 1.1. It is important to note that all of these vectors must necessarily be massless, as avector gauge term (∼−m2AµAµ ) would violate gauge-invariance. We will see where the masses for theobserved vector bosons come from in the next section.Of course, these vectors need something to interact with: these are the fermions of the StandardModel. They will transform as ψi j→UCULUYψi j, where i and j represent the SU(3)×SU(2) represen-tations for that particular state. The fermions can be broken up into two subgroups: quarks, which have1Note that because we only experimentally see the strong force at short ranges, we expect there to be no color neutral gluonwhich would mediate a long range force, thus we postulate that the strong force is SU(3) instead of U(3), which would have 9generators.5Vector Gauge Boson SU(3)C SU(2)L U(1)YGaµ 8 1 0W aµ 1 3 0Bµ 1 1 0Table 1.1: Vector gauge bosons and their gauge charges.SU(3)C charge, and leptons, which only have SU(2)L×U(1)Y . Both quarks and leptons come in threegenerations, with each copy being identical except for the masses of the quarks and leptons involved.Like the gauge bosons, we can explicitly write out the gauge charges of every fermion in the StandardModel, which we show in Table 1.2.Note again that here we have used 2-component Weyl spinors to represent the fermions. In reality,four-component Dirac spinors represent the physical fields, which are simply the sum of the two Weylspinors: ψD =(ψLψR). However, it is enlightening to write it this way as it emphasizes the chiral nature ofthe EW gauge force. Only the left-handed fermions are charged under SU(2)L, while the right-handedfermions are singlets, causing the EW force to be maximally parity violating. Because of this, wecannot explicitly write out a mass term, and these fermions must necessarily be massless to preservegauge invariance. Not only that, but in the SM there is no right-handed neutrino at all, and as we shallsee, this means that in the SM, the neutrinos should necessarily all have zero mass. This is not the case,and hints at problems with our model of physics. Nonetheless, armed with these gauge charges, we cannow write down the gauge portion of the SM Lagrangian:Lgauge = −14 Ga µνGaµν − 14W a µνW aµν − 14 BµνBµν +∑ψ ψ¯iσ¯µDµψ (1.15)where the vector kinetic terms are defined in Eq. (1.13), ψ ∈ {QL,uR,dR,LL,eR} and covariant deriva-tives are defined in Eq. (1.11) (where the correct representation must be used for each fermion, notingthat tar =0 under the trivial representation). For the SM, the covariant derivatives are explicitly given by:Dµ = ∂µ + igstarcGaµ + igtprLWpµ + ig′Y Bµ (1.16)where gs, g, and g′ are the Strong, Weak, and hypercharge (Y) couplings. tarc corresponds to the gen-erators of the fundamental representation of SU(3), which are given by the Gell-Mann matrices in Eq.(1.5), while tarL are the generators of SU(2), given by the Pauli matrices in Eq. (1.4). Y represents thehypercharge, the charge of the field under U(1)Y .So we see that, if the SM consisted solely of fermions charged under the SM gauge symmetries, andtheir corresponding vector gauge bosons, we could entirely fix the Lagrangian, but we would necessarilyrequire massless particles. This is not the case, and so we must introduce a new mechanism to providemass for the particles. This will be discussed in the next section.6Fermion SU(3)C SU(2)L U(1)YQL =(uLdL)3 2 +1/6uR 3 1 +2/3dR 3 1 −1/3LL =(νLeL)1 2 −1/2eR 1 1 −1Table 1.2: Fermions of the SM and their gauge charges. The top group corresponds to quarks,while the bottom group corresponds to leptons.Higgs SU(3)C SU(2)L U(1)YH 1 2 +1/2Table 1.3: Higgs Boson and its gauge charges.1.2.2 Spontaneous Symmetry BreakingUnder the SM symmetries SU(3)C×SU(2)L×U(1)Y , all of the vector gauge bosons must necessarilybe massless. Furthermore, the chiral nature of the EW symmetry, SU(2)L×U(1)Y , necessarily requiresthat the SM fermions must also be massless. However, this does not agree with what we see in nature,and we need some new mechanism to match observations. In particular, the W± and Z bosons of theWeak force have masses of ∼80 GeV and 91 GeV, respectively, while every fermion mass has beenmeasured to be non-zero as well.2 Although these two mechanisms are distinct, it turns out that we canreconcile both cases with a single new field. This is the Englert-Brout-Higgs-Guralnik-Hagen-Kibblemechanism, which gives rise to the more commonly named Higgs Boson [27–29] through spontaneoussymmetry breaking (SSB). The next two pieces of the SM Lagrangian are thus LHiggs and LYukawa,which provide the Higgs connection to both the weak vector bosons as well as (most of) the fermions.Here we will show that these are both necessary to give mass to the rest of the SM.To provide masses, we need to spontaneously break the EW gauge symmetries. We do this byintroducing a new complex scalar Higgs field, H, that is a doublet under Weak isospin:H =(φ+φ 0)(1.17)This will be the final field of the SM, and we write down its gauge representation in Table 1.3.Because this field is charged under SU(2)L×U(1)Y , it must include a covariant derivative withinthe kinetic term:LHiggs ⊃ |DµH|2 =∣∣∣∣(∂µ + igσ p2 W pµ + ig′ 12Bµ)H∣∣∣∣2 (1.18)2For up-to-date measurements of all of these masses, you can see, for example, Ref. [26]7We also include here the symmetry breaking scalar potential:LHiggs ⊃−(−µ2|H|2+ λ22|H|4)(1.19)This potential has a minima that is not at zero, but rather at v/√2 = µ/λ (where v = 246 GeV has beennormalized to match standard convention). We can expand around this minima to define excitations ofthe Higgs field out of this minima. However, because the Higgs is a complex scalar, we can rotate thisν by any phase and still be at the minima. Choosing a specific minima will force us to spontaneouslybreak our symmetries. To be concrete, we choose a gauge to work in (which can always be done byapplying an SU(2)L×U(1)Y rotation), and define the Higgs vacuum state to be:〈H〉=(0v/√2)(1.20)where v is chosen by construction to be real and positive. Working in this gauge, and expanding H asH =(0(v+h(x))/√2)(1.21)we can show that this vacuum state is trivially invariant under SU(3)C. Because it is an eigenstate ofweak isospin, it is also invariant under an abelian subgroup of SU(2)L×U(1)Y =U(1)EM, which hascharge Q = t3 +Y , where t3 is the third component of the weak isospin. We know that for each groupgenerator that does not leave the vacuum state invariant, we expect a Nambu-Goldstone Boson (NGB)[30–32]. In EW theory, there are 3 such generators. These modes would remain massless. However,what actually happens is that the NGB modes of H become the longitudinal polarizations of the now-massive gauge bosons. We can look at this by explicitly counting degrees of freedom. The original Higgsfield had 4 degrees of freedom (being a complex doublet). After symmetry breaking, we have chosena gauge in which it will have 1 degree of freedom, h. The other 3 degrees of freedom are ‘eaten’ by 3of the weak isospin bosons, giving them mass (and going from 2 massless, independent polarizations to3 massive, independent polarizations), leaving the fourth massless. These will, after diagonalization ofthe mass matrices, become the W± and Z modes of the weak theory, while the massless state becomesthe photon. These will be explicitly shown in the next section.As we mentioned above, the Higgs is again used to generate the masses of the fermions. This is alsodone through the broken symmetry of the Higgs field, but arises due to Yukawa connections betweenthe Higgs and the fermions, rather than the covariant derivative of the vector bosons. In the unbrokenphase, it is possible to write down gauge invariant operators involving the Higgs field and the fermionsas Yukawa interactions:LYukawa =−YuQ¯LH˜uR−YdQ¯LHdR−YeL¯LHeR+h.c. (1.22)where H˜ = iσ2H∗ and Yi are general 3x3 matrices that allow the generations of different quarks andleptons to mix. These are the most general interactions we can write down that obey the combined EW8gauge group SU(2)L×U(1)Y . After electroweak symmetry breaking (EWSB), H will be replaced byEq. (1.21) in the unitary gauge, which leads to terms of the form ∼ ν f¯L fR, which correspond to massterms for the fermions. Note that there is no equivalent −Yν L¯LH˜νR term, as there are no right-handedneutrinos in the Standard Model. This is ultimately the reason that neutrinos do not have mass in theSM.Now that we have all of the pieces required for the SM as laid out in Eq. (1.1), let us put this alltogether (in the broken phase) to see exactly what the physical world (nearly3) looks like today.1.2.3 The Physical Standard ModelLet us now write down the physically observed fields of the Standard Models, and the values of theircouplings and masses. An extensive review of this, as well as up-to-date values for all masses andcouplings quoted here, can be found in Ref. [26], and references therein. We will begin with determiningthe appropriate combinations of fields that have physical masses, before commenting on the effects ofconfinement on fields that have SU(3) charges.Fundamental FieldsWe begin with the simplest portion, the mass of the Higgs boson. If we expand around our chosenvacuum state, we find that the scalar potential in Eq. (1.19) reduces to:LHiggs ∼−v2λ 2h2+ (self couplings) (1.23)and so we identify the mass of the Higgs boson asmH =√2λv . (1.24)This has been experimentally measured to be mH = 125.18±0.16 GeV. The vacuum expectation value(vev) (v) for the Higgs is constrained to be v ∼ 246 GeV, which implies that the self-coupling value, λmust be ∼0.13.Next, we consider the gauge bosons in the broken phase. Although Eq. (1.18) will also containHiggs-gauge boson interactions in the broken phase, we again focus on the v2 terms that will contributeto the mass matrix. Writing this out, we find:|DµH|2 = 12g2v24(W 1 2µ +W2 2µ)+12v24(g′Bµ −gW 3µ)2+ (Higgs-gauge boson couplings) (1.25)3As we will show in a later section, there are problems with the Standard Model that still need to be explained. We havehinted at this already with the fact that neutrinos do not have mass in the SM, and yet they do in reality.9which gives us the mass matrixM =v24g2 0 0 00 g2 0 00 0 g2 −gg′0 0 −gg′ g′2 (1.26)Diagonalizing this, and re-arranging the W 1,2 bosons into two with orthogonal states that have charge±1 in the U(1)EM gauge, we find:W±µ =1√2(W 1µ ∓ iW 2µ ) (1.27)with massM2W =14g2v2 . (1.28)This has been measured to be 80.379±0.012 GeV. The W 3 and B bosons mix according to:(ZµAµ)=(cW −sWsW cW)(W 3µBµ)(1.29)where cW (sW ) are the cosine (sine) of the Weinberg angle (or weak mixing angle, θW ), defined bysW = sin(θW ) =g′√g2+g′2(1.30)Measurements of the Weinberg angle give sW = 0.23120± 0.00015. In this basis, the mass matrix isnow diagonal, with a massless photon (Aµ ), and the Z boson has mass:M2Z =(g2+g′22)v2 (1.31)The Z mass has been measured as MZ = 91.1876± 0.0021 GeV. Thus, the symmetry breaking energyscale set by v also sets the scale for the masses of the weak bosons. Because of this, we consider energiesO(100 GeV) to be at the weak scale.We complete our physical SM model by looking at the Yukawa terms in Eq. (1.22). When H isreplaced by its vev, we find:LYukawa =− v√2Yuu¯LuR− v√2Yd d¯LdR− v√2Yee¯LeR+h.c. (1.32)To get the physical mass eigenstates, we need to diagonalize the mixing matrices, Yu,d,e. This can be10done by choosing a unitary transform for each left or right handed fermion, V fL and VfR , such thatM fdiag =v√2V fL YfVf †R , (1.33)and sending each fermion to:fL→V fL fL, fR→V fR fR . (1.34)When we do this, we are choosing a new basis that corresponds to the physical mass eigenstates ofthe quarks and leptons, where the coupling between the fermion and the Higgs is explicitly realized as:y f =√2m fv(1.35)This is of course allowed, and if we apply these transformations to the rest of the SM Lagrangian, it turnsout that nearly all of the gauge interactions will be invariant under these rotations as well. However,the charged-current interactions of the quarks via the W± gauge-bosons will not be invariant. This isequivalent to the Yukawa couplings breaking an (SU(3))3 global symmetry under rotations of QL, uR,and dR between the different generations. Thus, by choosing to diagonalize the masses of the up anddown quarks, we are forced to include a mixing matrix elsewhere. This shows up in the charged currentinteraction, which mixes up and down quarks as:LCC =− g√2(u¯L, c¯L, t¯L)σ¯µW±µ VCKM dLsLbL+h.c. (1.36)where u, c, t correspond to the three ‘up’ generations of quarks, and d, s, b correspond to the ‘down’generations. VCKM = Vu†L VdL is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [33, 34]. This is aunitary matrix, characterized by three mixing angles and one charge-conjugation parity (CP) violatingphase.4 Note that, because we have no neutrino masses, we do not have this difficulty in the leptonsector, as we can always choose a neutrino mixing matrix V νL =VeL .ConfinementAlthough we can now define all the fundamental particles, and their masses, in the Standard Model, wecannot quite explain the particles that we physically observe at lower energies. Mesons and baryons, forexample, are confined states of 2 and 3 quarks, respectively. These are the physical fields that we observeat low energies, but they do not correspond directly to fundamental fields in the Lagrangian. We wishto briefly describe this phenomena here in the context of QCD, but we will make use of confinement inPart IV.4It is interesting to note that the CP violating phase only arises due to the fact that there are 3 generations of quarks. Withonly 2, we can always define a real unitary matrix via a clever choice of field redefinitions that will rotate the CP violatingphase away.11To understand why quarks and gluons exist in bound states, we must first look at how the strongcoupling, gs, depends on the energy that we are concerned with. This can be done by solving therenormalization group (RG) equations that link the coupling at one momentum scale, µ , to another. Fora non-Abelian gauge theory, the RG equation becomes [16]:dgdt:= β (t) =− b(4pi)3g3 (1.37)where g is the running coupling, t = ln(µ/µ0), and b is given by:b =113C2(A)−∑r, f23T2(r)−∑r′,c13T2(r′) (1.38)Where C2(A) is the Casimir of the adjoint (N for SU(N)) and T2(r) is the trace invariant of the rep-resentation (1/2 for the fundamental representation). The first (second) sum over r (r′) runs over all2-component fermions (complex scalars) in the theory with masses below the scale of interest. ForSU(3)C, which has 6 fermion flavours (12 2-component fermions), and no complex scalars, this reducesto:bQCD = 7 (1.39)for energies above the top quark mass. Solving the RG equation leaves us with the unusual running ofthe QCD coupling strength:αs(µ) =g2s (µ)4pi=αs(µ0)1+ bQCD2pi αs(µ0) log(µµ0) (1.40)Because bQCD is positive, the coupling strength actually decreases as the energy increases. This isknown as asymptotic freedom. On the other end, as our energy scales decrease, we approach non-perturbativity as αs→ 4pi . This occurs at the scale µ ≡ ΛQCD ∼ 214MeV. Above this scale, QCD canbe treated perturbatively, with free quarks and gluons. Below this, however, the strong force becomesnon-perturbative and other methods, such as Lattice QCD, must be used [35, 36].We can still make qualitative statements about the low energy theory based on confinement andglobal symmetries. Quarks and gluons become strongly-coupled at low energy, or equivalently, largedistance. If we consider a qq¯ pair, we can imagine attempting to pull these apart. Eventually thepair will be separated enough that the energy stored in the gluon flux tube will be large enough thatit becomes energetically favourable to break the tube and nucleate a new quark pair from the vacuum(as long as the quark masses are below the confinement scale). This will be a new meson state, andsignals that confinement is preferred to free states at low energies. However, these confined states arenot fundamental, and thus not well defined by the QCD Lagrangian. In principle, the QCD Lagrangianshould be able to calculate these states and their interactions, but it is unclear how to do so as it is highlynon-perturbative. To describe these states, we would like to have an effective field theory (EFT) for12QCD at low energies.One such approach is to write down an EFT that contains the dynamical fields of interest, togetherwith all possible operators consistent with the underlying symmetries of QCD. This is known as chiralperturbation theory, and provides an excellent starting point for an EFT of QCD [37]. The underlyingidea is that there is an SU(2)L × SU(2)R×U(1)V ×U(1)A symmetry associated with the QCD La-grangian in the massless phase when considering only up (u) and down (d) quarks. This correspondsto SU(2) transformations of the isospin doublets(uLdL)and(uRdR), as well as Baryon number conservationvia the vector current (U(1)V ) and a symmetry associated with the axial vector current (U(1)A). Allof these but U(1)A have cancellations in their anomalies, and so SU(2)L× SU(2)R×U(1)V is a goodglobal symmetry that should be obeyed in the low energy limit.One further useful feature of QCD comes from the quark condensate vacuum state:〈q¯RqL〉 6= 0 (1.41)This expectation value does not respect the global symmetry defined above, and thus we can expect tofind NGBs of a spontaneously broken symmetry. Here, the full global symmetry is broken down to asubgroup SU(2)V ×U(1)V , where SU(2)V is the subgroup corresponding to transformations that affectleft and right-handed states in the same way. Thus, we start with (3+3+1=7) generators of the full globalsymmetry, broken down to (3+1=4) generators that leave the vacuum state invariant. Thus we expectto see 3 NGBs; we can associate these with the lightest meson states, the pions: pi0, pi±. Althoughthe NGBs should be massless, in the full QCD theory the quark states are not massless. This explicitlybreaks the global symmetry above, and so the pions actually correspond to pseudo-Goldstone bosonswithin chiral field theory. This is in fact why this EFT is termed chiral perturbation theory; breaking theglobal chiral symmetry plays a large role in the creation of our new bound states.This can be extended formally to include other meson states, all the quark flavours (that fall belowthe confinement scale), as well as the full baryon spectrum. Bound states of gluons are even predicted toexist, in the form of glueballs [5, 6].5 Thus we have both chiral field theory and Lattice QCD methodsto attempt to understand not only the fundamental fields of the SM, but the physically observed fields aswell.Now we have all the pieces of the physical Standard Model, where everything can be neatly ex-pressed in terms of their gauge-boson couplings (gs, g, g′ and their RG evolutions) and representations,Higgs potential couplings (v, λ ), and the Yukawa couplings (yi, which if diagonalized, lead to the CKMmatrix terms as well). With all this in hand, we can move on to problems with the Standard Model thatmotivate our search for new physics beyond.5We will explicitly study a non-Abelian gauge theory in Part IV that looks at realizations of glueballs and their effects inthe early Universe.131.3 Problems with the Standard ModelDespite overwhelming evidence that the Standard Model does an exceptional job of modeling funda-mental physics, it is known to be incomplete. There are many problems in particle physics that cannot beexplained by the chosen gauge symmetries, generations of fermions, and the presence of a light Higgs.In this section we will lay out some of the most prominent problems with the SM today, in the hopes tomotivate searches for physics beyond it (also known as BSM physics). Here, we will focus on concreteproblems arising directly from the Standard Model. In a future section, we will extend our motivationsfor BSM physics by identifying problems that arise due to our coupling of knowledge of both the SMand cosmology (such as, for example, the need for dark matter and even more CP violation than thatpresent in the CKM matrix). Good references addressing and providing overviews of some of theseproblems can be found at [20, 38, 39].Neutrino MassesAs we have hinted at already, one problem with the SM is the lack of neutrino masses. Under the SM,right-handed neutrinos do not exist, and so we cannot produce neutrino masses via the Higgs mech-anism. However, the discovery of oscillations between neutrino flavours from atmospheric neutrinos[40], as well as solar neutrinos [41] necessitates that at least two of the neutrino species must have mass.Qualitatively, this can be seen by the argument that a truly massless particle must travel at the speed oflight, and by construction will thus not feel the passage of time. If that were the case, oscillations (whichhappen over large distances or equivalently timescales) would be impossible. This can be worked outquantitatively as well. In a simplified 2-state neutrino model, with flavour eigenstates |νe,µ〉 and masseigenstates |ν1,2〉, the probability of arriving in the same flavour state as the initial is [42]:Pee = 1− sin2(2θ)sin2(∆m2L4E)(1.42)where ∆m2 is the neutrino mass difference, L the distance travelled, E the neutrino energy, and θ themixing parameter that mixes flavour and mass states. For three generations of neutrinos, similar argu-ments will apply, although we must use a mixing matrix similar to the CKM matrix in the quark sector.For leptons, this is known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [43, 44].Thus the observation of oscillations provides concrete evidence for at least some of the SM neutrinosto have mass, which necessitates BSM physics. The easiest solution is to allow for the existence of threeright-handed neutrinos that are gauge singlets. This would allow for Yukawa couplings to the Higgs andthus give mass to the neutrinos. However, the extreme difference in masses between neutrinos (which,from cosmological observations must be less than an eV [45]) and charged leptons is enough to questionif there is some other mechanism providing neutrino masses.6 One other popular solution is the (Type I)see-saw mechanism [46]. In this model, the new neutrinos, N, being gauge singlets, are given Majorana6The top quark to electron mass ratio is ∼ 105, while the electron to neutrino mass ratio is at least the same order ofmagnitude. Although these are not dissimilar, it will still be satisfying to identify some mechanism for the mass hierarchies.14= + +Figure 1.1: Feynman diagrams contributing to the one-loop corrections to the Higgs mass inthe SM. From left to right, these include fermion loops (particularly the top), Higgs self-couplings, and massive gauge boson loops. All three diagrams are quadratically divergent.masses at a scale MN :L ⊃−12N¯RMNNR (1.43)When combined with the Yukawa masses of Eq. (1.35), this results in a mass matrix of the form:Mν =(0 yνv/√2yνv/√2 MN)(1.44)If the scale at which the new physics is present is much larger than the Yukawa coupling and Higgs vev,MN >> yνv/√2, then the left and right handed neutrino masses essentially decouple:mν ,L ∼ yνv√2MN, mν ,R ∼MN (1.45)and we are left with our three SM neutrinos that naturally have small masses, suppressed by v/MN , aswell as three (mostly singlet) heavy neutrino states. These are effectively sterile neutrinos, as the singletstates are extraordinarily hard to detect. Although new neutrinos have not yet been discovered, therehave been hints of a fourth neutrino at experiments such as Liquid Scintillator Neutrino Detector (LSND)and MiniBoone [47, 48], and as such models that explain the neutrino masses and add new neutrino-likestates are well motivated to study and understand.Hierarchy ProblemAs we have now seen, new physics at higher scales is well motivated. An elegant solution to theneutrino mass problem relies on heavy sterile neutrinos. Many other models, such as Grand UnifiedTheories (GUTs) and quantum gravity also point to natural energy scales much higher than the EWscale.7 However, introducing new energy scales above the EW scales leads to a new problem. One ofthe most interesting problems with the SM is this hierarchy problem, and the desire for a natural modelof physics.In particular, when computing quantum corrections to the Higgs mass, diagrams of the form shown7GUTs typically require energy scales around 1014 GeV, while quantum theories of gravity are naturally expected at thePlanck Mass, MPL ∼ 1019 GeV.15in Fig. 1.1 must be considered. In all three cases, the superficial degrees of divergence of the loopintegrals are D = (power of p in numerator)-(power of p in denominator) = 2, which naively leads toquadratic divergences [16].8 For example, the first diagram corresponding to a loop from the top quarkwill have a loop integral contribution of the form:loop integral ∝∫d4 pγµ pµ +mp2−m2γµ p′µ +mp′2−m2 ∝ p2 , (1.46)where p and p′ correspond to the internal propagator momenta of the top quark, and m its mass. Thiswill naively lead to a quadratic divergence in the integral if we integrate over all possible momenta. Inthis case, all the naive expectations align with reality, and contributions to the Higgs mass are of theform:δm2H ∼Λ232pi2[−6y2t +14(9g2−3g′2)+6λ](1.47)where the loops have been regularized using a momentum cut-off at Λ. If Λ is much larger than theweak scale, where the Higgs and weak boson masses appear, then in order to realize a physical Higgsmass of mH = 125 GeV we would require a very fine tuned value of the bare Higgs mass to nearlyexactly cancel any large contributions from new physics at scale Λ. For example, new physics is almostcertainly expected to arise at the Planck scale, where quantum gravity is expected to emerge. WithΛ= MPL ∼ 1019GeV , then mH,bare must be chosen with a precision of one part in 1032. This fine-tuningis the hierarchy problem, as there is naively no reason for these parameters to show such remarkablecancellations.We might worry that these quadratic divergences are simply a relic of using a cut-off regulator, andthey should disappear when renormalizing the theory to remove UV divergences. However, it is moreappropriate to understand Λ as a new physical scale corresponding to the mass of the new heavy particle,and the corrections that appear in Eq. (1.47) can be considered as finite contributions from heavy parti-cles (such as the top and gauge bosons) that are proportional to the scale of new physics. For example,integrating out the massive sterile neutrinos of the previous section leaves a non-renormalizable operatorin the SM EFT that will necessarily include corrections of the form of Eq. (1.47), with Λ∼MN . Thus,the hierarchy problem is truly a problem that the SM Higgs is sensitive to new scales - if there is nonew UV physics there is no hierarchy problem. But as UV physics is well motivated, this problem is afundamental one that many physicists have attempted to explain.It is also interesting to note that this problem is unique to elementary scalars, and does not arisewhen we consider either fermions or gauge bosons. If we pose the exact same problem but consider8Note that sometimes the superficial degrees of divergence do not lead to the correct UV divergent behavior. This is thecase, for example, with the photon self-energy correction, which naively expects a quadratic divergence but is only logarithmic.Symmetries in the theory (such as the gauge symmetry which leads to the Ward Identity [49]) can reduce the number of degreesof freedom, changing the UV behavior. Nevertheless, it is still a good reference point to understand where in our theories thesedivergences may arise.16corrections to the fermion mass, these corrections will scale as:m f ∝ m f log(Λm f)(1.48)which is a far slower divergence, and protects the masses of the fermions (and gauge bosons) from thedivergence we see in scalars. This arises due to the fact that there is an approximate symmetry in thechiral theory. If m f → 0, this symmetry becomes exact, which serves to suppress large corrections tothe masses.Many solutions to the hierarchy problem have been proposed. One possible option is that thisis not a problem at all, and we simply require fine-tuning for life to exist. This anthropic argument isunsatisfying to many, as it is still possible to create ‘habitable Universes’ for Higgs masses that are muchlarger [50]. More popular solutions have included supersymmetry (SUSY) and Higgs extension models.In supersymmetry, (a good primer for which can be found in Ref. [51]), the Poincare´ symmetries ofspace-time are extended to include particles of different spins. In doing this, an entire complement ofnew particles to the SM are predicted, with every elementary SM particle gaining a superpartner withspin differing by half.9 This cleanly solves the hierarchy problem, as the superpartners will contributeto the quadratic divergence of the Higgs with the same couplings (modulo their broken masses) butopposite signs as their SM counterparts. As long as SUSY is broken softly, this leads to δm2H ∼ m2so f t ,and so new superpartners near the TeV range would help to explain the hierarchy problem.With no hint yet of such physics at the Large Hadron Collider (LHC) experiments, other solutionsto the hierarchy problem are also gaining interest. Most of these involve changing the elementary natureof the Higgs in some way. These are collectively known as Composite Higgs models, including LittleHiggs, Twin Higgs, Technicolor, and more [52–57]. In these models, the common motivation is that theHiggs is no longer an elementary scalar, but rather a composite bound state of some more fundamentalobjects. In these cases the Higgs is then an NGB of an underlying global symmetry, G. To allow theHiggs a quadratic mass term, the underlying symmetry must not be exact, but rather an approximatesymmetry. In this way, G is explicitly broken, making Higgs a pseudo-Nambu-Goldstone-Boson. Theunderlying shift-symmetry associated with NGBs allows for mH to naturally be small, providing anothersolution to the hierarchy problem.Strong CP ProblemThe last major issue that we will comment on here is the strong CP problem [58–61]. When we wrotedown the gauge boson kinetic terms for the SM Lagrangian in Eq. (1.14), we ignored a number ofpossible gauge invariant operators. We could also have included terms of the form:L ⊃−(θQCD32pi2GµνG˜µν +θL32pi2WµνW˜ µν +θY32pi2Bµν B˜µν)(1.49)9In standard SUSY models, the masses of the superpartners are predicted to be identical to their counterparts. As we havenot discovered any such particles, if SUSY exists, it must be a spontaneously broken form to lift the mass degeneracies.17where G˜µν = εµναβGαβ/2 is the dual field strength tensor. It can be shown that these terms are equiv-alent to total derivatives of the fundamental fields. These would then vanish in the action as boundaryterms if the fields have non-trivial windings. This is true for Abelian fields (and so we do not worryabout θY ), but the theta terms cannot be removed for non-Abelian theories. While θL can be removedby transformations in lepton number (L) and baryon number (B) [62], the term cannot be removed fromQCD, and so we should include its effects when considering standard physics models. Even if we wereto naively set θQCD = 0, rotations of the quark fields to try to produce diagonal mass matrices results ina re-introduction of the theta term, and so it would appear to be the case that we should naturally expectsome non-zero value of the phase.However, there is an important effect that is predicted if this term exists. In particular, CP violatingcouplings of pions to neutrons are expected, that give rise to a calculable neutron electric dipole moment[63]. This is calculated to be:dN ∼ 5.2×10−16θ cm (1.50)Current limits on the (non)-existence of such a dipole moment imply extremely small values for θ <5×10−11 [64]. Understanding why this value is so small is the strong CP problem.The most popular solution to this is the axion, a new particle that arises in the Peccei-Quinn theory[65]. In this theory, an extra U(1) symmetry is added to the Standard Model in such a way so as tobalance the problematic term. Spontaneous breaking of this symmetry, together with explicit break-ing associated with low-energy QCD effects lead to a pseudo-NGB. This new particle is the axion, apseudoscalar field, a, that couples to the SM as:L ⊃− g2s32pi2afaGG˜ , (1.51)with a potential:V (a)∼ m2a f 2a(1− cos(θ¯ +a/ fa))(1.52)where θ¯ corresponds to the value of θQCD chosen by nature, which is arbitrarily expected to be non-zero. After SSB, the axion gains a vacuum expectation value that exactly cancels off θ¯ , thus completelyeliminating the strong CP problem. Other solutions to the strong CP problem involve invoking discretesymmetries instead of the continuous U(1) Peccei-Quinn symmetry. Invoking a P symmetry at highenergies between left-handed SM particles and right-handed mirror particles will eliminate the thetaterm, once again solving the CP problem [66].These are only a few of the problems associated with the SM, (with some more to be discussed laterin the context of cosmology) but already we can see that there are plenty of well motivated reasons tosearch for new physics. However, when doing so, we need to remember that any new physics modelsmust remain consistent with the rest of our understanding of how the Universe appears to work andinteract with itself. Of interest to us will be how new physics may interact with the SM in the early18Universe, and so next we will provide a short overview of useful cosmological theory.19Chapter 2Cosmology and Dark Matter2.1 IntroductionWhile the SM of particle physics has done a remarkable job on small scales, predicting new particlesand explaining their interactions, the ‘Standard Model’ of cosmology has been doing the same thingon larger scales, successfully modelling the evolution and expansion of the observable Universe. Thismodel has successfully predicted the Cosmic Microwave Background (CMB) power spectrum [67], aswell as the abundances of light elements created during Big Bang Nucleosynthesis (BBN) [68, 69].However, just like the SM, there are problems that cannot be solved in cosmology with only the knownparticles, and so we must once again turn to new physics to attempt to explain the unexplainable.In this chapter, we will build up a working knowledge of the cosmology needed to incorporatenew physics into the early Universe. Some good reviews and texts for the following discussion canbe found at Refs. [70–72]. This will include looking at how we currently understand the expansionof the Universe is driven, as well as a detailed look into the thermodynamics of the early Universe.Following this, we will address problems that cannot be explained by our current models. We will focuson dark matter (DM) and dark sectors, as this is the frontier where particle physics directly coincideswith cosmology, and understanding how dark sectors interact with the SM will be the main focus ofthis thesis. In particular, in this chapter we will address how DM (and other new particles that may berelevant in the early Universe) can leave an observable imprint that can be used to constrain new physics.Through this chapter, we will be using the mostly negative convention for the metric: ηµ,ν = diag(1,-1,-1,-1). Also note that we will use the reduced Planck mass, MPL = 1√8piG whenever it is referenced.2.2 The ΛCDM ModelWe begin our survey of the ‘Standard Model’ of cosmology by considering the expansion of the Uni-verse. On large scales, all astrophysical evidence points towards the idea that the observable Universeis homogeneous and isotropic. The most general metric (gµν ) that satisfies these conditions is the20Friedmann-Robertson-Walker (FRW) metric, which can be written in the form:ds2 = dt2−a2(t)(dr21− kr2 + r2dθ 2+ r2 sin2 θdφ 2)= gµνdxµdxν (2.1)where (t, r, θ , φ) are comoving coordinates, a(t) is the scale factor, and k = -1, 0, 1 correspond tospaces of negative, zero, or positive spatial curvature, respectively. When k = 0 (as is suggested frommany large scale observations), r is the standard radial comoving coordinate. In flat space, then, we canfind physical distances between objects by simply scaling the comoving distances:~xphysical = a(t)~xcomoving (2.2)It is standard to assume that the present-day scale factor, a(t0) = 1. Often, distances to astrophysicalobjects are quoted in terms of redshift, z, instead of the scale factor. This is a direct measurement ofhow much a wavelength of light has stretched before reaching us: λ0 = (1+ z)λi. These are related by:z(t) =a(t0)a(t)−1 (2.3)Thus, in order to study the evolution of the Universe we simply need to understand how the scale factorchanges with time. To do this, we solve the Einstein equations:Gµν ≡ Rµν − 12gµνR = 8piGTµν (2.4)where Gµν is the Einstein tensor, Rµν the Ricci tensor (which depends on the metric and its derivatives),R the Ricci Scalar = gµνRµν , G the Gravitational constant, and Tµν the stress-energy tensor describingall the fields and energy that are present. For simplicity (and to match the symmetries, isotropy, andhomogeneity of the metric), we consider perfect fluids as the source of all energy. In this case, thestress-energy tensor can be written in terms of the energy density, ρ , and pressure, p, of the fluid, as:Tµν =ρ 0 0 00 −p 0 00 0 −p 00 0 0 −p (2.5)When we impose conservation of the stress-energy tensor (under vanishing covariant derivative,∇µT µν=0),we find the first law of thermodynamics in an expanding Universe:∂ρ∂ t+a˙a(3ρ+3p) = 0 (2.6)21where dots are used to denote time derivatives. If we take this stress-energy tensor, together with theFRW metric, we find, for the time-time component of Einstein’s equations, the Friedmann equation:(a˙a)2+ka2=8piG3ρ (2.7)This is usually written in terms of the Hubble rate, H = a˙/a and a critical energy density, ρC = 3M2PLH2such that we find:ρρC= 1+kH2a2, (2.8)or equivalently, in dimensionless form:Ω= 1−Ωk (2.9)where Ω gives the dimensionless fractional energy density with respect to the critical, and Ωk =− kH2a2 .In a closed Universe, k = +1 andΩ> 1. If the Universe is open, k = -1 andΩ< 1. Finally, a flat Universewill find that the total energy density will exactly match the critical density, and so the curvature mustvanish. Experimentally, there is a growing amount of evidence that supports this scenario, and so fromhere on we will consider a flat Universe only, and explicitly set k = 0.The spatial components of the Einstein equations can similarly be used to determine the secondFriedmann equation. When taken together with Eq. (2.6), this is typically written as an equation gov-erning the acceleration of the scale factor:a¨a=−4piG3(ρ+3p) . (2.10)Let us now explain why the standard cosmological model of expansion is called ΛCDM. To take theFriedmann equations any further, we require knowledge of the type of fluid that is driving expansion.This can be done using the equation of state,p = wρ (2.11)where w parametrizes the type of fluid present. The three main parametrizations that we will considerare w = 1/3 (radiation), 0 (matter), and -1 (vacuum energy, or a cosmological constant, Λ). Using theFriedmann equations, we can show that the evolution of the energy densities must be:ρ ∝ a−3(1+w) (2.12)This makes intuitive sense: matter scales like a−3, which is equivalent to the volume of space expanding,as expected. Radiation scales like a−4, which has the volume of space expanding, plus an additionalfactor for the redshifting of each wavelength. Finally, the cosmological constant does not scale, butremains constant as the Universe expands. We can take it a step further to determine how the scale22factor must change with time given a particular equation of state as well:a(t) ∝t2/3(1+w) w >−1eH0t w =−1 (2.13)Because the energy densities scale at different rates, we can define unique Cosmological epochs withinwhich different types of energy density are dominant. Early in the history of the Universe, whena << a(t0), radiation energy density will be the dominant form. As the Universe continues to expand,eventually we will reach the matter dominated era, signalled by the time of matter-radiation equality:ρr,0a4=ρm,0a3(2.14)This can be solved for the cross-over scale factor, aeq =Ωr/Ωm, where it is more standard to work usingthe dimensionless energy fractions, Ωx = ρx/ρC.After a period of matter domination, the Universe will eventually be dominated by the cosmologicalconstant term, provided by Λ. When we are within each epoch, it is fairly numerically safe to solvethe Friedmann equations using only one of the three types of energy densities. If we are crossing overmultiple epochs, however, it is best to use the full equation for the Hubble rate, which is best expressedas:H2 = H20(Ωr,0a4+Ωm,0a3+ΩΛ)(2.15)where H0 is the present Hubble rate, and Ω0 = Ωr,0 +Ωm,0 +ΩΛ is the present day (dimensionless)energy density. Typically, H0 is parametrized in terms of a dimensionless number, h, as well, and ishistorically given as H0 = 100 h km/s/Mpc. The current observational limits on the above values aregiven by the latest CMB results from the Planck telescope, together with lensing reconstruction andbaryonic acoustic oscillation (BAO) results [67]:h = 0.6766±0.0042Ωm = 0.3111±0.0056 (2.16)ΩΛ = 0.6889±0.0056and Ωr is negligible today. These are enough to fully inform the evolution of the Universe. Thus, theΛ comes from the cosmological constant, and CDM comes from the cold dark matter (that we willdelve into in a future section) that makes up the dominant portion of the matter density of the Universe(although Ωm still accounts for the baryonic portion as well). Together these make up effectively all ofthe energy density in the Universe today.232.3 Thermal Evolution2.3.1 ThermodynamicsWith our understanding of how the Universe evolves in hand, we can now turn to thermodynamics to un-derstand how evolution will move forward for species that are in thermal equilibrium. Before discussingthe early Universe and the radiation-dominated phase, we first review some basic thermodynamics.In order to track the evolution of a species i, we will wish to know the number density ni, energydensity, ρi, and pressure, pi. For a given gas of particles with gi internal degrees of freedom, this onlydepends on the phase space distribution fi(p):ni =gi(2pi)3∫fi(~p)d3 p (2.17)ρi =gi(2pi)3∫E(~p) fi(~p)d3 p (2.18)pi =gi(2pi)3∫ |~p|23Efi(~p)d3 p (2.19)where E2 = |~p|2+m2. In kinetic equilibrium, we can identify the phase space occupancy by the Fermi-Dirac or Bose-Einstein distribution:fi(~p) =1e(Ei−µi)/Ti±1 (2.20)where +1 pertains to fermions, -1 to bosons, and µi is the chemical potential of the species. Furthermore,if we have chemical equilibrium as well as kinetic, then the chemical potential of each species can berelated to those it interacts with. For example, for an interaction of the form:i+ j↔ k+ l , (2.21)we must have:µi+µ j = µk +µl . (2.22)In the relativistic limit, these equations can be solved exactly for both sets of statistics, giving:n(0)i =[34]ζ (3)pi2giT 3i (2.23)ρ(0)i =[78]pi230giT 4i (2.24)p(0)i =ρ(0)i3(2.25)where the factor in square brackets [] must be included for Fermi statistics. Shown here are the limitswhere µ = 0 (which will often be referred to as the equilibrium value). In the non-relativistic limit, both24sets of statistics reduce to the same form:n(0)i = gi(miTi2pi)3/2e−mi/Ti (2.26)ρ(0)i = mini (2.27)p(0)i = niTi ∼ 0 (2.28)If we wish to relate these solutions back to the full solution including the chemical potential, we canneglect quantum statistics, and approximate the phase space distribution by the Maxwell-Boltzmanndistribution ( f ∼ exp(−(Ei−µi)/Ti)). This is typically reasonable for T < m/3 [73]. The full numberdensity, for example, would then be given as ni = eµi/Tin(0)i .We can see parametrically that, due to the Boltzmann suppression of non-relativistic species, thenumber and energy densities for relativistic species is typically much larger. As such, it is a goodapproximation to treat the entire energy density as a relativistic bath of particles, and essentially ignoreany particle whose temperature has dropped below its mass. In this way, we can rewrite the entireradiation energy density as:ρr =pi230g∗T 4 (2.29)where we assume that T is the photon bath temperature, and the effective massless degrees of freedomis a sum over all relativistic (Ti > mi) species:g∗ = ∑bosonsgi(TiT)4+78 ∑f ermionsgi(TiT)4(2.30)In reality, the drops in g∗ are not quite this discrete, as the change from relativistic to non-relativistic is asmooth process. The full evolution of g∗(T ) as a function of temperature in SU(3)C×SU(2)L×U(1)Yis shown in Fig. 2.1.Using Eq. (2.29) as the entire energy density in the early Universe, we can then write down anexplicit form for the Hubble rate we started with in Eq. (2.7):H =√g∗pi290T 2MPL(2.31)Furthermore, we can show that the temperature will scale as 1/a (up to slight variations as g∗ drops).As we will often be concerned with changes that are occuring during the radiation epoch, these relationswill prove to be useful when we wish to solve for the number density evolution of interacting particles.The final piece of thermodynamics that we will need to introduce is the idea of entropy. In anexpanding Universe, the second law of thermodynamics can be written as:T dS = d(ρV )+ pdV −µd(nV )≥ 0 (2.32)25��� ��� ��� ��� ��-� ��-� ��-� ��-� ��-��������/���� *(� *�)�*�*�Figure 2.1: Evolution of the relativistic degrees of freedom, g∗ and g∗s as a function of temperaturein the SM.where S is the entropy, and V is the volume of the region in question. In an expanding Universe, suchthat Eq. 2.6 describes conservation of energy and the vast majority of the energy density is in thermalequilibrium, we can actually take this inequality to be exact, and thus have an adiabatically expandingUniverse such that dS = 0. If we take this together with the first law of thermodynamics given in Eq.