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On the Yukawa interaction as a slow, gravity-like force Loggia, Elizabeth 2018

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On the Yukawa interaction as a slow, gravity-like forcebyElizabeth LoggiaA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2018c© Elizabeth Loggia, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:On the Yukawa interaction as a slow, gravity-like forcesubmitted by Elizabeth Loggia in partial fulfillment of the requirements for thedegree of Master of Science in Physics.Examining Committee:Dr. Kris Sigurdson, PhysicsSupervisorDr. Graham White, PhysicsSupervisory Committee MemberiiAbstractThe nature of dark matter continues to be one of the most elusive mysteries inphysics. The astrophysical and cosmological support for dark matter seems over-whelming, but all of the current observational evidence is from only the gravita-tional influence on baryonic matter. According to the standard cosmology, darkmatter is five times as prevalent as baryonic matter, where, taking the contributionfrom dark energy in to account, only 5% of our universe is made of baryonic matter.Ongoing experimental searches for particle dark matter have provided only con-straints without direct detection. As such, alternative theories to dark matter needto be explored. One such alternative idea is an emergent gravity theory. Gravity, nolonger a fundamental interaction, emerges from thermodynamic principles in theform of an entropic force. When this theory is applied to cosmology, the gravita-tional effect that we observe and attribute to dark matter is rather a memory effectfrom the emergence of space; it is an intrinsic property of the spacetime itself. Asit is unclear how to proceed from this theory in general, a proper framework isrequired so that we can eventually make testable predictions. We propose that theaddition of a slow, gravity-like force to general relativity is such a framework. Weestablish that the Yukawa interaction is gravity-like in certain limits, from both aparticle physics and a general relativity perspective, where the massless Yukawafield has infinite range. Exploring spherical collapse in Einstein-de Sitter cosmol-ogy, we show that the addition of the Yukawa interaction does not affect the overallevolution of the density contrast, except to decrease the time to collapse. We con-sider the equations of motion for a massive scalar field coupled to a massless scalarYukawa field, and plot the solutions as functions of the scale factor. The resultingplots have distinct behaviour before and after the scale factor is of the same mag-iiinitude as the coupling. Finally, we consider the effects of a slow, gravity-like forceand derive the Lagrangian density for a slow, massless scalar field.ivLay SummaryBased on gravitational evidence, it appears that there exists extra matter in theuniverse, called dark matter, that is five times as prevalent as ordinary matter. Ex-periments have been searching for particle dark matter for decades without anydirect detection. Therefore, it is important to study alternative theories to darkmatter. One such theory postulates that gravity might not be the fundamental inter-action we think it is, but rather an emergent phenomenon. In this theory, the darkmatter that we observe is actually just a memory effect from the emergent gravity.To develop a framework for this theory, we propose adding an extra, gravity-likeforce that propagates at a slower speed than the speed at which gravity propagates.I demonstrate that a particular force, called the Yukawa force, is gravity-like andwork toward developing the proper mathematics to slow its propagation speed.vPrefaceThis dissertation is original, unpublished, independent work by the author, E. Loggia.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 The search for dark matter . . . . . . . . . . . . . . . . . 31.1.2 Dark matter production in ΛCDM cosmology . . . . . . . 81.1.3 Alternative theories . . . . . . . . . . . . . . . . . . . . . 141.2 The homogeneous and isotropic expanding universe . . . . . . . . 191.2.1 Friedmann-Robertson-Walker metric . . . . . . . . . . . 201.2.2 General relativity . . . . . . . . . . . . . . . . . . . . . . 211.2.3 Stress energy tensor . . . . . . . . . . . . . . . . . . . . 221.3 Cosmological perturbation theory . . . . . . . . . . . . . . . . . 241.3.1 Metric perturbations . . . . . . . . . . . . . . . . . . . . 251.3.2 Matter perturbations . . . . . . . . . . . . . . . . . . . . 31vii2 The gravity-like Yukawa interaction . . . . . . . . . . . . . . . . . . 352.1 From a particle perspective . . . . . . . . . . . . . . . . . . . . . 352.2 From the Einstein-Hilbert action . . . . . . . . . . . . . . . . . . 392.3 Spherical collapse . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Coupling matter to the Yukawa force . . . . . . . . . . . . . . . . . 473.1 The equations of motion . . . . . . . . . . . . . . . . . . . . . . 473.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 Toward a slow force . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1 Concepts and motivation . . . . . . . . . . . . . . . . . . . . . . 664.2 A slow, massless scalar field . . . . . . . . . . . . . . . . . . . . 695 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1 The gravity-like Yukawa interaction . . . . . . . . . . . . . . . . 735.2 Toward a slow force . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A Feynman rules for the Yukawa interaction . . . . . . . . . . . . . . 87B Additional plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90viiiList of FiguresFigure 1.1 Universe content according to Planck. . . . . . . . . . . . . . 6Figure 1.2 Dark matter freeze out. . . . . . . . . . . . . . . . . . . . . . 13Figure 2.1 Tree level Feynman diagram for the Yukawa interaction. . . . 36Figure 3.1 Background solutions for a massive scalar field coupled to amassless scalar Yukawa field for large and small coupling. . . 54Figure 3.2 Background solution for the derivative of the massless scalarYukawa field for large and small coupling. . . . . . . . . . . . 55Figure 3.3 Background solutions for the Hubble parameter for large andsmall coupling. . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.4 Background solutions for a massive scalar field coupled to amassless scalar Yukawa field, as well as for the Hubble param-eter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.5 First order perturbation solutions for a massive scalar field cou-pled to a massless scalar Yukawa field, as well as for the New-tonian gravitational potential. . . . . . . . . . . . . . . . . . . 58Figure 3.6 Background solutions for the normalized total energy densityand pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.7 Background solution for the equation of state. . . . . . . . . . 60Figure 3.8 First order perturbation solutions for the density, pressure, andequation of state. . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.9 Density contrast plots. . . . . . . . . . . . . . . . . . . . . . 64Figure 4.1 The effect of a slow, gravity-like force on a massive point particle. 67ixFigure 4.2 Light cone diagram for a slow force. . . . . . . . . . . . . . . 68Figure 4.3 Baryon acoustic oscillations. . . . . . . . . . . . . . . . . . . 69Figure B.1 First order perturbation solutions for different Fourier modesfor a massive scalar field coupled to a massless scalar Yukawafield, as well as for the Newtonian gravitational potential. . . . 91Figure B.2 First order perturbation solutions for different Fourier modesfor the density, pressure, and equation of state. . . . . . . . . 92Figure B.3 Density contrast plots for different Fourier modes. . . . . . . 93xAcknowledgmentsI would like to thank my supervisor, Kris Sigurdson, for his ongoing help andguidance. I would also like to thank my family and friends, in particular Davideand Luna, for their continued support and encouragement.xiDedicationTo Luna, for always keeping me company and forcing me outside every day.xiiChapter 1IntroductionThe nature of dark matter is one of the most pressing mysteries in cosmology today.Historically, dark matter emerged to explain inconsistencies between Einstein’sgeneral relativity and galaxy rotation curves [1–3]. Since then, dark matter has hadvarious successes through indirect detection, with some of the best evidence com-ing from the power spectrum of temperature anisotropies in the cosmic microwavebackground (CMB), a background radiation that provides a snapshot of the earlyuniverse. Thanks to experiments like Planck, standard cosmology tells us that thecontent of the universe is made up of approximately 5% baryonic matter, 27% darkmatter, with the rest going to dark energy [4]. Therefore, only a small fraction ofthe matter content in the universe is well understood. The search for particle darkmatter has lasted several decades without any direct detection, and the parameterspace in which dark matter could live continues to shrink as constraints increasewith ongoing experiments. For these reasons, it is important to study alternativetheories to dark matter.Alternative theories to dark matter are not new. For example, modifying New-tonian gravity on large scales has been used to try to fit the data with varyingdegrees of success. One interesting idea is to consider gravity as an emergentphenomenon driven by entropic differences, rather