(2.6), it is possible to define an entropy density, s = S/a3 as:s =ρ+ p−µnT(2.33)In standard cosmology in the early Universe, we only need to consider radiation energy density, asit will be much larger than contributions from any other sources.1 In that case, the entropy density canthen be written as:s =2pi245g∗sT 3 (2.34)Here, we have introduced a new effective degree of freedom, g∗s:g∗s = ∑bosonsgi(TiT)3+78 ∑f ermionsgi(TiT)3(2.35)At the hottest temperatures, this will map directly onto g∗, as all SM particles are in thermal equilibriumand share a common temperature. However, as particles cool and decouple from the photon bath so thatTi 6= T , this will deviate from g∗. Both degrees of freedom are shown in Fig. 2.1. As we will discuss in1We keep the full entropy density here as reference, however, as it will prove to be useful in Part IV, where we will need toconsider entropy conservation in a sector entirely dominated by out of equilibrium, non-relativistic particles26a future section, the main reason for the deviation is the decoupling of neutrinos at T ∼MeV, at whichpoint they are no longer heated by further photon interactions, such as the decoupling of electron andpositron annihilations.Conservation of S implies that s ∝ a−3, and that g∗sT 3a3 remains constant as the Universe expands.Thus, in periods when g∗s is constant (or varying slowly), T ∝ 1/a, as we predicted before. The evolutionof s with a−3 also helps us define a useful quantity that will be used throughout later chapters. For aparticle that is no longer interacting with the thermal bath such that the total number of particles hasstopped changing, it is straightforward to see that the quantity:Yi ≡ nis (2.36)will remain constant. This is known as the yield for species i. Thus, if we can determine n(T ) at the lastpoint of interaction, we can determine the density we would expect to see today, which can be used tomake predictions and provide constraints based on observations of the Universe as we view it today:n0 = nis0si(2.37)where s0 is the current entropy density, given by the temperature of photons today, T = 2.7255±0.0006K [74], such that s0 ∼ 3000 cm−3.We now have enough understanding of the thermodynamics of the early Universe to move ontoparticle interactions, which will be governed by Boltzmann equations.2.3.2 Boltzmann EquationsAs particles are interacting and maintaining thermal equilibrium, their number densities naturally be-come suppressed as the Universe expands. When the equilibriating reactions responsible for particleinteractions drop below the Hubble rate, departures from thermodynamic equilibrium will occur. Thisdeparture from equilibrium can be described as a modification in the phase space distribution, fi, of aspecies. The modification of the distribution for a single species is governed by the Boltzmann equation:Lˆ[ f ] = C[ f ] (2.38)where Lˆ is the Liouville operator, and C the collision operator. For a spatially homogeneous andisotropic distribution (such that f (pµ ,xµ) = f (|~p|, t)) in the FRW metric, the Liouville operator is:2Lˆ =∂∂ t−H p ∂∂ p(2.39)where p corresponds to the usual magnitude of the momentum 3-vector. If we integrate both sides ofEq. (2.38) by d3 p, and include the appropriate normalizing constants, this can be transformed to a2Note that we should actually take derivatives with respect to the affine parameter [71, 72]. However, we can adjust ourdefinition of the collision operator to account for the extra factor of E that this would involve.27Boltzmann equation for the number density:3∂n∂ t+3Hn =∫gd3 p(2pi)3C[ f ] = C˜[ f ] (2.40)Integrals over higher moments can be taken as well to get Boltzmann equations for the energy density,pressure, etc. In general, the collisional term will be a complex expression, involving the matrix ele-ment associated with the forward and reverse reactions, as well as the phase space distributions of eachspecies. However, for 2→ 2 interactions of the form in Eq. (2.21), the equation can be vastly simpli-fied. We present this case here, although we will be interested in generalizations of this in later chapters(for example, we will consider particles starting out of equilibrium in Part II, and 3→ 2 interactions inPart IV). We begin by writing down a general collision equation for the 2→ 2 reaction, where we areinterested in the evolution of species i:C[ f ] =12Ei1S∫dΠ jdΠkdΠl(2pi)4δ (4)(pi+ p j− pk− pl)×{|M |2kl→i j fk fl(1± fi)(1± f j)−|M |2i j→kl fi f j(1± fk)(1± fl) }(2.41)where Ei is the energy of particle i, |M |i j→kl is the standard unpolarized amplitude, obtained with theusual Feynman rules, and we average over initial and sum over final spins. S is a symmetry factor thataccounts for identical particles in the initial or final state. dΠi corresponds to the differential Lorentz-invariant phase space:dΠ j = gid3 pi(2pi)32Ei(2.42)The 1± fi terms correspond to Bose enhancement and Pauli blocking factors. Reversibility and unitarityof the interaction will imply that |M |i j→kl = |M |kl→i j ≡ |M |. For many cases of interest, we will typi-cally consider systems at temperatures smaller than Ei−µ . In this regime, we can make a further approx-imation that the quantum statistics can be ignored. In particular, we can ignore the Bose enhancementand Pauli blocking factors, and further replace the equilibrium distributions by fi,eq(E)→ eµi/T e−Ei/T .Making these adjustments, we can write our modified collision operator C˜ as:C˜[ f ] =∫dΠidΠ jdΠkdΠl(2pi)4δ (4)(pi+ p j− pk− pl)1S |M |2( fk fl− fi f j) (2.43)To proceed further, we need to make some assumptions about the form of the phase space distributions.In most cases, we will be interested in particles annihilating into SM particles, or particles coupledtightly to the plasma, such that kinetic and chemical equilibrium are immediately obtained. Thus, wecan replace fk fl → fk,eq fl,eq. Furthermore, detailed balance allows us to replace these with the initial3Strictly speaking, the term Boltzmann equation only applies to Eq. 2.38, but this terminology is widely accepted in thecosmological community.28state distributions: fk,eq fl,eq = fi,eq f j,eq ∼ e−(Ei+E j)/T .4 Because of this, we can do the final state phasespace integrals independent of their distributions, and replace them with the cross section correspondingto the interaction:σv =14EiE j∫dΠkdΠl(2pi)4δ (4)(pi+ p j− pk− pl)1S |M |2 (2.44)where v is the Møller velocity:v =√(pi · p j)2−m2i m2jEiE j(2.45)Using all of this, we can write a simple version of the collision operatorC˜[ f ] = 〈σv〉(n(0)i n(0)j −nin j) (2.46)where n(0)i = ni(µ = 0) as defined in Eq. (2.17) and we have taken the thermal average of the crosssection:〈σv〉= 1n(0)i n(0)j∫gid3 pi(2pi)3g jd3 p j(2pi)3fi,eq f j,eqσv (2.47)Using the thermal average here allows us to use truncation schemes for the cross-section. For smalltemperatures, and thus low relative velocities, we can compute σv in the center of mass frame, and thenexpand in terms of the relative velocity, v. Most interactions will have an s-wave term that is velocityindependent. Then, 〈σv〉 ∼ σ0. If a symmetry forbids the lowest angular momentum term, then thep-wave cross section will become relevant: 〈σv〉 ∼ σ1T/m. In general, we can use:〈σv〉 ∼ σn(mT)−n(2.48)where σn will be constant, and n corresponds to the lowest allowed angular momentum state. n = 0 fors-wave, 1 for p-wave, and so on to higher angular momentum states. Putting everything together, thisgives us the basic Boltzmann equation for a 2→ 2 interaction in the early Universe:∂ni∂ t+3Hni =−〈σv〉(nin j−n(0)i n(0)j ) (2.49)We will make use of this equation when we discuss DM in section 2.5, and similar modified versionsthroughout later sections.29Inflation> 109GeV< 10−32sEW Phase Transition∼ 100GeV∼ 10−12sQCD Phase Transition∼ 1GeV∼ 10−6sν Decouple∼ 1MeV∼ 1sBBN∼ 0.1MeV∼ 3minm-r Equality∼ 1eV∼ 50,000yrsRecombination∼ 0.1eV∼ 380,000yrsStructure Formation∼ 0.01eV∼ 105yrsReionization< 0.01eV∼ 106yrsToday∼ 10−4eV∼ 14GYrFigure 2.2: Brief timeline of the evolution of the Universe. Note that inflation could in principlehappen at much lower energies.2.4 The Universe TimelineNow that we have tools to describe particle interactions and the evolution of the Universe, we wish toreconstruct the thermal history of the Universe. In doing so, we will highlight key problems that we donot yet fully understand, as well as epochs in our history that lead to key pieces of observational evidencethat can be used to constrain new physics. A brief outline is given in Fig. 2.2 as reference. Note thatin this upcoming section, we will simply assume that there is some form of dark matter making up theprimary abundance of matter. In a future section we will discuss this puzzling detail that is necessaryfor standard Big Bang cosmology in more detail.The Early UnknownAbove temperatures around ∼ 5MeV, not much is known experimentally about what were the ex-act conditions of the early Universe. However, well-founded theories typically extend to energiesmuch higher than this, often involving multiple different phase transitions early on. For example,our current vacuum state is well described by SU(3)C ×U(1)em, which is the unbroken remnant ofSU(3)C× SU(2)L×U(1)Y . It is expected that we would have gone through a phase-transition at tem-peratures around 100 GeV, at which point the Higgs potential would be spontaneously broken and allrelevant particles would gain mass. There is also an expected QCD phase transition at around 1 GeV(the scale of non-perturbativity in QCD, ΛQCD), above which quarks and gluons would be free, butbelow we expect the usual bound states of baryons and mesons.4This is typically always true, as scattering processes that enforce kinetic equilibrium will stay relevant for longer thanthose that enforce chemical equilibrium.30However, even before this, we expect a phase transition corresponding to a period of inflation. Inorder to postulate the existence of a spatially flat, homogeneous and isotropic metric, a period of cosmicinflation is almost a requirement [75]. This epoch can solve many different problems in cosmology,including the flatness and horizon problems. However, to do so we must postulate the existence of somenew scalar field to drive inflation.5To see how inflation could occur, we can consider a scalar field with a symmetry breaking potentialsimilar to the Higgs,V (φ) =14λφ 4− 12µ2φ 2+V (0) (2.50)Although this has a vacuum expectation value at 〈φ〉 = µ√λ , at finite temperatures there are radiativecorrections of the form T 2φ 2. These would restore the symmetry, and 〈φ〉 → 0. As the Universe cools,the symmetry would be broken either through a first order phase transition (and the field would tunnelthrough a barrier), or via a second order transition (and the field can transform smoothly from the localto global minima). In either case, the field can slowly transform from the false vacuum to the truevacuum if the properties of the potential are appropriate. As the field ‘slow rolls’ from the false vacuumto the true vacuum, it will be dominated by the constant energy density of the potential, V (0). This canbe large enough to overcome the energy density of any radiation present, and so dominate the energydensity of the Universe. As we saw in Eq. (2.13), a constant energy density leads to exponentiallydriven expansion of the scale factor.During this exponential expansion, many desired effects will happen. First, the observed isotropyand homogeneity of the Universe would be fixed, as one small causally connected region would beblown up, and the entire visible Universe would have been in thermal contact early on. This solves thehorizon problem, as evolving backwards from our current point would imply that the Universe we seetoday would never have been causally connected, without this period of inflation. Furthermore, we canalso solve the flatness problem. This is best seen by looking at the evolution of the net energy density:1−Ω= a2(1−Ω0)Ωr +Ωma+ΩΛa4(2.51)Before ΩΛ came to be relevant, 1−Ω was always increasing. However, current measurements of Ω0imply that the Universe is still incredibly flat, and so the initial curvature must have been fine-tuned to1 part in 1060 to see the flatness we do today [77]. However, having an early period of inflation drivesthe flatness to zero automatically, thus bypassing the fine-tuned flatness problem altogether.Once inflation has finished, the inflaton field can transfer its energy density to the visible, radiativesector through a process termed reheating. As the scalar field reaches the global minima of its potential,it will oscillate around this point. The kinetic energy associated with these oscillations will be transferredvia decays to SM (or hidden sector) coupled particles. When the SM has had enough energy transferred5For some time, it was thought that perhaps the Higgs field itself could drive inflation, but after the discover of the Higgsboson at its observed mass and self-interaction values, this is no longer considered to be the case, as it would require anextremely large coupling in |H|2R. So the question of what new field is required is still open [76].31and rethermalized, we can continue with the standard hot big bang model, where radiation energy willdominate. In future sections, we will often refer to a period of reheating as an initial condition for themodels we consider, but will not attempt to model the UV physics that would be required to describehow the reheating came to occur.After this period of inflation, there is a new puzzle we must address: baryogenesis. This is thecosmological problem associated with the observed baryon asymmetry in the Universe. From our per-spective, anti-matter appears to be extremely rare, only seemingly detected in accelerators or cosmicrays. In both cases, the presence of anti-matter is expected to be a result of collisions of high-energymatter, and not a primordial source. Understanding why there are more baryons than anti-baryons to-day is a puzzle that has not yet been explained by the SM or cosmology. Explicitly, we are trying tounderstand why the parameter:η =nb−nb¯nγ= (6.09±0.06)×10−10 (2.52)is close to, but not exactly zero [9, 75]. 6 Typically, we cannot even use inflation as a source ofthe asymmetry: any initial fluctuations in the baryon symmetry are diluted by the expansion of theUniverse, while entropy is being produced, implying we would expect a symmetric Universe directlyafter inflation. Thus, we need a mechanism by which we can dynamically produce the asymmetry. Todo so, one must postulate three conditions for baryogenesis: violation of Baryon number, B, violation ofC (charge conjugation) and CP (charge conjugation-parity), and a departure from thermal equilibrium.These are known as the Sakharov conditions for baryogenesis [78], and are necessary to produce anasymmetry. Baryon number must be violated so that an asymmetry can be produced from a symmetricstate. C and CP must be violated so that baryons will be produced preferentially over antibaryons.Finally, the process must occur out of equilibrium, so that the reverse process does not have the samerate.Although all three of these conditions are met in the Standard Model, they are not strong enoughto explain the entirety of the asymmetry that we see today. As such, some new physics must be incor-porated to explain our observations. One such solution arises in the see-saw mechanism we consideredearlier in the context of neutrino masses. In this context, an asymmetry in leptons (aptly named leptoge-nesis [79]) due to decays of the heavy right-handed neutrino is transferred to the baryon sector throughSM sphalerons, which conserve B-L [70]. There have been many other models of baryogenesis pro-posed [80], all of which must invoke some new physics at scales above ∼ 5 MeV. However, once wereach 5 MeV, we can take this asymmetry as a given parameter in our models, and continue the evolutionwith this asymmetry in mind.6It is actually better to use the entropy instead of photon number density in the ratio, such that the whole ratio is actuallyconserved as particles become non-relativistic and freeze-out of the relativistic energy density. This is given by nb/s = η /7.04today.32Neutrinos DecoupleNow that the Universe has cooled to a few MeV, we can start making concrete predictions based on theinteractions of the SM. One of the first important things that will happen is the decoupling of neutrinosfrom the rest of the visible sector. At hot enough temperatures, neutrinos are kept in equilibrium throughweak interactions such as ν¯ν ↔ e+e− and νe↔ νe. If we compare the interaction rates to the Hubblerate, we find:ΓH=n〈σv〉H' G2FT5T 2/MPL'(TMeV)3(2.53)where GF is the Fermi constant describing weak interactions, and factors that are of order unity havebeen left out. At temperatures above an MeV, the interaction rates are strong enough to drive the neutri-nos into thermal contact with the plasma. Below this, however, the Hubble rate is such that the Universeexpands faster than the neutrinos can interact, freezing them out of thermal contact. After this point, theneutrino temperature will scale as 1/a, independent of anything occurring in the photon plasma. Thisis the first of many times we will consider ratios of interaction rates to the Hubble rate, a useful tool inunderstanding the parametrics of the early Universe.Shortly after neutrinos freeze-out, the temperature drops below the mass of electrons, and the en-tropy that was in e± pairs will be transferred to photons, but not neutrinos. We can estimate the effectthis will have on the temperature of photons by counting relativistic degrees of freedom before and af-ter e± annihilations. Directly before, we have photons (g=2), e± pairs (g=4), and neutrinos (g=6), andeverything shares the same temperature. However, directly after, the e± pairs will no longer contribute,and the photon temperature will be raised above the neutrinos as they gain the electron entropy. Byequating the entropy density before and after the electron annihilations, we can find that the ratio oftemperatures between photons and neutrinos must be:TTν=(114)1/3' 1.40 (2.54)We make use of this temperature ratio when considering the effects of new physics on BBN, the nextstep in the evolution of the Universe.Big Bang NucleosynthesisAfter neutrinos have decoupled, the next major step in the evolution of the Universe comes from thecreation of light elements during Big Bang Nucleosynthesis (BBN). These elements include D, 3He,4He, and 7Li. This takes place primarily at temperatures just below an MeV, when the Universe hascooled enough to allow for reactions involving heavier elements to proceed.The main output of BBN is the relic abundance of 4He, which can be estimated extremely simplyby considering the parametric evolution of all the reactions involved [81]. There are three main phasestypically considered in BBN: the freeze-out of neutrons, the Deuterium bottleneck, and the nuclearreactions that drive BBN. Before the onset of BBN, neutrons and protons are in equilibrium, due to33weak interactions of the form n+e+↔ p+ ν¯e and n+νe↔ p+e−. Because the abundances are drivento equilibrium, the ratio of neutrons to protons will simply be given by:np∣∣eq = e−∆m/T (2.55)where ∆m = mn−mp = 1.293 MeV is the neutron-proton mass difference. Similar to neutrinos, theweak interactions keeping this abundance in equilibrium will freeze-out around 1 MeV, which leads toa freeze-out ratio of ∼ 1/6. This sets the amount of neutrons that are initially available to take part inthe creation of heavier elements. However, because neutrons decay (n→ p+ e−+ ν¯e), there will beslightly fewer available at the onset of BBN. This reduces the freeze-out ratio from ∼ 1/6 to ∼ 1/7.The main reason for the delay is the Deuterium bottleneck; many of the interactions that occur duringthe creation of light elements involve Deuterium. For example, the first link in the nucleosynthesischain is p+ n→ d+ γ . It is also required for the production of 3He (D+D→3 He+ n), as well as 4He(D+T→4 He+n). Because the ratio of baryons to photons is so small, the temperature must cool wellbelow the binding energy of Deuterium (EB = 2.2 MeV) before the Deuterium destruction rate fallsbelow the production rate. Thus, BBN is delayed until η−1e−EB/T ∼ 1. This is at approximately 0.1MeV.Once we have passed this bottleneck, all nuclear reactions proceed fairly quickly. To a good approx-imation, nearly all of the neutrons present at this point will end up in 4He, the most stable light element.Thus the primordial mass fraction of Helium, Yp ≡ ρHe/ρb can be estimated by:Yp =2 np1+ np∼ 0.25 (2.56)This turns out to be a remarkably good estimate of the full numerical solution. The numerical resultswill also estimate the final yields of D and 3He to ∼ 10−5 relative to H, while 7Li is reduced further to∼ 10−10 H. These can be compared to the present day observations of the abundances of these isotopes.Deuterium is measured using absorption lines of distant quasars through damped Lyman-α (DLA)systems that have low metallicities [82]. These are good sites as they have high column densities ofneutral gas, which implies a good optical depth for many of the Lyman series. Low metallicity is alsouseful as there will be negligible D astration from sources such as dust within the region [83, 84].Researchers have also studied whether deuterium is formed in appreciable amounts, such as via stellarprocesses [85, 86], and found that deuterium is essentially not created, but only destroyed, and thus onlymonotonically decreases over time. Thus, any measurement of D made today will be a lower bound onthe D present at BBN. That is: (DH)DLA≤(DH)p(2.57)One group (Cooke), continually updates their result every time a new observation for a Ly-α systemis completed that matches their selection criteria [87–91]. Their most recent observation/analysis, [91]34uses results from 7 systems, and is:(DH)p= (2.527±0.030)×10−5 (2.58)Another group has also recently completed an analysis of 13 systems, with a slightly smaller uncertainty[92]: (DH)p= (2.545±0.025)×10−5 (2.59)If we take the weighted average of these two analyses, we get:(DH)p= (2.538±0.019)×10−5 (2.60)The consensus on the Helium-3 values are much more vague, as it is a much more difficult isotopeto track. Unlike when searching for D, which uses absorption lines, 3He is observed in emissions fromregions of ionized gas (HII regions) [93]. Thus far, only successful searches have been done withinthe galaxy, specifically in planetary nebulas, HII regions, and within the solar system. There is also asecondary issue that does not arise with D: stellar processes are more than happy to create helium in lowmass stars, or destroy it in high mass stars. As such, it is hard to decipher the true primordial amount,and thus searches are typically done in locations that have not been too disturbed by galactic/stellarprocesses (such as protosolar clouds).However, Kawasaki ([94]) and Sigl ([95]) both point out that helium cannot be used to overproduceD, and so, even though 3He is much more sensitive to its surroundings, the ratio 3He/D should alwaysmonotonically increase, and so you should be able to get upper bounds on the primordial abundancefrom regions where both He and D are measured. It also appears to be the case that the quantity (D+ 3He)/H is stable through galactic evolution [96], and so measuring this, together with D/H can giveratios such as 3He/H and 3He/D.Some of the most recent observations are from Bania [97], who measure the upper limit on theabundance from planetary neblua:3HeH= (1.1±0.2)×10−5 (2.61)Mahaffy provide results from the Galileo probe that measured the ratio of He isotopes in Jupiter’satmosphere [98]:3He4He= (1.66±0.06)×10−4 , (2.62)35and Geiss provide results from protosolar clouds [96]:D+3 HeH= (3.63±0.35)×10−5 . (2.63)Note that if we use the results from Eqs. (2.58) and (2.63), we get:3HeH= (1.10±0.35)×10−5 (2.64)Which agrees with the results from Eq. (2.61). We can also combine the Eqs. (2.58) and (2.63) to arriveat:3HeD< 0.44±0.14 (2.65)Note that this is much lower than the value cited in Kawasaki [94]: this is because they don’t use theobserved value of D/H from Cooke, but rather those from Geiss. Using this, you arrive at:3HeD< 0.83±0.27 (2.66)Like 3He, 4He measurements are coming from ionization regions such as HII, the sun, planetarynebulas, and regions within the galactic disk [99]. Again, they look for regions that have low metallicityso that the region is relatively unaffected by stellar processes, giving the best indication of the initialprimordial abundance. Although the fit is done by determining the abundance vs. metallicity and thenextrapolating to low metallicity [100, 101], having as much data as possibile at low metallicities is thecurrent observational goal.There appear to be three current measurements of Yp, the helium fraction. The first is from thegroup that did the actual observations [102], who update their old measurement by incorporating bothoptical and IR measurements of the helium emission lines to help constrain the density by breaking adegeneracy in the parameter space with temperature. They have 45 low metallicity HII regions from‘primitive galaxies.’ The Izotov group determines:Yp = 0.2551±0.0022 (2.67)Two other groups have used the data from the observations. The first is Aver [101], who use updatedemissivities from Porter [103] to arrive at:Yp = 0.2449±0.0040 (2.68)And the second is Peimber [104], who also use the updated emissivities, and arrive at:Yp = 0.2446±0.0029 (2.69)36Note that Eqs. (2.68) and (2.69) agree with each other, but not with (2.67). This appears to be dueto the methods used, and selection criteria used for including/discarding various sources from the dataset. The consensus seems to be that the Aver and Peimber results are most reliable [67]. The combinedresult from both of these analysis is:Yp = 0.2447±0.0023 (2.70)Finally, there is a puzzle associated with the abundance of Lithium. The best systems for observationof the Lithium abundance are metal-poor stars in the spheroid of the Galaxy [105]. Here, observationshave shown that Lithium does not appear to vary significantly with low metallicity [106]. However,systematics and uncertainties in different measurement techniques have led to a host of different mea-surements for the observed abundance of Lithium [107–109]. Furthermore, in some very metal poorstars, Lithium is not even detected at all [110, 111]. Thus best estimates appear to come from themid-range metallicities, with a value of:LiH= (1.6±0.3)×10−10 (2.71)taken as the safest estimate [111]. However, while observations of Helium and Deuterium are in agree-ment with BBN predictions, the measurement of Lithium is not, leading to the Lithium puzzle. BBNpredictions tend to overpredict the amount of Lithium that should be produced compared to observa-tions. However, there is much more uncertainty on the observational abundance and its relation to theprimordial abundance, due to the possibility of both creating and destroying Lithium in different astro-physical environments. Although we do not attempt to resolve this puzzle here, it is worth noting as itcould be incorporated into future works, and is often considered as a possible source of new physics.Because the predictions of BBN agree so well with the primordial abundances inferred from presentday observations, any new physics that may be present at this time is highly constricted [112]. Indirectalterations to the radiation energy density (typically denoted through the parameter Ne f f , the number ofeffective degrees of freedom in the neutrino sector) will alter when neutrons freeze-out, which in turnalter the final amount available to produce 4He. Furthermore, direct dissociation of the elements canoccur via the decay or annihilation of new particles during or following BBN.If the energy is injected via hadronic channels, it is possible to affect both the process of BBNas well as the outputs. For short lifetimes (τ < 104s), hadronic decays will produce partons that willhadronize into stable particles (such as pions and nucleons) that will directly alter the final abundances.Long-lived partons may interact with background nuclei, and charged current interactions with pions canalter the neutron-proton ratio, all of which will alter the final abundances of light elements. However,more stringent constraints can be placed on decays after the production of 4He. Once 4He has beenproduced, hadrodissociation can occur, breaking up the light elements. Energetic nucleons can alsocause spallation processes on Helium, once again reducing its abundance and interfering with presentday observations.Energy can also be injected via electromagnetic channels as well (or into the EM channels via sec-37ondary EM showers produced by hadronic decays). In this case, photodissociation of the light elementscan take place. However, these tend to occur at later lifetimes, as high energy decays can efficientlyscatter off the background photons, dissipating the energy that could be used for photodissociation. It isonly when energies drop below a critical energy:Ec =m2e22T∼ 2 MeV(6 keVT)(2.72)that photodissociation typically begins to take place [7]. This tells the temperature (and equivalently,the time), at which electromagnetic decays become important. In particular, we can equate the bindingenergies of the light elements with the critical energies to find the relevant temperatures [112]:Tph '7 keV 7Be+ γ →3 He+4 He Eb = 1.59 MeV5 keV D+ γ → n+ p Eb = 2.22 MeV0.6 keV 4He+ γ →3 He+n Eb ∼ 20 MeV(2.73)As the temperature drops below each of these scales, the light element begins to be destroyed, whichhighly constrains the presence of any new particles decaying at these temperatures. Note that as thetemperature falls below 0.6 keV, photodisintegration of 4He actually implies net production of D and3He as well. We can make use of this to constrain new physics, and will use the remarkable accuracy ofBBN and the primordial abundances of the light elements in future chapters.Matter-Radiation EqualityOnce the light element abundances have been formed, the Universe continues to evolve relatively un-interrupted. At some point, the Hubble expansion rate has caused the scale factor to reach a point atwhich matter and radiation contribute equally to the radiation density. If we ignore any contributions tomatter except for the baryon component, we would find that this occurs at a redshift of z∼550, whichwill have consequences for structure formation [113].However, we expect there to be a large dark matter component to the matter density today (in fact,this is one of the reasons why). If we plug the observed densities today into Eq. (2.14), we find that theactual epoch of matter-radiation equality physically corresponds to a scale factor of a ∼ 2×10−4, or aredshift of z ∼ 3600. This is when the Universe is ∼ 50,000 years old. After this occurs, matter willbe the dominant energy form, and the expansion speed will actually increase relative to the radiationdominated epoch, with scale factor increasing like t2/3 instead of t1/2. At this point, dark matter over-densities, which have already formed out of primordial fluctuations, will begin the slow push to non-linear growth that will signal the true start of structure formation.Recombination and the CMBAs the Universe continues to expand, the photon-baryon plasma is initially tightly coupled, as Comptonscattering of free electrons efficiently thermalizes photons to the baryonic matter. However, as the38temperature cools below the ionization energy of Hydrogen, neutral atoms form and photons no longerhave anything to scatter with. This point of recombination is the ‘last scattering’ surface for photons,after which they free-stream towards us. This occurs at temperatures of about 0.1 eV, or 380,000 years(or a redshift of 1100) into the evolution of the Universe.Because photons free-stream from this point on, they give us a unique snapshot of what the Universelooked like when it was still relatively young. Careful study of this epoch can give us a rich and detailedaccount of the parameters that make up the standard model of cosmology. Light from this time is knownas the Cosmic Microwave Background (CMB), and its spectral form is a main supporting pillar of thehot Big Bang model for the Universe. In particular, we can use the CMB to study the evolution ofdensity perturbations that would eventually give rise to the structure (galaxies, clusters, etc) that wesee today. A good review can be found at Ref. [114], while a more up-to-date review, with currentparameter values, can be found at Ref. [26].Although the physics that must be incorporated to completely model the shape of the CMB spectrumis complicated, we can explain qualitatively the main features that we expect to see. These can beexplained by the evolution of two different types of length scales (or equivalently, the Fourier modes,described by wavenumber, k): those that were in causal contact at the time of last scattering, and thosethat were not. Because the over-densities at this point are still small (O(10−5)), they can be modelledas linear perturbations, and thus the separate modes will evolve independently. These length scales canbe directly related to angular distances on the sky: smaller causally connected regions (larger k) mapdirectly onto smaller angular distances.With this in mind, we can lay out the main features of the CMB spectrum:• The ‘monopole’ of the CMB. This corresponds to the overarching blackbody spectrum and asso-ciated temperature that is predicted for an isotropic and homogeneous fluid. Because inflation (orsomething like it) provided a causally connected surface that later rapidly expanded out of con-tact, it is expected that the temperature of the CMB should be uniform in all directions. Indeed,the fact that we observe such a uniform backbody is a strong motivation for inflation.7• The ‘dipole’ of the CMB. This is another large, overarching feature due to Doppler boosting ofthe underlying spectrum caused by the motion of the Earth relative to the isotropic background.This is not a ‘primordial’ feature of the CMB, but rather an effect that must be removed to see thetrue underlying anisotropies.• CMB anisotropies. As we mentioned already, the seeds of density perturbations that eventuallygive rise to structures should also be present in the CMB. These density perturbations would, forexample, arise naturally as quantum fluctuations in the gravitational potential during inflation.These lead to four main features in the anisotropies of the CMB:1. The Sachs-Wolfe plateau. For modes that were not yet back in causal contact at the time ofrecombination, the spectra should be a direct measure of the primordial density fluctuations,7Late decays of non standard particles could distort the near perfect blackbody frequency spectrum of the CMB. We discussthe possible effects of such decays in Parts III and IV.39as only gravity is able to affect these modes. By this point, the density fluctuations inmatter are closely following these initial perturbations, and so the large scale anisotropiesin the temperature of CMB photons should follow the over-densities present in matter. Asthe Universe is matter dominated at the time of recombination, the gravitational potentialis constant, and so there should be a plateau in the power spectrum as all different scalescrossing into causal contact see the same potential [71]. Interestingly enough, although atthe time of recombination, the over-densities would correspond to hotter, denser regions, thephotons that reach us have actually redshifted (to climb out of the gravitational wells), andso we see these as colder regions in the CMB at large scales.2. The Integrated Sachs-Wolfe (ISW) rise. At the largest scales, we expect a deviation from theprimordial fluctuations due to the presence of the cosmological constant, Λ. This is due tothe change in the comoving distance that must be considered when modes come into causalcontact closer to the time of dark energy dominating the energy density. Thus, the largestscales are slightly raised above the Sachs-Wolfe plateau.3. Acoustic Peaks. For small scales that are back in causal contact before recombination,the photons were tightly coupled to the electrons present in the plasma. As a result ofthis coupling, the photon-baryon plasma can effectively be treated as a driven harmonicoscillator, with sound speed cs ∼ 1/√3, driven by the perturbations of the gravitationalpotential. As such, the photons and baryons will oscillate around the over-densities, causingthe photon temperature spectrum to fluctuate accordingly. At the time of recombination, thephases of the oscillations are frozen in, which become projected onto the sky today as peaksand troughs in the temperature power spectrum. The locations of the peak will vary withthe sound speed (which depends on the baryon abundance), and the strength of each peakdepends on the baryon abundance as well. Thus, the size and location of these peaks tellsus unique information about the baryon density instead of the entire matter density. This isa strong piece of evidence for dark matter as well, as it is indicative of the baryon densitybeing much smaller than the net matter density required to explain the Universe as we see ittoday.4. Silk Damping. Because the photon-electron plasma coupling is not perfect, there will besome diffusion associated with the coupling. In particular, it takes time for the ionizationfraction of electrons to be reduced to zero, and so the last scattering surface will actuallyhave some thickness associated with it. Any modes that are shorter than this scale will havetheir anisotropies washed out, and the peaks in the oscillations are damped.Taken together, these peaks and plateaus provide a strong theoretical prediction for the CMB that hasbeen tested extensively over the years, from the COBE satellite, to WMAP, to the third generation CMBtelescope Planck [115–117]. In this era of precision cosmology, the CMB continues to agree remarkablywell with predictions, and stands as a true pillar of cosmology. Because of the remarkable agreementbetween the CMB and the standard model of cosmology, any new physics that we might introduce must40not interfere too much with this spectrum. For example, increasing the amount of radiation energy inthe early Universe (through the parameter Ne f f , the effective degrees of freedom in the neutrino sector)will delay the onset of matter-radiation equality, damping the peaks in the CMB anisotropy oscillations.We will use the stringent behaviour of the CMB to place limits on new models in future chapters.Structure FormationFollowing the formation of the CMB, matter over-densities will continue to evolve. During the radiationdominated epoch, these perturbations will be partially damped by pressure waves in the radiative bath.However, as the over-dense regions grow large enough, they overcome the radiative effects, growinglogarithmically with the scale factor during this epoch [71]. After matter-radiation equality, the over-densities continue to grow, but now linearly with scale factor. For density modes that have only justrecently entered into the cosmic horizon, the growth will be slightly altered due to the presence of thecosmological constant. In a low mass-density Universe, the growth of over-densities is reduced to:d lnδd lna∼Ωγm (2.74)where δ is the matter over-density, and the gamma parameter is ∼0.55, independent of vacuum density[118]. Thus, perturbations grow fastest during matter dominated epochs. This is strong evidence fordark matter being ‘cold.’ If dark matter was hot, or relativistic, for a long enough time, it would takelonger for the Universe to reach matter-radiation equality, thus delaying the start of structure formation,at which point we would not expect to see the structures that we do today. This actually implies a lowerbound on the mass of thermal dark matter [119]:mX >∼ keV (2.75)Eventually, these densities will grow non-linear, and the full structure of the Universe will begin to berealized. The earliest stars will form, expected to be extremely large and bright Population III stars[120, 121]. The furthest galaxy yet recorded has been at a redshift of z∼ 11 [122], signalling the onsetof structure formation at around 105−6yrs.ReionizationAfter the earliest stars and galaxies have formed, the Universe may go through a period of reionization.In this epoch, the non-thermal light from stars, quasars, early supernovae, etc., will travel through neutralHydrogen that has not collapsed yet, partially reionizing the matter. This reionization shows up in theCMB today as an increase in the damping of small scale anisotropies and peaks. The presence of thisastrophysical effect shows up in the measurement of the reionization optical depth from CMB data,which is taken as a fundamental parameter that must be measured. The optical depth, τ is a measure ofthe mean free path of photons between now and reionization, and causes the amplitude of the anisotropypeaks to be damped by e−2τ [123].41This can be used as a useful tool if we wish to constrain new physics. Any new particles that injectenergy into the inter-galactic medium (IGM) during the cosmic ‘dark ages,’ while photons are free-streaming and the ionization fraction is low can alter the reionization optical depth, which in turn wouldappear in the temperature spectrum of the CMB. This can occur if particles decay or annihilate to SMparticles during this epoch. Although the effect may persist throughout the cosmic dark ages, ionizationof the IGM has the strongest effect on the CMB if energy injections actually occur near recombination,ionizing neutral hydrogen and broadening the last scattering surface. As the CMB is so well measuredand understood, this can place strong constraints on the types of new physics that may be allowed.The Present DayAfter reionization, we finally arrive at the present day. At this point, structure formation has successfullybuilt up a vast network of galaxies, galactic clusters, filaments, and voids to create the enchantingUniverse that we live in. As recently as redshifts of z ∼ 0.3, the Universe is expected to have becomedominated by the cosmological constant, Λ. During this current epoch, there are a variety of differentmethods that may be used to constrain new physics, as we can physically look out and search for signalsthat are being created today. These include looking at gamma ray bursts from high density regions,searches for weakly-interacting massive particles (WIMPs) in various experimental setups, and evenmore recently using gravitational waves to limit new models. Thus we have now seen the evolution ofthe Universe, and have seen many different tools that we can use to constrain new physics. In the bulkof this work, we will focus on constraints that pertain to BBN and the CMB, although when applicablewe will consider others as well. In particular we are interested in how new physics in a dark sector willinteract with the rest of the SM. This dark sector may contain DM, which we will discuss in more detailnow.2.5 Dark MatterOver the last few decades, evidence for the existence of some source of non-luminous matter has becomeoverwhelming. We discuss here the evidence for such dark matter, as well as possible candidates andtheir production mechanisms. Excellent references and reviews can be found at [124–127].2.5.1 EvidenceEvidence for dark matter began to amass as early as the 1930s. Some of the earliest and most convincingevidence comes from the velocity dispersion of galaxies. This was first observed in the Coma cluster,which had galactic velocities within the cluster not supported by the amount of luminous matter alone[128]. This was further reinforced by the measurement of galactic rotation curves [129–131]. We mayconsider a single object that is orbiting at some radius, r, from the center of a galaxy. Assuming sphericalsymmetry and a stable Keplerian orbit, the velocity of this satellite should be:v(r) =√GM(r)r(2.76)42where M is the amount of mass inside the orbit. Observations of spiral galaxies indicate that most ofthe luminous matter lies within the central bulge, and so, if this makes up the bulk of matter, velocitiesshould be ∝ 1/√r. However, galactic rotation curve observations show that the velocity is roughlyconstant at large r, implying that M ∝ r in this region, or ρ ∝ r−2. This cannot be explained by thevisible matter alone, and so the standard adopted theory is that of a spherical dark matter halo, withinwhich the baryonic matter of stars, dust, and gas is embedded.The dark matter distribution of galaxies, clusters, and even larger scales can also be mapped usinggravitational lensing. This technique uses background galaxies to map the density of matter by corre-lating distortions in their images to the amount of matter present in the foreground. One of the mostfamous examples of this is the Bullet Cluster, which is the remnants of the collision of two clustersin recent (cosmological) times [132]. The result of the collision was the collapse of baryonic matteras particles interacted and dissipated kinetic energy as they fell into the gravitational well. This canbe measured directly using x-ray emissions from the hot gases. However, when gravitational lensingis used, the map clearly shows that the bulk of matter associated with the cluster is separate from thevisible matter, having not felt the strong dissipative effects associated with the baryons.8These astrophysical observations are supported by many other independent sources of proof. Mostof these we have discussed already in Section 2.4, and come from a larger cosmological scale. Forexample, BBN and the CMB together put strict constraints on the overall abundance of baryons that canbe present to produce both the abundance of light elements we see today, and the peaks of the oscillationsin the CMB temperature anisotropies. However, the CMB also needs a specific overall matter densityso that the correct scales are sub-horizon at the time of recombination, to match the observed patternswe see today. Between these two values, we can place stringent limits on both matter densities [67]:Ωch2 = 0.1198±0.0012 (2.77)Ωbh2 = 0.02233±0.00015 (2.78)where Ωb is the baryon abundance and Ωc is the excess abundance required by cold dark matter. Thisimplies that we need 5-6 times more dark matter than we have baryons, a ratio that is in agreementwith the amount required by velocity dispersions in galaxies. This is also in agreement with structureformation arguments for dark matter. The presence of early universe galaxies is only possible if non-linearities have had time to grow before the redshift of galactic observations. Because we know fromthe CMB that the density perturbations were still very linear at the time of recombination, there needs tobe an extended period of time between then and galactic formation in which matter over-densities cangrow faster. Because these over-densities grow linearly (vs logarithmically) during matter dominatedepochs, and cold dark matter would push the matter dominated epoch back further, the presence of suchold galaxies is once again compelling evidence for cold dark matter.While there is extremely compelling evidence for the existence of DM on scales the size of galaxies8In fact, this is strong evidence that dark matter can only be weakly self-interacting, otherwise the dark matter would nothave remained separated as the clusters collided. This can be used to place limits on the mass and self-interactions of darkmatter candidates, as we shall consider in later chapters.43and larger, there are still some troubling conflicts between astrophysical observations and simulations onsmaller scales. These challenges to the DM hypothesis include the missing satellite problem, cusp-vs-core, diversity, and the ‘too-big-to-fail’ ([133, 134]) problem. In the missing satellite scenario, N-bodysimulations of DM in galaxies tend to predict far more substructure and satellite/sub-haloes than whatwe actually observe [135]. Similarly, these same simulations often predict ‘cuspy’ profiles for the darkmatter, with the density scaling like 1/r near the core of galaxies. However, galactic observations includemany cored inner profiles, with densities nearly constant over the core [136]. This ties into the diversityproblem, which implies that there should be remarkably little scatter in density profiles for a given halomass. In nature, this does not seem to be evident, as there seems to be a large scatter in the profiles ofobserved galaxies [137]. Finally, the ‘too-big-to-fail’ problem is that the largest sub-haloes of the MilkyWay should be dense enough to host star formation and their own observable galaxies, but this is notobserved [138]. All of these issues stem from the overabundance of substructure in dark matter structuresimulations. There is ongoing work to try to solve these issues: these include incorporating baryons orself-interactions of dark matter into the simulations [137, 139, 140].2.5.2 Candidates and their Production MechanismsIn order to have a candidate for dark matter, it must satisfy multiple conditions. It must be stable oncosmological timescales, have extremely weak EM interactions, and have the correct relic abundance.There are many such candidates to be found in BSM models, including those models which we havealready discussed when attempting to solve other problems in the SM. For example, the axion, originallypostulated to solve the strong CP problem, may produce DM via the oscillation of the axion field aroundits minima. These coherent oscillations redshift the same way as matter, and could have the correct relicabundance if the free parameters of the model are tuned appropriately [125, 141]. Supersymmetry,proposed to solve the hierarchy problem, may also produce dark matter candidates, if there is an exactR-parity that forces the lightest super-partner (LSP) to be stable [142]. Little Higgs models also produceviable DM candidates if a T-parity is imposed, such that the lightest T-odd particle is stable [125].Finally, sterile neutrinos (proposed both to give mass to SM neutrinos and perhaps solve the baryonasymmetry) could also be DM candidates, with the added bonus that light sterile neutrinos could makeup a small portion of warm dark matter that could help to alleviate the core-vs-cusp problem [143, 144].To address the question of whether or not these (and many other) candidates have the correct relicyield, we must consider the various production mechanisms. These fall into two broad categories:thermal and non-thermal production.Thermal ProductionIn thermal production of dark matter, the new BSM candidate is in thermal equilibrium with the SM athigh temperatures. This is the standard WIMP scenario, which has very weak couplings to the SM, butlarge enough to equilibriate at high temperatures. WIMPs will typically have interactions of the formχχ ↔ SM SM, where χ is the DM candidate. As the temperature cools, the reaction rates keeping thisequilibrium will fall below the Hubble rate, causing the particle to freeze-out of equilibrium, at which44YxYeq� � �� �� ��� �����-����-����-���-���-������� = ��/�� �=� �/� <σ�>Figure 2.3: Generic freeze-out of a WIMP, the most commonly considered production mechanismfor dark matter. Going to lower branches is equivalent to moving to larger cross-sections.point it will evolve independently with constant yield, as defined in Eq. (2.36). The numerical solutionfor such a yield is shown in Fig. 2.3. This constant yield can be estimated semi-analytically in the caseof WIMPs. We can begin at Eq. (2.49), where the standard lore is to use the relic yield, Yx, instead ofparticle number, and an inverse temperature (x=mx/T ), where mx is the DM candidate mass, instead oftime, as the evolution variable. Recast in these variables, the freeze-out interaction Boltzmann equationbecomes:dYxdx=− xsH(mx)〈σv〉(Y 2x −Y (0)2x ) (2.79)We can estimate the temperature at which the particle freezes out by equating its interaction rate to theHubble rate:H ' 〈σv〉n(0)x (2.80)where we can use the equilibrium number density as an approximate solution before freeze-out. Solvingthis equation for an arbitrary partial wave (Eq. (2.48)) in the cross-section, we find the approximatefreeze-out temperature [70]:x f o ' ln[√908pi3gxg1/2∗mxMPLσn]− (n+1) ln(ln[√908pi3gxg1/2∗mxMPLσn])(2.81)where n is the power of the temperature in the partial-wave being considered and gx the degrees offreedom of the DM particle. This corresponds to x f o ∼ 20 for a large range of masses, as there is only alogarithmic dependence on the parameters involved. Following freeze-out, the yield will stop following45equilibrium, and Eq. (2.79) can be solved exactly, as Yx  Y (0)x . Doing this results in an approximatesolution for the final yield of:Y∞ ∼ 452pi√90g1/2∗g∗sn+1MPLmxσnxn+1f o (2.82)or in dimensionless units (for n=1):Ωxh2 = mxYxs0ρC∼ 0.1(3×10−26cm3/s〈σv〉)(x f o20)(2.83)A new particle with weak-scale couplings, such that〈σv〉 ∼ g4(4pi)21m2x∼ 10−23cm3/s(100 GeVmx)2(2.84)has nearly the exact cross-section required for the correct relic abundance for masses at the weak scale.9This is known as the WIMP miracle, and has driven the search for WIMP-like DM for many years.Although many thermal interactions will follow this approximate form, there are some deviationsthat must also be considered. These become important when the DM candidate is interacting with otherstates present in the plasma at the time of decoupling. These effects include coannihilations, in whicha species with a similar mass efficiently depletes the DM density, and resonant enhancements if themass of the DM particle is close to that of a mediator [145]. Sommerfeld enhancement due to thepresence of a light boson mediator will also cause changes to the present day cross-section (relative tothe cross-section at freeze-out), which has significant implications when constructing detectors [146].Finally, another thermal method to produce dark matter may be through a 3↔ 2 self-interaction ofthe DM candidate [147]. If kinetic equilibrium with the visible sector is maintained, these so-calledself-interacting massive particles (SIMPs) require large self-interactions in order to deplete the yield toan acceptable level. We consider a variation of this production mechanism in Part IV, in which we donot require thermalization with the visible sector, but make use of this self-interacting number changinginteraction to set the relic abundance in a disconnected dark sector.Non-Thermal ProductionDark matter can also be produced via non-thermal mechanisms, in which the thermally averaged cross-section connecting the SM to DM is not the main contributing factor. As we briefly discussed above,axions are a classic example of a non-thermal dark matter candidate, as it is simply the coherent oscil-lations of the axion field that contribute to dark matter, and not the axion itself.Another non-thermal production mechanism includes gravitational production of WIMPZillas, su-perheavy DM states [148]. These are produced similarly to the inflationary generation of gravitational9Although there are a few orders of magnitude difference between the cross section required to get the correct relic densityand the naive weak-scale estimate, the fact that these two values are still so close is the true miracle. There are many sourcesof uncertainties, such as the true coupling values, form of the cross section, current Hubble rate, etc., that could easily pushthis closer to the desired value.46perturbations that seed the formation of large scale structure. Effectively, for very large masses, a verysmall number of particles must be produced. This is done via the decay of the inflaton, which effectivelyreheats the SM sector, which in turn can produce a very small amount of the WIMPZilla candidate. Thisis a specific case of a more general class of massive particle decays in the early Universe. If the originalmassive particle has very weak couplings to the SM, it may freeze-out of equilibrium very early witha large relic abundance. This could lead to an early period of matter domination, which could have in-teresting implications for the evolution of structure in the Universe. When the massive particle decays,the Universe goes through a (possibly secondary) period of reheating, as the massive particle transfersall of its energy to radiative energy density. This could partially transfer energy to ‘superWIMPs’ at thesame time, leading to a non-thermal production of the DM candidate [149, 150].There is also the possibility that dark matter is set by an asymmetry in the dark sector, similar to theunknown mechanism producing the baryon abundance [151]. This would require that the DM particle,ψ , have a distinct antiparticle, with which it could efficiently annihilate. If these annihilations are strongenough (typically much stronger than WIMP interactions), the symmetric densities will be washed out,leaving behind the asymmetric abundance of only ψ particles. This would have the unfortunate side-effect of being very difficult to detect today, as there would no longer be any annihilations occurring.However, it does have an appealing symmetry, in that the asymmetry setting the dark matter abundancecould also, in principle, be setting the baryonic abundance. This could happen if dark matter has aconserved charge related to baryon number. In this case, we get a nice relationship between the mass ofdark matter and baryonic matter. As there is ∼5 times more dark matter than baryonic, then we wouldexpect the masses to be ∼5 times higher as well.Finally, there is the possibility of freeze-in (FI) of dark matter. This corresponds to the productionof DM through a thermal scattering off SM particles that was too feeble to ever thermalize the twosectors [152], but strong enough to allow some energy-leakage into the hidden sector. Freeze-in canoccur if two conditions are met: the dark matter particle starts with a negligible density, nx  n(0)x ,and the particle interacts weakly with the SM so that thermodynamic equilibrium is never attained.When renormalizable operators govern the interaction, this is typically called feebly-interacting massiveparticle (FIMP) dark matter, as the coupling strengths must be much smaller than even a standard WIMPscenario. An excellent review of FIMPs can be found at Ref. [153]. Although renormalizable operatorshave been well studied, less emphasis has been placed on non-renormalizable operators [154]. We studyeffects associated with this explicit scenario in Part II.2.5.3 Observational StatusNow that we have built up models to explain the nature of dark matter (and other phenomena), we canattempt to observe this new physics. Within the context of dark matter, this is typically done in three dif-ferent ways: through direct detection, indirect detection, and collider searches, depicted schematicallyin Fig. 2.4. Direct detection attempts to measure the small recoil energy of a nucleus in undergrounddetectors that have scattered off the ambient dark matter present that makes up the galactic halo. Thesehave been mostly aimed at the detection of WIMP DM, as the thermal annihilation cross-section that47DM SMDM SMIndirect DetectionCollider SearchesDirectDetectionFigure 2.4: Three fundamental methods to detect dark matter.gives rise to the relic yield should also give rise to nucleon scattering interactions today. Although therehave been some hints at detections (see for example, [155, 156]), there have mostly been null results.An excellent review of the current status of direct detection experiments can be found in Ref. [26]. Al-though direct detection WIMP searches have historically focused on the mass range 1-1000 GeV, morerecent electron scattering experiments have aimed to explore the low mass regime [157], while the upperbound has been reconsidered in multi-scatter models [158].While direct detection considers the scattering cross-section, indirect detection looks explicitly forthe flux of final stable particles produced by annihilations and decays of dark matter. Gamma-rayobservations of dwarf galaxies and the Galactic center, which should have high densities of dark matter,provide robust limits on the cross-sections and masses of dark matter that could be present. Althoughthere have been some reported excesses (such as the Galactic center excess [159], the 3.5 keV line inclusters [160], an antiproton excess [161], and a positron excess [162]), it is not clear yet whether theseexcesses are BSM in nature, or if they can be explained by more natural astrophysical sources. A reviewof the current status of indirect detection experiments and limits can be found in Ref. [163].A more direct way to attempt to see dark matter is at colliders, where we do not rely on the ambientpresence of dark matter on galactic scales, but rather attempt to produce it via the collision of SMparticles. There has been a great deal of work in constraining dark matter in this way, with more nullresults complementing the direct detection searches [164–166].The majority of these DM searches hinge upon having a strong enough interaction in Fig. 2.4 fora detection to occur. This has lead to the search being mostly focused on WIMPs, as they have themost appealing interactions to search for today. The lack of evidence for WIMP DM thus far has leadto growing interest in the non-thermal production mechanisms listed above. With the lack of WIMPDM, we have also had many null results in the search for physics to explain any of the other currentproblems in the SM. Because of this, combined with an unprecedented era of precision Cosmology, wehave begun to place more indirect constraints on new physics using all of the different Universal epochswe discussed above. These constraints are complementary to those provided by direct detection and48Visible Sector Dark SectorRealization, Part IVProduction, Part IIObservation, Part IIIFigure 2.5: Visualization of the connections possible between the visible and dark sectors.colliders, and provide invaluable information as we attempt to narrow down the possibilities for newphysics in our Universe.2.6 Beyond the Standard ModelsNow that we have built up a solid foundation of well understood physics, between the Standard Model ofparticle physics and the Standard Model of cosmology, we wish to learn how new physics will interactwith these models. As we have seen, there are problems that cannot be explained by the SM, includingthe nature of DM, the hierarchy problem, and more. This has motivated many different possible exten-sions, with each problem spawning a plethora of unique solutions. However, well motivated physicsmodels need to do more than just answer the questions they were proposed to solve. In particular, newmodels must not contradict currently known and well understood effects in other areas. In this thesis,we study exactly how new physics may interact with old, from a variety of angles. Concretely, we shallfocus on dark (or hidden) sectors, that may contain dark matter, decaying dark particles, or more. Thesedark sectors may have weak, feeble, or non-existent interactions with the SM. The very nature of thesefeeble interactions suggest that we must move beyond the conventional collider searches, and so weapproach this problem using cosmological observables as our basis for constraining new physics. As weare now in an era of unprecedented precision cosmology, these observables will prove to be invaluabletools moving forward.This is depicted schematically in Fig. 2.5. Within each sector, there will be interactions between allthe particles present, while at the same time energy may flow between the two sectors during differenteras of the Universe, complicating the responses. In this thesis, we look to break this down, and focuson different components of these interactions in specific, realized scenarios. In Part II, we will startwith the transfer of energy from the visible to the dark sector only, and see how this energy inflow willhave a role as the dark sector continues to evolve. Following this, in Part III, we will look at the reverseprocess, and consider how energy flowing from the dark to the visible sector will have consequences forthe evolution of SM particles and their abundances, which we can physically observe and thus constrain49this interaction. Finally, in Part IV, we will consider a fully realized model of a dark sector, that couldpossibly contain energy transfer in both directions, as well as a complex network of interactions in thedark sector. In breaking it up this way, we identify important information along each of the variousarrows within Fig. 2.5, that can be applied to more models than just those considered here.50Part IIFrom the Visible to the Dark51Chapter 3Infrared Effects of Ultraviolet Operatorsand Dark Matter Freeze-In3.1 IntroductionWe begin our exploration of the relationship between the visible and dark sectors by considering theeffects that the visible sector can have on the hidden, which we schematically depict in Fig. 3.1. Inthis scenario, the visible sector will directly inject energy into the dark sector, which will concurrentlybe interacting with itself. These dark interactions may have self-interactions or couplings between thedifferent components of the hidden sector. The relationship and interplay between the visible sector andthe interactions of the dark sector are what we wish to explore.This relationship is most evident when the visible sector is used to directly produce dark particles,such as dark matter (DM). It has long been known that the SM does not provide a complete descriptionof the universe, with a key missing element being DM. This is a pressing issue, as DM has been observedcosmologically to make up the majority of matter today [67]. However, very little is known about DMbeyond its gravitational influence, such as its particle properties or how its density was created in theearly universe [124, 142, 167]. In this chapter, we focus on the latter in our effort to explore how thevisible sector can influence a dark sector, even (especially) if the coupling between the sectors is smallcompared to the individual sector’s self-interactions1.Many theories of DM coupled directly to the SM rely on thermal production, with the most-studiedparadigm being thermal freeze-out, as discussed in Part I [70, 168, 169]. This simple mechanism forDM production has many attractive features: it is insensitive to the state of the very early universe, andit yields the correct relic abundance (to within a couple orders of magnitude) for a generic WIMP withmass near the weak scale[170].Despite these enticing features of thermal freeze-out, the lack of discovery in direct detection ex-0This chapter is based on L. Forestell and D. E. Morrissey, Infrared Effects of Ultraviolet Operators on Dark MatterFreeze-In, [arXiv:1811.08905] [1].1Here, I do not necessarily mean the standard definition of self-interaction in which a single particle interacts with itself,but rather all the reactions between the (possibly many) different particles that only live in one sector or the other.52VS: H DS: ψ , XµFreeze-InInteractionFigure 3.1: Flow of information considered in this chapter. The visible sector transfers energy andnumber density to the dark sector, which may go through further self-processing effects thatinterplay with the inflow of energy. In this chapter, the visible sector provides an interac-tion via the Higgs boson, while the dark sector consists of a fermionic DM candidate and amassless vector boson.periments and collider searches for WIMPs has motivated the study of other DM production mecha-nisms [171, 172]. As was mentioned previously, a promising alternative is freeze-in (FI) [152], in whichthe DM species is assumed to interact only very feebly with the SM and to have an initial abundancewell below the value it would obtain in equilibrium with the SM plasma. Transfer reactions of theform SM+SM → DM+DM then create a sub-equilibrium abundance that evolves to the DM densityseen today. The feeble coupling together with this sub-equilibrium abundance means that FI is an idealenvironment for studying the visible to dark sector connection, as it implies a one-way connection fromvisible to hidden, while the reverse is generally not true. Freeze-in can arise as a production mecha-nism in sterile neutrino models[173, 174], as well as the production of gravitinos[175–178], and fullyunderstanding this method of production will be useful for many future avenues of BSM models.Within this paradigm, there are two general classes of connectors between DM and the SM withvery different cosmological behaviours, both of which are shown in Fig. 3.2. The first and most studiedhas DM connected to the SM through a renormalizable operator with small coupling. Production ofDM for this class is dominated by temperatures near the DM mass, T ∼ mψ [152, 153, 179–188]. Forthis reason, it is usually categorized as infrared (IR) freeze-in. This FI mechanism retains much of theattractive insensitivity to initial conditions as WIMP freeze-out aside from the assumption of a verysmall initial DM density. On the flip side, the renormalizable couplings needed for IR freeze-in must beextremely feeble, which would need further explanation if one wishes to retain naturalness.The second class of connectors leading to FI are non-renormalizable operators connecting the DMto the SM, whose interaction strength is naturally very small at low temperatures. Dominant DM pro-duction typically occurs at the highest SM temperatures attained during the radiation era, such as thereheat temperature TRH post inflation[189]. Because of this early temperature dependence, this is whatis called ultraviolet (UV) freeze-in [154, 189–192], with a well-known example being the gravitino inSUSY [193–196]. A less attractive property of this paradigm, is that the DM abundance depends onthe state of the universe very early in its history. However, because of the natural mass suppression thatarises due to the dimensionality of these operators, no further feeble (unnatural) coupling constant isrequired.In this chapter we demonstrate that both UV and IR freeze-in can play a role in determining theDM relic abundance through a single, non-renormalizable connector operator. This contrasts with thestandard expectation that non-renormalizable operators decouple once and for all at higher temperatures.53���� ���� � �� �����-����-���-���-���-�� = ��/�� �=� �/�������-����� ������-���� ������-��Figure 3.2: General behaviour for the yield Y = nx/s of various production mechanism for darkmatter. The red curve shows the classical freeze-out behaviour (with the dotted line follow-ing equilibrium), while the blue and green curves show UV and IR freeze-in, respectively.Freeze-out occurs near x∼ 20, while IR freeze-in is dominant around x∼ 1 and UV freeze-inoccurs almost completely at xmin ∼ xRH  1.We illustrate this feature in a concrete dark sector model consisting of a stable Dirac fermion ψ withmass mψ that is charged under an unbroken U(1)x gauge force with vector boson Xµ and couplingstrength αx = g2x/4pi . The only connection between the dark sector and the SM is assumed to be throughthe fermionic Higgs portal operator[197, 198],−L ⊃ 1M|H|2ψψ . (3.1)Here, M defines a very large mass scale of new physics above the energy and temperature ranges weconsider. This can be done using a variety of UV completed models, as described in Refs. [199, 200].For example, a singlet scalar S can interact with the Higgs via a renormalizable coupling. If we alsoinclude a Yukawa coupling between S and the fermionic DM, the scalar S can be integrated out to leaveus with the Higgs portal. This is shown schematically in Fig. 3.3. Although the mediator creating theHiggs portal is not of interest in this work, it is still interesting to note that direct searches can be donefor the mediators, such as discussed in Ref. [197, 200].Note that we assume no gauge kinetic mixing between U(1)x and hypercharge, which can be en-forced by an exact charge conjugation symmetry in the dark sector [201]. We also do not considerthe pseudoscalar partner to this operator, 1/M|H2|ψ¯iγ5ψ . This is done by imposing CP symmetry.Although higher order operators could be considered, such as the dimension-6 term that comes fromthe four-fermion interaction, L ∼ 1/M2(Q¯LγµQL + q¯RγµqR)(ψ¯γµψ), these will all be suppressed by(mψ/M)2(d−4). Because of this factor, the lowest dimension (i.e. Higgs) portal will provide the strongesteffects, and thus we focus here in order to highlight the visible to hidden effects.54Figure 3.3: Feynman diagrams for a possible UV completion of the Higgs portal. The left diagramis the full UV model, mediated by a scalar particle. On the right, the scalar particle has beenintegrated out, and its propagator and couplings combine to create a new coupling, withapproximate strength 1M .The UV connector operator of Eq. (3.1) can generate both UV and IR freeze-in effects over a broadrange of parameters when three plausible conditions are met. First, reheating after inflation is assumedto populate only the SM sector with visible reheating temperature TRH well below the connector massscale M. The dominant source of dark sector particles then comes from visible-to-dark transfer reac-tions through the connector operator (UV freeze-in). Second, for moderate to large values of the darksector gauge coupling αx the dark sector can self-thermalize to a temperature Tx less than the visibletemperature T but greater or similar to the dark fermion mass mψ . And third, if the DM annihilationcross section is sufficiently large the DM abundance can track the equilibrium abundance (at temper-ature Tx < T ) for long enough that transfer reactions from the non-renormalizable connector operatorreturn as the dominant contributor to the DM abundance. This allows us to further examine the under-lying behaviour of a dark sector that is directly being influenced by an inflow of energy from the visiblesector, while also providing a counterexample to the standard lore that non-renormalizable operatorsdecouple in the early universe.The combined UV and IR freeze-in behavior we focus on in this chapter is only one of a number of“phases” of freeze-out and freeze-in possible within this dark sector model. These phases are analogousto the four phases studied in Ref. [182] for a similar dark sector consisting of a charged complex scalarDM particle connected to the SM Higgs field through the standard renormalizable Higgs portal operator,but tilted towards the UV. When the mass scale M in the fermionic connector operator of Eq. (3.1) islarge relative to the weak scale and αx → 0, the theory reduces to standard UV freeze-in of ψ darkmatter as studied in Ref. [191] with no significant dark self-thermalization or later annihilation. Incontrast, for much smaller M near the TeV scale the dark and visible sectors are thermally coupled (viathe connector) throughout ψ freeze-out, and this operator can control the freeze-out process even whenαx is very small [197, 202, 203]. We focus on the scenario between these relative extremes with largerM and αx.This chapter is structured as follows. Following the introduction, we discuss in Sec. 3.2 the UVfreeze-in transfer of number and energy density through the connector operator of Eq. (3.1) as well asdark self-thermalization. Next, in Sec. 3.3 we compute the interplay between freeze-out and IR freeze-in in determining the relic abundance ψ particles and determine the conditions under which both UVand IR freeze-in can be relevant. In Sec. 3.4, we comment briefly on the astrophysical implications of55the new dark force from DM self-interactions. Finally, Sec. 3.5 is reserved for our conclusions. Sometechnical details related to thermally-averaged cross sections and the calculation of freeze-in transferrates are contained in Appendix A. This chapter is based on work published in Ref. [1] in collaborationwith David Morrissey.3.2 Populating the Dark Sector through UV Freeze-InWe begin by investigating the transfer of energy and number density to the dark sector by UV freeze-inthrough the connector operator of Eq. (3.1). For this, we make the standard freeze-in assumption thatonly the visible SM sector is populated significantly by reheating after inflation with reheating tem-perature TRH  M [152, 154].2 The dark sector is then populated by transfer reactions of the formH +H† → ψ +ψ (assuming unbroken electroweak) mediated by the operator of Eq. (3.1). Once thenumber density ofψ grows large enough, the dark sector may also thermalize to an effective temperatureTx through further reactions such as ψ+ψ↔ Xµ +Xν . In this section we study the creation of ψ parti-cles from SM collisions during and after reheating as well as the conditions for the self-thermalizationof the dark sector.3.2.1 Transfer without the Dark VectorIt is convenient to study first the creation of ψ fermions by SM collisions in the absence of dark vectors(αx→ 0) [191]. The number and energy transfer via H +H†→ ψ+ψ is described bydnψdt= −3Hnψ −〈σtrv(T )〉(n2ψ −n2ψ,eq(T )) (3.2)dρxdt= −3Hρx−〈∆E ·σtrv(T )〉(n2ψ −n2ψ,eq(T )) (3.3)where ρx is the total energy density in the dark sector and ∆E is the energy transfer per collision.Starting with number transfer, in the limit of nψ  nψ,eq and T  mψ the collision term is approxi-mately−T (T ) =−〈σtrv(T )〉(n2ψ −n2ψ,eq(T )) '14pi5T 6M2. (3.4)Details of the calculation are given in Appendix A. Assuming radiation domination up to the reheatingtemperature TRH  mψ , this gives the simple solution for the yield of ψ (and ψ) ofYψ(T ) ' Yψ(TRH)+Yψ,eq(T )√5/22ζ (2)pi4g−1/2∗MPlTRHM2[1−(TRHT)−1]. (3.5)This solution only holds in the limit Yψ  Yψ,eq, corresponding to a consistency condition of (for2Obtaining such an asymmetric reheating between different sectors has been studied recently in Refs. [204, 205].56Yψ(TRH)→ 0 and T  TRH)3TRH  2ζ (2)pi4√5/2g1/2∗M2MPl. (3.6)Larger reheating temperatures imply thermalization between the dark and visible sectors at reheatingwith Yψ(T )→ Yψ,eq(T ) for T ∼ TRH . In this chapter we focus on the non-thermalization scenario.Turning next to energy transfer, the transfer term is computed in Appendix A and for mψ  T Mand nψ  nψ,eq reduces to−U (T ) =−〈∆E ·σtrv(T )〉(n2ψ −n2ψ,eq(T )) '32pi5T 7M2. (3.7)Solving as above, we find(ρxρψ,eq)'(ρxρψ,eq)TRH+180√107pi8g−1/2∗MPlTRHM2[1−(TRHT)−1]. (3.8)Again, this is only valid for Yψ  Yψ,eq. For sufficiently large TRH , ρx→ ρψ,eq(T ) at T ∼ TRH .Comparing Yψ and ρx found above for Yψ Yψ,eq, we see that the mean momentum of the fermionsproduced near reheating is on the order p ∼ TRH . At later times, these momenta simply redshift as1/a provided T  mψ . Indeed, the detailed analysis of Ref. [191] shows that (in the absence of darkvectors) the dark fermions obtain an approximate Bose-Einstein distribution with effective temperatureTx ' (1.155)TRH(aRH/a).3.2.2 Thermalization with the Dark VectorNow that we have considered the effect that the visible sector can have on our dark particle, let usinclude a dark vector boson Xµ coupling to ψ with strength αx = g2x/4pi . This interaction allows thedark fermions to scatter with each