than a fundamental interaction,as posited by Verlinde [5]. This theory is attractive as a thermodynamic formal-ism, where the equations governing gravity can be derived from first principles.By applying his emergent gravity idea to cosmology, Verlinde explains observa-1tions attributed to particle dark matter as rather an intrinsic property of spacetimerealized by an extra force exhibiting memory effects [6]. This theory serves as mo-tivation for the work presented here. From his paper, it is unclear how to proceed infull generality. In order to explore further, we consider replacing dark matter witha slow, gravity-like force while keeping general relativity the same. This forcecouples to baryonic matter and acts in much the same way that gravity does, butwith an important distinction: speed. Where gravitational interactions propagateat the speed of light, the interactions from this new force propagate more slowly,providing the ‘memory’ effect. The hope is to create a framework wherein thistheory is fully testable and to explore how this delayed gravity-like force affectsthe dynamics of baryonic matter that were originally explained via conventionaldark matter.We propose that coupling baryonic matter to a slow Yukawa force will provideboth the desired gravity-like and memory effects and is therefore a good frameworkin which to study a toy theory version of Verlinde’s emergent gravity. Therefore,it is important that the Yukawa interaction is well understood; its exploration isthe purpose of this thesis. We begin with the relevant background material includ-ing dark matter in section 1.1, the Friedmann-Robertson-Walker (FRW) universein section 1.2, and cosmological perturbation theory in section 1.3. Chapter 2 ex-plores the gravity-like nature of the Yukawa interaction. We show the Yukawainteraction is analogous to gravity in certain limits from both a particle perspectivein section 2.1 and a general relativity perspective in section 2.2. In section 2.3, weexplore how the addition of a Yukawa force affects spherical collapse. In chapter 3,we consider a massive scalar field coupled to a massless Yukawa field. We derivethe equations of motion in section 3.1 and solve them numerically in section 3.2.Finally, we work towards a slow force in chapter 4. In section 4.1, we outline thegeneral concepts and motivation for a slow force. In section 4.2, we provide theLagrangian density for a slow, massless scalar field. We end with a discussion ofthis work, including future directions, in chapter 5.Throughout this thesis we use the natural units c = h¯ = 1.21.1 Dark matterAs tests of general relativity improve with the increasing precision of astronomicaland cosmological data, it is evident that there are inconsistencies between generalrelativity and the data. Most of these inconsistencies can be solved by adding someextra, unseen matter, called dark matter, to the universe. Dark matter has a long andstoried history that is looked at in depth in [7]. The search for dark matter, includingthe current observational evidence and ongoing experiments, is presented in section1.1.1. An overview of the standard ΛCDM model for cosmology, including darkmatter production, is presented in section 1.1.2. In section 1.1.3 we cover leadingtheories alternative to dark matter. In preparing this section, we consulted [8–11].1.1.1 The search for dark matterIn this section, we discuss the search for dark matter, giving an incomplete listof the observational evidence, providing two possible candidates for particle darkmatter, and touching on the ongoing experimental searches.Observational evidence for dark matterGalaxy clusters In 1931, Edwin Hubble and Milton Humason published data onthe various galaxy clusters, where one cluster, the Coma cluster, displayed largevelocity dispersion among the constituent galaxies [12]. In 1933, Fritz Zwickyused the virial theorem, which relates the average kinetic energy 〈T 〉 to the averagepotential energy 〈U〉 via〈T 〉=−12〈U〉 (1.1)to calculate the mass of the Coma cluster [1]. He found that this calculated masswas much greater than the mass calculated from photometry, and therefore thegalaxies only accounted for a small percentage of the overall mass of the cluster.In the 1970s, Vera Rubin and her collaborators studied galaxy rotation curves[13], which had also been suggested by Zwicky as a means to infer mass [2]. Byassuming the stars in a galaxy would behave similarly to planets in a solar system,3the expected velocity distribution is Keplerian and given byv(r) =√Gm(r)r(1.2)where v(r) is the rotation speed of the star at radius r, G is the Newtonian grav-itational constant, and m(r) is the mass enclosed in a radius r. Without anydark matter, the expected curve decreases with radius. However, the observationsshowed that the velocity curve increased until reaching a point where it flattenedout. Therefore, the mass distribution cannot follow the light distribution [3], imply-ing the presence of a dark matter halo. Similar results had been found previouslyusing 21 cm data [14].Gravitational lensing In general relativity, mass bends spacetime which affectsthe path that light takes when travelling through space. If we observe a brightobject behind a massive object, the light from the bright object will be observedin rings around the massive object. This phenomenon is known as gravitationallensing. The radius of the ring, called the Einstein radius, is given byΘE =√4GMdLSdLdS(1.3)where G is the Newtonian gravitational constant, M is the mass of the source, dLSis the angular diameter distance between the lens and the source, dL is the angulardiameter distance to the lens, and dS is the angular diameter distance to the source.Therefore, by measuring the Einstein radius and the angular diameter distances, wecan calculate the mass of an object from lensing. By looking at the gravitationallensing of clusters, Bergmann, Petrosian, and Lynds found that the calculated massfrom lensing was much higher than that inferred from photometry [15], providingmore evidence for dark matter.As the evidence for dark matter grew, physicists started speculating on possi-ble candidates. One such class of candidates were called MAssive Compact HaloObjects (MACHOs), such as brown dwarfs, neutron stars, black holes, and planets.The idea was to explain dark matter using very faint baryonic matter sources. To4search for these sources, the MACHO and EROS collaborations statistically ana-lyzed millions of stars in the sky looking for microlensing. However, only a handfulof possible lensing events were found, greatly constraining MACHOs as the sourcefor dark matter and indicating that dark matter was nonbaryonic in nature [16, 17].Cosmic microwave background Some of the best evidence for dark matter comesfrom the CMB, first discovered by Penzias and Wilson [18], since it is an excellentsource from which to learn about the composition of the universe, as explainedbelow.In order to explore the CMB, the COsmic Background Explorer (COBE) waslaunched in 1989. Data collected from this mission showed that CMB was a re-markably uniform perfect blackbody with a temperature of 2.73 K. The fluctu-ations away from this temperature were measured to be of order 10−5. Fluctua-tions from uniformity are the seeds for the structure formation we see in the uni-verse today, as fluctuations in density cause matter to clump in gravitational wellsand eventually produce stars and galaxies. However, the fluctuations measured byCOBE were not big enough to account for the observed structure of our universewithout the addition of dark matter [19].The anisotropies of the CMB arise from two sources. For large scales, thephotons we observe that come from denser parts of the universe have less energydue to losing it escaping the gravitational well. This is known as the Sachs-Wolfeeffect. On small scales, anisotropies arise from the acoustic oscillations. In theplasma of the early universe, the photon-baryon fluid compresses and expands.As the fluid falls into gravitational wells it is compressed, and pressure builds un-til the fluid starts expanding. This cycle repeats itself until the photons decouplefrom the baryons at the time of last scattering. These are the photons we measureas the CMB, and their temperature is affected depending on where they were inthe cycle of acoustic oscillations as they decoupled. Therefore, CMB tempera-ture anisotropies and the amount of baryonic matter in the universe are intimatedconnected.The Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001,followed by the Planck satellite in 2009, to obtain more precise CMB data. Thesemissions would provide the most convincing evidence for dark matter through5measurements of the power spectrum of the temperature anisotropies in the CMB[4, 20]. In particular, measuring the height of the second acoustic peak determinesthe amount of baryonic matter in the universe. The universe content is summarizedin a pie chart in figure 1.1, showing that dark matter is approximately five times asprevalent as baryonic matter.Dark matter26.8%Dark energy68.3%Baryonic matter4.9%Figure 1.1: Content of the universe according to Planck assuming the ΛCDMmodel [4].Large scale structure The large scale structure of the universe is uniform overlarge distances. Like the CMB, we can gain information from studying the fluctu-ations to uniformity. The Sloan Digital Sky Survey (SDSS) and 2dF