other, annihilate to vector bosons, and emit vectors as radiation,as shown in Fig. 3.4. If these reactions are strong enough, the dark fermion and vector species canthermalize with each other to yield an effective temperature Tx ≤ T .The self-thermalization of heavy dark particles coupled to a massless dark vector was investigatedin Refs. [182, 189]. As in these works, we only make parametric estimates of the very complicated fullthermalization processes. We identify self-thermalization in the dark sector with the conditionΓth(Tth) = H(Tth) , (3.9)where Γth is an effective thermalization rate to be discussed below and this relation defines the visiblethermalization temperature Tth implicitly. Note that Tth ≤ TRH , and we set Tth = TRH if Γth(TRH) ≥H(TRH).3The number and energy density produced through thermal transfer prior to reheating by the operator of Eq. (3.1) is a verysmall fraction of that produced at reheating [191, 192].57Figure 3.4: Interactions involved in the dark sector. The left hand S-channel (and the equivalentT and U channels) will contribute to the thermalization of the hidden sector. The right handT-channel (as well as a corresponding U-channel) will contribute to thermalization as well aseventual ψ freeze-out.It is convenient to classify the thermalization processes contributing to Γth into: i) 2→ 2 processeswith hard momentum exchange; ii) 2→ 3 inelastic processes together with 2→ 2 with soft momentumexchange. The first class includes annihilation ψ+ψ→ Xµ+Xν and hard scatterings such as ψ+ψ→ψ+ψ for which we estimate the rate to be [182]Γel(T ) ∼ pi α2xT 2nψ(T ) , (3.10)where nψ(T ) is the number density of ψ prior to dark self-thermalization. Using Eq. (3.5) (withYψ(TRH)→ 0), for T  mψ it is given bynψ(T ) ' 3√5/22pi6g1/2∗MPlTRHM2T 3 . (3.11)The second class of soft and inelastic processes was studied in Ref. [189] with the net resultΓin(T ) ∼ min{α3x nψ(T )µ2IR, α2x√nψ/T}, (3.12)where µIR an effective infrared cutoff given byµIR = max{√αxnψ/T , H, mψ}. (3.13)We take the full thermalization rate to be the sum of the hard and inelastic rates, Γth = Γin+Γel .If thermalization occurs with Tth mψ , a smaller number of ψ and ψ fermions with typical energyT are redistributed into a larger number of ψ , ψ , and Xµ particles in equilibrium with each other attemperature Tx. Treating the thermalization as instantaneous, the resulting dark sector temperature canbe obtained from energy conservation and the result of Eq. (3.8):Tx(Tth)Tth≡ ξ (Tth) '[180√1011pi8g1/2∗MPlTRHM2]1/4. (3.14)589 10 11 12 13 14 15 16 17 18 19log10(M/GeV)23456log10(mψ/GeV)9 10 11 12 13 14 15 16 17 18 19log10(M/GeV)234569 10 11 12 13 14 15 16 17 18 19log10(M/GeV)23456-4-3-2-10log10(ξmin)Figure 3.5: Minimum consistent values of ξ (TRH) in the M–mψ plane for αx = 10−1 (left),10−2 (middle), 10−3 (right). The black line indicates where ξ (TRH)→ 1 and our assumptionof non-thermalization with the SM breaks down.At later times, separate conservation of entropy in the dark and visible sectors impliesξ (T ) ' ξ (Tth)[g∗S(T )g∗S(Tth)· g∗S,x(Tth)g∗S,x(T )]1/3, (3.15)where g∗S(x) refers to the number of visible (hidden) entropy degrees of freedom.The analysis leading to the temperature ratio of Eq. (3.14) has three assumptions built into it, andtheir consistency implies maximal and minimal allowed values of ξ (Tth). First, the assumption of non-thermalization between the visible and dark sectors implies ξ (Tth) 1. Second, the validity of theeffective connector operator description of Eq. (3.1) requires TRH  M corresponding to a maximumvalue of ξ (Tth) . (10−3 MPl/M)1/4. And third, we have so far neglected the mass of the ψ fermion.Demanding that Tx(Tth)& mψ then leads to a lower bound on ξ (Tth) that we use to defineξmin ≡ mψTth . (3.16)This also defines an implicit lower bound on the thermalization temperature for given values of mψ , M,and αx, and correspondingly a lower limit on the reheating temperature TRH .In Fig. 3.5 we show the values of ξmin in the M−mψ plane for αx = 10−1 (left), 10−2 (middle), and10−3 (right). Larger M and αx and smaller mψ lead to smaller ξmin. The white regions in the upper leftcorners of the plots (bounded by black lines) have ξmin → 1 corresponding to thermalization betweenthe visible and dark sectors when dark self-thermalization is achieved. As stated above, in the analysisto follow we focus on the lower right region where this does not occur.3.3 Freeze-Out and Late Transfer in the Dark SectorIf the dark sector is populated by UV freeze-in and is self-thermalized at temperature Tx &mψ , the darkfermion will undergo freeze-out by annihilation to dark vectors when Tx falls below mψ . While freeze-59out in a dark sector with Tx T has been studied in Refs. [189, 206–208], we identify a qualitativelynew feature in the present context. Specifically, we show that the UV connector operator responsible forinitially populating the dark sector at reheating can drastically change the freeze-out dynamics at muchlater times. This interplay between visible effects and dark sector self-interactions are precisely whatwe wish to study in this chapter.3.3.1 Evolution EquationsThe evolution of the ψ dark fermion number density at Tx . mψ is described bydnψdt+3H nψ ' −〈σv(Tx)〉ann(n2ψ −n2ψ,eq(Tx))+ 〈σtrv(T )〉n2ψ,eq(T ) (3.17)In writing this expression we have assumed self-thermalization in the dark sector with Tx T and noasymmetry between ψ and ψ .The first term on the right side of Eq. (3.17) describes annihilation ψ+ψ→ Xµ+Xν with a thermalaverage at temperature Tx. The leading-order perturbative result for the cross section at low velocityis [209]σann,pv =pi α2xm2ψ. (3.18)However, the full cross section receives independent non-perturbative enhancements from the Sommer-feld effect [210–212] and bound state formation [206, 213]. The full cross section can be written in theform [206, 214]σannv = [Ssomm(v)+Srec(v)] σann,pv , (3.19)where v is the relative velocity andSsom(v) =2piz1− e−2piz , (3.20)Srec(v) = Ssom(v)293z4(1+ z2)2e−4z tan−1(1/z) , (3.21)with z = αx/v, and which have the limitsSi(v)→ 1 for v αx.The second term on the right side of Eq. (3.17) corresponds to transfer reactions of the formH +H† → ψ +ψ , and has all relevant quantities evaluated at the visible temperature T .4 An explicitexpression for this transfer term is given in Appendix A, which reduces to〈σtrv(T )〉n2ψ,eq(T ) ≡ T (T ) '14pi5T 6M2 ; T  mψ332pi4m2ψT4M2 e−2mψ/T ; T  mψ. (3.22)4Since Tx T , we can neglect the reverse reaction.60For Tx < mψ but T  mψ , the standard annihilation term in Eq. (3.17) receives an exponential suppres-sion in temperature while the transfer term is only suppressed by a power. We show below that this canallow the transfer term derived from a UV connector operator to play a significant role in the IR.3.3.2 Analytic EstimatesIt is instructive to estimate the relic density of ψ particles analytically to understand the effect of late-time transfer by the UV connector. To do so, we treat the annihilation cross section as being power-lawin velocity: 〈σannv〉 → σ0 x−nx where x≡ mψ/T and xx ≡ mψ/Tx = ξ−1 x.Freeze-Out Without the Transfer TermConsider first the relic density of ψ with no transfer term but a definite value of ξ  1. This can becomputed by a simple generalization [206–208] of the analytic freeze-out approximation of Refs. [70,73, 168, 169, 215]. Freeze-out occurs when the mass to dark temperature ratio isxx, f o ' ln[(0.192)(n+1)(gψ/g1/2∗ )MPl mψ σ0 ξ 2]− (n+ 12) ln(xx, f o) , (3.23)which can be solved iteratively for x f ox . This translates into an approximate relic density of5Ωψh2 ' (2.07×108 GeV−1)ξ (n+1)xn+1x, f o(g∗S/g1/2∗ )MPlσ0. (3.24)Relative to the freeze-out of a species in thermodynamic equilibrium with the visible sector with thesame mass and cross section, these relations implyx f ox ' x˜ f o+(2−1/x˜ f o) lnξ , Ωψh2 ' ξ(1+2lnξ/x˜ f o)Ω˜ψh2 , (3.25)where x˜ f o and Ω˜ψh2 are the values for these quantities if the species were thermally coupled to the SM.The most important change is a reduction of the relic density by a factor of about ξ  1.Freeze-Out With the Transfer TermLet us now include the transfer term from Eq. (3.17) in the evolution of the density of ψ . As Tx fallsbelow mψ , annihilation is expected to dominate and keep nψ close to its equilibrium value at temperatureTx. However, since the corresponding annihilation rate falls exponentially in this regime, it decreasesmore quickly than the Hubble and transfer rates, and thus the near-equilibrium regime ends when oneof these other rates catches up. We show here that late-time transfer reactions can significantly modifythe final ψ relic density when the annihilation rate meets the transfer rate before reaching Hubble.Define Tx,= to be the value of the dark temperature Tx that solves the equation〈σannv(Tx)〉n2ψ,eq(Tx) =T (Tx/ξ ) , (3.26)5Note that we use MPl = 2.43×1018 GeV, and the full DM relic density is the sum of equal ψ and ψ densities.61where T (T ) is the transfer rate given in Eq. (3.22). If the solution has T= = Tx,=/ξ < mψ , an approxi-mate expression for it isxx,= ' 12 ln(pi22g2ψ σ0M2 ξ 6)+(3−n2)ln(xx,=) , (3.27)which can be solved iteratively for xx,= provided it is greater than unity. When xx,= is greater than thefreeze-out temperature without transfer, xx, f o given in Eq. (3.23), the transfer operator does not signifi-cantly alter the ψ relic density. In particular, the condition xx,= > xx, f o implies that the evolution of theψ density is dominated by Hubble dilution rather than transfer for all xx > xx, f o since the expansion termdecreases less quickly than the transfer term in this regime. In contrast, transfer effects are importantfor xx,= < xx, f o.When xx,= < xx, f o, the transfer and annihilation terms in Eq. (3.17) can reach a balance with eachother for xx > xx,= until the Hubble term catches up. The number density of ψ is then approximatelynψ,=(Tx) '√T (Tx/ξ )σ0xn/2x (3.28)→ 12pi5/2m3ψ√σ0M2ξ−3 x−3+n/2x(Tx/ξ  mψ)(3.29)where the expression in the second line only applies for Tx/ξ mψ . Note that the density in this regimeis always greater than the equilibrium density nψ,eq(Tx), even when Tx/ξ < mψ . This follows logicallyfrom the fact that in this regime, the sourcing of new particles from the visible sector is compensatingfor the annihilations that are occurring, producing a net number density that is above equilibrium.If the balance regime is achieved, xx,= < xx, f o, it ends when the Hubble term in Eq. (3.17) catches upto the annihilation and transfer terms. This later decoupling corresponds approximately to the condition〈σannv(Tx)〉nψ,=(Tx) ' H(Tx/ξ ) . (3.30)Defining Tx,dec as the dark temperature that satisfies the relation above, an approximate solution forTx,dec/ξ  mψ isxx,dec '[(0.086)mψMPl√σ0g1/2∗ Mξ−1]1/(1+n/2). (3.31)The solution for Tx,dec/ξ .mψ is more complicated but can be obtained similarly. The final relic densitycan be written in a form very similar to standard freeze-out via Eq. (3.30):Ωψh2 ' (2.07×108 GeV−1)ξ xn+1x,dec(g∗S/g1/2∗ )MPlσ0. (3.32)Since nψ,=(Tx)> nψ,eq(Tx) we must have xx,dec > xx, f o whether or not Tx,dec/ξ is larger or smaller than62mψ , and therefore the relic density of Eq. (3.32) is bigger than the pure freeze-out result of Eq. (3.24).3.3.3 Numerical Results for Freeze-OutTo confirm the analytic estimates derived above and map out the parameter space of the theory, weperform a full numerical analysis of the dark matter freeze-out process. In Fig. 3.6 we show the evolutionof the relevant rates in the upper panels and the ψ density in the lower panels for αx = 0.1, ξ = 0.1,mψ = 104 GeV, and M = 1012 GeV (left) and 1015 GeV (right). The rate plots show the rates for Hubble,annihilation, and late transfer defined according toHubble = H(T ) , Annihilation = 〈σannv(Tx)〉nψ , Transfer =T (T )/nψ , (3.33)where nψ is the number density obtained from solving Eq. (3.17) and T (T ) is the transfer rate ofEq. (3.22). The value of M is smaller in the left panels of this figure, and late-time transfer becomesmore important. In the ψ number density plots, we show the densities in equilibrium (dashed line), andwith and without the transfer operator (upper and lower solid lines).Late transfer by the fermionic Higgs portal operator is seen to increase significantly the final relicdensity in the left panels of Fig. 3.6, while its effect is negligible in the right panels. The differencecorresponds to the larger transfer rate for M = 1012 GeV in the left panels versus M = 1015 GeV in theright. Following the rates for M = 1012 GeV, transfer is seen to catch up to annihilation before Hubbleleading to a regime of balanced rates and enhanced number density. In contrast, the Hubble rate catchesup to annihilation before transfer in the right panels with M = 1015 GeV and never plays a significantrole in the evolution of nψ .In Fig. 3.7 we show the enhancement of the relic density in the M–mψ plane for αx = 0.1 (left) and0.01 (right) with ξ = ξmin as computed previously. The contours in both panels indicate the relic densitywe find to the value that would be obtained without late transfer effects, Ωψ/Ωno−trψ . Late transferby the connector operator initially increases as M decreases and the transfer operators becomes moreeffective. However, as M continues to decrease we find a competing effect between the efficiency oftransfer and the increasing value of ξmin. As the dark and visible temperatures approach each other,transfer is more likely to occur while T → mψ and the effect becomes exponentially suppressed, asseen in Eq. (3.22). Transfer effects are also reduced at αx = 0.1 relative to αx = 0.01 due to the non-pertubative enhancements in the annihilation cross section at low velocities for the larger value of thegauge coupling.Ultimately, we are interested in the parameter space where ψ can make up all the dark matter. InFig. 3.8 we show the values of mψ for which this occurs as a function of M for αx = 0.1 (left) and0.01 (right) for various values of ξ (solid lines). The lines in these plots are cut off at smaller Mwhen ξ falls below ξmin. As expected from the annihilation cross section, larger values of αx coincidewith larger dark matter masses. In the right part of both panels the allowed DM mass mψ reaches avalue that is independent of M for fixed ξ . This region corresponds to late transfer being negligible forthe freeze-out process, with the relic density scaling approximately as ξ−1α2x /m2ψ . Going to smaller63����������������������������� � �� ��� ���� �����-����-����-���-���������(���)� = ��� ���� = ���� ���ξ = �/��α� = �������������������������������� � �� ��� ���� �����-����-����-���-���������(���)� = ��� ���� = ���� ���ξ = �/��α� = ������ � �� ��� ���� �����-����-����-����Δ������+������ = ��� ���� = ���� ���ξ = �/��α�= ������ � �� ��� ���� �����-����-����-����Δ������+������ = ��� ���� = ���� ���ξ = �/��α�= ���Figure 3.6: Evolution of the relevant rates in the upper panels and the ψ density in the lower panelsfor αx = 0.1, ξ = 0.1, mψ = 104 GeV, and M = 1012 GeV (left) and 1015 GeV (right).M, transfer eventually becomes important and the relic density increases. Correspondingly, the massmψ that produces the correct relic density decreases. As M decreases further, the lines for differentξ values in Fig. 3.8 come together. This can be understood from Eqs. (3.31) and (3.32), where thedirect dependence on ξ is seen to cancel for cross sections 〈σannv〉 = σ0x−n with n→ 0, as we havehere (up to the Sommerfeld and bound state enhancements). The upper shaded region in both panels isexcluded because the resulting relic density of ψ is always greater than the observed DM density for anyconsistent value of ξ . Going from αx = 0.1 to 0.01, lower ψ masses are needed to produce the correctrelic density. Also shown in this figure are bounds from DM self-interactions to be discussed below.3.4 Dark Matter Self-InteractionsDark matter in our theory is charged under an unbroken U(1)x gauge force implying long-range selfinteractions among DM particles that can modify their behavior in collapsed systems. Such interactionshave been suggested as a way to resolve several apparent discrepancies between simulations of DMstructure formation and observations [137, 216]. However, these interactions are also constrained to not64���������������Ω � Ω ���-��Figure 3.7: Enhancement of the ψ relic density due to late transfer effects relative to the valuewithout this effect, Ωψ/Ωno−trψ for αx = 0.1 (left) and 0.01 (right) and ξ = ξmin.be so large as to overly disrupt cosmic structures [217, 218].An upper bound on DM self-interactions can be derived from the observed ellipticity of galactichalos such as NGC720 [219, 220]. For charged DM coupled to an unbroken U(1), Refs. [206, 208]derived limits on the gauge coupling of the formαx . {0.35, 2.5}×10−6( mψGeV)3/2, (3.34)where the two numbers in brackets correspond to the analyses of Refs. [206] and [208], respectively.While the limit derived in Ref. [206] is considerably stronger, Ref. [208] (and Ref. [221]) argue for aweaker one based on the application of the ellipticity constraint only at larger galactic radii and a numberof smaller factors. We show both upper bounds on αx in Figs. (3.8). These favor smaller temperatureratios ξ and larger DM masses mψ , well above the weak scale.The limits on αx from the ellipticity of NGC720 correspond to an effective transfer cross sectionper mass below about σT/mψ . 1cm2/g in this system with a velocity dispersion on the order of v '300km/s. Dark matter self-interactions in this regime are described by a Rutherford-like transfer crosssection [206, 208, 221]:σT ' 8pi α2xm2ψ1v4lnΛ , (3.35)where lnΛ∼ 45−75 is a collinear enhancement factor cut off by the typical interparticle spacing in thesystem [208]. Since this cross section has a very strong velocity dependence, the DM self-interactionsin systems with lower velocity dispersions such as dwarf halos can be much stronger. Using typical65�� �� �� �� �� �� �� ������������(�/���)��� ��(�/���)ξ = �ξ = �/�ξ = �/��ξ = �/��ξ = �/���ξ = �/����ξ = ξ���Ω���= �����������������σ�/� = ���/���α� = ���T m = 10 cm2/g<latexit sha1_base64=" 1er1La8Pu8+FFdJPZzoyeuAJ9eA=">AAACG3icbVDL SsNAFJ3UV62vqEs3g0VwIW1SBbsRCm5cVugLmhgm02 k7dCYJMxOhhPyHG3/FjQtFXAku/BsnbQRtPTBw5px7 ufceP2JUKsv6Mgorq2vrG8XN0tb2zu6euX/QkWEsMG njkIWi5yNJGA1IW1HFSC8SBHGfka4/uc787j0RkoZB S00j4nI0CuiQYqS05Jk158yRdMSR16pyL3EiSVN4BW 1L6xypseAJ5uldrfrzG6WeWbYq1gxwmdg5KYMcTc/8 cAYhjjkJFGZIyr5tRcpNkFAUM5KWnFiSCOEJGpG+pg HiRLrJ7LYUnmhlAIeh0C9QcKb+7kgQl3LKfV2ZbSgX vUz8z+vHalh3ExpEsSIBng8axgyqEGZBwQEVBCs21Q RhQfWuEI+RQFjpOEs6BHvx5GXSqVXs84p1e1Fu1PM4 iuAIHINTYINL0AA3oAnaAIMH8ARewKvxaDwbb8b7vL Rg5D2H4A+Mz29GeKDw</latexit>�� �� �� �� �� �� �� ������������(�/���)��� ��(�/���)ξ = �ξ = �/�ξ = �/��ξ = �/��ξ = �/���Ω���= �����������������σ�/� = ���/���α� = ����ξ = ξ���T /m = 10 cm2/g<latexit sha1_base64="1er1La8Pu8+FFdJPZzoyeuAJ9eA=">AAACG3icbVDLSsNAFJ3UV62vqEs3g0VwI W1SBbsRCm5cVugLmhgm02k7dCYJMxOhhPyHG3/FjQtFXAku/BsnbQRtPTBw5px7ufceP2JUKsv6Mgorq2vrG8XN0tb2zu6euX/QkWEsMGnjkIWi5yNJGA1IW1HFSC8SBHGfka4/uc787j0RkoZBS00j4nI0CuiQYqS05Jk15 8yRdMSR16pyL3EiSVN4BW1L6xypseAJ5uldrfrzG6WeWbYq1gxwmdg5KYMcTc/8cAYhjjkJFGZIyr5tRcpNkFAUM5KWnFiSCOEJGpG+pgHiRLrJ7LYUnmhlAIeh0C9QcKb+7kgQl3LKfV2ZbSgXvUz8z+vHalh3ExpEsSIBn g8axgyqEGZBwQEVBCs21QRhQfWuEI+RQFjpOEs6BHvx5GXSqVXs84p1e1Fu1PM4iuAIHINTYINL0AA3oAnaAIMH8ARewKvxaDwbb8b7vLRg5D2H4A+Mz29GeKDw</latexit>Figure 3.8: Values of mψ that give the correct relic density of ψ dark matter as a function of M forαx = 0.1 (left) and 0.01 (right) for various fixed values of ξ . Each solid line corresponds tothe correct ψ relic density for the corresponding value of ξ . The red shaded upper region isexcluded due to overproduction of ψ relic density for any consistent value of ξ . The lowerblue shaded regions indicate exclusions from the effects of ψ dark matter self-interactionsfrom the observed ellipticity of galactic halos, with the dark blue indicating a conservativeexclusion and the light blue showing a more aggressive one. The dotted line indicates a DMself-scattering transfer cross-section per mass in dwarf halos of σT/mψ = 10cm2/g.velocities and densities for dwarf halos, this translates intoσT/mψ ' 18cm2/g( αx0.1)2(5×104 GeVmψ)3(10km/sv)4( lnΛ50)(3.36)Interaction cross sections of this size are expected to lead to the formation of cores in dwarf halos, withRefs. [222, 223] suggesting a better agreement between simulations and data for σT/mψ ∼ 10cm2/g.On the other hand, it is not clear what the upper bound on σT/mψ is from these systems, with thesimulations of Ref. [224] finding reasonable behavior for σT/mψ = 50cm2/g (the largest value studied)and Ref. [208] arguing that much larger values can work as well. Indeed, the results of Ref. [224] appearto be consistent with the approximate duality between σT/mψ and mψ/σT about Knudsen number closeto unity suggested in Ref. [208] based on the analyses of Refs. [225, 226]. For reference, we also showdashed contours indicating σT/mψ = 10cm2/g in Figs. ConclusionsThis chapter has focused on the interplay between a dark sector interacting with itself and small con-nections between the visible and dark sectors. As a first step into our exploration of the two sec-tors, this small connection only allowed for energy to flow from the visible to the hidden, and not66in reverse. Specifically, this was realized in a model consisting of a freeze-in transfer effect from anon-renormalizable connector operator together with a self-thermalizing hidden sector. The standardexpectation for non-renormalizable operators in the early universe is that their effects are greatest athigh temperatures and that they decouple at lower temperatures. For this reason, DM creation from SMcollisions connecting to a secluded dark sector through a non-renormalizable operator is referred to asUV freeze-in [152, 154]. In this work we showed that such operators can also contribute importantly atlower temperatures when combined with freeze-out in a dark sector.To illustrate the effect, we studied a concrete dark sector consisting of a massive Dirac fermion ψDM candidate and a massless Abelian dark vector Xµ , with the only connection to the SM through thedimension-five fermionic Higgs portal operator of Eq. (3.1). At the end of reheating, the dark sectorcan be populated by transfer reactions SM+SM→ ψ + ψ¯ mediated by the non-renormalizable portaloperator to a density below the value it would have in full equilibrium with the SM. As the universe coolsfurther, the population of dark fermions can equilibrate with the dark vectors at temperature Tx belowthe visible SM temperature T provided the dark gauge coupling and the initial fermion density are largeenough. Freeze-out occurs in the dark sector when Tx falls below the fermion mass mψ . For a broadrange of parameters in this theory, the relic density of ψ fermions can receive a significant additionalenhancement from late transfer reactions through the non-renormalizable portal operator during thecourse of the freeze-out process for T down and below the fermion mass. The UV connector operatorof Eq. (3.1) is therefore seen to play an important role in the IR.The dark sector theory we have considered here also has interesting implications for DM self-interactions, which are motivated by a number of puzzles in cosmic structure [137, 216]. Such in-teractions were investigated for this theory in Refs. [206, 208, 221] and suggest that to be viable largerDM masses and smaller temperature ratios ξ = Tx/T are required to avoid bounds from the observedellipticity of NG720. These bounds, and the dependence of the self-interaction cross section on theDM velocity, could potentially be softened by extending the theory to include a small mass for the darkvector [218]. The calculations presented in this work can be carried over to such a massive vector sce-nario provided its mass is much smaller than the decoupling temperature of the dark fermion so that itprovides a relativistic thermal bath during this process. Furthermore, the vector mass would also haveto be small enough to avoid too much vector boson DM [227, 228].While this work focused on a specific dark sector theory and non-renormalizable connector operator,it is expected to generalize to other connections. Specifically, a similar IR contribution from a UVoperator that produces the relic density of dark-sector DM is expected to occur as well for other darksectors or connector operators. For the effect to arise, the DM candidate in the dark sector must undergosignificant annihilation to allow the power-suppressed transfer reactions (relative to reheating) of theconnector operator to catch up. Other non-renormalizable connector operators can also lead to late IRtransfer contributions to the DM relic density, although initial estimates suggest that the effect becomesless important as the operator dimension increases. Late-time transfer of a symmetric density could alsobe relevant in scenarios of secluded asymmetric DM.Dark matter arising from a dark sector that is colder than the SM in the early universe has been67investigated in a wide range of scenarios of new physics [3, 4, 154, 187, 187, 206–208, 229–238]. Insome of these works, the dark temperature Tx is taken as an input to the calculation of the DM relicdensity without reference to how the dark sector was populated initially. Our results show that such anassumption is not always justified, and the nature of the connector operators that mediate transfer fromthe SM to the dark sector can play an important role in determining the relic density of DM.Now that we have thoroughly investigated the effect that the visible sector can have on the evolutionof a dark sector, we turn to the reverse effect. In particular, we will consider how the dark sector canleave a visible imprint on the relics of BBN.68Part IIIFrom the Dark to the Visible69Chapter 4Limits from BBN on Light Decays andAnnihilations4.1 IntroductionIn the previous chapter we explored in detail how the visible sector can play a role in the subsequentevolution of a dark sector. We now turn to the reverse scenario, and look at how dark particles can play arole in the visible. This turns out to be a very powerful technique. We can place stringent constraints onvarious new models of physics, because we can see the effects of this scenario explicitly through morethan just gravitational potentials. The flow of energy is once again shown in Fig. 4.1. In particular, wewill focus on low energy transfers that will ultimately affect the outcomes of an early universe process,Big Bang Nucleosynthesis (BBN).BBN is one of the most powerful probes of the very early universe [112, 239–241]. Over the courseof BBN, free protons and neutrons assemble into a handful of light elements [242–244]. Assuminga standard Λ cold dark matter (ΛCDM) cosmological history, the primordial abundances of these el-ements can be predicted using known nuclear reaction rates in terms of a single input parameter, theoverall baryon density. These predictions agree well with observational determinations of primordialabundances up to plausible uncertainties in astrophysical determinations and nuclear rates [81].1The success of BBN gives very strong evidence for the ΛCDM cosmological model up to radiationtemperatures near the MeV scale [248–250], which extends much earlier than other known tests [67].BBN also places stringent constraints on new physics beyond the Standard Model that injects energyinto the cosmological plasma or influences the expansion rate at early times. This includes the decaysof massive particles with lifetimes greater than τ ' 0.1s [94, 194, 196, 251–261], dark matter (DM)annihilation with an effective cross section near the critical value for thermal freeze-out [262–265], andany new thermalized species with mass below a few MeV [10, 266–268].Limits from BBN on the decays of long-lived massive particles have been studied in great detail [94,0This chapter is based L. Forestell, D. E. Morrissey, and G. White, Limits from BBN on Light Electromagnetic Decays,JHEP, 1901, (2018), 074, [arXiv:1809.01179] [2].1The extrapolated densities of 6Li and 7Li give a particularly acute puzzle in this regard [111, 245–247].70VS:4He, 3He,T, D, n, p,e, ν , γDS:XDecayAnnihilationInteractionFigure 4.1: Flow of information considered in this chapter. The dark sector now transfers en-ergy via decays and annihilations to the visible sector, which will go through further self-processing effects that interplay with the inflow of energy. In this chapter, the visible sectorconsists of relevant BBN particles, while the dark sector consists of a single species X .258–261]. In the majority of this work, often motivated by new physics connected to the electroweakhierarchy puzzle or weakly-interacting massive particle (WIMP) dark matter, the energy injected bythe decay has been assumed to be close to or greater than the weak scale. Thus, the decay productstypically have initial energies that are much larger than the thresholds for nuclear reactions relevant toBBN which are typically on the order of several MeV. Weak-scale decay products typically also haveboth hadronic and electromagnetic components, if only through radiative effects.Hadronic energy injection can modify the light element abundances at times as early as t ∼ 0.1s [258,259]. Initially, these products scatter with protons and neutrons and alter the ratio of these baryons andthus the resulting helium abundance. At later times, injected hadrons destroy and modify the abun-dances of helium and other light elements through hadrodissociation. Since the initial hadronic energiesare usually assumed to be much larger than the MeV scale, thresholds for these reactions are easilyovercome.Electromagnetic (EM) energy – photons, electrons, and positrons – injected into the cosmologicalplasma does not have a significant effect on the light element abundances until much later. The main ef-fect of electromagnetic injection on the light elements is photodissociation (unless the amount of energydeposited is enormous). However, being much lighter than hadrons, photons and electrons lose theirenergy very efficiently by scattering off the highly-abundant photon background. The electromagneticcascade initiated by this scattering is strongly suppressed for energies above Ec, given by [7, 269]Ec ' m2e22T' (2 MeV)(6 keVT), (4.1)where T is the cosmological photon temperature. As a result, even for initial energies orders of magni-tude above the MeV-scale thresholds for photodissociation, the fraction of energy available for photodis-sociation is tiny until the background temperature falls below T . 10 keV, corresponding to t ∼ 104 s.While much of the focus on new sources of energy injection during BBN has been on decays orannihilations at or above the weak scale, there exist many well-motivated theories that also predict newsources well below the weak scale. Specific examples include dark photons [270, 271], dark Higgsbosons [271, 272], dark gluons and glueballs [3, 4, 235], light or strongly-interacting dark matter [273,274], and MeV-scale neutrino decays [275, 276]. As the injection energy falls below the GeV scale,hadronic decay channels start to become kinematically unavailable and disappear entirely below the pion71threshold. This leaves electromagnetic and neutrino injection as the only remaining possibilities. Evenmore importantly, it was shown in Refs. [8, 277] that the development of the electromagnetic cascade atthese lower energies can differ significantly relative to injection above the weak scale. Furthermore, asthe injection energy falls below a few tens of MeV, photodissociation reactions begin to shut off.In this chapter we expand upon the analysis of Refs. [8, 277] and investigate the effects of electro-magnetic energy injection below 100 MeV on the primordial element abundances created during BBN.One focus of this study is the development of the electromagnetic cascade from initial photon or electron(e+e−) injection. For high energy injection, the resulting spectrum of photons is described very wellby the so-called universal spectrum rescaled by a temperature- and energy-dependent relaxation rate.This spectrum is used widely in studies of photodissociation effects on BBN, it can be parametrizedin a simple and convenient way, and has the attractive feature that it only depends on the total amountof electromagnetic energy injected. However, for lower-energy electromagnetic injection, the universalspectrum does not properly describe the resulting electromagnetic cascades.The universal spectrum fails for lower-energy injection in two significant ways. First, the universalspectrum is based on a fast redistribution of the initial energy EXEc to a spectrum populated at E ≤Ecthrough Compton scattering and photon-photon pair production. As shown in Refs. [8, 277], this picturedoes not hold for initial injection energies EX < Ec, which can easily occur for smaller EX and largerdecay lifetimes. And second, as argued in Ref. [271] the Compton scattering with background photonsthat dominates electron interactions is qualitatively different at high energies compared to low. At higherenergies, s ∼ E T  m2e , electrons scatter in the Klein-Nishina limit and typically lose an order unityfraction of their energy in each scattering event. In contrast, lower energy scattering with s∼ E T m2eenters the Thomson regime where the fractional change in the electron energy per collision is very smalland the up-scattered photon energy is much less than the initial electron energy.To address the breakdown of the universal spectrum for lower-energy electromagnetic injection, wecompute the full electromagnetic cascade for photon or electron (e+e−) injection with initial energiesEX ∈ [1, 100] MeV following the methods of Ref. [7]. Our work expands upon Refs. [8, 277] that studiedthe photon portion of the cascade for photon injection. We compare and contrast our results to theuniversal spectrum, and study their implications for BBN. In addition to finding important differencesfrom the universal spectrum at these lower energies, we also demonstrate that final-state radiation (FSR)from electron injection can have a very significant impact on the resulting photon spectrum. For verylow injection energies approaching the MeV scale, we also study the interplay of the spectrum with thethresholds for the most important nuclear photodissociation reactions.Although energy injected directly into electromagnetic cascade products will have the most dras-tic effect on the visible sector via the photodissociation of BBN products, there is another, more subtlemethod by which the dark sector can influence the visible at these low energies. This is via the alterationof Ne f f , the effective number of neutrino species[10, 157, 278–282]. This can be done either via thepresence of a new relativistic species, such as a sterile neutrino or a low mass WIMP[10, 276, 283–287],or via energy injection through particle decays and annihilations[288]. The energy injected into the vis-ible sector can serve to alter the relative ratio of the background photon temperature to the temperature72of neutrinos. For the Standard Model with 3 generations of neutrinos, Ne f f = 3. 2 The altered valueof Ne f f will in turn affect the Hubble expansion rate as the total radiation energy density increases (ordecreases), relative to that obtained via a standard Ne f f :ρR = ργ[1+78(Tν ,0Tγ,0)4Ne f f](4.2)where ρR is the total radiative energy density contributing to the Hubble rate, ργ the portion comingfrom photons, and the rest represented by Ne f f , where the factors out front have been included to makeNe f f = 3 for the standard 3 neutrino generations.Both BBN and the CMB can be used to place limits on the value of Ne f f . During BBN, increasingNe f f leads to a faster freeze-out of the neutron population, which will result in more 4He and deu-terium being produced with the extra neutron availability[81, 284, 291–293]. This is thus constrainedby the same present day measurements that also place limits on the effects of photodissociation. TheCMB will also provide constraints, as increasing the effective number of relativistic degrees of free-dom increases the small-scale (Silk) damping of the CMB power spectrum[268, 280, 294, 295]. Withprecise measurements from telescopes such as the Atacama Cosmology Telescope (ACT)[296], SouthPole Telescope (SPT)[297] and Planck[9], we can provide a limit on the possible values of Ne f f . Theunique thing about the Ne f f limit is it will apply to both electromagnetic interactions, as well as neutrinoeffects. Although neutrinos have frozen-out and do not interact with the electromagnetic backgroundat the time of BBN, and thus do not affect BBN directly, they are still a viable, kinematically allowed,decay/annihilation candidate at these light energies. Annihilations and their effects on Ne f f have beenbroadly studied previously in the literature (see, for example, Refs. [10, 157, 268, 279–281]). However,the constraints that arise from decays have not been as thoroughly examined, so we provide a calcuationhere to complete our examination of low energy constraints.The outline of this chapter is as follows. After this introduction, we present our calculation of theelectromagnetic cascade in Sec. 4.2. Next, in Sec. 4.3 we study the impact of such electromagneticinjection on the light element abundances. In Sec. 4.4 we contrast the bounds from photodissociationof light elements with other limits on late electromagnetic injection, including those derived from Ne f f .Finally, Sec. 4.5 is reserved for our conclusions. Some technical details can be found in our paper [2],Appendix A for completeness. This chapter is based on work published in Ref. [2] in collaboration withDavid Morrissey and Graham White, with extended sections covering the effects on Ne f f .4.2 Development of the Electromagnetic CascadeIn this section we compute the electromagnetic cascade in the early universe following the injection ofphotons or electrons (e+e−) with initial energy EX < 100 MeV.2More precisely, the value predicted for Ne f f is slightly above this, ∼ 3.046, due to the fact that neutrinos have notcompletely decoupled at the time of e± annihilations [289, 290].73Figure 4.2: Most important reactions for the development of the electromagnetic cascade. Toprow: high energy photons scatter off background photons or nuclei. Bottom row: Comptonscattering for either high energy photons or e± (left), as well as final state radiation (right).The fastest processes tend to be 4P (top left) and IC (bottom left).4.2.1 Computing the Electromagnetic CascadeEnergetic photons or electrons injected into the cosmological plasma at temperatures below the MeVscale interact with background photons and charged particles leading to electromagnetic cascades thatproduce spectra of photons and electrons at lower energies. Since the development of the cascade ismuch faster than the typical interaction time with the much more dilute light elements created in BBN,these spectra can be used as inputs for the calculation of photodissociation effects.The most important reactions for the development of the electromagnetic cascade in the temperaturerange of interest T ∈ [1 eV, 10 keV] are show in Fig. 4.2, and are given by: [7]:• photon photon pair production (4P): γ+ γBG→ e++ e−• photon photon scattering (PP): γ+ γBG→ γ+ γ• pair creation on nuclei (PCN): γ+NBG→ NBG+ e++ e−• Compton scattering (CS): γ+ e−BG→ γ+ e−• inverse Compton (IC): e∓+ γBG→ e∓+ γ• final state radiation (FSR): X → e++ e−+ γOf these processes, IC and 4P are typically the fastest provided there is enough energy for them to occur.We define Na = dna/dE to be the differential number densities per unit energy of photons (a = γ)and the sum of electrons and positrons (a = e). The Boltzmann equations for the evolution of thesespectra take the formdNadt(E) =−Γa(E)Na(E)+Sa(E) , (4.3)74where Γa(E) is a relaxation rate at energy E, and Sa(E) describes all sources at this energy. Sincethe relaxation rates are typically much faster than the Hubble rate, the Hubble dilution term has beenomitted. Furthermore, the relaxation rate is also much larger than the mean photodissociation rateswith light nuclei, so a further quasistatic approximation can made with dNa/dt→ 0 [7]. This gives thesolutionNa(E) =Sa(E)Γa(E). (4.4)Note that Na(E) evolves in time in this approximation through the time and temperature dependencesof the sources and relaxation rates. The source terms are discussed in more detail below while explicitexpressions for the contributions to the relaxation rates are given in Appendix A of Ref. [2].Monochromatic Photon InjectionFor monochromatic photon injection at energy EX from a decay with rate per volume R, the source termsareSγ(E) = ξγR δ (E−EX)+∑b∫ EXEdE ′Kγb(E,E ′)Nb(E ′) , (4.5)Se(E) = 0+∑b∫ EXEdE ′Keb(E,E ′)Nb(E ′) , (4.6)where ξγ is the number of photons injected per decay, and the Kab(E,E ′) functions describe scatteringprocesses that transfer energy from species b at energy E ′ to species a at energy E ≤ E ′. Explicitexpressions for these transfer functions are given in Ref. [2]. Note that in the case of decays of species Xwith lifetime τX , the rate is R = nX(t)/τX . These equations can also be applied to annihilation reactionsof the form X + X¯ → nγ with cross section 〈σv〉 by setting R = 〈σv〉nX nX¯ and ξγ = n.It is convenient to describe the cascades resulting from the initial monochromatic (delta function)injection with smooth functions that are independent of the injection rate. To this end, we definef¯γ(E) =1RNγ(E)− ξγΓγ(EX) δ (E−EX) (4.7)f¯e(E) =1RNe(E) . (4.8)Using this form in Eq. (4.4) with the sources of Eqs. (4.5,4.6), we obtain the relationsΓγ(E) f¯γ(E) = ξγKγγ(E,EX)Γγ(EX)+∑b∫ EXEdE ′Kγb(E,E ′) f¯b(E ′) (4.9)Γe(E) f¯e(E) = ξγKeγ(E,EX)Γγ(EX)+∑b∫ EXEdE ′Keb(E,E ′) f¯b(E ′) (4.10)The functions f¯γ and f¯e are expected to be smooth, and can be used to reconstruct the full spectraNγ(E)75andNe(E) uniquely for any given injection rate R.Determining the electromagnetic cascade from monochromatic photon injection is therefore equiv-alent to solving Eqs. (4.9,4.10). We do so using the iterative method of Ref. [7], with an importantmodification to account for the Thomson limit of IC scattering. In this method, the spectra f¯a(E) aredetermined on a grid of energy points Ei given byEi = E0(ENE0)i/N, (4.11)where we use E0 = 1 MeV, EN = EX , i= 0,1, . . . ,N, and N 1. For the top point i=N, Eqs. (4.9,4.10)givef¯γ(EN) = ξγKγγ(EN ,EN)/Γ2γ(EN) , f¯e(EN) = ξγKeγ(EN ,EN)/Γγ(EN)Γe(EN) . (4.12)To compute the spectra at lower points, we use the fact that the transfer integrals at a given energyE only depend on the spectra at energies E ′ > E. Thus, at any step i the integrals in Eqs. (4.9,4.10)can be approximated numerically (e.g. Simpson’s rule) using the spectra already determined at pointsj = i+1, . . . ,N. Relative to Ref. [7] we also apply a finer grid to compute the top two energy points.This approach to computing the cascades works well for ye =EeT/m2e 1, but becomes numericallychallenging for ye . 0.1. The problem comes from the contribution of inverse Compton (IC) scatteringto Kee. As ye becomes small, IC scattering enters the Thomson regime in which the cross section islarge but the fractional change in the electron energy per scattering is much less than unity, and thusthe function Kee(E,E ′) develops a strong and narrow peak near E ′ ' E. To handle this we followRefs. [269, 298] and treat the electron energy loss due to IC in the Thomson limit as a continuousprocess by replacing−Γe(E) f¯e(E)+∫ ENEdE ′Kee(E,E ′) f¯e(E ′) → ∂∂E[E˙ f¯e(E)]. (4.13)Here, E˙ is the rate of energy loss from IC of a single electron in the photon background, given by [298]E˙E= −43[3ζ (4)ζ (3)](ETm2e)σT nγ , (4.14)where σT = (8pi/3)α2/m2e is the Thomson cross section, nγ = [2ζ (3)/pi2]T 3 is the thermal photondensity, and ζ (z) is the Riemann zeta function. The approximation of Eq. (4.13) is valid provided thefractional energy loss rate E˙/E is much smaller than the total scattering rate σT nγ , which coincideswith ye 0.1. In this limit, the two terms on the left-hand side of Eq. (4.13) are much larger than theirdifference leading to a numerical instability in the original iterative approach.When computing the electromagnetic spectra, we use the iterative method described above withEqs. (4.9,4.10) until y j = E jT/m2e < 0.05 is reached. For lower energy bins we keep Eq. (4.9) for f¯γ but76apply the replacement of Eq. (4.13) for f¯e, yielding the solutionf¯e(E) =(E jE)2f¯e(E j)+1aT E2∫ E jEdE′′S ′e(E′′) , (4.15)withS ′e(E′′) = ξγKeγ(E′′,EN)Γγ(EN)+∫ EXE ′′dE ′Keγ(E′′,E ′) f¯γ(E ′) , (4.16)andaT =E˙E2=4pi245σTT 4m2e. (4.17)Again, this can be evaluated iteratively, from high to low. While we use the specific value ye < 0.05to match from one method to the other, we find nearly identical results from matching within the rangeye ∈ [0.001,0.1].Monochromatic Electron InjectionMonochromatic injection of electrons (and positrons) at energy EX can be treated nearly identically tomonochromatic photon injection, with the only major change being in modifying the sources toSγ(E) = SFSRγ (E)+∑b∫ EXEdE ′Kγb(E,E ′)Nb(E ′) , (4.18)Se(E) = ξeR δ (E−EX)+∑b∫ EXEdE ′Keb(E,E ′)Nb(E ′) , (4.19)where R is the decay (or annihilation) rate per unit volume, ξe is the number of electrons plus positronsinjected per decay, and SFSRγ (E) is a contribution to photons from final-state radiation to be discussed inmore detail below. For decays of the form X → e++ e− we have R = nX(t)/τX and ξe = 2, while forannihilation X + X¯ → e++ e− the rate is R = 〈σv〉nX nX¯ and ξe = 2.Given these source terms, it natural to define the reduced spectra f¯a(E) byf¯γ(E) =1RNγ(E) (4.20)f¯e(E) =1RNe(E)− ξeΓe(EX) δ (E−EX) (4.21)Applying this to Eq. (4.4) with the sources of Eqs. (4.18,4.19), we obtain the relationsΓγ(E) f¯γ(E) =SFSRγ (E)R+ξeKγe(E,EX)Γe(EX)+∑b∫ EXEdE ′Kγb(E,E ′) f¯b(E ′) (4.22)Γe(E) f¯e(E) = ξeKee(E,EX)Γe(EX)+∑b∫ EXEdE ′Keb(E,E ′) f¯b(E ′) (4.23)77These equations can be solved using the same methods as described above for photon injection, includ-ing a matching in the Thomson limit using Eq. (4.13).A new feature that we include for electron injection is a contribution to the photon spectrum fromfinal-state radiation (FSR) off the injected electron; SFSRγ (E) in Eq. (4.18). For processes of the formX→ e++e− or X+ X¯→ e++e− with X uncharged and EX me, this new source can be approximatedby [299, 300]SFSRγ (E)'REXαpi1+(1− x)2xln[4E2X(1− x)m2e]Θ(1− m2e4E2X− x), (4.24)where x = E/EX . To be fully consistent, a corresponding subtraction should be made from the electronsource. However, we find that this modifies the spectra by less than a percent. In contrast, we showbelow that the direct contribution to the photon spectrum from FSR can be the dominant one at higherenergies when EX T/m2e 1, when the initial electrons scatter via IC with the photon background mainlyin the Thomson regime.4.2.2 Review of the Universal SpectrumMany studies of the effects of electromagnetic energy injection on BBN approximate the photon spec-trum with the so-called universal spectrum. This is a simple parametrization of the full calculations ofthe photon spectrum in Refs. [7, 269]. It replaces the source terms (direct and cascade) in Eq. (4.4) witha zeroeth generation spectrum Sγ(E)/R→ pγ(E) based on the assumption that 4P and IC processesinstantaneously reprocess the initial injected electromagnetic energy.The standard parametrization used for the zeroeth generation spectrum is [112, 257, 269]pγ(Eγ) '0 ; Eγ > EcK0(EγEm)−2.0; Em < Eγ < EcK0(EγEm)−1.5; Eγ < Em, (4.25)where Ec ' m2e/22T and Em ' m2e/80T are derived from Ref. [7], and K0 is a normalization constant.For monochromatic injection of ξ photons, electrons, and positrons each with energy EX , it is fixed bythe requirementξ EX =∫ EX0dE E pγ(E) , (4.26)implying K0 = ξEX/[E2m(2+ ln(Ec/Em)] for EX > Ec. An important feature of the spectrum is that it isproportional to the total injection energy (for either photons or electrons) provided EX  Ec, up to anoverall normalization by the total amount of energy injected.Within the universal spectrum approximation, the final spectra are given byfγ(E) =pγ(E)Γγ(E), fe(E) = 0 , (4.27)78102010251030103510400.001 0.01 0.1 1 10 100f¯ γ(GeV−2)Eγ (GeV)EX = 1000GeVUniversal SpectrumKawasaki + MoroiT = 1 eVT = 10 eVT = 100 eV102010251030103510400.001 0.01 0.1 1 10 100f¯ γ(GeV−2)Eγ (GeV)EX = 100GeVUniversal SpectrumKawasaki + MoroiT = 1 eVT = 10 eVT = 100 eVFigure 4.3: Photon spectrum f¯γ(E) for single photon injection with energy EX = 1000 GeV (left)and 100, GeV (right), for temperatures T = 1, 10, 100 eV. Also shown are the predictionsof the universal spectrum (solid) and the parametrizations of Kawasaki and Moroi given inRef. [7].where fγ(E) = Nγ(E)/R, and the relaxation rate Γγ(E) accounts for the further reprocessing of thespectrum by slower processes like Compton scattering, pair creation on nuclei, and photon-photon scat-tering.3 These spectra have no residual delta-function parts since the initial injection is assumed to befully reprocessed into the zeroeth-order spectrum by 4P and IC scatterings.4.2.3 Results for Photon InjectionTo validate our electromagnetic spectra, we compare our results to previous calculations and the uni-versal spectrum at high injection energies. In Fig. 4.3 we show our photon spectra f¯γ(E) for singlephoton injection with EX = 1000 GeV (left) and 100 GeV (right) at temperatures T = 1, 10, 100 eV.Also shown in the figure are the predictions from the universal spectrum and parametrizations of theresults of Kawasaki and Moroi listed in Ref. [7]. In all cases here, EX  Ec and the universal spectrumis expected to be a good approximation. Our spectra agree well with the results of Ref. [7] but are some-what larger than the universal spectrum. We have also checked that our spectra scale proportionally tothe total energy injected provided EX  Ec. In all cases shown in the figure, the electron spectra aresmaller than the photon spectra by orders of magnitude due to efficient IC scattering. Also visible is thestrong suppression of the photon spectra for E > Ec where the 4P process is active.In contrast to electromagnetic injection at high energies with EX  Ec, injection at lower energieswith EX . Ec has received much less attention. In Fig. 4.4 we show our computed photon spectra forsingle photon injection with EX = 100 MeV (left), EX = 30 MeV (middle), and EX = 10 MeV (right)for T = 1, 10, 100 eV. Also shown are the predictions of the universal spectrum (normalized accordingto Eq. (4.26)) and the prescription by Poulin and Serpico of Ref. [8]. Since EX < Ec, the assumptionsthat go into the universal spectrum are not met and it is not expected to be accurate in this regime, asfirst pointed out in Ref. [8]. Our spectra agree fairly well with the results of Ref. [8], which only kept3In practice, this Γγ (E) is effectively equal to the full relaxation rate that also includes 4P scattering since this process isvery strongly Boltzmann-suppressed for E < Ec.7910281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 100MeVUniversal SpectrumPoulin+SerpicoT = 1 eVT = 10 eVT = 100 eV10281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 30MeVUniversal SpectrumPoulin+SerpicoT = 1 eVT = 10 eVT = 100 eV10281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 10MeVUniversal SpectrumPoulin+SerpicoT = 1 eVT = 10 eVT = 100 eVFigure 4.4: Photon spectrum f¯γ(E) for photon injection with EX = 100 MeV (left), EX =30 MeV (middle), and EX = 10 MeV (right), with T = 1, 10, 100 eV. Also shown are thepredictions of the universal spectrum (solid) and the low-energy prescription of Ref. [8].the photon part of the spectrum. Some deviations are seen at lower energies where photon regenerationby IC becomes significant. Note as well that the full cascade also contains a moderately damped delta-function part that is not shown here (and was explicitly removed in our definition of f¯γ in Eq. (4.7)).4.2.4 Results for Electron InjectionFor electron and positron (e+e−) injection with energies EX  Ec, we find the same photon (and elec-tron) spectra as from photon injection with an equal total input energy, and thus our results agree rea-sonably well with Ref. [7] and the universal spectrum in this limit. However, for EX . Ec we findvery significant variations from the universal spectrum as well as from pure photon injection. Photonspectra f¯γ resulting from e+e− injection are shown in Fig. 4.5 for input energies EX = 100 MeV (left),EX = 30 MeV (middle), and EX = 10 MeV (right) and temperatures T = 1, 10, 100 eV. The solid linesshow the full spectra, while the dashed lines show the corresponding result when FSR off the initial de-cay electrons is not taken into account. Also shown is the universal spectrum for the same total energyinjection (normalized according to Eq. (4.26)). Let us also mention that the photon spectra do not havea delta function component for electron or positron injection.The strong suppression of the photon spectrum from electron injection at lower energies in theabsence of FSR was pointed out in Ref. [271]. As argued there, this suppression can be understoodin terms of the behavior of IC scattering at low energy, which is the main mechanism for electronsto transfer energy to photons in this context. For smaller EX and T , the dimensionless combinationye = EeT/m2e  1 is small, and IC scattering lies in the Thomson regime where each collision onlyslightly reduces the initial electron energy. Correspondingly, the maximal scattered photon energy E ′γin the Thomson limit is E ′γ ≤ 4(Ee/me)Eγ , where Eγ is the energy of the initial photon. Since the initialphoton comes from the CMB, Eγ ∼ T is expected so thatE ′γ . 4(Ee/me)2T (4.28)∼ 15 MeV(Ee100 MeV)2( T100 eV).8010281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 100MeVUniversal SpectrumT = 1 eVT = 10 eVT = 100 eV10281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 30MeVUniversal SpectrumT = 1 eVT = 10 eVT = 100 eV10281030103210341036103810400.001 0.01 0.1f¯ γ(GeV−2)Eγ (GeV)EX = 10MeVUniversal SpectrumT = 1 eVT = 10 eVT = 100 eVFigure 4.5: Photon spectrum f¯γ(E) for electron plus positron (e+e−) injection with energies EX =100 MeV (left), 30 MeV (middle), and 10 MeV (right), with T = 1, 10, 100 eV. The solidlines show the full spectrum, while the dashed lines show the result when FSR is not takeninto account. Also shown is the universal spectrum for the same total injected energy.Higher scattered photon energies are possible, but they come at the cost of an exponential Boltzmannsuppression.In this regime, FSR from the injected electrons and positrons can be the dominant contributionto the photon spectrum, as illustrated in Fig. 4.5. Relative to the rest of the cascade, the distribution ofphotons from FSR is hard, falling off roughly as 1/E instead of as 1/E2. Despite the suppression of FSRby (α/pi)× log (with log ∼ few), it can easily overcome the exponential suppression of IC for photonenergies above the bound of Eq. (4.28). We show below that this has a very important implication for theeffects of lower-energy electron injection on the primordial light element abundances. Note, however,that FSR has only a very minor effect on the spectra for photon injection or when EX  Ec.4.3 Effects of Electromagnetic Injection on BBNHaving computed the electromagnetic cascades from lower-energy injection, we turn next to investigatethe effects of such injection on the primordial element abundances from BBN.4.3.1 Photodissociation of Light ElementsPhotodissociation of light element begins when the temperature of the cosmological plasma falls lowenough for MeV photons to populate the electromagnetic cascade. From Eq. (4.1), this does not beginuntil temperatures fall below about 10 keV (corresponding to t ∼ 104 s). By this time element creationby BBN has effectively turned off, and thus we can compute the effects of photodissociation as a post-processing of the outputs of standard BBN [241, 257].The effects of photodissociation on the light element abundances can be described by a set of cou-pled Boltzmann equations of the formdYAdt=∑iYi∫ ∞0dEγNγ(Eγ)σγ+i→A(Eγ)−YA∑f∫ ∞0dEγNγ(Eγ)σγ+A→ f (Eγ) , (4.29)81Process Threshold (MeV) Peak value (mb)D+ γ → p+n [301] 2.220 2.473He+ γ → D+ p [302] 5.490 1.183He+ γ → p+ p+n [302] 7.718 1.02T+ γ → n+D [303, 304] 6.260 0.818T+ γ → n+n+ p [304] 8.480 0.8784He+ γ → T+ p [305] 19.81 1.314He+ γ → 3He+n [306, 307] 20.58 1.284He+ γ → D+D [257] 23.85 0.00514He+ γ → n+ p+D [305] 26.07 0.182Table 4.1: Processes included in our calculation of photodissociation effects from electromagneticinjections, as well as their threshold energies and peak cross sections.where Nγ(Eγ) are the photon spectra calculated above, A and the sums run over the relevant isotopes,and YA are number densities normalized to the entropy density,YA =nAs. (4.30)Note that we do not include reactions initiated by electrons because the electron spectra are alwaysstrongly suppressed by IC scattering.In our analysis we include the nuclear species hydrogen (H), deuterium (D = 2H), tritium (T = 3H),helium-3 (3He), and helium (He = 4He). Heavier species including lithium isotopes could also beincluded, but these have much smaller abundances and they would not alter the results for the lighterelements we consider. The nuclear cross sections included in our study are listed in Table 4.1, for whichwe use the simple parametrizations of Ref. [257]. All these cross sections have the same general shapeas a function of energy, with a sharp rise at the threshold up to a peak followed by a smooth fall off. Welist the threshold energies and peak values of the cross sections in the table to give an intuitive pictureof their relevant strengths and ranges of importance. Of the nine cross sections listed, it is helpful togroup them into processes that destroy helium and create deuterium and helium-3 with thresholds above20 MeV, and processes that destroy the lighter isotopes with significantly lower thresholds.It is straightforward to solve the evolution equations of Eq. (4.29) numerically following the standardconvention of converting the dependent variable from time to redshift. For standard BBN values of theprimordial abundances, we use the predictions of PArthENoPE [308, 309]:Yp = 0.247 ,nDnH= 2.45×10−5 , n3HenH= 0.998×10−5 . (4.31)In the analysis to follow, we compare the computed output densities to the following observed val-ues, quoted with effective 1σ uncertainties into which we have combined theoretical and experimental824 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3He4log 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3He4Helog 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3Helog 10Figure 4.6: Limits on EX YX from BBN on the monochromatic photon decay of species X as afunction of the lifetime τX for photon injection energies EX = 10 MeV (left), 30 MeV (mid-dle), and 100 MeV (right). Bounds are given for the effects on the nuclear species D, 3He,and 4He.uncertainties in quadrature:Yp = 0.245±0.004 (Ref. [101]) (4.32)nDnH= (2.53±0.05)×10−5 (Ref. [91]) (4.33)n3HenH= (1.0±0.5)×10−5 (Ref. [310]) . (4.34)For the helium mass fraction Yp, the value we use is consistent with Ref. [104] and previous deter-minations but significantly lower than the determination of Ref. [102]. The quoted uncertainty on theratio nD/nH is dominated by a theory uncertainty on the rate of photon capture on deuterium fromRef. [311]. For n3He/nH, we use the determination of (nD + n3He)/nH of Ref. [310] together with thevalue of nD/nH from Ref. [91]; the resulting upper bound (with uncertainties) is similar to but slightlystronger than what is used in Ref. [94]. The uncertainties quoted here are generous, and in the analysisto follow we implement exclusions at the 2σ level.4.3.2 BBN Constraints on Photon InjectionFollowing the methods described above and the electromagnetic cascades computed previously, wederive BBN bounds on monochromatic photon injection from late decays with lifetime τX and initialinjection energy EX . In Fig. 4.6 we show the resulting limits on the combination EX YX , where YX isthe predecay yield of the decaying species X (assumed to produce one photon per decay) for injectionenergies EX = 10, 30, 100 MeV. The bounds coming from D, 3He, and 4He are shown individually, andcorrespond to 2σ exclusions. Early on, when Ec is small, the dominant effect is destruction of D sinceit has the lowest photodissociation threshold. Later on, as Ec increases, it becomes possible to createexcess D and 3He through the destruction of 4He provided the injection energy is larger than the 4Hethreshold of about 20 MeV. Destruction of D is the dominant effect at all times for EX below the heliumthreshold, as can be seen in the leftmost panel of Fig. 4.6.833 4 5 6 7 8 9 10 11 12102030405060708090100EXMeVlog10 τ(s)-14-13-12-11-10-9-8log10 EXYX (GeV)Figure 4.7: Combined limits on EX YX as a function of τX and EX for the decay of a species X withlifetime τX injecting a single photon with energy EX .In Fig. 4.7 we show maximal values of EXYX from monochromatic photon injection at energy EXfrom the decay of species X as a function of τX and EX . The combined exclusion is based on theunion of 2σ exclusions of the individual species. Clear features are visible in this figure at τX ' 106 sand EX ' 20 MeV. These coincide with the structure of the exclusions shown in Fig. 4.6, with bothcorresponding to where the photodissociation of 4He turns off, either because Ec or EX is too small.4.3.3 BBN Constraints on Electron InjectionIn Fig. 4.8 we show the limits for e+ e− injection from the decay of a species X with lifetime τX onEX YX , where YX is the predecay yield of the decaying species X (assumed to produce one e+e− pairper decay) for injection energies for each electron of EX = 10, 30, 100 MeV (from left to right). Thebounds coming from D, 3He, and 4He are shown individually, and correspond to 2σ exclusions. Theelectromagnetic spectra used in this calculation include FSR from the injected e+e− pair. The resultingbounds are somewhat weaker than for photon injection and follow a similar pattern, and remain quitestrong even down to EX = 10 MeV. For comparison, we show the corresponding results when FSReffects are not included in Fig. 4.9. As expected, the exclusions are significantly weaker, particularly forlarger τX and lower EX where the relevant IC scattering is deep in the Thomson regime.In Fig. 4.10 we show maximal values of EXYX from monochromatic e+e− injection at energy EXfrom the decay of species X as a function of τX and EX , with FSR effects included in the electromagneticcascade. The combined exclusion is based on the union of 2σ exclusions of the individual species.Again, the exclusions become weaker for τX . 106 s or EX . 20 MeV where the photodissociation of4He turns off. The bounds on e+e− injection are also typically weaker than for photon injection, but notdrastically so when FSR is taken into account.844 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3He4log 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3He4log 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3Helog 10Figure 4.8: Limits on EX YX from BBN on the monochromatic e+e− decay of species X as afunction of the lifetime τX for individual electron injection energies EX = 10 MeV (left),30 MeV (middle), and 100 MeV (right). Bounds are given for the effects on the nuclearspecies D, 3He, and 4He, and contributions to the electromagnetic cascades from FSR areincluded.4 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3Helog 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3Helog 104 6 8 10 12-15-14-13-12-11-10-9log10τ(s)EXYX(GeV)D3He4log 10Figure 4.9: Same as Fig. 4.8 but without FSR effects.4.4 Other Constraints on Low Energy DecaysIn addition to directly modifying the primordial light element abundances, energy injection in the earlyuniverse can produce other deviations from the standard cosmology. Electromagnetic decays near orafter recombination at trec ' 1.2×1013 s can modify the the temperature and polarization power spectraof the CMB [123, 312–314]. Since current CMB observations are found to constrain such decays muchmore strongly than BBN [80, 315], we focus here on decays prior to recombination. The best limitsin this case, aside from BBN, come from alterations to Ne f f and modifications to the CMB frequencyspectrum. Ne f f can be constrained both before the beginning of BBN, as while as much later dueto its effect on the CMB as well. Early alterations of Ne f f will alter the radiation energy density, inturn leading to an altered neutron abundance available for the main BBN processes. Ne f f will also beconstrained by the CMB as it increases the amount of small-scale Silk damping present in the CMBpower spectrum[268, 280, 294, 295]. In this section we estimate these other limits on late energyinjection and compare them to our results for BBN.853 4 5 6 7 8 9 10 11 12102030405060708090100EXMeVlog10 τ(s)-14-13-12-11-10-9-8log10 EXYX (GeV)Figure 4.10: Combined limits on EX YX as a function of τX and EX for the decay of a species Xwith lifetime τX injecting an electron-positron pair each with energy EX , with FSR effectsincluded.4.4.1 Constraints from Ne f fRecall that we have defined Ne f f to be a proxy for the total contribution of radiative species to theradiation energy density that are not photons, as shown in Eq. (4.2). This includes standard modelneutrinos, such that without the presence of new physics, Ne f f = Nν = 3.046, which is slightly largerthan 3 due to reheating during the non-instantaneous decoupling of electron-positron annihilations. TheTν ,0/Tγ,0 factor is the temperature ratio of neutrinos to photons assuming no new physics. It accountsfor the fact that neutrinos have a different temperature from photons after e± annihilation, and is givenby [70, 71]:Tν ,0Tγ,0=1 T  me(4/11)1/3 T  me (4.35)As was dicussed in Ch. 2. If we allow for particles to decay either electromagnetically, or into neutrinos,we inject extra energy into the visible sector. This energy will effectively heat up all particles that are inthermal equilibrium with the decay products. As such, we can also write out the total radiation energydensity during times of interest as:ρR = ργ +ρν +ρX (4.36)Note that here, we do not include e± as radiative species, as we will typically be concerned only withtemperatures below their mass. In theory, the decaying X particles could contribute to the radiationdensity as well, if they are light enough and have a thermal temperature Txmx. This type of effect has86been well studied for particles in thermal equilibrium with standard model particles [10, 268, 279, 280].Instead, we focus on the case of a cold dark particle, that will not itself contribute to the radiation energydensity, and thus ρX = 0, and it instead contributes indirectly via the alteration of either ργ or ρν .Comparing Eqs. (4.2) and (4.36), and using Eq. (2.24), with gi = 2 (6) for photons (neutrinos), wesee that a working definition of Ne f f is given by:Ne f fNν=(Tγ,0Tν ,0× TνTγ)4(4.37)To calculate changes to Ne f f , we consider conservation of both energy and entropy. We begin byassuming that the decay will happen instantaneously, when the Hubble rate has dropped to match thedecay rate, 1/τ . This occurs at a decay temperature, Td of:Td =(MPlτ)1/2(g∗(Td)pi290)−1/4(4.38)with g∗ defined as in Eq. (2.30). This not only allows us to produce analytic estimates, but it will beaccurate to within a few percent. If the decay occurs to electromagnetic species, the photon temperaturewill increase, and Ne f f will be reduced. However, if the the decay occurs to neutrinos, then Ne f fincreases. Note that in both cases, we assume explicitly that the decay is occurring after the neutrinoshave decoupled, at temperature T ∼ 2 MeV. Otherwise, any energy introduced into the visible sector isequilibriated between both photons and neutrinos, and both Tν and Tγ will change by the same amount.In particular, we assume that the decaying particle has some steady comoving number density beforeit decays. If the decaying particle has an initial mass weighted yield, mxYx, then the energy density ofthe particle right before it decays is:ρX = mxnx = mxYxs(Td) (4.39)with s the relativistic entropy density, given in Eq. (2.34). We can use this to calculate the energy thatis transferred to either ργ or ρν . In particular, we find that for decays to electromagnetic species, thisresults in an Ne f f of:Ne f fNν=1 T > Td[1+ 23 g∗s(Td)(mxYxTd)]−1T < Td(4.40)while for decays to neutrinos, we find:Ne f fNν=1 T > Td[1+ 1663 g∗s(Td)(Tγ,dTν ,d)4(mxYxTd)]T < Td(4.41)Note that the factor Tγ,d/Tν ,d is 1 for decays before e± annihilations, and (11/4)1/3 otherwise.87-�� -� -� -� -�-�-�-����������(����/���)Δ� ���τ = ����τ = ����τ = ����� → νν� → ��Δ���� = ���� ± ����Δ�������� = �Figure 4.11: Effects of low energy decays on Ne f f for decays to neutrinos (top, blue), and electro-magnetic species (bottom, red). Shown are effects for various particle lifetimes. The centralgrey band corresponds to the conservative estimate given in Eq. (4.42), in agreement withPlanck and BBN estimates[9, 10]. The enlarged green region shows the extra phase spacethat could be allowed if a sterile neutrino with ∆Ne f f = 1 is included.The results for decays to light visible species are given in Fig. 4.11. The blue, increasing linescorrespond to decays to neutrinos, while the red, falling lines correspond to decays to γ and e±, as weexpect. We also include a grey band to show the region that is consistent with present day observations.Currently, the 2018 Planck results constrain Ne f f to a central value of 3.27±0.15, when considered inconjunction with other astrophysical observations [67]. Furthermore, if we consider BBN constraints,then Ne f f = 3.56±0.23 is favoured [10]. This gives us combined results of:∆Ne f f = Ne f f −3.046 = 0.31±0.16 (4.42)which implies that cosmological observations favour a slightly larger value of Ne f f than is present in theSM. In Fig. 4.11, we use 95% confidence intervals to estimate a conservative region that is consistentwith observations.It’s also interesting to note that the decay to EM species has an interesting caveat. If there is a fullythermalized sterile neutrino, this would correspond to a ∆Ne f f = 1. As such, decays to EM species canbe compensated by the presence of a sterile neutrino, thus allowing slightly larger decays to remainconsistent with current observations [316]. This is given by the extended green band in Fig. 4.11.884.4.2 Constraints from the CMBLate decays releasing electromagnetic energy can also distort the frequency spectrum of the CMB [317,318], which is observed to be a nearly-perfect blackbody [319]. The effect depends on the decay timeτX relative to the times τdC ' 6.1× 106 s when double-Compton scattering freezes out and τC ' 8.8×109 s when Compton scattering turns off [317, 318]. Decays with τdC < τX < τC yield products thatthermalize through Compton scattering and generate an effective photon chemical potential µ givenby [317, 318, 320]µ ' 5.6×10−4(∆E YX10−10 GeV)(τ106 s)1/2e−(τdC/τ)5/4. (4.43)For τX > τC, electromagnetic injection produces a distortion that can be described by the Comptonparameter y = ∆ργ/4ργ , with the approximate result [317, 318, 320]y ' 5.7×10−5(∆E YX10−10 GeV)(τ106 s)1/2C (τ) , (4.44)where C (τ) = 1 for τ < teq and C (τ)' (τ/teq)1/6 for τ > teq. The current limits on µ and y are [319]µ < 9×10−5, |y|< 1.5×10−5 , (4.45)while the proposed PIXIE satellite is to have sensitivity to constrain [321]µ < 1×10−8, |y|< 2×10−9 . (4.46)In the left and right panels of Fig. 4.12 we show the limits from Ne f f and CMB spectral distortions.The solid (dashed) red line shows Ne f f = 0.31±0.16 for decays to EM (neutrinos), with a conservative95% estimate included. Note that the constraint arising from EM decays is much stronger, due to thefact that these decays lower Ne f f , while cosmologically evidence preferentially raises it. This constraintcan be mediated by the presence of sterile neutrinos, as discussed above. For CMB spectral distortionswe show bounds on the µ and y parameters in blue based on the approximate estimates above based onmeasurements by COBE/FIRAS (solid) and the projected sensitivity of PIXIE (dotted). For comparison,we show in green the limits derived above for monochromatic photon injection (left) and monochromatice+e− injection (right). In both panels, the dotted, dashed, and solid lines correspond to injection withEX = 10, 30, 100 MeV. Even for low injection energies, BBN constraints currently dominate for τ &104 s until being replaced by bounds from either CMB frequency or power spectrum variations. Evenwith the vast improvement expected from PIXIE, BBN will continue to provide the strongest limit onelectromagnetic decays in the early universe with lifetimes 104 s . τX . 106 s and energy injectionsabove a few MeV.89� � � � � �� ��-��-��-��-������(τ/�)��� ��(Δ���/���) χ → ννχ → ��Δ����������� ��� ��������� �� ������������ ��� ��������� �� ��� (�����)���� � � � � �� ��-��-��-��-������(τ/�)��� ��(Δ���/���) χ → ννχ → ��Δ����������� ��� ��������� �� ������������ ��� ��������� �� ��� (�����)���Figure 4.12: Other bounds on electromagnetic decays in the early universe as a function of thelifetime τX and the total electromagnetic injection ∆E YX relative to limits derived fromBBN. In both panels, the red line shows ∆Ne f f = 0.31±0.16, while the solid (dotted) bluelines show the current and projected CMB frequency bounds from COBE/FIRAS (PIXIE).The left panel also indicates the limits derived from BBN for photon injection with energyEX = 10, 30, 100 MeV with green dotted, dashed, and solid lines. The right panel showsthe corresponding BBN bounds from monochromatic e+e− injection.4.5 ConclusionsIn this chapter, we have focused on the interplay between the dark sector and the visible sector fromthe perspective of decaying dark species. As in the previous chapter, we have focused specifically ona one-way energy flow, as indicated in Fig. 4.1. In this section, the energy flow was from the dark tothe visible, so that we can isolate the effect that dark decays and annihilations may have on various SMparticles and how we can observe this influential effect. Also similar to the previous chapter, we focuson ‘small’ interactions. In Ch. 3, we focus on small interactions via a feeble coupling, such that thetwo sectors never fully thermalize. In this chapter, however, we focus on ‘small’ in the sense that weconsider low energy transfers, typically below 100 MeV. Although early, well motivated theories tendedto focus on new physics at or above the weak energy scale, new models can still be well motivated muchbelow this, and these low energy ranges have not yet been fully explored. We are able to place stringentconstraints on new physics, simply by modelling how the visible sector, which is full of well-understoodand strongly coupled particles, responds to the energy inflow. This response can be compared to presentday observations to identify models and parameter spaces that are in conflict.To set the stage for dark energy deposition, we have investigated the electromagnetic cascades in-duced by electromagnetic energy injection in the range EX = 1−100 MeV and we have studied itseffects on the light elements abundances created during BBN. As in Ref. [8], we find significant de-viations from the universal photon spectrum for monochromatic initial photon injection with energyEX . Ec = m2e/22T . Our study also expands on previous work by computing the full electromagnetic90cascade including electrons.Photon and electron injection produce very similar electromagnetic cascades for EX  Ec but differin important ways for EX .Ec. Initial hard photons induce a smooth population of lower-energy photonsthrough Compton and photon-photon scattering. In contrast, electrons injected with EX . m2e/10Tinteract mainly through inverse Compton (IC) scattering off the CMB, which lies in the Thomson regimeat such energies. The upscattered photons from Thomson scattering have much lower energy than theinitial electron, and can easily fall below the MeV scales needed to induce photodissociation. However,in this regime we find that photons radiated off the initial hard electrons can populate and dominate theinduced photon spectrum up to near the initial electron energy. To our knowledge, the contribution ofFSR to the photon spectrum has not been considered before in this context since its effects are verysmall at the higher initial injection energies that have been investigated in the greatest detail.We also study how this impacts BBN, both directly through photodissociation of the light elements,and indirectly through changes to Ne f f . For either photon or electron injection, we find that BBNprovides the strongest constraint on late-decaying particles with lifetimes between 104 s . τ . 1013 sfor electromagnetic energies nearly all the way down to the photodissociation threshold of deuteriumnear Eth ' 2.22, MeV. For earlier lifetimes, the indirect effects obtained via Ne f f provide the strongestconstraints. When considering Ne f f , we gain the ability to consider decays to the other kinematicallyallowed species, neutrinos. While decays to neutrinos typically raise their temperature and thus increaseNe f f , decays to EM species do the opposite. Because current observations favour a slight increase inNe f f , the EM decays are thus more constrained (although they do allow for the possibility of a sterileneutrino to compensate for the Ne f f reduction).While this work has concentrated on decays, our results for electromagnetic cascades are also ap-plicable to annihilation in the early universe. Our results could also be used to investigate potentialsolutions to the apparent anomalies in the lithium abundances, which was studied in Refs. [276, 322]using the universal spectrum.At this point, we have studied elementary models that move energy from either the visible sectorto the dark, or the dark sector to the visible. In both cases, we have considered dark sectors that haveminimal interactions, and focused on understanding how the two sectors might interact with each other.Let us now turn to a more concrete and complex model of a hidden sector, in which we can apply thetools we have learned to study it in greater detail.91Part IVA Complete Dark Model92Chapter 5Non-Abelian Dark Forces and the RelicDensities of Dark Glueballs5.1 IntroductionUp to this point, we have split the dark and visible sectors of the Universe, in an effort to understand howthe two sectors may interact and play roles in the evolution of each other. In doing this, we have utilizedfairly simple models of the dark sector, so that the inter-sector interactions would be highlighted andnot masked by effects in the hidden sector. In the next few chapters we will focus on a more concreterealization of a dark sector, in which we can find a rich spectrum of interactions. To begin, we willfocus explicitly on the dark sector isolated by itself, and attempt to understand how the self-interactionsof this sector play a role in its own evolution, as shown in Fig. 5.1. In the following chapter, we willincorporate Standard Model connections, thus creating a full, rich, and connected dark sector that willincorporate what we have learned in the previous chapters.To motivate the dark sector we will consider, we note that gauge invariance under the SU(3)c×SU(2)L×U(1)Y group of the Standard Model provides a remarkable description of the non-gravitationalforces of Nature. Yet, our knowledge of the Universe is incomplete and new gauge forces beyond thoseof the SM may be crucial to describing the laws of physics. The existence of such forces is highlyconstrained if they couple significantly to SM matter unless they have an associated mass scale (such asfrom confinement or the Higgs mechanism) well above a TeV [323, 324]. In contrast new dark gaugeforces, with only feeble connections to the SM, can exist at energy scales much less than the TeV scale(or even be in a massless phase) and still be fully consistent with existing experimental bounds [325–327]. Such dark forces may also be related to the cosmological dark matter [146, 209, 213].Abelian dark forces have been studied in great detail and have the novel property that they canconnect to the SM at the renormalizable level through gauge kinetic mixing [328, 329]. Limits onthe existence of such a kinetically-mixed dark photon have been obtained from existing experimental0This chapter is based on L. Forestell, D. E. Morrissey, and K. Sigurdson, Non-Abelian Dark Forces and the Relic Densitiesof Dark Glueballs, Phys. Rev. D, 95, (2016), 015032, [arXiv:1605.08048] [3]93VSDS: X →0++, 1+−,0−+, 2++,. . .InteractionFigure 5.1: Flow of information considered in this chapter. We start our inspection of a non-Abelian dark force with a completely hidden sector, entirely isolated from the visible. How-ever, even though isolated, the dark sector will still have a complex set of interactions, withmany stable glueball states that can be involved in self-interactions, annihilations, and so on.searches and astrophysical and cosmological observations for a range of dark photon masses spanningmany orders of magnitude [325–327]. An exciting dedicated experimental program to search for darkphotons is also underway [326, 327].Non-Abelian dark forces have received somewhat less attention. As gauge invariance forbids thesimple kinetic-mixing interaction with the SM, it is less clear how they might connect to the SM. Evenso, non-Abelian dark forces are well motivated and arise in many contexts including string theory con-structions [330], in models of dark matter [229, 231, 232, 235, 331–338], baryogenesis [339–341],theories of neutral naturalness [54, 342], and within the hidden valley paradigm [11, 12, 343]. Non-Abelian dark forces can also lead to very different phenomenological effects compared to their Abeliancounterparts owing to the requisite self-interactions among the corresponding gauge bosons and theirpotential for a confining phase transition at low energies.The minimal realization of a non-Abelian dark force is a pure Yang-Mills theory with simple gaugegroup Gx. Such a theory is expected to confine at the characteristic energy scale Λx, with the elementarydark gluons binding into a spectrum of colour-neutral dark glueballs (GBs) of mass m ∼ Λx [344].These dark states may have significant cosmological effects even when their connection to the SM istoo small to be detected in laboratory experiments. For very small values of Λx, dark gluons can actas self-interacting dark radiation [345–348], and can be consistent with existing constraints providedtheir effective temperature is somewhat lower than the SM plasma. With larger Λx, the glueballs willcontribute to the density of dark matter if they are long-lived [229, 231, 232, 235, 336], or they may leadto observable astrophysical or cosmological signals if they decay at late times [229, 235, 336].Assessing the cosmological impact of massive dark glueballs requires a precise knowledge of theirrelic abundances. The primary goal of this chapter is to compute these abundances and map out theranges of parameters where one or more dark glueball states might constitute all or some of the observeddark matter. We focus mainly on Gx = SU(3), but we also comment on how our results can be applied toother non-Abelian gauge groups. In the next chapter we will describe in detail the cosmological effectsof both stable and unstable primordial glueball populations and use them to constrain the existence ofnon-Abelian dark forces.Starting from an early Universe containing a thermal plasma of dark gluons with temperature Tx >Λx, typically different than the temperature of the SM plasma, dark glueballs will be formed in a phase94transition as the temperature of the dark sector falls below the confinement scale, Tx.Λx. Since glueballnumber is not conserved, the number densities of the glueball states will then track their equilibriumvalues so long as their 2→ 2 and n→ 2 interaction rates are fast relative to the Hubble expansion rate.The key difference compared to standard freeze-out is that without direct annihilation or rapid decaysto SM or lighter hidden states, the overall chemical equilibrium of the dark glueballs will be maintainedprimarily by 3↔ 2 number-changing reactions [147, 349, 350]. Moreover, if the hidden glueballs donot have a kinetic equilibration with the SM or a bath of relativistic hidden states, the energy released bythe 3→ 2 annihilations will cause the remaining glueballs to cool much more slowly than they wouldotherwise [349]. Together, these two effects produce freeze-out yields with a much different dependenceon the underlying parameters of the theory than the typical freeze-out paradigm of annihilation into lightrelativistic particles.Previous works have studied the effects of 3→ 2 annihilation and self-heating in general massiveself-coupled sectors [147, 349, 350]. The specific application of these processes to dark glueballs hasalso been studied in Refs. [231, 232, 235]. We expand upon these works in two ways. First, we in-vestigate possible effects of the confining phase transition on the final glueball yields.1 And second,we compute the freeze-out abundances of the heavier glueball states in addition to the lightest mode.We also comment on the importance of including some of the heavier states, as when the glueballs areconnected to the SM, the heavier relic glueball states can sometimes have a greater observational effectthan the lightest mode.Following this introduction, we discuss the general properties of dark glueballs in Section 5.2. Next,we study the freeze-out of the lightest glueball in Section 5.3 and investigate the effects of the confiningphase transition. In Section 5.4 we extend our freeze-out analysis to include the heavier glueball states.The possibility of dark glueball dark matter is studied in Section 5.5, as well as a brief introduction toadditional constraints that may be placed on general dark forces when a connection to the SM is added.We give brief concluding remarks in Section 5.6. This chapter is based on work published in Ref. [3] incollaboration with David Morrissey and Kris Sigurdson.5.2 Glueball Spectrum and InteractionsThe spectrum of glueballs in pure SU(N) gauge theories has been studied extensively using both analyticmodels and lattice calculations [352]. Stable glueballs are classified according to their masses and theirquantum numbers under angular momentum (J), parity (P), and charge conjugation (C). The lighteststate is found to have JPC = 0++ [6, 353, 354], as expected based on general grounds [355], but a numberof stable states with other JPC values are seen as well. In this section we summarize briefly the expectedspectrum of glueballs and we estimate how they interact with each other.1These effects were studied in a slightly different context in Ref. [351].95JPC mr0 (N = 2) mr0 (N = 3)0++ 4.5(3) 4.21(11)2++ 6.7(4) 5.85(2)3++ 10.7(8) 8.99(4)0−+ 7.8(7) 6.33(7)2−+ 9.0(7) 7.55(3)1+− − 7.18(3)3+− − 8.66(4)2+− − 10.10(7)0+− − 11.57(12)1−− − 9.50(4)2−− − 9.59(4)3−− − 10.06(21)Table 5.1: Masses of known stable glueballs in SU(2) [5] and SU(3) [6].5.2.1 Glueball MassesMuch of what is known about the spectrum of glueballs in SU(N) gauge theories comes from lat-tice calculations. It is conventional to express these masses in terms of a length scale r0 correspond-ing to where the gauge potential transitions from Coulombic to linear [356, 357], or in terms of theconfining string tension√σ . Both of these quantities can be related to the energy scale ΛMS wherethe running gauge coupling becomes strong [358]. For SU(3) (with zero flavors), they are given byr0ΛMS = 0.614(2)(5) [358] and r0√σ = 1.197(11) [5, 357]. To facilitate connections with modernlattice calculations, we will express the glueball masses in terms of 1/r0 and define the strong couplingscale as the mass of the lightest 0++ glueball, Λx ≡ m++0 .Assuming conserved P and C in the dark sector, the dark glueballs will have definite JPC quantumnumbers. In Table 5.1 we list the spectra of SU(N) glueballs for N = 2 and N = 3 determined inlattice studies in units of r0. The N = 3 glueballs in the table correspond to all the known stable states,with the masses listed taken from Ref. [6]. Listings for the N = 2 case are based on Ref. [5], havesignificantly larger fractional uncertainties, and may not give a complete accounting of all the stablestates. Note that the absence of C-odd states is expected for SU(2) and other Lie groups with a vanishingdabc = tr(ta{tb, tc}) symbol (where ta is the generator of the fundamental representation) [11, 352, 359].Glueball spectra for SU(N > 3) have also been investigated on the lattice [360, 361]. The (lowest-lying) glueball masses are found to scale with N according tor0 m(N)' P+Q/N2 , (5.1)with P and Q on the order of unity. These corrections are found to be numerically modest for N > 3,and the glueball spectrum for larger N appears to be similar to N = 3. Extrapolations to large N alsofind that r0√σ ' 1.2 remains nearly constant [5], while the strong-coupling scale decreases smoothlyto r0ΛMS ' 0.45 [362]. A further variation on SU(N) theories is the addition of a non-zero topological96theta term. This violates P and T explicitly, shifts the string tension√σ and glueball masses [363], andinduces mixing between glueball states with different P quantum numbers [363, 364].The glueballs for other non-Abelian gauge groups have not been studied in as much detail on thelattice, but a few specific features are expected based on general arguments. As mentioned above, thereare no C-odd states for SU(2), SO(2N+1), or Sp(2N) due to their vanishing dabc coefficient [11, 352,359]. For SO(2N), SO(4) ∼= SU(2)× SU(2) and SO(6) ∼= SU(4) reduce to previous cases, while for2N > 6 the C-odd states are expected to be significantly heavier than the lowest C-even glueballs [11].This follows from the fact that the minimal gluon operators giving rise to the C-odd states for the groupshave mass dimension 2N [359], and higher-dimension gluon operators are generally expected to lead toheavier glueball states [6, 11, 359].In this study we concentrate on SU(N) glueballs with P and C conservation in the dark sector.However, other non-Abelian gauge groups could be realized in nature [330], and we comment on thesemore general scenarios when they lead to important phenomenological distinctions.5.2.2 Glueball CouplingsDark glueball freeze-out in the early Universe depends on the cross sections for 2→ 2 and 3→ 2glueball reactions. Glueball self-couplings and transition matrix elements are thus needed to computetheir cosmological evolution. These quantities have not been studied in as much detail on the latticeas the glueball mass spectrum. Here, we collect the relevant existing lattice results, and we use naivedimensional analysis (NDA) [365–367] and large-N scaling [368, 369] to make estimates when nolattice data is available.Glueball interactions are expected to be perturbative in the limit of large N (for an underlying SU(N)gauge group), and this motivates writing an effective Lagrangian in terms of glueball fields. Combiningthe N scaling of gluon n-point functions with dimensional analysis suggests the formLe f f =(N4pi)2m4x F(φ/mx,∂/mx) , (5.2)where φ represents a glueball field interpolated by a single-trace gluon operator, mx is a characteristicglueball mass scale, and F(x,y) is a smooth function that is finite as N→∞. Expanding this function ina power series and rescaling to obtain a canonical kinetic operator, the effective Lagrangian becomesLe f f =12(∂φ)2−∑nann!m4−nx(4piN)n−2φ n+ . . . (5.3)where the coefficients an are expected to be of order unity. This form matches the NDA scaling ofRef. [231] as well as the 1/N counting of Ref. [235]. Note that shifting the gluon field to remove thelinear term does not alter this general form. In the analysis to follow, we identify mx =m0 with the massof the lightest glueball.This gives rise to diagrams such as those shown in Fig. 5.2, which allows for both 2→ 2 as well as3→ 2 interactions. Applying this form to 2→ 2 elastic scatterings of the 0++ state with mass mx, we97Figure 5.2: Feynman diagrams for the scalar 0++ glueball state, which include a typical self-interaction (left), as well as a number changing interaction (right).estimateσ2→2v' A4pi(4piN)4 βs, (5.4)where A is dimensionless and close to unity, s is the square of the center-of-mass energy, and β =√1−4m2x/s. The same arguments applied to 3→ 2 processes at low momentum giveσ3→2v2 ' B(4pi)3(4piN)6 1m5x, (5.5)with B also close to unity. These cross sections are at the limit of perturbative unitarity for small N butbecome moderate for N & 4pi , reflecting the expected transition to weak coupling in this regime [369].In the analysis to follow we set A = B = 1, and we generalize the cross section estimate for 2→ 2interactions to more general processes involving other glueball states using the same NDA and large-Narguments.5.3 Freeze-out of the Lightest GlueballHaving reviewed the properties of glueballs, we turn next to investigate their freeze-out dynamics in theearly Universe. In this section we study the thermodynamic decoupling of the lightest 0++ glueball in asimplified single-state model. We also discuss the confining transition in which the glueballs are formedand investigate how it might modify the glueball relic density. The freeze-out of heavier glueballs willbe studied in the section to follow.Throughout our analysis, we assume that the dark glueballs are thermally decoupled from the SMduring the freeze-out process but maintain a kinetic equilibrium among themselves. This implies thatthe entropy of the dark sector is conserved separately from the visible sector, up to a possible increaseduring a first-order confining phase transition. This motivates the definitionR≡ sxs= constant , (5.6)where sx is the entropy density of the dark sector after the confining transition and s is that of the98visible. The value of R is an input to our calculation, and may be regarded as an initial condition setby the relative reheating of the dark and visible sectors after inflation if they were never in thermalcontact [204, 231], or by the thermal decoupling of the sectors if they once were [350]. Since inflationcan potentially reheat the dark and visible sectors very asymmetrically, we consider a broad range ofR ∈ [10−12,10−3]. For Tx Tc and Gx = SU(N), the entropy ratio is related to the temperatures in thetwo sectors byR =2(N2−1)g∗S(TxT)3, (5.7)where g∗S is an effective number of degrees of freedom in the visible sector at temperature T . This ratiowill be maintained through the confining transition provided it is not too strongly first order [351].5.3.1 Single-State ModelConsider first a dark sector consisting of a single real scalar φx with mass mx, 2→ 2 and 3→ 2 self-interaction cross sections given by Eqs. (5.4,5.5), and no direct connection to the SM. We show belowthat this is often an accurate simplified model for the freeze-out of the lightest 0++ glueball, even whenthe heavier glueballs are included.The freeze-out dynamics of this model coincide with the general scenario of Ref. [349]. Chemicalequilibrium of the φx scalar is maintained by 3→ 2 transitions. These transitions also transfer energyto the remaining φx particles in the non-relativistic plasma causing them to cool more slowly than theywould if there was a relativistic bath to absorb the input heat [349, 350]. Freeze-out occurs when the3→ 2 transition rate becomes too slow to keep up with the Hubble expansion. While this happens,kinetic equilibrium is maintained by 2→ 2 elastic scattering of glueballs, which is parametrically muchfaster than the 3→ 2 processes at dark-sector temperatures below the scalar mass.Kinetic equilibrium implies that the number density of φx particles takes the formnx =∫ d3 p(2pi)3[e(Ex−µx)/Tx−1]−1, (5.8)where E =√~p2+m2x , and Tx and µx refer to the temperature and chemical potential of the φx plasma.Analogous to our derivation of the 2→ 2 Boltzmann equation in chapter 2, this number density evolvesin time according to [147, 349]n˙x+3Hnx =−〈σ32v2〉(n3x−n2x n¯x) , (5.9)where H is the Hubble rate (sourced by both the visible and dark sectors), n¯x = nx(µx→0) is the number99�� ������������������������������� ���� ���� � � ����������� �Figure 5.3: Temperature evolution of the hidden sector while the 0++ is freezing-out. Shown arethe inverse of the dark and visible sector temperatures (xx(x) = mx/Tx(T )). Before freeze-out, the temperature drops (xx rises) much slower than the visible sector, as the glueballsreheat themselves through the 3→ 2 process (dashed red line). After freeze-out, the darktemperature scales as Tx ∝ a−2 (solid red line). The dashed black line shows the comparisonwith how the temperatures would evolve if there was no reheating.density in the limit of zero chemical potential, and the thermally-averaged cross section is〈σ32v2〉 = 1n¯3x∫dΠ1dΠ2dΠ3 e−(E1+E2+E3)/Tx σ32v2 (5.10)' 1(4pi)3(4piN)6 1m5x,where dΠi = gid3 pi/(2pi)32Ei and we have used Eq. (5.5) in going to the second line. The dark-sectorentropy isTxsx = ρx+ px−µxnx , (5.11)with the energy density ρx and pressure px determined by the same distribution function as nx inEq. (5.8). Together, Eqs. (5.6,5.9) provide two equations for the two unknowns Tx(t) and µx(t) thatcan be solved in conjunction with the Friedmann equation for H(t) [70]. Prior to freeze-out of the 0++mode (and after dark confinement), the dark temperature falls as Tx ∝ 1/ ln(a) due to the energy injectedby 3→ 2 annihilations [349]. After 0++ freeze-out, the dark temperature falls as Tx ∝ a−2. This effectis illustrated in Fig. 5.3.While the results we present below are based on the numerical evaluation of Eqs. (5.6,5.9), it isinstructive to derive an approximate solution for the non-relativistic freeze-out process [349]. For mx100Tx, µx, the dark-sector entropy density issx '(mxTx)nx . (5.12)This relation is maintained with zero chemical potential until freeze-out occurs, after which the numberdensity just dilutes with the expansion of spacetime. Matching these limits and applying Eq. (5.6), thefreeze-out yield isYx =nxs' Rx f ox, (5.13)where x f ox =mx/Tf ox and Tf ox is the dark temperature at which chemical equilibrium is lost. To determinex f ox , we follow Ref. [349] and identify freeze-out with the point at which the equilibrium 3→ 2 rate fallsbelow the fractional rate of change of nxa3, which gives3H ' x f ox 〈σ32v2〉n¯2x . (5.14)Assuming visible radiation dominates the total energy density during freeze-out, this implies a visible-sector freeze-out temperature ofT f o ' n¯xx f ox MPl〈σ32v2〉√g f o∗ pi2/101/2 , (5.15)where MPl is the reduced Planck mass and gf o∗ is the number of effective energy degrees of freedom inthe visible sector [70] at glueball freeze-out. Combining this with the entropy relation of Eq. (5.6) andthe explicit form of n¯x in the non-relativistic regime, we find(x f ox )5/2 e2xf ox =g f o∗S180piRm4xMPl〈σ32v2〉√g f o∗ pi2/103/2 , (5.16)with g f o∗S the number of effective entropy degrees of freedom in the visible sector [70] at glueball freeze-out. This relation can be solved iteratively for x f ox . Numerically, we find xf ox ∈ [5,20] for R∈ [10−12,0.1]and m0 ∈ [10−3,109] GeV.In Fig. 5.4 we show the mass-weighted relic yield mxYx of φx with N = 3 as a function of the massof the lightest glueball Λx = mx and the dark-to-visible entropy ratio R. If the lightest glueball is stable,the mass-weighted yield is related directly to the relic density byΩxh2 = (0.1186)×(mxYx4.322×10−10 GeV). (5.17)We also indicate on the plot where the relic yield coincides with the observed dark matter relic density,101-� � � � � �-��-��-�-�-������(Λ�/���)�����(�) Ω� � �= ������ � ��� <������(����/���)-��-���Figure 5.4: Mass-weighted relic yields in the single-state simplified model discussed in the textwith N = 3 as a function of the mass Λx = mx and entropy ratio R. The solid white lineindicates where the glueball density saturates the observed dark matter abundance Ωxh2 =0.1186 [9]. The dark masked region at the lower right indicates where freeze-out occurs forx f ox < 5 and our freeze-out calculation is not applicable due to the unknown dynamics of theconfining phase transition.Ωxh2 = 0.1186 [9]. The dark shaded region at the lower right corresponds to x f ox < 5. As will be dis-cussed below, there is an additional uncertainty in the relic abundance in this region when this simplifiedmodel is applied to dark glueballs, and the present calculation might not be applicable here.5.3.2 Dynamics of the Confining TransitionDark glueballs are first formed in the early Universe in a confining phase transition. At dark temperaturesmuch larger than the confinement scale, Tx Λx, the dark sector can be described as a thermal bath ofweakly interacting dark gluons with g∗ = 2(N2−1) degrees of freedom. As Tx cools below Λx a phasetransition occurs with the gluons binding to form glueballs. Depending on the nature of the transitionand the interaction rate of the resulting glueballs, this transition can affect the glueball relic density.The nature of the confining transition in pure SU(N) gauge theories has been studied in detail onthe lattice [370–379] and in a number of semi-analytic models (e.g. Refs. [380–385]). The transitionis found to be second order for N = 2, weakly first order for N = 3, and increasingly first order forN ≥ 4 [372, 373]. The dark-sector critical temperature Tc for N = 2−8 is fit well by the relation [378]Tc/√σ = 0.5949(17)+0.458(18)/N2 , (5.18)where√σ ' 1.2/r0 [357] (or√σ ' 2.5ΛMS [358, 362]). Note that this is about a factor of five smallerthan the mass of the lightest glueball in Tab. 5.1. For N > 2 where the transition is found to be first-order,102the latent heat Lh scales according to [376]Lh(N2−1)T 4c= 0.388(3)−1.61(4)/N2 , (5.19)while the interface tension between the phases is consistent with [373]σcdT 3c= 0.0138(3)N2−0.104(3) . (5.20)In the confined phase just below the critical temperature, 0.7Tc . Tx < Tc, the entropy and pressure aresignificantly larger than what is predicted from the known glueball states [377, 386]. Interestingly, thisdiscrepancy can be explained by additional glueball states with a Hagedorn spectrum corresponding tothe excitations of a bosonic closed string [386–388], in agreement with the model of Ref. [389]. Thelattice studies of Refs. [390, 391] also suggest that the lowest-lying glueball pole masses persist nearlyunchanged up to Tc (although see Ref. [392] for a different conclusion).Much less is known about the non-equilibrium properties of the SU(N) confining transition suchas the nucleation temperature and rate. An estimate of the nucleation rate in the early Universe forSU(3), valid in the limit of small supercooling, is given in Ref. [351]. For supercooling by an amountTx = (1−δ )Tc, they find a decay per unit volume ofΓ/V ' T 4x e−∆Fc/Tx (5.21)with∆FcTx' 16pi3σ3cdL2hTcδ−2 (5.22)' 2.92×10−4δ−2N2(1−7.54/N2)3(1−4.15/N2)2, (5.23)where ∆Fc is the difference between the free energies of the two phases, and in the second line wehave generalized the result of Ref. [351] to SU(N ≥ 3) using the central lattice values of Lh and σcdlisted above. For moderate N, this suggests that nucleation occurs at Tx extremely close to Tc (providedTx/T ∼R1/3 is not too small) with only a very small injection of entropy. For very large N, the nucleationrate becomes small and the assumption of small supercooling made above breaks down. This suggeststhat significant supercooling can occur at large N, although a full non-perturbative calculation of thenucleation rate would be needed to verify this.To apply these results to the calculation of relic glueball abundances, we assume that the phasetransition completes with Tx = Tc ' m0++/5 [393] and that the mass spectrum of stable glueballs justafter the transition is the same as at Tx→ 0. The simplified model discussed above can then be used withinitial conditions at xx = xcx ≡mx/Tc, which can be specified completely in terms of R= sx/s and µx(xcx).If the 3→ 2 depletion process is fast relative to the Hubble rate at xx = xcx, the initial chemical potentialrelaxes quickly to zero and the final relic density is specified completely by the choice of R. However,103�=� �=� �=�� �=�����-� ��� ��� ��� ��� �����-����-����-���-���-���-���-�Λ� (���)� ���(���)��� ��� �����-���-���-�Ω���= ������Figure 5.5: Mass-weighted relic yields in the single-state simplified model with the initial densityset by f = Yx(xcx)/Yx(xcx,µx = 0) at Tx = Λx/5 with R = 10−9 and N = 3.if full chemical equilibration does not occur at xx = xcx, a range of µ(xcx) values can be consistent withthe equilibration rate relative to Hubble, and there is an additional uncertainty in the final glueball relicdensity for a given value of the entropy ratio R.To investigate the potential dependence of the relic yield on the initial glueball density following thephase transition, we repeat the freeze-out calculation described in the previous section for Gx = SU(3)with different initial glueball densities at Tx = Tc defined by the ratio f = Yx(xcx)/Yx(xcx,µx = 0). Ourresults are shown for a range of values of Λx with R= 10−9 and N = 3 in Fig. 5.5. For most of the rangeof Λx and R of interest, dark freeze-out occurs with x f ox > xcx ' 5 and the final glueball relic density isinsensitive to the initial value after the phase transition. Even when x f ox < xcx, some residual annihilation(or creation) typically occurs, and the final density tends to be similar to f = 1. The region in the Λx–Rplane in which this additional uncertainty is present is indicated by the shaded area in Fig. Freeze-Out with Multiple GlueballsWe turn next to the heavier glueballs above the lightest state. Recall from Section 5.2 that multiplestable glueballs are expected in a confining Yang-Mills theory, with the spectra found for SU(2) andSU(3) groups listed in Table 5.1. These heavier states lead to new annihilation channels involving thelightest glueball, and their relic densities can be of cosmological interest.The freeze-out of the full glueball spectrum involves many states and a network with numerousreaction channels. Despite this complexity, we find that the glueball relic densities follow a relatively104i JPC mi/m0++1 0++ 1.002 2++ 1.393 3++ 2.134 0−+ 1.505 2−+ 1.796 1+− 1.707 3+− 2.058 2+− 2.409 0+− 2.7410 1−− 2.2311 2−− 2.2712 3−− 2.39Table 5.2: List of stable glueball states and mass ratios for SU(3), from Ref. [6].simple pattern with three main features. First, the relic density of the lightest glueball is described verywell by the simplified one-state model presented above provided it freezes out while it is significantlynon-relativistic. Second, the relic densities of the heavier C-even states are typically extremely smallrelative to the lightest glueball. And third, the total relic density of C-odd states (for SU(N ≥ 3) gaugegroups) is dominated by the lightest C-odd mode and is much smaller than the lightest 0++ state buttypically larger than all the other C-even states. This significant difference arises from the conservedC number in the dark sector, which allows coannihilation of the heavier C-even states with the lightestglueball but forbids it for C-odd states.In this section we investigate the relic densities of the full set of glueballs for the dark gauge groupSU(3). We begin by determining which 2→ 2 glueball reactions are allowed by JPC conservation in thedark sector, and we estimate their rates. Next, we study a simplified reaction network of C-even statesthat we argue captures the most important features of the full dynamics. Finally, we perform a similaranalysis for the C-odd states.5.4.1 Glueball ReactionsTo discuss glueball reactions for Gx = SU(3), it will be convenient to label the modes in the spectrumby i = 1,2, . . . ,12 as in Table 5.2. This table also lists their relative masses and JPC quantum numbers.The specific interactions between glueballs are not known, but all possible processes consistent withdark-sector J, P, and C conservation are expected to be present. For a 2→ 2 glueball reaction of theform i+ j→ k+ l, conservation of C requiresC jC j =CkCl . (5.24)105This is trivial to apply and rules out a number of reactions. Conservation of P impliesPiPj = (−1)LPkPl , (5.25)where L is the relative orbital angular momentum of the reaction channel. When identical particles arepresent, they must also be symmetrized. In general, it can be shown that there always exists a value of Lsuch that both parity and total angular momentum are conserved unless either Ji = J j = 0 or Jk = Jl = 0.If the process i+ j↔ k+ l is allowed, it contributes to the collision term in the Boltzmann equationfor glueball i according to∆n˙i = −〈σv〉i jklnin j + 〈σv〉kli jnknl , (5.26)where 〈σv〉i jkl is the thermally-averaged cross section and ni refers to the number density of the i-thspecies. Assuming kinetic equilibrium is maintained among the glueballs, we haveni = gi eµi/Tx(4pi)m2i Tx K2(mi/Tx) (5.27)' gi(miTx2pi)3/2e−(mi−µi)/Tx , (5.28)where Tx is the temperature of the glueball bath and gi, mi, and µi are the number of degrees of freedom,mass, and chemical potential of the type-i glueball. The thermally-averaged cross-section is given by〈σv〉i jkl = 1nin j∫ d3 pi(2pi)3∫ d3 p j(2pi)3gi e(µi−Ei)/Txg j e(µ j−E j)/Tx(σv)i jkl (5.29)=gig jn¯in¯ j∫ d3 pi(2pi)3∫ d3 p j(2pi)3e−(Ei+E j)/Tx (σv)i jkl , (5.30)where Ei =√m2i +~p2i and n¯i = ni(µi = 0). Note that the chemical potentials cancel in this expression.The reaction i+ j → k+ l is either exothermic (mi +m j ≥ mk +ml) or endothermic (mi +m j <mk +ml). Equilibration of this process implies µi+µ j = µk +µl . Combined with detailed balance, wemust have〈σv〉i jkl n¯in¯ j = 〈σv〉kli j n¯kn¯l . (5.31)Using these relations, the thermally-averaged rates of endothermic reactions can be estimated based onthose of exothermic reactions.Thermal averaging of cross sections was studied in detail in Refs. [73, 215]. Generalizing theirresults slightly and using the large-N and NDA estimates of interaction strengths, we estimate thethermally-averaged cross section of an exothermic process i+ j→ k+ l that proceeds at lowest orbital106angular momentum level L by〈σv〉i jkl ' (4pi)3N4βi jklsi jcL(2xi+x j)L, (5.32)where xi = mi/T ,si j =(1+3xi+ x j)(mi+m j)2 , (5.33)along withβi jkl =2p′kl√si j (5.34)=1si j(s2i j +m4k +m4l −2si jm2k−2si jm2l −2m2l m2k)1/2,and the coefficients cL are [73]c0 = 1, c1 = 3/2, c2 = 15/8, c3 = 35/16, c4 = 315/128 . (5.35)The first factor in Eq. (5.32) contains the couplings, the second factor describes the kinematics nearthreshold in the non-relativistic limit, while the third is the velocity suppression for a process that goesat the L-th partial wave.The cross-section estimates of Eq. (5.32) can be used to judge which reactions are most significantduring freeze-out. The relative effect of the process i+ j→ k+ l (with j, k, l 6= i) on the number densityof glueball species i is|∆n˙i|ni= 〈σv〉i jkl n j . (5.36)In general, this reaction is cosmologically active for |∆n˙i|/ni > H. Scanning over all possible 2→ 2reactions of SU(N = 3) glueballs, we find that in full equilibrium with xx > 5 and for every glueballspecies i > 1 there exist multiple number-changing 2→ 2 reactions down to lighter states with |∆n˙i|/nisignificantly larger than the corresponding quantity for 3→ 2 annihilation of the lightest glueball. Thisimplies that relative chemical equilibrium is maintained among the glueballs during and for some timeafter 3→ 2 freeze-out, withnin j=n¯in¯ j' gig j(mim j)3/2e−(mi−m j)/Tx . (5.37)Equivalently, the number densities of all species immediately after 3→ 2 freeze-out are given by theirequilibrium values with a common chemical potential.Relative chemical equilibrium after 3→ 2 freeze-out implies further that the relative importance ofdifferent 2→ 2 reactions on the subsequent freeze-out of the heavier glueballs can be estimated using107their equilibrium number densities. This allows us to greatly simplify the set of reaction networks bykeeping only the dominant processes and concentrating exclusively on a few key states. It turns out to beconsistent and convenient to study the C-even and C-odd states independently, and this is the approachwe take below.5.4.2 Relic Densities of C-Even StatesThe lightest C-even glueballs above the lowest mode have JPC = 2++, 0−+, 2−+ and correspond to i =2, 4, 5, in our labelling scheme. Scanning over all possible reactions for these states and estimating theirrelative effects on the number densities as above, the dominant interactions near relative equilibrium arefound to form a minimal closed system. The reaction network for the system is described byn˙1+3H n1 = −〈σ32v2〉n21(n1− n¯1) (5.38)−12〈σv〉2111[(n¯2n¯1)n1n2−n22]−〈σv〉2211[(n¯2n¯1)2n21−n22]−12〈σv〉2214[(n¯22n¯1n¯4)n1n4−n22]−12〈σv〉2415[(n¯2n¯4n¯1n¯5)n1n5−n2n4]n˙2+3H n2 = +12〈σv〉2111[(n¯2n¯1)n1n2−n22](5.39)+〈σv〉2211[(n¯2n¯1)2n21−n22]+〈σv〉2214[(n¯22n¯1n¯4)n1n4−n22]+12〈σv〉2415[(n¯2n¯4n¯1n¯5)n1n5−n2n4]−12〈σv〉1512[(n¯1n¯5n¯1n¯2)n1n2−n1n5]n˙4+3H n4 = −12〈σv〉2214[(n¯22n¯1n¯4)n1n4−n22](5.40)+12〈σv〉2415[(n¯2n¯4n¯1n¯5)n1n5−n2n4]n˙5+3H n5 = −12〈σv〉2415[(n¯2n¯4n¯1n¯5)n1n5−n2n4](5.41)+12〈σv〉1512[(n¯1n¯5n¯1n¯2)n1n2−n1n5]108��++��++��++��++��-+��-+��-+��-+� �� �� �����-����-����-����-����-����-����� ���(���)Λ� = ��� ���� = ��-���++��++��++��++��-+��-+��-+��-+� �� �� �����-����-����-����-���-���� ���(���)Λ� = ��� ���� = ��-���++��++��++��++��-+��-+��-+��-+� �� �� �����-����-����-����-���� ���(���)Λ� = ��� ���� = ��-���++��++��++��++��-+��-+��-+��-+� �� �� �����-����-����-����-���-������ ���(���)Λ� = ��� ���� = ��-�Figure 5.6: Mass-weighted relic yields of the four lightest C-even glueballs in SU(3), JPC =0++, 2++, 0−+, 2−+, as a function of the dark glueball temperature variable xx =mx/Tx com-puted using the simplified reaction network discussed in the text. The solid lines show theyields derived from the reaction network while the dashed lines indicate the yields expectedif the states were to continue following equilibrium with µi = 0. Top left: (Λx/GeV, R) =(1, 10−9). Top right: (Λx/GeV, R) = (105, 10−9). Bottom left: (Λx/GeV, R) = (1, 10−3).Bottom right: (Λx/GeV, R) = (105, 10−3).The factors of 1/2 appearing here are symmetry factors for initial states that are not included in thestandard definition of the thermally-averaged cross section [215]. They ensure that the summed numberdensity n1+n2+n4+n5 is conserved in the absence of 3→ 2 reactions. In addition to these evolutionequations, the ratio of entropies R = sx/s is conserved after the confining transition at T cx ' mx/5, withthe dark sector entropy now extended to include all (relevant) glueball modes.Numerical solutions of this system of equations for SU(3) dark glueballs are shown in Fig. 5.6 for109� = ��-� � = ��-�� = ��-� � = ��-����-� ��� ��� ��� ��� �����-����-����-����-����-���-�Λ� (���)� �-+ � �-+ (���) Ω���= ������Λ� = ���Λ� = ���Λ� = ���Λ� = ���Λ� = ��-���-�� ��-�� ��-� ��-� ��-���-����-����-����-����-���-��� �-+ � �-+ (���) Ω���= ������Figure 5.7: Mass-weighted relic yields of the 0−+ dark glueball in SU(3) as functions of Λx = mxand R, computed using the simplified C-even reaction network discussed in the text. Forreference, we also indicate the yield corresponding to the observed dark matter density. Notethat the yield of the 0++ state is much larger.the parameter values (Λx/GeV, R) = (1, 10−9), (105, 10−9), (1, 10−3), (105, 10−3). In each panel, theevolution of the mass-weighted yields miYx with xx =mx/Tx are given by the solid lines, while the dashedlines show the mass-weighted yield of each species with µi = 0. In all four panels, the lightest 0++ modeis seen to dominate the total glueball relic abundance for xx & 10. This abundance is found to matchclosely with the value determined by the one-glueball simplified model discussed above. The muchsmaller relic abundances of the heavier glueball modes is due to the efficient coannihilation reactionsthey experience. Since these 2→ 2 processes are parametrically faster than the 3→ 2 annihilationssetting the 0++ density, relative chemical equilibrium is maintained to large values of xx. This implies astrong exponential suppression of the heavier glueball densities as in Eq. (5.37).Let us also point out that the 0−+ state freezes out (of relative chemical equilibrium) well before the2++ and 2−+ modes, even though it is heavier than the 2++. This can be understood by examining therelative rates of the depletion reactions for the 0−+ state; for xx& 20 it is found to be 0−++0++→ 2+++2++. Comparing masses, this reaction is found to be endothermic and thus it receives an additional ratesuppression as discussed in Ref. [394]. The dependence of the 0−+ (i = 4) glueball relic density onΛx = mx and R is also shown in Fig. Relic Densities of C-Odd StatesFreeze-out of the C-odd glueballs is qualitatively different from that of the C-even modes due to theconservation of C number in the dark sector. This forbids coannihilation reactions of the C-odd stateswith the relatively abundant lightest 0++ glueball into final states with only C-even modes, and can lead110to a significant relic density for the lightest C-odd 1+− state.To see how this comes about, let us split up the labels of the state indices defined in Table 5.2according toi, j = 1,2,3,4,5 =C-even , a,b = 6,7, . . .12 =C-odd , (5.42)and let us also define the net C-odd density byn− =12∑a=6na . (5.43)The net collision term in the Boltzmann equation for n− is∆n˙− = ∑a∆n˙a (5.44)= − ∑ab,i j〈σv〉abi j nanb+ ∑i j,ab〈σv〉i jab nin j . (5.45)The key feature of this expression is that all processes contributing to the rate of change of n˙− havetwo C-odd particles either in the initial or the final state [215]. Using detailed balance, we can rewriteEq. (5.45) in the form∆n˙− = −〈σv〉6611n2−[∑abi j( 〈σv〉abi j〈σv〉6611nanbn2−Θ++〈σv〉i jab〈σv〉6611nanbn2−n¯in¯ jn¯an¯bΘ−)](5.46)+〈σv〉6611n21(n¯−n¯1)2[∑abi j( 〈σv〉abi j〈σv〉6611n¯an¯bn2−nin jn¯in¯ jn¯21n21Θ++〈σv〉i jab〈σv〉6611nin jn2−n¯21n21Θ−)],where Θ+ = Θ(ma +mb−mi−m j) and Θ− = Θ(mi +m j−ma−mb) are step functions to select outexothermic reactions as appropriate.Consider the relative sizes of the individual terms in Eq. (5.46) when relative equilibrium is main-tained. In the first line, the first term is on the order of unity for a = b = 6 but has an exponentialsuppression otherwise from the factor of nanb/n2−, while the second term has an additional exponen-tial suppression from the factor n¯in¯ j/n¯an¯b (mi +m j > ma +mb). Similar arguments also apply to theterms in the second line of Eq. (5.46), noting that n¯in¯ jn21 = nin jn¯21 in relative equilibrium, and only thea = b = 6 portion of the first term avoids an exponential suppression. Indeed, a numerical evaluationof these contributions, assuming relative equilibrium and moderate xx & 10, confirms that the a = b = 6terms of the Θ+ pieces dominate the collision term.The total C-odd density begins to deviate appreciably from the relative equilibrium value for〈σv〉6611 n¯2−(n1/n¯1)2 ∼H. This occurs well before the C-even states freeze out, and also well before C-odd transfer reactions, such as 7+1↔ 2+6, turn off. The latter result implies that the relative densitiesof C-odd states are maintained among themselves (but not the C-even states) even after the net C-odddensity has frozen out. Therefore we also expect n6/n−→ 1 and na>6/n−→ 0 provided these processes111� = ��-� � = ��-�� = ��-� � = ��-����-� ��� ��� ��� ��� �����-����-����-����-���-����Λ� (���)� �+- � �+- (���) Ω���= ������Λ� = ��� Λ� = ���Λ� = ��� Λ� = ���Λ� = ��-���-�� ��-�� ��-� ��-� ��-���-����-����-����-���-������ �+- � �+- (���) Ω���= ������Figure 5.8: Mass-weighted relic yields of the 1+− dark glueball in SU(3) as a function of Λx =mxand R, computed in the simplified two-state network discussed in the text. For reference, wealso indicate the yield corresponding to the observed dark matter density. Note that the yieldof the 0++ state is much larger.turn off at moderate xx & 10.The net result of this analysis is that it is generally a good approximation to compute the freeze-outof the C-odd density using a simplified two-state system consisting only of the i = 1, 6 (0++ and 1+−)glueballs. Correspondingly, the system of Boltzmann equations isn˙1+3Hn1 = −〈σ32v2〉n21(n1− n¯1) (5.47)+〈σv〉6611[n26−(n1n¯1)2n¯26]n˙6+3Hn6 = −〈σv〉6611[n26−(n1n¯1)2n¯26]. (5.48)Corrections to this estimate are expected to be of order unity, which is well within the uncertainties onthe cross sections.The mass-weighted yields of the lightest 1+− C-odd SU(3) glueball based on this analysis are shownin Fig. 5.8. Like for the C-even states, the inclusion of additional heavier C-odd glueballs generally hasa negligible effect on the final abundance of the lightest 0++ mode relative to the one-state modeldiscussed previously. Furthermore, the 0++ state dominates the total glueball density, and the relicabundance of the 1+− state is smaller by several orders of magnitude. However, the 1+− density canbe considerably larger than any of the C-even states, even though it is heavier than the 2++ and 0−+glueballs. As discussed above, this can be understood by the absence of relevant coannihilation reactions112involving the much more abundant 0++ glueball. Let us also point out that C-odd dark glueballs providean explicit realization of the scenario discussed in Ref. [395] consisting of a stable dark matter statefreezing out in the background of a massive bath.5.5 Dark Matter Scenarios and Connections to the SMStable dark glueballs will contribute to the dark matter (DM) density of the Universe. However, if thedark sector has a connection to the SM, some or all of the dark glueballs will be able to decay [229].Possible SM connections include heavy matter charged under both the dark and SM gauge groups [11,12], a Higgs portal [337], or a Yukawa connection [12]. In all of the above scenarios, the decay rates ofglueballs through the various operators can span an enormous range. For lifetimes beyond the age of theUniverse the glueballs will contribute to the DM density and the considerations discussed above applyhere as well. In addition, for lifetimes τ . 1026 s there will also be constraints from energy injectioninto the CMB near recombination [123, 270, 396], x-ray and gamma-ray fluxes [235, 397], and energyrelease during primordial nucleosynthesis [258–260, 270], such as that discussed in Part III. Given theparametrically similar decay rates and the much larger relic density of the lightest 0++ glueball relativeto the others, these bounds apply primarily to this state, although it is possible for the 1+− (and toa lesser extent, the 0−+) to also contribute to the constraints if certain symmetry conditions are met.These will be discussed in detail in the following chapter.These SM operators can also be relevant for the glueball freeze-out abundances. At high temper-atures they can lead to the thermalization of the dark and visible sectors, although the specific detailsdepend on the reheating history after primordial inflation. They may also help to further populate thedark sector through inverse decays [270], or induce decays before freeze-out occurs, although this typi-cally requires relatively larger values ofΛx/M. This is a modification of the scenario discussed in Part II,and so we must keep in mind the connection between the two sectors when moving forward.With no connection to the SM, all the states in the glueball spectrum discussed in Section 5.2 willbe stable and contribute to the net DM density2. As reported in Sections 5.3 and 5.4, the total glueballcontribution will be dominated by the lightest 0++ state. The DM scenario in this case coincides with theglueball scenarios considered in Refs. [231, 232, 235] in which only the lightest glueball was considered.Avoiding overclosure by the glueball relic density bounds Λx and R from above, as can be seen inFig. 5.4. If the lightest glueball makes up all the DM, Λx is bounded from below by the requirement thatits self-interaction cross section not be too large, σ2→2/m. 10cm2/g, which translates into [231, 232,235]Λx & 100 MeV(3N)4/3. (5.49)Smaller Λx can also interfere with cosmic structure formation [207, 235, 398].2Decays to gravitons are possible, but the corresponding lifetime is much longer than the age of the Universe for Λx .107 GeV [235].1135.6 ConclusionsIn this chapter we have investigated the freeze-out dynamics of SU(3) dark glueballs in the earlyUniverse. This is our first step towards creating a complex hidden sector (that will eventually haveSM connections). Such glueballs arise from confinement in theories with a new non-Abelian gaugeforce decoupled from the SM and all charged matter significantly heavier than the confinement scale.Our results expand upon previous studies of the cosmological history of dark glueballs in two keyways [231, 232, 235]. First, we studied potential new effects on the glueball relic density due to theconfining phase transition itself. And second, we performed a detailed analysis of the freeze-out dy-namics of the heavier glueballs in the spectrum. We also briefly discussed connections to the SM aswell as some of the implications of the heavier glueballs on dark matter, astrophysics, and cosmology,with a more detailed study of these effects to follow in the next chapter.When the glueballs are unable to decay efficiently through connectors to the SM (or other lighterstates), we find that the lightest 0++ state dominates the total glueball relic abundance, and the abun-dance we calculate is in agreement with previous studies that only considered the lightest state [232,235]. The relative relic densities of heavier glueballs in the spectrum are orders of magnitude smaller,with the largest contributions coming from the 0−+ and 1+− modes (for SU(3)). Even though the abun-dances of these states are much smaller than the lightest 0++, they can also be parametrically long-livedcompared to the 0++. This opens the possibility of the 0++ mode decaying away early, and the heaviermodes making up the DM density today or leading to the most stringent constraints on dark Yang-Millstheories. A detailed study of these effects based on the results determined here for the freeze-out relicabundances will be the focus of the next chapter.Our results are also be applicable to other non-Abelian gauge groups with some straightforwardmodifications. The lightest glueball, which is generically expected to have JPC = 0++ [355], will havethe largest relic yield. This yield can be computed reliably in the single-state model of Section 5.3,provided 3 → 2 annihilation processes are active after the confining transition. The relic yields ofthe heavier glueballs will depend on their specific masses and quantum numbers, but can be computedfollowing the general methods of Section 5.4. For a given confinement scale, their masses will be similarto those of SU(3) for general SU(N) groups, while the C-odd states are expected to be considerablyheavier for SO(2N) groups and absent for SU(2), SO(2N + 1), and Sp(2N) groups with a vanishingdabc symbol. The different properties of the more massive glueballs will only be relevant to cosmologywhen they have lifetimes that are parametrically much longer than the lightest 0++ mode.Now that we have developed a solid foundational understanding of how glueballs will interact andevolve when treated independently from external influences, we relax that assertion. As has been alludedto already, in the next chapter we build up our non-Abelian gauge force to include SM connections. Thiswill entail incorporating all of the previous chapters. For example, we will need to be careful to workin regimes that will not be affected by the presence of extra freeze-in transfer (or include those effectsif necessary), as we found in Ch. 3. After building up the model, we will identify the implications thatthe glueballs will have on astrophysics and cosmology, using constraints very closely related to thosederived in Ch. 4. As such, the next chapter will be a culmination of everything we have studied thus far.114Chapter 6Cosmological Bounds on Non-AbelianDark Forces6.1 IntroductionIn previous chapters, we have explored the many different facets of SM interactions with a dark sector.To begin, we started with only SM influence on a dark sector, and learned about how the evolution of thedark sector may change, even in the presence of UV operators. Next, we considered low-energy decaysin the context of BBN constraints, which explicitly considers how energy injected from a hidden sectorcan alter the abundances of the products of BBN. Finally, we moved away from simple dark sectors toa more complex model, motivated by many different areas of study. This new model, a non-Abeliangauge force, gives rise to a full spectrum of stable glueballs. We studied this glueball spectrum in anisolated environment, and learned how the various states will interact, ultimately leading to solutions forthe relic yields of the different glueballs. Now, we put all of this information to use, and consider howthis more complex model will interact with the Standard Model. Although we focus more explicitlyon the effect that glueball decays will have on the visible sector, we still make mention of the rolethat SM connectors might have on their production, before moving on to constraining the full theory.Schematically, we depict this wholistic view of the two sectors in Fig. 6.1.As we motivated previously, new gauge forces may be realized in nature beyond the SU(3)c ×SU(2)L×U(1)Y structure of the Standard Model (SM). If a new gauge force connects directly to SMmatter, it must have a characteristic mass scale above about a TeV to be consistent with experimentaltests of the SM [399–401]. On the other hand, new dark gauge forces that couple only very weaklyto the SM can be significantly lighter [325–327]. Such dark forces can be very challenging to probedirectly in experiments, and in many scenarios the strongest bounds on them come from astrophysicaland cosmological observations [270–272, 402].In this chapter we investigate the cosmological evolution and constraints on new non-Abelian dark0This chapter is based on L. Forestell, D. E. Morrissey, and K. Sigurdson, Cosmological Bounds on Non-Abelian DarkForces, Phys. Rev. D, 97, (2018), 075029, [arXiv:1710.06447] [4]115VS: SM DS: 0++, 1+−Figure 6.1: Flow of information considered in this chapter. We now have a fully realized darksector, with complex interactions. The dark sector may also transfer energy to the visiblesector via decays, while the visible sector may influence the production of dark glueballs.forces. The requirement of gauge invariance in theories of non-Abelian dark forces implies that the newgauge vector bosons can only couple to the SM through non-renormalizable operators [11, 12]. Thisstands in contrast to Abelian dark forces that can connect to the SM at the renormalizable level throughkinetic mixing with hypercharge. As a result, direct low-energy searches for non-Abelian dark forcesare very difficult, and cosmological observations usually provide the most powerful tests of them [231,232, 235–237, 336–338, 351, 403–407].The particle spectrum in theories of non-Abelian forces is diverse and complicated, and depends onboth the gauge group and the representations of the matter fields charged under it. We continue to focuson the minimal realization of a non-Abelian dark force consisting of a pure Yang-Mills theory with asimple gauge group Gx. If the visible and dark sectors do not interact, they evolve independently withdistinct temperatures T and Tx. This was thoroughly investigated in the previous chapter. After con-finement at Tx = Tc, the dark glueballs undergo a complicated freeze-out process. The energy density ofthe dark sector is dominated by the lightest glueball state, which on general grounds is expected to haveJPC = 0++ [355]. The lightest 0++ number density changes mainly through (3→ 2) self-annihilationprocesses [235], as we demonstrated explicitly in the previous chapter. While these reactions are active,the dark temperature changes very slowly, only falling off as the logarithm of the cosmological scalefactor [349, 408]. As a result, the lightest glueballs form a massive thermal bath in which the otherheavier glueballs annihilate through 2→ 2 processes and eventually freeze out [395, 409]. In the end, acollection of relic glueball densities is left over, dominated by the 0++ with exponentially smaller yieldsfor the heavier states [409].The process of glueball freeze-out can change drastically if there are operators that connect thevisible and dark sectors. Such operators are always expected at some level; quantum gravitational effectsare thought to induce gauge-invariant operators involving both SM and dark sector fields suppressed bypowers of the Planck mass [410–413]. Even stronger connections can arise if there exist new matterfields that couple directly to both the visible and dark sectors [11, 12]. As long as the new physicsgenerating these operators is much larger than the confinement scale, their effects can be parametrizedin terms of a set of non-renormalizable connector operators.With connectors, energy can now be transferred between the dark and visible sectors [229, 235–237].After confinement, connector operators can also modify the glueball freeze-out dynamics and inducedecays of some or all of the dark glueballs to the SM. If one of the glueballs is long-lived or stable, it will116contribute to the density of dark matter (DM) [231]. However, glueball lifetimes that are not exceedinglylong will inject energy into the cosmological plasma and modify the standard predictions for big bangnucleosynthesis (BBN) [259, 260] and the cosmic microwave background (CMB) [123, 312], as well asact as astrophysical sources of cosmic and gamma rays [402].The aim of this chapter is to estimate the bounds on pure non-Abelian dark forces in the presenceof connector operators from cosmology and astrophysics. We focus mainly on the dark gauge groupGx = SU(3) with glueball masses above m0 ≥ 100 MeV, and we study the leading connector operatorsbetween the dark vector bosons and the SM with characteristic mass scale M m0. As an initial con-dition, we assume inflation (or something like it) followed by preferential reheating to the visible sectorto a temperature above the confinement scale but below that of the connectors. With these assumptions,we find very strong limits on non-Abelian dark forces.Cosmological effects of dark gluons and glueballs were studied previously in Refs. [229, 231, 232,235–237, 336, 351, 402, 406], including the detailed analysis of the relic yield we presented in Ch. 5.We extend these earlier works with a more detailed analysis of the leading (2-body) connector operatorsand their effects on energy transfer between the visible and dark sectors. We also investigate the effectsof heavier glueballs in the spectrum beyond the lightest mode, and we show that the lightest C-oddglueball can play an important role in some cases and even make up the observed DM density when itis long lived or stable.Following this introduction, we discuss and review the general properties of glueballs relevant to thisanalysis in Sec. 6.2. Next, we present the leading connector operators to the SM and investigate theirimplications for glueball decays in Sec. 6.3. In Sec. 6.4 we study the cosmological evolution of the darkgauge theory and we compute glueball yields both with and without connector operators. These resultsare then applied to derive cosmological constraints on dark glueballs in Sec. 6.5. Finally, Sec. 6.6is reserved for our conclusions. Some technical details about gluon thermalization are collected inAppendix B. This chapter is based on work published in Ref. [4] in collaboration with David Morrisseyand Kris Sigurdson.6.2 Glueball PropertiesGlueballs have been studied using a variety of methods for a wide range of non-Abelian gauge groups [352,359]. In this section we review and derive some general results for SU(N) glueballs that will be essen-tial for the analysis to follow. The basic properties, such as masses and self-interaction strengths can befound in Ch. 5, while extra connections relevant to the standard model will be given here.6.2.1 Glueball Matrix ElementsIn Ch. 5, we described the glueball self-interactions. However, we will also need glueball matrix el-ements in the analysis to follow. Specific glueball states can be identified with gauge invariant gluonoperators, in the sense that the operators can create one-particle glueball states from the vacuum. For117example [11],S = tr(XµνXµν) → 0++P = tr(Xµν X˜µν) → 0−+Tµν = 12 tr(XµαXαν )− 14ηµνS → 2++, 1−+, 0++Ω(1)µν = tr(XµνXαβXαβ ) → 1−−, 1+−Ω(2)µν = tr(X αµ Xβα Xβν) → 1−−, 1+−(6.1)Here, Xµν = Xaµνta is the dark gluon field strength contracted with the generators of the fundamentalrepresentation of the group normalized to tr(tatb) = δ ab/2.The two matrix elements of greatest interest to us areαxFS0++ ≡ αx〈0|tr(XµνXµν)|0++〉 ∼ m3x (6.2)α3/2x M1+−0++ ≡ α3/2x 〈0++|(Ω(1)µν − 514Ω(2)µν)|1+−〉 ∼ √4piNm3x , (6.3)where the estimates on the right hand sides are based on large-N and NDA, and αx = g2x/4pi is the darkgauge coupling. In the second line, we have also suppressed the Lorentz structure of the matrix element,εµναβ pαεβ , where pα is the outgoing momentum and εβ is the polarization of the initial state [11]. Thefirst of these matrix elements, FS0++ , has been computed on the lattice for N = 3 with the result [354, 414]4piαxFS0++ = 2.3(5)m3x , (6.4)which agrees reasonably well with our large-N and NDA estimate and is scale independent. In contrast,the second matrix element has not been calculated on the lattice. We use the lattice value of FS0++ andthe NDA estimate α3/2x M1+−0++ =√4pi/N m3x in the analysis to follow.6.3 Connections to the SM and Glueball DecaysWith the SM uncharged under the dark gauge group Gx, gauge invariance forbids a direct renormalizableconnection of the dark gluons to the SM. However, massive mediator states that couple to both sectorscan generate non-renormalizable operators connecting them. If the characteristic mass scale of themediators is MΛx, the leading operators have mass dimension of eight and six, and take the form [11,12]O(8a) ∼ 1M4tr(FSMFSM) tr(XX) , (6.5)O(8b) ∼ 1M4Bµν tr(XXX)µν , (6.6)O(6) ∼ 1M2H†H tr(XX) , (6.7)where X and FSM refer to the dark gluon and SM field strengths. If present, these operators allow someor all of the glueballs to decay to the SM. In this section we illustrate mediator scenarios that generate118Figure 6.2: Diagrams that contribute to the effective Lagrangian of Eqs. (6.9) and (6.10). Effectiveoperators are created via integration of loops of the heavy mediator fermions[11]. The lefthand diagram will loosely correspond to decays of 0++ (Eq. (6.9)), while the right handdiagram will also contain terms that describe 1+− decays (Eq. (6.10)).these operators, and we compute the glueball decay rates they induce. This is an explicit realization ofthe unstable glueball scenarios hinted at in Ch. Dimension-8 OperatorsDimension-8 operators of the form of Eqs. (6.5,6.6) lead to glueball decays with characteristic rateΓ8 ∼ m9xM8. (6.8)Here, we present an explicit scenario of mediator fermions that generates these operators and we com-pute the glueball decay rates they induce.Before proceeding, it is helpful to organize the dimension-8 operators according to a dark chargeconjugation operation Cx under which Xaµ → −η(a)Xaµ , where η(a) is the sign change of the funda-mental generator ta under charge conjugation [415], with the SM vector bosons being invariant. Theoperators of Eq. (6.5) are even under Cx and those of Eq. (6.6) are odd. Furthermore, Cx coincides withthe Cx-number assignments of the glueball states. Correspondingly, the operators of Eq. (6.5) only allowdirect decays of Cx-even glueballs to the SM, or glueball transitions from even to even or odd to odd. Inparticular, at d = 8 the operator of Eq. (6.6) is required for the lightest Cx-odd 1+− glueball to decay.Consider now a set of massive vector-like fermions with masses Mr ∼M Λx, each transformingas a fundamental or antifundamental under Gx = SU(N) and the representation r of the SM gaugegroup (defined with respect to the left-handed component of the fermion). Direct collider and precisionelectroweak limits on such fermions imply Mr & 100 GeV if they only have electroweak charges, andMr & 1000 GeV if they are charged under QCD [11, 12, 416]. The diagrams that will be relevant forthe effective operators are shown in Fig. 6.2. The effective Lagrangian generated by integrating the119fermions out is [11]:Le f f ⊃ αxM4(α1χ1BµνBαβ +α2χ2W cµνWcαβ +α3χ3GaµνGaαβ)×(160Sηµνηαβ +145Pεµναβ + . . .)(6.9)+α3/2x α1/21M4χY Bµν1445(Ω(1)µν − 514Ω(2)µν). (6.10)Here, the dark gluon operators S, P, and Ω(1,2)µν correspond to Eq. (6.1), and the coefficients χi aregiven byχi = ∑rd(ri)T2(ri)/ρ4r (6.11)χY = ∑rd(ri)Yr/ρ4r , (6.12)where the sums run over the SM representations r of the fermions, and ρr = Mr/M. For each suchrepresentation, we define sub-representations r = (r1,r2,r3) with respect to the SM gauge factors Gi =U(1)Y , SU(2)L, SU(3)c. The quantity d(ri) is the number of copies of the i-th sub-representation withinr, and T2(ri) is the trace invariant for that factor (normalized to 1/2 for the N of SU(N) and Y 2 forU(1)Y ).1Generic representations of mediator fermions break the dark charge conjugation number Cx ex-plicitly and generate both operator types of Eqs. (6.5,6.6). This is explicit in Eq. (6.9), with botheven (χi 6= 0) and odd operators (χY 6= 0). However, there exist mediator fermion combinations thatpreserve Cx [201] and yield χY = 0. From Eq. (6.12), we see that this requires a specific combination offermion charges as well as masses. The presence of masses also implies that Cx can be broken softly. Incontrast, the χi coefficients of Eq. (6.11) are positive semi-definite and not subject to cancellation.The Cx-preserving operator of Eq. (6.9) allows direct decays of the 0++ glueball to pairs of SMvector bosons. The corresponding decay widths are [11]Γ(0++→ gg) = (N2c −1)α2316pi(260)2χ23m30(αxFS0++)2M8, (6.13)Γ(0++→ γγ)Γ(0++→ gg) =1(N2c −1)(αχγα3χ3)2(6.14)Γ(0++→ ZZ)Γ(0++→ gg) =1(N2c −1)(α2χZα3χ3)2(1−4m2Zm20)1/2(1−4m2Zm20+6m4Zm40)(6.15)Γ(0++→W+W−)Γ(0++→ gg) =2(N2c −1)(α2χ2α3χ3)2(1−4m2Wm20)1/2(1−4m2Wm20+6m4Wm40)(6.16)Γ(0++→ γZ)Γ(0++→ gg) =2(N2c −1)(√αα2 χγZα3χ3)2(1− m2Zm20)3(6.17)1Note that due to our normalizations, our χ2,3 are smaller by a factor of 1/2 than the corresponding terms in Ref. [11].120where m0 = mx is the 0++ glueball mass, FS0++ is given by Eq. (6.2), N2c − 1 = 8, the χi are definedin Eq. (6.11), χγ = χ1 + χ2, χZ = (s4Wχ1 + c4Wχ2)/c2W , and χγZ = (c2Wχ2− s2Wχ1)/cW , with sW beingthe sine of the weak mixing angle. Note that the decay width to gluons in Eq. (6.13) only applies form0  1 GeV; at lower masses the final states consist of hadrons. We do not attempt to model thishadronization, and instead we apply a factor of√1− (2mpi/m0)2 to the decay width. In evaluating thewidth of Eq. (6.13), we take α3 at scale m0 since the corresponding gluon operator is renormalized (atone-loop) in the same way as the standard field strength operator.Decays of the lightest 1+− glueball occur through the Cx-odd operator term in Eq. (6.9), with theleading decay channels expected to be 1+−→ 0+++{γ,Z}. The widths are [11]Γ(1+−→ 0++γ) = α24piχ2Y(1− m2xm21)3 m31 (α3/2x M1+−0++)2M8(6.18)Γ(1+−→ 0++Z) = α24pit2Wχ2Y[(1+m2xm21− m2Zm21)2−4 m2xm21]3/2m31 (α3/2x M1+−0++)2M8(6.19)with m1 = m1+− , and M1+−0++ defined in Eq. (6.3).The total decay lifetimes τ = 1/Γ of the 0++ and 1+− glueball states from the dimension-8 operatorsabove with χi = χY = 1 and Gx = SU(3) are shown in the left and right panels of Fig. 6.3. In theupper left of both plots, we mask out the regions with m0 > M/10 where our treatment in terms ofeffective operators breaks down. The dotted, solid, and dashed lines indicate reference lifetimes ofτ = 1/Γ = 0.1s, 5×1017 s,1026 s. These lifetimes correspond to decays that occur early in the historyof the Universe, at the present day, and long lived glueballs, respectively. Both decay rates follow theapproximate scaling of Eq. (6.8). All other known (SU(3)) glueballs can decay through these dimension-8 operators as well with parametrically similar rates, although there can be numerically significantdifferences due to coupling factors and phase space [11].6.3.2 Dimension-6 OperatorsGlueball decays through the dimension-6 operator of Eq. (6.7) proceed with characteristic rateΓ6 ∼ m50M4. (6.20)We present here two mediator scenarios that generate the operator of Eq. (6.7) and we compute thedecay rates they induce.Our first mediator scenario follows Ref. [12] and consists of mediator fermions with Yukawa cou-plings to the SM Higgs boson, as shown in Fig. 6.4. A minimal realization contains a vector-like SU(2)Ldoublet P with gauge quantum numbers (rx,1,2,−1/2), and a vector-like singlet N with quantum num-bers (rx,1,1,0) together with the interactions [12, 416]−L ⊃ MPP¯P+MNN¯N+λ P¯HN+(h.c.) . (6.21)121��������������(τ/�)-��������������Figure 6.3: Decay lifetimes τ = 1/Γ of the 0++ (left) and 1+− (right) glueball states due to thedimension-8 operators as a function of M and m0 for χi = χY = 1 and Gx = SU(3). Themasked regions at the upper left show where m0 > M/10 and our treatment in terms ofeffective operators breaks down, while the white dotted, solid, and dashed lines indicatereference lifetimes of τ = 0.1s, 5×1017 s, 1026 s.Figure 6.4: Diagram that contributes to the effective Lagrangian of Eq. (6.22). Effective operatorsare created via integration of loops of the heavy mediator fermions[12].For MN , MP  mh, the leading glueball effective operator from integrating out the fermions can beobtained using the low-energy Higgs theorem [417],Le f f ⊃ αx6pi T2(r)λ 2M2H†HXaµνXa µν , (6.22)where M2 ' MPMN and T2(rx) = 1/2 is the trace invariant of the fermion representation rx under thedark gauge group Gx. In addition to the dimension-6 operator above, the massive fermions also generatedimension-8 operators of the form of Eq. (6.9).A second mediator scenario consists of a complex scalar Φx charged under the dark gauge groupwith a Higgs-portal coupling,−L ⊃M2Φ|Φx|2+κ|Φx|2|H|2 (6.23)122Applying the low-energy Higgs theorem to this state (for MΦ mh), we find−Le f f ⊃ − αx48pi T2(r)κM2ΦH†H XaµνXaµν . (6.24)In passing, we note that the Higgs portal coupling of Eq. (6.23) respects dark Cx number.The operator generated in either mediator scenario can be written in the form−Le f f ⊃αxy2e f f6piM2H†HXaµνXaµν , (6.25)with the dimensionless coefficient ye f f . Since this operator is even under Cx, it only allows direct decaysof Cx-even glueballs to the SM, or even-to-even or odd-to-odd glueball transitions. It was shown inRef. [12] that this is sufficient to allow all known SU(3) glueballs to decay, except for the 1+− and 0−+modes. The absence of a 1+− decay follows from Cx considerations, while the conclusion for 0−+ is aresult of spin and parity, rather than Cx. This mode can decay at the dimension-6 level if a topologicaldark gluon term is added to the UV Lagrangian or by extending to a two-Higgs doublet model [12].Using the parametrization of Eq. (6.25), the direct decay of the 0++ glueball to the SM has rate [12]Γ(0++→ SM) =(y2e f f3pi)2(√2〈H〉)2 (αxFS0++)2M4 [(m20−m2h)2+(mhΓh)2]Γh(mh→ m0) , (6.26)where√2〈H〉 = 246 GeV is the electroweak vacuum expectation value, FS0++ is defined in Eq. (6.2),mh = 125 GeV is the Higgs mass, Γh = 4.1 MeV is the Higgs width, and Γh(mh→m0) is the total widththe SM Higgs would have if its mass were m0 (and includes decays to Higgs final states for m0 > 2mh).We evaluate this width using the expressions of Refs. [418, 419].In Fig. 6.5 we show the decay lifetime τ = 1/Γ of the 0++ glueball from the dimension-6 (anddimension-8) operators above with ye f f = 1 and Gx = SU(3). The upper region of the plot is maskedout since it corresponds to m0 >M/10 where our treatment in terms of effective operators breaks down.The dotted, solid, and dashed lines indicate lifetimes of τ = 0.1s, 5× 1017 s,1026 s. For m0 mh, the0++ lifetime scales according to Eq. (6.20), while for m0 < mh there is an additional suppression fromsmall Yukawa couplings. Comparing to the 1+− lifetime in Fig. 6.3, we see that it is parametricallylong-lived compared to the 0++ when both dimension-6 and dimension-8 operators are present.6.3.3 Decay ScenariosBased on the discussion above, we present four glueball decay scenarios organized by the dimensionsof the relevant decay operators and the dark conjugation charge Cx:1. Dimension-8 decays with broken CxIn this scenario glueballs decay exclusively through the dimension-8 operators of the form ofEq. (6.9). All glueballs are able to decay with parametrically similar rates. To realize this scenario,we use the effective interactions in Eq. (6.9) with χi = χY = 1.123Figure 6.5: Decay lifetime τ = 1/Γ of the 0++ glueball due to the combined dimension-6 anddimension-8 operators as a function of M and m0 for χi = χY = 1, ye f f = 1, and Gx = SU(3).The masked region at the upper left shows where m0 > M/10 and our treatment in termsof effective operators breaks down, while the dotted, solid, and dashed white lines indicatelifetimes of τ = 0.1s, 5×1017 s, 1026 s.2. Dimension-8 decays with exact CxThis scenario is similar to the first, but now with χY = 0. Conservation of Cx implies that thelightest 1+− glueball is stable. The other glueballs are all able to decay with parametricallysimilar rates.3. Dimension-6 decays with broken CxGlueball decays occur through the dimension-6 operator of Eq. (6.25) and the dimension-8 op-erators of Eq. (6.9). We realize the scenario by setting ye f f = 1 together with χi = χY = 1.With the exception of the 1+− mode (and possibly the 0−+), glueballs decay primarily throughthe dimension-6 operator. In contrast, the 1+− glueball only decays through the Cx-breakingdimension-8 operator with a parametrically suppressed rate, making it much longer-lived thanthe other glueballs, which in turn leads to different cosmological scenarios when considering theconstraints we can place on this model.4. Dimension-6 decays with exact CxDecays occur through the dimension-6 operator of Eq. (6.25) and the Cx-conserving terms inEq. (6.9). We realize the scenario by taking ye f f = 1, χi = 1, and χY = 0. The 1+− glueball isstable, while the other glueballs decay mainly through the dimension-6 operator.We study the cosmological implications of these four decay scenarios in the analysis to follow.1246.4 Glueball Densities in the Early UniverseGlueballs are formed in the early universe in a confining transition as the dark sector temperature Txfalls below a critical temperature Tc ∼ m0. After they are created, the glueballs undergo a complicatedfreeze-out process involving a range of 2→ 2 and 3→ 2 reactions. These dynamics become even morecomplicated when the dark sector connects to the SM through the operators discussed above, with neweffects such as energy transfer between the visible and dark sectors and glueball decays. In this sectionwe briefly review the formation and freeze-out of glueballs in the absence of connectors to the SM, aswas studied in detail in Ch. 5, and we investigate how this picture changes when connectors are present.6.4.1 Glueball Formation and Freeze-Out without ConnectorsThis section is a review of Ch. 5, and serves to emphasize the key points and parameters that are usedin estimating glueball yields. This will be important when we move on to consider the effects of thedecay scenarios discussed previously. Although parts of this are repetitions of the previous chapter, weinclude it to highlight the key differences and results that will be important in further analysis.In the absence of operators that connect to the SM, the visible and dark sectors do not thermalizewith each other. We assume that enough energy is liberated by reheating following primordial infla-tion (or something similar) that both sectors are able to thermalize independently with temperatures Tand Tx [204], and furthermore that Tx ≥ Tc at this point.2 Following the recipe for glueball formationprovided in Ch. 5, entropy is conserved independently in both sectors while kinetic equilibrium is main-tained. This implies that the ratio of entropy densities s and sx in the two sectors remains constant, andcan be parametrized by a single value, R. We again take R as an input to our calculation. However, wedo assume R < 1 corresponding to preferential reheating to the visible sector.Once formed, dark glueballs interact with each other and undergo a freeze-out process in whichthey depart from thermodynamic equilibrium and develop stable relic densities. This process is whatwe studied in detail in Ch. 5. In the last chapter, the evolution of glueball numbers was computednumerically using a network of Boltzmann equations containing the most important 2→ 2 and 3→ 2reactions, with thermally averaged cross sections estimated using the glueball effective Lagrangian ofEq. (5.3). There, we considered various different subsets of glueballs, in which we discovered that the0++ relic yield is predominately determined by the 3→2 reaction, and including other glueballs waslargely irrelevant to the overall yield. However, because some of our decay scenarios explicitly allowfor stable 1+− states, we consider the two-state model of Sec. 5.4.3 for the rest of this chapter so thatwe can track both the 0++ and 1+− yields for further analysis.In Fig. 6.6 we show the relic yields of 0++ (left) and 1+− (right) glueballs in the absence of connec-tors to the SM in the m0–R plane for Gx = SU(3). The white lines in both panels indicate where the relicdensity of that species coincides with the observed DM density, ΩDMh2 = 0.1188(10) [9]. The shadedregions at the lower right of both panels show where x f ox < 5 implying the glueball densities are set bythe non-perturbative dynamics of the confining phase transition. As expected, the 1+− yield is always2If not, the glueball relic density is set by the details of inflationary reheating.125��������������(����)-��-��-��-��-�����Figure 6.6: Mass-weighted relic yields of the 0++ (left) and 1+− (right) glueballs in the m0–Rplane in the absence of connectors for Gx = SU(3). The solid white lines in each panel indi-cate where the relic density saturates the observed dark matter abundance. The dark maskedregion at the lower right of both panels shows where 0++ freeze-out occurs for x f ox < 5 andour freeze-out calculation is not applicable due to the unknown dynamics of the confiningphase transition.much lower than the 0++ yield.Going beyond the two-state model, our arguments regarding the exponential suppression of the 1+−density relative to the 0++ also apply to the other heavier glueball modes, as we showed in Ch. 5. Thetotal glueball relic density is strongly dominated by the 0++ density, while 2→ 2 annihilation reactionspush the heavier glueball densities to much smaller values. In fact, these reactions tend to be much moreefficient for the other heavier glueballs than the 1+− due to coannihilation with the 0++. As a result,the 1+− state generally develops the second largest relic density, with the densities of the other darkglueballs being much smaller. This, combined with the unique decay properties of the 1+− glueballwhen connectors are included, is the reason why we only consider the effects of the 0++ and 1+−glueballs in our analysis of glueball cosmology.6.4.2 Glueball Freeze-Out with ConnectorsConnector operators can modify the freeze-out of glueballs in a number of ways. As we explicitlyexamined in Ch. 3, we realize that it is important to fully understand these connector operators beforemoving forward with our models. Scattering and decay reactions mediated by such operators transferenergy between the visible and dark sectors, and may allow them to thermalize. Decays through theconnector operators after confinement also deplete glueballs, and can occur before or after the freeze-out of the various (3→ 2) and (2→ 2) reactions. We investigate these effects here, both before andafter confinement, with a focus on the 0++ and 1+− glueballs. Our goal is to compute the yields of thesespecies prior to their decay.As in the freeze-out analysis without connectors, we take as an initial condition primordial inflation(or something like it) with preferential reheating to the visible sector characterized by a temperatureTRH that is larger than the confinement transition temperature Tc ' m0/5.5. With connectors, we also126assume TRH M. Reheating above the connector scale M is likely to thermalize the dark and visiblesectors at TRH , and can produce a relic abundance of the connector particles themselves. These can haveinteresting cosmological effects in their own right, acting as quirks if they carry Gx charge [342, 420,421], and potentially creating dark glueballs non-thermally [406, 407, 422, 423]. By taking TRH M,the production of connector particles in the early universe is strongly suppressed allowing us to focuson the effects of the glueballs.Energy Transfer before ConfinementConsider first the transfer of energy at temperatures T well above the confinement temperature Tc. In theabsence of connectors, preferential reheating to the visible sector produces Tx T . Connector operatorsallow reactions of the form SM+SM↔ X +X that transfer energy from the visible sector to the darksector. For Tx > Tc, the evolution equation for the energy density of the dark sector is [424, 425]dρxdt+4Hρx = −〈∆E ·σv〉(n2x− n˜2x), (6.27)where 〈∆E ·σv〉 is the thermally averaged energy transfer cross section for X +X→ SM+SM, nx is thedark gluon number density, and n˜x = g˜x(ζ (3)/pi2)T 3 is the value it would have in full equilibrium withthe visible sector with g˜x dark gluon degrees of freedom (equal to g˜x = 2(N2−1) for Gx = SU(N)).3For Tx T , the n˜2x term on the right side above dominates and leads to a net energy transfer to the darksector. This transfer saturates and ceases when Tx→ T and nx→ n˜x.For visible radiation domination with constant g∗, Eq. (6.27) can be rewritten asddT( ρxT 4)=1HT 5〈∆E ·σv〉(n2x− n˜2x) . (6.28)With the connector operators of Eqs. (6.5,6.7) and T  Tx, the right side of Eq. (6.28) takes the para-metric form∆C ≡ 〈∆E ·σv〉(n2x− n˜2x) (6.29)∼ −Dn MPl Tn−2Mn, (6.30)where n = 4, 8. Integrating from temperature T to the reheating temperature TRH , the approximatesolution is( ρxT 4)−( ρxT 4)RH∼ Dn(n−1)MPlT n−1RHMn[1−(TTRH)n−1], (6.31)This expression is dominated by the contribution near the reheating temperature, and represents thecontribution to the dark energy density from transfer reactions.3Implicit in Eq. (6.27) is the assumption of self-thermalization of the energy injected into the dark sector to a temperatureTx > Tc. Thermalization of non-Abelian gauge theories tends to be efficient [426], and we expect this assumption to be validprovided the total energy transfer is not exceedingly small.127The approximate forms of Eqs. (6.30,6.31) are only valid for T < TRH and T > Tx ≥ Tc. The firstof these conditions corresponds to the upper limit on the era of radiation domination. An even higherradiation temperature can be achieved prior to reheating, but for standard perturbative reheating andn < 29/3' 9.67 we find that the energy transfer before the radiation era is also dominated by reactionsnear T ∼ TRH . The second condition T > Tx ≥ Tc is needed to justify our neglect of the n2x term onthe right side of Eq. (6.28) and our assumption of a deconfined phase. As Tx approaches T due to theenergy transfer, this term becomes important and the net energy transfer goes to zero, corresponding tothe thermalization of the two sectors.Motivated by these considerations, let us define∆( ρxT 4)≡∫ T·dT ′(∆CHT ′5)(6.32)∼ Dn(n−1)MPlT n−1Mn. (6.33)This represents the contribution to the dark sector energy from thermal transfer in the vicinity of tem-perature T . Thermalization occurs when∆( ρxT 4)≥ pi230g˜x , (6.34)where g˜x is the number of dark gluon degrees of freedom. Let Tth be the temperature that solvesEq. (6.34) as an equality. If Tth < Tc, the visible and dark sectors remain thermalized at least untilconfinement. Conversely, if Tth > Tc thermalization is lost at T = Tth and the dark and visible sectorsevolve independently thereafter with separately conserved entropies.The dark to visible entropy ratio R is constant for T < Tth and depends on reheating. If Tth < TRH ,thermalization occurs after reheating and is maintained until T = Tth. The entropy ratio R (for Tth > Tc)after thermalization ceases is thenR = Rmax ≡ g˜xg∗S(Tth) . (6.35)Thermalization need never have occurred after reheating if TRH < Tth. In this case, (for Tth > Tc) we candefineTxRH = TRH[30pi2g˜x∆( ρxT 4)RH]1/4. (6.36)This implies a lower bound on the entropy ratio ofR≥ g˜xg∗S(TRH)(TxRHTRH)3. (6.37)In general, lower reheating temperatures allow for smaller values of R. We define Rmin to be the value of128��������������(����)-��-��-��-���Figure 6.7: Values of the minimal entropy ratio Rmin in the M–m0 plane for energy transfer viadimension-8 (left) and dimension-6 (right) operators for Gx = SU(3). The black shadedregion at the upper left indicates where our treatment in terms of effective operators breaksdown. The diagonal black dotted, solid, and dashed lines show reference values of Rmin =10−3, 10−6, 10−9. In the cyan region in the right panel, thermalization between the visibleand dark sectors is maintained at least until confinement.R such that TxRH = Tc, the lowest possible reheating temperature given our assumption of TxRH ≥ Tc.4When Tth > Tc, the range of R values is therefore Rmin ≤ R≤ Rmax.In Appendix B we present explicit expressions for the collision term ∆C needed to compute theenergy transfer ∆(ρx/T 4) via Eq. (6.32). The results obtained for Rmin are shown in Fig. 6.7 in the m0–M plane for energy transfer via dimension-8 (left) and dimension-6 (right) operators for Gx = SU(3).The shaded region at the upper left has m0 > M/10 and indicates where our treatment in terms ofeffective operators breaks down. The black dotted, solid, and dashed lines show reference values ofRmin = 10−3, 10−6, 10−9. In the cyan region in the right panel, thermalization between the visible anddark sectors is maintained at least until confinement, corresponding to Tth < Tc.Evolution of the 0++ DensityGlueballs form at Tx = Tc and undergo freeze-out, transfer, and decay reactions. In the absence ofconnectors, the dominant glueball species is the lightest 0++ mode. To track its evolution with connectoroperators, it is convenient to organize the analysis according to the thermalization temperature Tth,computed above in the unconfined phase, relative to the confinement temperature.Tth < Tc: This condition implies that thermalization is maintained at least until confinement, and thuswe expect T = Tx = Tc as an initial condition for the glueball evolution. To compute the 0++ densityand thermal transfer after confinement we adapt the analysis of Refs. [350, 428] based on Refs. [424,425, 429], which is applicable here since T, Tx ≤ Tc ' m0/5.5. If thermal equilibrium is maintained4Even lower values of R are possible for TxRH < Tc, but this also implies that reheating can interfere with the freeze-outprocess [427], and goes beyond the scope of this work.129independently within both the dark and visible sectors, the dark temperature evolves as [350, 428]dTxdt' −2HTx+ 23n0 (Cρ −m0Cn) (6.38)where Cρ and Cn are the collision terms appearing in the evolution equations for the 0++ energy andnumber densities. The Hubble term in Eq. (6.38) gives the usual 1/a2 redshifting of the effective temper-ature of an independent massive species, while the second term describes energy transfer from scatteringand decay processes.The explicit forms of the collision terms areCn ' −〈σ32v2〉n20(n0− n¯0)−Γ0 [n0(1−3Tx/2m0)− n˜0(1−3T/2m0)] , (6.39)where n¯0 = n0(Tx) and n˜0 = n0(T ), as well asCρ ' n0nSM〈σelv·∆E〉−m0Γ0(n0− n˜0) . (6.40)The only new piece in these expressions is the elastic scattering term n0nSM〈σelv·∆E〉 in Eq. (6.40). Itcorresponds to reactions of the form SM+ 0++→ SM+ 0++, and was studied in detail in Refs. [424,425].Combined in Eq. (6.38), the (3→ 2) scattering term from Eq. (6.39) tends to heat the dark glueballs,and the elastic scattering and decay terms tend to drive Tx→ T . Applied to the 0++ glueball with eitherthe dimension-8 or dimension-6 connector operators, we find that thermalization below confinementimplies Γ0 > H(T = m0). Thus, the 0++ density simply tracks the equilibrium value with temperatureT following confinement.5Tth > Tc: With Tth > Tc, the visible and dark sectors are not thermally connected at confinement, andthus T ≥ Tx at this point with a well-defined entropy ratio in the range Rmin ≤ R ≤ Rmax. Using thescaling arguments applied above, it can be shown that R ≥ Rmin implies T ≤ m0 when the 0++ decaysset in at Γ0 ' H(T ).6 The evolution equations for the 0++ number density and temperature can thus bewritten as (to leading order in Tx/m0)dn0dt= −3Hn0−〈σ32v2〉n20(n0− n¯0)−Γ0(n0− n˜0) (6.41)dTxdt= −2HTx+ 23m0〈σ32v2〉n0(n0− n¯0)+Γ0Tx(1− n˜0n0TTx)(6.42)where again n¯0 is the equilibrium value at temperature Tx and n˜0 is the equilibrium value at temperatureT . Note that we have neglected the elastic scattering term because it can be shown to be parametricallysmall relative to the Hubble term for T < Tth and R≥ Rmin.5In the absence of decays, massive glueballs with connectors would give an explicit realization of the SIMP [147] orELDER [350, 428] DM scenarios.6Our numerical analysis confirms this as well.130When the decay terms are neglected, the evolution equations of Eqs. (6.41, 6.42) are equivalent tothose we used previously with no connector operators (to leading order in Tx/m0). Glueball decays onlybecome significant when Γ0'H(T ), and quickly drive Tx→ T and n0→ n˜0. It follows that our previousanalysis without connectors can be applied to compute the 0++ relic yield prior to decay (which mayoccur before freeze-out). The only significant effect of energy transfer on this calculation is to limit therange of the initial entropy ratio to Rmin ≤ R≤ Rmax.Evolution of the 1+− DensityEven though the lightest 0++ glueball dominates the total glueball density and controls the dark temper-ature prior (and even after) its decay, the heavier 1+− glueball can also be relevant for cosmology dueto its longer lifetime. Recall that the 1+− is parametrically long-lived relative the 0++ in the decay sce-narios 2–4 listed in Sec. 6.3.3, where the 0++ decays through a dimension-6 operator while the 1+− isstable or only decays at dimension-8. Even in decay scenario 1, where both states decay at dimension-8,the 0++ decay rate tends to be larger than the 1+− by a factor of (N2c −1)α3/α .The evolution of the 1+− density is sensitive to the 0++ density in several ways. Prior to decay, the0++ density acts as a massive thermal bath that cools very slowly relative to the visible temperature,thereby delaying the freeze-out of the 1+− state. This thermal bath collapses and disappears whenthe 0++ decays, which can hasten 1+− freeze-out. If the 0++ density is large when it decays, theentropy transferred to the visible sector can also dilute the densities of the remaining 1+− glueballs. Weinvestigate these effects here, dividing the analysis into Tth < Tc and Tth > Tc cases.Tth < Tc: Recall that this case is only realized for dimension-6 transfer operators, and implies thatthe 0++ decay rate is larger than Hubble following confinement. This means the 0++ density tracksits equilibrium value with effective temperature Tx = T , and there is no longer a separately conservedentropy in the dark sector. The evolution of the 1+− number density in this context isdn1dt+3Hn1 =−〈σ22v〉(n21− n˜21)−Γ1(n1− n˜1) , (6.43)where n˜1 denotes the equilibrium density of the 1+− at temperature T . Note that Eq. (6.43) assumesthe 1+− mode also thermalizes with the visible sector. This is expected prior to freeze-out since theequilibrium density of the 1+− is smaller than that of the 0++, and elastic scattering between these twospecies is at least as efficient as the annihilation reaction.Tth > Tc: This second case implies Tx ≤ T at confinement, with 0++ decays inactive (Γ0 < H) untilT < m0. To compute the resulting 1+− relic density, we treat the 0++ decay as instantaneous andmatch the density evolution immediately before and after it occurs. Prior to the decay, the dark andvisible entropies are conserved independently with ratio R, and the glueball densities evolve accordingto Eqs. (5.47,5.48). Decays of the 0++ are implemented at Γ0 = H, where the Hubble rate includescontributions from both the visible and dark energy densities. If Ti < m0 is the visible temperature prior131to the decay, the visible temperature afterwards is obtained from local energy conservation,ρ(Tf ) = ρ(Ti)+ρx(Ti) , (6.44)where we have neglected the exponentially subleading contribution of the 1+− mode to the energydensity. Note that Tf > Ti is always smaller than m0 as well. The evolution of the 1+− number densityafter the 0++ decays is given by Eq. (6.43). Since the 1+− number density is not changed by the decays,n1(Tf ) = n1(Ti) is used as the initial condition at T = Tf .The interplay of glueball annihilation, transfer, and decays leads to many different qualitative be-haviours. These were investigated in Ref. [395, 409] for a simplified model consisting of an unstablemassive bath particle and a heavier DM state. Dark glueballs provide an explicit realization of thisscenario, with the 0++ making up the massive bath and the 1+− acting as (metastable) dark matter.Compared to the simple model studied in Ref. [395, 409], the 0++ massive bath particle always freezesout (or decays) before the would-be 1+− dark matter, corresponding to the chemical or decay scenariosdiscussed there. A potential further behavior that we have not captured in our approximations is thelate production of 1+− glueballs through transfer reactions while T > m1 but after 1+− freeze-out hasoccurred in the dark sector. We estimate that this is potentially relevant in a very limited corner of theparameter space, and will only increase the limits we find.6.4.3 Comments on Theoretical UncertaintiesBefore applying our results for dark glueball lifetimes and densities to derive cosmological and astro-physical constraints on them, it is worth taking stock of the theoretical uncertainties in our calculations.It is also useful to identify how some of these uncertainties might be reduced with improved latticecalculations.The glueball lifetimes computed in Sec. 6.3 rely on glueball masses and transition matrix elements.Masses for Gx = SU(3) have been obtained to a precision greater than 5% in Refs. [6, 354], while thematrix element relevant for 0++ decay was determined to about 20% in Refs. [354, 414]. Thus, weexpect our determination of the 0++ decay width to be reasonably accurate. The situation is less clearfor the 1+− width, which relies on a 1+− → 0++ transition matrix element that we were only able toestimate using NDA. In the absence of lattice calculations for this matrix element, we estimate that our1+− width is only reliable to within a factor of a few.Turning next to the cosmological evolution of the dark gluons and glueballs, we implicitly treatedtheir interactions as being perturbative. This is a good approximation at temperatures well above theconfinement scale, but significant deviations can arise as the temperature falls to near confinement [377].For the range of entropy ratios R due to energy transfer computed above, this implies that values ofRmax with Tth  Tc are reliable, but the specific values of Rmin and Rmax for TRH ∼ Tc could receivelarge corrections. Similarly, the glueball interactions used to compute the (3→ 2) and (2→ 2) crosssections are quite strong for N = 3. It is difficult to quantify how this affects the pre-decay glueball relicdensities, but we do note that the densities typically depend roughly linearly on R and the annihilation132cross sections. Our naive estimate is that the pre-decay glueball densities we find are accurate to withinabout an order of magnitude.6.5 Cosmological ConstraintsIn the analysis above we showed that dark glueballs can have a wide range of decay rates and a varietyof formation histories in the early universe. Very long-lived dark glueballs can potentially make up thecosmological dark matter. On the other hand, shorter-lived glueballs are strongly constrained by themodifications they can induce in the standard predictions for big bang nucleosynthesis (BBN) [94, 259,260], the cosmic microwave background (CMB) [123, 430] and the spectrum of cosmic rays [402]. Weinvestigate the bounds from cosmology and astrophysics on dark glueballs in this section for the fourdecay scenarios discussed in Sec. 6.3. Throughout the analysis, we focus on Gx = SU(N=3), and weassume reheating such that TRH  M and TxRH ≥ Tc. Details of how we implement the bounds fromBBN, the CMB, and cosmic rays are collected in the following section.6.5.1 Decay Constraints from BBNParticle decays during or after big bang nucleosynthesis (BBN) can modify the primordial abundancesof light elements including tritium, deuterium, helium, and lithium [81, 112, 259, 260]. The observedabundances of these light elements (with the exception of lithium) agree well with the predictions ofstandard BBN when the baryon density deduced from the CMB is used as an input [81]. If there wasnon-standard physics present during the era of BBN, such as the decays of dark glueballs to SM fields,the predictions the elemental abundance would be altered. Thus, constraints can be placed upon decaysof glueballs after the onset of BBN.Hadronic decays of a long-lived relic after t ' 0.05s can modify the neutron (n) to proton (p)ratio and increase the helium fraction through charge exchange reactions such as pi−+ p→ pi0 + n, ordestroy light elements through spallation reactions like n+4He→D+ p+2n [81, 112]. Electromagneticdecays are only constrained at later times, after about t ∼ 104 s, since energetic electromagnetic decayproducts emitted before this thermalize with the photon-electron plasma before can they can destroylight elements by photodissociation [7, 81, 112].The combined effects of hadronic and electromagnetic decays on BBN have been studied in a num-ber of works, including Refs. [94, 259, 260]. We apply the exclusions derived in Ref. [94] to placelimits on decaying glueballs, using an interpolation to generalize their results to arbitrary relic massvalues between the range 30 GeV≤ mx ≤ 106 GeV they studied, and matching to the appropriate set offinal states. For masses outside these ranges, we apply the constraint for the nearest mass boundary.6.5.2 Decay Constraints from the CMBParticle decays during or after recombination at t ' 1.2×1013 s can modify the temperature and polar-ization spectra of the CMB. They do so by injecting energy that increases the ionization fraction andtemperature of the cosmological plasma. In turn, this broadens the last scattering surface and alters the133correlations among the temperature and polarization fluctuations [123].Detailed studies of the impact of such energy injection on the CMB have been performed in Refs. [80,313–315, 430–432]. Corresponding limits on particle decays based on the CMB measurements ofPlanck [9] were extracted in Refs. [80, 430]. Given the theoretical uncertainties in our calculationof the pre-decay glueball yields, we apply a very simple parametrization of the results of Ref. [430]:miYi < (4.32×10−10 GeV)( τ1024 s)F (τ) , (6.45)where F (τ) accounts for the effects of early decays. It is obtained by fitting to the curve of Fig. 4of Ref. [430], and is normalized to unity for τ  1.2× 1013 s. The form of Eq. (6.45) neglects milddependences on the mass of the decaying glueball and the specific final state, but these effects aresmaller than the uncertainties in the calculation of the pre-decay yield. We also apply this limit torelic masses well above the largest value of mx ∼ 10 TeV studied in Ref. [430] (and elsewhere). Suchlarge masses lead to injections of highly energetic photons and electrons that deposit their energy veryefficiently in the cosmological plasma [315]. As a result, we do not expect any major loss of sensitivityfor glueball masses well above 10 TeV.Bounds on glueball decays can also be obtained from their effects on the CMB frequency spec-trum [318, 433]. We find that these are subleading compared to those derived from BBN and the CMBpower spectra.6.5.3 Decay Constraints from Gamma RaysGlueballs with lifetimes greater than the age of the universe t0 ' 4.3× 1017 s can produce observablesignals in gamma ray and cosmic ray telescopes, even if their density is only a small fraction of thetotal DM value. Limits on the lifetimes of decaying DM were derived in Ref. [402] for dimension-6 glueball decays and other final states over a broad range of masses using galactic gamma ray datafrom Fermi [434]. With the theoretical uncertainty on glueball yields in mind, we use the followingparametrization of the limits on the glueball yield:miYi < (4.32×10−10 GeV)(τ5×1027 s)et0/τ e(10 GeV/mi) , (6.46)where the last two factors account for the depletion of the signal if the decay occurs before the presenttime and the loss of sensitivity of Fermi at lower masses [435]. This limit is fairly conservative and canbe applied safely to all dominant 0++ decays, which occur on their own or shortly after being created ina 1+− decay.6.5.4 Application to GlueballsIn this next section we apply the constraints collected above to the specific decay scenarios presented insection 6.8: Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 1with dominant dimension-8 operators and broken Cx. The upper two panels have R =Rmin, Rmax, while the lower three panels have fixed R = 10−9, 10−6, 10−3. The grey shadedregion in each panel indicates where our theoretical assumptions fail, while R < Rmin to theleft of the dashed line.Decay Scenario 1: Dimension-8 Decays with Broken CxThis scenario has all the dimension-8 operators of Eq.(6.9) with χi = χY = 1. Both the 0++ and 1+−glueballs decay with parametrically similar rates, as shown in Fig. 6.3.The cosmological constraints on this scenario are shown in Fig. 6.8 in the M-m0 plane for vari-ous values of the entropy ratio R. The upper two panels have R = Rmin, Rmax respectively,7 and thelower three panels show R = 10−9, 10−6, 10−3. The grey shaded regions indicate where our theoret-ical assumptions break down. The rising diagonal portion of the gray shaded region corresponds tom0 > M/10; we demand smaller values of m0 to justify our treatment in terms effective operators sup-pressed by powers of M. The upper part of the grey shaded region indicates Tx f o > Tc, correspondingto glueball densities set by the non-perturbative dynamics of the confining phase transition. To the leftof the diagonal dotted lines in the lower three panels, the given fixed value of R is less than Rmin and isinconsistent with minimal energy transfer by the connector operators for TxRH > Tc.We see from Fig. 6.8 that dark glueballs are strongly constrained by cosmological and astrophysicalobservations. When the 0++ is long-lived, corresponding to small m0/M, its relic density tends to betoo large unless the entropy ratio R is much less than unity. With sufficiently small R the 0++ can makeup all the dark matter corresponding to the white line in the left panel of Fig. 6.6. Such a DM candidate7Recall from Eq. (6.35) that Rmax corresponds to thermalization after reheating, while from Eq. (6.37) Rmin is the lowestpossible entropy ratio consistent with energy transfer and TxRH > Tc.135Figure 6.9: Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 2with dominant dimension-8 operators and conserved Cx. The upper two panels have R =Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. The grey shadedregion indicates where our theoretical assumptions fail, while to the left of the dashed linewe find R < Rmin.would be very difficult to probe, with the most promising avenues being high energy gamma rays andmodifications to cosmic structure from glueball self interactions. Using large-N and NDA, the 2→ 2self-interaction cross section of 0++ glueballs is [231, 235]σ2→2/m0 ' (10cm2/g)(3N)4(100 MeVm0)3. (6.47)This is at (or slightly above) the current limit for N ≥ 3 and m0 ≥ 100 MeV and could have observableeffects close to these values [137], but falls off very quickly with higher mass or if the 0++ glueball isonly a small fraction of the full DM density. For larger m0/M ratios, the 0++ and 1+− glueballs bothdecay quickly enough to alter BBN or the CMB or create high energy gamma rays. Not surprisingly,the bounds from glueball decays in this scenario come primarily from the 0++ which has a much largerrelic yield prior to decay.Decay Scenario 2: Dimension-8 Decays with Exact CxOur second decay scenario has dominant dimension-8 operators with χi = 1 and a conserved Cx chargethat implies χY = 0 and a stable 1+− glueball. The cosmological bounds on this scenario are shown inFig. 6.9 for various values of the entropy ratio R. The upper two panels have R=Rmin, Rmax respectively,and the lower three panels show R = 10−9, 10−6, 10−3. As above, the grey shaded regions indicate136Figure 6.10: Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 3with dominant dimension-6 operators and broken Cx. The upper two panels have R =Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. The black shadedregion indicates where our theoretical assumptions fail, while to the left of the dashed linewe find R < Rmin.where our theoretical assumptions are not satisfied, and the diagonal dashed lines have R< Rmin to theirleft.The cosmological exclusions on this scenario are nearly identical to those on scenario 1 except forthe new bounds from the 1+− relic density. At the lower edge of the cyan excluded region, the 1+−glueball can make up all the dark matter. This occurs primarily when the 0++ decays relatively quickly,since otherwise it tends to dilute the 1+− relic density too strongly. Note as well that the 1+− glueballcan make up the dark matter for a wide range of values of the entropy ratio R, and for masses well abovethe weak scale, between about 102 GeV. 105 GeV. For smaller values of m0/M, the 0++ is long-livedand remains the dominant species as in scenario 1.Decay Scenario 3: Dimension-6 Decays with Broken CxThe third decay scenario 3 has both dimension-6 and dimension-8 operators with ye f f = 1 and χi = χY =1, and broken Cx. This leads to 0++ decays dominated by the dimension-6 operator, but decays of the1+− only through the dimension-8 operators. As a result, the 1+− glueball is parametrically long-livedrelative to the 0++ (and the other glueball states).We show the cosmological and astrophysical bounds on this scenario in Fig. 6.10 for various valuesof the entropy ratio R. The upper two panels have R = Rmin, Rmax respectively, and the lower threepanels show R = 10−9, 10−6, 10−3. As above, the grey shaded regions indicate where our theoretical137Figure 6.11: Cosmological constraints on dark glueballs in the M–m0 plane for decay scenario 4with dominant dimension-8 operators and conserved Cx. The upper two panels have R =Rmin, Rmax, and the lower three panels have fixed R = 10−9, 10−6, 10−3. The black shadedregion indicates where our theoretical assumptions fail, while to the left of the dashed linewe find R < Rmin.assumptions are not satisfied, and the diagonal dashed lines have R < Rmin to their left, except in theR = Rmax panel. Here, thermalization is maintained all the way to confinement (and beyond) to the leftof the line.Decays of both the 0++ and 1+− glueballs lead to relevant exclusions in this scenario. The 0++relic density tends to be much larger than the 1+− prior to decay, and produces the strongest constraintsfor small values of m0/M when it is long-lived. For very long lifetimes and small R, it can make upall the DM as before. However, larger values of m0/M lead to relatively short-lived 0++ glueballsthat decay before the start of BBN. In this case, the longer-lived 1+− can decay late enough to disruptnucleosynthesis or the CMB in an unacceptable way. Note as well that the region in which the 1+− relicdensity is potentially large, it decays too quickly to make up the dark matter.Decay Scenario 4: Dimension-6 Decays with Exact CxOur final decay scenario 4 has has both dimension-6 and dimension-8 operators with ye f f = 1 and χi = 1,together with conserved Cx (and χY = 0). The 0++ mode decays as in the previous scenario, but nowthe 1+− is stable.The cosmological bounds on this scenario are shown in Fig. 6.11 for various values of the entropyratio R. The upper two panels have R = Rmin, Rmax respectively, and the lower three panels show R =10−9, 10−6, 10−3. As above, the grey shaded regions indicate where our theoretical assumptions are notsatisfied, and the diagonal dashed lines have R < Rmin to their left, except in the R = Rmax panel. Here,138thermalization is maintained all the way to confinement (and beyond) to the left of the line.The exclusions on this scenario from the 0++ are identical to those on scenario 3. However, theconstraints from the 1+− are now from its relic density rather than the effects of its decays on BBN andthe CMB. This state can make up the dark matter for a range of values of its mass and the entropy ratioR. Compared to the analogous scenario 2, the relic density of the 1+− tends to be larger here because itexperiences less dilution from the more rapid decay of the 0++.6.6 ConclusionsIn this chapter we culminated all of the individual pieces that were considered in previous chapters. Wehave investigated the cosmological constraints on non-Abelian dark forces with connector operators tothe SM. We have focused on the minimal realization of such a dark force in the form of a pure Yang-Millstheory. In the early universe, the dark gluons of such theories confine to form a set of dark glueballs.Connector operators allow the transfer of energy between the visible (SM) and dark sectors, modify thefreeze-out dynamics of the glueballs, and induce some or all of the dark glueballs to decay. Late decaysof glueballs can modify the standard predictions for BBN, the CMB, and cosmic ray spectra, while verylong-lived or stable glueballs must not produce too much dark matter. Using these considerations, wehave derived strong constraints on the existence of new non-Abelian dark forces.A significant new feature of our work compared to previous studies [229, 231, 232, 235–237, 336,351, 406] is the inclusion of the heavier 1+− glueball species. This state can be parametrically long-livedor stable relative to the other glueballs. It freezes out in conjunction with the 0++, with the 0++ densityforming a massive thermal bath, leading to a rich array of freeze-out and decay dynamics [395, 409].In general, the (pre-decay) relic density of the 1+− mode is much smaller than the 0++. Even so, the1+− can sometimes yield the strongest cosmological bounds due to its longer lifetime. Specifically, the0++ could decay before impacting standard cosmological processes such as BBN, while the 1+− decayslate enough to directly interfere. In some cases, the 1+− glueball could even make up the observed DMdensity.Our study also concentrated on the dark gauge group Gx = SU(N=3) with a lightest 0++ glueballmass above m0 ≥ 100 MeV. The constraints found here could also be generalized to other dark gaugegroups and lower masses. A very similar glueball spectrum is expected for SU(N > 3) [5], but theconfining phase transition will be more strongly first-order and its effect on glueball freeze-out deservesfurther study [351, 378]. For Gx = SU(2), SO(2N+1), Sp(2N) there are no Cx-odd glueballs [11, 359],but otherwise we expect the constraints based on the 0++ glueballs to be applicable here. In the caseof SO(2N > 6), the Cx-odd states are expected to be significantly heavier than the 0++, and thus theadditional constraints on the lightest Cx-odd mode would typically be weakened.Thus, we have now completed a comprehensive overview of how the visible and dark sectors ofthe Universe may interact with one another. We have used our knowledge and intuitions, built up overthe previous chapters, to create a fully realized hidden sector in the form of a non-Abelian dark gaugefield. The sectors own self-interactions were studied extensively, before allowing for SM connections,completing the energy transfer circle we first outlined in Fig. 2.5, and made explicit in Fig. 6.1.139Part VConclusions140Chapter 7Conclusions and Future OutlookIn this thesis we have investigated how new physics beyond the Standard Model may interact with thatwhich we understand already. This was depicted in Fig. 2.5. In particular, we have focused on how adark sector may be influenced by, or influence, the visible sector that we know and understand. Thishas involved studying and constraining new physics in different epochs in the Universe, ranging fromBig Bang Nucleosynthesis to how new physics could alter galactic formation. These are effects that thedark sector could have on the hidden. However, it is also possible to affect the dark sector directly viathe visible sector, and so we have also studied how energy transfer in this direction can play a factor aswell.In Chapter 3, we started our overview of the visible-dark dynamic by looking specifically at the visi-ble to hidden energy transfer, in a modified freeze-in scenario. We wished to understand how ultravioletfreeze-in may continue to play an important role in the relic abundance of dark matter, well after it istypically considered complete. Specifically, we realized this scenario by including a freeze-in transfereffect from a non-renormalizable, UV operator, coupled with a self-thermalizing hidden sector. To beconcrete, we focused on the dimension-five fermionic Higgs portal operator to connect the two sectors,but note that this effect should extend to any UV operator. The freeze-in transfer serves to initially min-imally populate the hidden sector (although we leave open the possibility that during reheating, someother mechanism could have also produced hidden sector particles, but to a temperature less than thevisible sector). Within the hidden sector there is contained a candidate dark matter fermion, as well asa massless Abelian dark vector that couples to the dark matter. This typically results in what wouldbe a standard, hidden freeze-out scenario as the Universe continues to expand and cool in both sectors.However, the extra freeze-in coupling serves to halt the freeze-out process. Instead, the dark mattercandidate reaches a new pseudo-equilibrium state, where the freeze-out rate becomes balanced by thefreeze-in rate, even well after the freeze-in rate should have become negligible. Thus, we find that theUV connector operator continues to play an important role, even into the IR.While this effect is suppressed if there is no dark sector annihilation subsequent to the initial trans-fer, it is nevertheless expected to generalize to any other connection. Thus, we expect IR contributionsfrom any UV operators that produce relic abundances of dark-sector DM, as long as there is some formof annihilation that continues to occur in the hidden-sector. This can lead to a significant additional en-141hancement of the relic density over a broad range of parameters in the theory. As we continue to expandour searches for new physics, dark-sector scenarios grow more complicated, and it is thus important tokeep in mind this visible energy transfer dynamic. An interesting extension would be to consider a mas-sive vector boson in the hidden sector, and consider the interplay between the freeze-in energy transferand subsequent evolutionary dynamics. Other more complicated hidden sectors could be consideredas well; any model of new physics that maintains a hidden annihilation will have to keep this effect inmind. As our direct detection experiments continue to probe smaller DM-nucleon cross-sections (withno evidence for WIMP-like DM), we need to build up a robust understanding of more production mech-anisms than just standard freeze-out, and as such it becomes important to understand how these feebleinteractions can have an effect on the evolution of hidden sector fields. Furthermore, as these interac-tions may never be directly probe-able with various experiments, it becomes crucial to understand thoseparts of new theories that will have long-lasting evolutionary effects, as observations of the DM relicdensity may be one of the only mechanisms by which we may study BSM physics.Following this study of the visible energy transfer, we moved on to a study of how dark sector energymay affect the visible in Chapter 4. In particular, we have focused on decaying dark species with SMby-products, in an effort to understand how this energy injection can alter the outcomes of Big BangNucleosynthesis. Although this has been studied in the literature before, it had not yet been done forthe lowest range of masses that may affect the outputs of BBN. Specifically, we have focused on the1-100 MeV range. This range has received less attention as the earliest, well-motivated BSM modelswere typically at the weak scale, ∼ TeV range. However, new models can still be motivated belowthis, and as these low energy ranges have not yet been fully explored, we wished to fill this gap in ourunderstanding. Even at these low energy transfers, we are still able to place stringent constraints on newphysics, by modelling how the SM responds to the energy inflow.Because we constrain ourselves to such low energy ranges, nearly all of the energy must be trans-ferred via electromagnetic energy injection, as the threshold for hadronic energy injection lies at thepion mass (∼ 134 MeV). Because of this, we model the electromagnetic cascades induced by EM en-ergy in this range. This leads to a photon spectrum that is different from the historical universal photonspectrum that is typically adopted for photon injection. The spectrum is altered due to the inclusion ofThomson scattering for low energy electrons, as well as including final state radiation in the decay ofthe dark sector particle to electron-positron pairs.The photon spectrum determined from electromagnetic decays of the dark sector particles is thenused to determine photodissociation rates of light elements in the epoch following their production. Forenergies below the threshold photodissociation energy of 4He, the most stringent constraint comes fromdestruction of Deuterium below the levels we observe today. For energies above the 4He threshold, thereis instead an interplay between the destruction of 4He and the overproduction of deuterium, as it getsproduced as a by-product of 4He dissociation. Because our present-day limits on 3He are much moreuncertain than the other two elements, this is typically a weaker limit than the other two, although thereare still some regions of parameter space where it may provide the limiting constraint. Future studiesmay wish to also include the Lithium observations, and perhaps attempt to explain (or contribute to) the142Lithium puzzle that has still not been solved.While direct energy injection can be used to directly dissociate the light elements, we also look atindirect effects on BBN. This was done by identifying the effect that energy injections may have onNe f f , which is a measure of the radiation energy density in the early Universe. Decays to electrons,photons, and even neutrinos can affect Ne f f , either increasing or decreasing the energy density, whichin turn effects when BBN occurs, altering the ratio of all the final elements present at the end of BBN.These indirect effects are the dominant constraint on particles with lifetimes < 104s, while lifetimes upto 1013s are most strongly constrained by the direct EM injections and photodissociation of the lightelements. For later lifetimes, constraints from the CMB and gamma-rays today become more relevant.Nevertheless, this chapter has provided a unique look into how low-energy transfers from a hiddensector can affect the elements that we know and understand.Now that we have developed a complete understanding of the constraints that may be placed onnew physics that is active during the epoch of BBN, we may begin to apply these constraints to specificrealizations of new physics. Although many early, well motivated theories of new physics typicallyinvolved much higher energy scales, the lack of evidence from direct detection and collider experimentshas pushed our studies to expand and look in different energy regimes. Moving forward, as new modelsdevelop that are continually pushing to lower energies, we can apply this BBN constraint to supplementour bounds on such new physics. For example, various light vector bosons often appear in new models.These could include dark photons, B-L symmetry theories, and even the dark glueballs consideredhere[270, 436]. Thus the future outlook for cosmological probes is bright, across many different energyscales.After studying cases of energy injected into either the hidden-sector or into the visible-sector, wemoved on to a more complicated hidden-sector in Part IV. Here, we investigated the dynamics associatedwith a hidden SU(3) sector of dark glueballs that would have formed and evolved in the early Universe.These glueballs arise from confinement in non-Abelian gauge sectors, and typically have masses near,or slightly above, the confinement scale.In Chapter 5, we began with a focused study of isolated glueballs, with no transfer allowed betweenthe glueballs and the visible sector. The primary focus was to perform a detailed analysis of the freeze-out dynamics associated with a complex dark sector. In particular, we modelled the 3→ 2 interactionthat dominates the freeze-out of the lightest 0++ state, while also determining what effects the heavierglueballs may have on the overall density. The heavier glueball states are included in the analysis viaan expanded set of Boltzmann equations, that include the most dominant and relevant 2→ 2 interac-tions. We found that the heavy C-even states do not have a large effect on the lightest state, althoughtheir individual yields are in fact affected by the presence of other glueballs, through effects such ascoannihilation. However, when the glueballs are unable to decay efficiently through SM connectors,these yields are always subdominant to the lightest 0++ state, and have a negligible effect on the relicabundance constraint associated with the glueball spectrum making up the cosmological dark matter.The C-odd states have a similar effect. However, these are modelled separately, as dark-sector symme-tries could stabilize the lightest C-odd states while the C-even states decay to SM particles, making the143lightest C-odd state, the 1+− state, of cosmological interest as well.This was studied explicitly in Chapter 6, in which we relaxed the assertion that these glueballs wereentirely isolated. We considered the two most relevant non-renormalizable decay operators that mightconnect the hidden SU(3) sector to the Standard Model. These are dimension-6 operators, which onlyallow for C-even (and so, in particular, the 0++ state) decays, as well as dimension-8 operators, whichallow all glueball modes to decay (although even here, only some of the vector portals allow for decayof the 1+− state). This leads to a broad host of cosmologically interesting constraints, including boundsfrom BBN, the CMB, and gamma-ray constraints for lifetimes close to the age of the Universe. Whenthe dimension-6 operator is not present, the 0++ and 1+− states have parametrically similar decay rates,and so the larger yield associated with the 0++ state causes this to have the strongest constraints overall epochs. If we consider still only the dimension-8 operator, but stabilize the 1+− state via a dark Csymmetry, then we gain constraints on the yield of 1+− being larger than the cosmological dark matterabundance today. Including the dimension-6 operator lifts the parametric similarities of the 0++ and1+− decay rates: the 0++ will now decay earlier. This causes the decays of both states to contribute tothe overall constraint picture, allowing us to rule out vast portions of the parameter space, regardless ofthe type of transfer operator being considered.Thus, we have now seen a comprehensive overview of how the visible and dark sectors of theUniverse may interact, and explored a rich and diverse hidden sector in the form of a full non-Abeliangauge force that is realized as a massive glueball spectrum. We have explored both visible and darksector energy injections in detail, while addressing how these energy injections can lead to constraintson new physics. Moving forward, these new constraints and insights should prove invaluable to futurestudies, and can be expanded upon by building more complex models, and updating the bounds as ourcosmological measurements grow ever-more precise.The outlook for the future is certainly bright. Although we are currently in an era of physics wherewe are not sure what the future holds or where BSM physics could be hiding, we have developed anextraordinary set of tools to guide our analyses moving forward. As the LHC goes through upgrades tocome back at higher luminosities, cosmological surveys probe further into the past history of the Uni-verse with evermore precision, and new technologies such as gravitational wave detectors are produced,theoretical models and constraints stand ready to be applied to whatever new signals lurk around thecorner. And if no signal presents itself immediately, we can still close the doors on, or at least narrowthe windows for, many new models of physics by simply continuing to develop our understanding ofthe cosmological evolution of known and unknown fields based on the immense amount of data wehave amassed and continue to cultivate. 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The squared matrix elementfor ψ + ψ¯ → H +H† derived from this interaction and summed over both initial and final degrees offreedom is|˜M |2 = 4M2(s−4m2ψ) , (A.1)with s= (p1+ p2)2. Note that we assume implicitly that the Higgs is in the electroweak unbroken phaseand can be treated as a massless SU(2)L scalar doublet.Number TransferThe relevant number transfer term via ψ(1)+ψ(2)→ H(3)+H†(4) isT (T ) ≡ 〈σtrv(T )〉(n2ψ −n2ψ,eq(T )) (A.2)≡∫dΠ1∫dΠ2∫dΠ3∫dΠ4 (2pi)4δ (4)(pi)|˜M |2( f1 f2− f3 f4) ,where dΠi = d3 pi/2Ei(2pi)3. To make the calculation tractable, we approximate the distribution func-tions by the Maxwell-Boltzmann form fi = ζie−Ei/T , where ζi is the rescaling needed to get the correctnumber densities relative to equilibrium at temperature T . We expect that the Maxwell-Boltzmannapproximation used here is correct up to factors very close to unity.For nψ  nψ,eq(T ) we have f1 = f2 ' 0, while Higgs fields in full thermodynamic equilibrium withthe SM (in the electroweak unbroken phase) imply f3 = f4 = 1. The transfer term then reduces toT (T ) =∫dΠ1∫dΠ2(4g2ψE1E2σtrv)e−(E1+E2)/T , (A.3)165with gψ = 2 being the number of fermion spin states. Note that the combination in brackets is Lorentzinvariant and can depend only on the variable s. It is given by(4g2ψE1E2σtrv)=∫dΠ3∫dΠ4 (2pi)4δ (4)(pi)|˜M |2 (A.4)=18pi(14pi∫dΩ |˜M |2)CM=12pi1M2(s−4m2ψ) .To integrate this over the initial states, we follow Refs. [73, 215] and use the fact that the integranddepends only on s and E+ = (E1+E2) to write∫dΠ1∫dΠ2 =14(2pi)4∫ ∞4m2ψds∫ ∞√sdE+√1−4m2ψ/s√E2+− s . (A.5)Since the only E+ dependence of the integrand is in the Boltzmann exponential, integrating using aBessel function identity1 givesT (T ) =14(2pi)4∫ ∞4m2ψds(4g1g2E1E2σtrv)√1−4m2ψ/s (A.6)=12(2pi)5T 6M2F (x) ,where x = mψ/T andF (x) =∫ ∞2xduu(u2−4x2)3/2 K1(u) (A.7)'16 ; x 16pi x2e−2x ; x 1Energy TransferWe are also interested in the net rate of energy transfer between the visible and dark sectors. Therelevant energy collision term for ψ+ψ→H+H† is identical to Eq. (A.2) but with an additional factor1Kν (z) =√pizν2νΓ(ν+1/2)∫ ∞1 dt (t2−1)ν−1/2e−zt .166of ∆E = (E1+E2) = E+ in the integrand. The result is2U (T ) ≡ 〈∆E ·σtrv(T )〉(n2ψ −n2ψ,eq(T ))≡ T4(2pi)4∫ ∞4m2ψds(4g1g2E1E2σtrv)√s−4m2ψ√sK2(√s/T ) (A.8)=12(2pi)5T 7M2G (x)with x = mψ/T andG (x) =∫ ∞2xduu2 (u2−4x2)3/2 K2(u)(A.9)'{96 ; x 112pi x3e−2x ; x 12∫ ∞1 dt t√t2−1e−zt =− ddz [K1(z)/z] = K2(z)/z.167Appendix BThermalization Rates for GlueballsIn this appendix we calculate expressions for the collision term, ∆C needed to compute the energytransfer ∆(ρx/T 4), in Eq. 6.27.The collision term relevant for thermalization corresponds to the process X +X → SM+SM, and isgiven by∆C = 〈σ v·∆E〉 n˜2x (B.1)=∫dΠ1∫dΠ2 f1 f2W (s)∆E ,where E1 and E2 are the initial-state energies, ∆E = (E1 + E2) is to be evaluated in the comovingframe, dΠi = gi d3 pi/(2pi)32Ei, fi are the equilibrium distribution functions at temperature T , and thescattering kernel is defined by [73, 215]W (s) = 4E1E2σv (B.2)=Sg1g2∫ d3 p3(2pi)32E3∫ d3 p4(2pi)32E4(2pi)4δ (4)(p1+ p2− p3− p4) ∑{int}|M |2 .Here, S is the symmetry factor for identical particles, gi are the numbers of degrees of freedom of theinitial-state particles, the sum runs over all internal degrees of freedom, and |M |2 is the squared matrixelement for the reaction. Note that this quantity is Lorentz invariant, and can therefore only depend ons = (p1+ p2)2.Following Refs. [73, 215], the expression of Eq. (B.1) can be reduced to a single integral if weapproximate the distribution functions by Maxwell-Boltzmann forms, fi = exp(−Ei/T ):∆C =g1g2T 232pi4∫ ∞(m1+m2)2ds p12F (√s/T )W (s)(B.3)=g1g232pi4T 5∫ ∞x+dx√(x2− x2+)(x2− x2−)F (x)W (s = x2T 2) ,whereF (x) =(K1(x)+ x2 [K0(x)+K2(x)])= [2K1(x)+ xK0(x)] and x± = (m1±m2)/T .168Cross Sections for Dimension-8 OperatorsThe relevant operator has the general formO8 =AM4XaαβXaαβ FCµνFCµν , (B.4)where FCµν is a SM field strength. This operator generates XX → AA transfer reactions for T  mA, m0,as well as XA→ XA elastic scattering. Concentrating on XX → AA, the corresponding matrix elementfor (p1,a)+(p2,b)→ (p3,C)+(p4,D) isM = 4As2M4δ abδCD(ε∗1 · ε∗2 )(ε3 · ε4) , (B.5)where a,b,C,D are “colours” and εi are polarization vectors. From this expression, we find (neglectingpossible masses)W (s) =1pi(g˜Ag˜x)A2s4M8, (B.6)where g˜x and g˜A = 2(N2A− 1) are the dark and visible numbers of degrees of freedom. The energy-transfer collision term is then∆C =g˜xg˜A32pi5A2[∫ ∞0dx x10F (x)]T 13M8. (B.7)The integral is dominated by x =√s/T ∼ 10, corresponding to scattering at the high end of the thermaldistribution.The coupling A can be obtained by matching to our previous results for dark gluon connector opera-tors. While there are several operators that can contribute, we keep only the S component correspondingto the operator listed above, which yieldsAi =αiαx120χi , (B.8)with Ai = Y,2,3 for each of the SM gauge factors. For χi→ 1, the gluon contribution dominates withg˜A = 2(N2c −1), and we focus on it exclusively. Note that since we are considering T & m0 & 100 MeVand the integration is dominated by√s∼ 10T , we should always be safely above the QCD confinementscale.Cross Sections for Dimension-6 OperatorsThe operator of interest is nowO6 =BM2|H|2 XaαβXaαβ , (B.9)169withB =αxy2e f f6pi. (B.10)To treat scattering through this operator, we should distinguish between temperatures above and belowthe electroweak phase transition at TEWPT ' 100 GeV. Above the transition, all the SM states aremassless and we can treat the Higgs field as a complex scalar doublet. Below the transition, we mustaccount for masses.For T > TEWPT , the dominant transfer reaction is X +X → H +H†, for which the scattering kernelisW (s) =1pi1g˜xB2M4s2 . (B.11)This yields the collision term∆C =g˜x32pi5B2[∫ ∞0dxx6F (x)]T 9M4, (B.12)where now the integral is dominated by√s∼ 6T .Below the transition temperature, we have f f¯ , hh, W+W−, and ZZ final states at leading order.Their contributions to the scattering kernels areWf (s) =N( f )cpi1g˜xB2M4s2(m2fs)(ss−m2h)2(1− 4m2fs)3/2(B.13)Wh(s) =14pi1g˜xB2M4s2(1− 4m2hs)1/2(B.14)WZ(s) =14pi1g˜xB2M4s2(ss−m2h)2(1− 2m2Zs+12m4Zs2)(1− 4m2Zs)1/2(B.15)WW (s) =12pi1g˜xB2M4s2(ss−m2h)2(1− 2m2Ws+12m4Ws2)(1− 4m2Ws)1/2(B.16)These results only apply for√s > 2mi; they are zero otherwise. Note that for√s 2mh, 2m f , the sumof these kernels is equal to the result of Eq. (B.11).170


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