are large red-shift surveys [21, 22]. By measuring large scale structure, the power spectrum fordensity perturbations can be measured to gain information about the matter contentof the universe. Again, dark matter is needed to explain the observed fluctuationspectrum.Bullet Cluster The Bullet Cluster is a pair of merging galaxy clusters. Since themajority of the baryonic mass exists in the gas of galaxy clusters, this gas heatsrapidly and emits X-ray radiation as these clusters merge. The X-ray radiation6then acts as a tracer for the location of the baryonic mass in the merging galaxyclusters. By comparing the location of the X-ray radiation with weak lensing, itbecomes apparent that the mass distributions do not match [23]. While dark matterwas able to pass through the collision without interacting, the baryonic matter wasslowed down due to friction. This is sometimes referred to as the ‘smoking gun’evidence for dark matter, as it is difficult to explain via alternative theories.Dark matter candidatesWe briefly introduce two popular candidates for particle dark matter.Weakly interacting massive particles Weakly Interacting Massive Particles (WIMPs)are perhaps the most well-studied candidate for particle dark matter. These neu-tral particles have a large mass and interact with the Standard Model through theweak force. They are also predicted by supersymmetry, a popular extension of theStandard Model of particle physics.Axions The axion was first postulated as a resolution to the ‘strong CP prob-lem’ [24]. According to quantum chromodynamics, which is the theory behind thestrong force, it is possible to violate charge-parity (CP). However, this violation isnot observed in nature. The axions that solve this problem, could potentially be thesame particle solution to dark matter.Experimental searches for particle dark matterAs described above, dark matter has seen much success through indirect detection.However, since all of the dark matter evidence is through its gravitational effect onvisible matter, this is not enough to determine the nature of dark matter. Therefore,the search continues for a direct detection of dark matter.Although there are dark matter experiments at particle accelerators such as theLarge Hadron Collider (LHC) [25], most direct searches for dark matter happen indeep underground laboratories. Since dark matter does not interact with the mat-ter in the earth, burying detectors underground will not affect the amount of darkmatter travelling through the detector. However, any particle that is not dark matter7must travel through the large layer of earth before hitting the detector. In order tobe able to detect dark matter, these detectors are very large and very sensitive, suchas [26–28]. The idea is to have a dark matter particle collide with the nucleus ofa detector atom. The larger the detector, the greater the chance for an event. Inturn, the detector must be sensitive enough to record energy deposited on the nu-cleus from the collision. A detection of this kind would give direct evidence of theparticle nature of dark matter. However, these searches have lasted several decadeswithout any direct detection.1.1.2 Dark matter production in ΛCDM cosmologyThe model that best fits the observational data is called the ΛCDM model, whereΛ represents the cosmological constant and CDM refers to cold dark matter. Thisis sometimes also referred to as the standard model of cosmology. Our universeis homogeneous and isotropic on large scales. Thankfully, this homogeneous andisotropic property of our universe allows us to make observations from our singlevantage point that are representative of the whole universe and to use these obser-vations to test cosmological models. This is known as the cosmological principle.The ΛCDM model is built on the cosmological principle, as well as the assumptionthat general relativity is the correct description of gravity. It is mainly supportedby the experimental observations that the universe is expanding, the CMB, andbig bang nucleosynthesis (BBN). In this section, we present a brief overview of themain components of ΛCDM cosmology and its relation to dark matter. Amazingly,the ΛCDM model describes the universe with only six free parameters [4], makingit the simplest model that can explain the observations.The fact that the universe is expanding according to the Hubble law is wellestablished by observation. In an expanding homogeneous and isotropic universe,the relative velocities of comoving observers also obey the Hubble law, with theHubble parameter H(t) independent of spatial coordinates. This Hubble parameteris defined via a scale factor a(t) according to1H(t)≡ a˙a(1.4)1Throughout this work we use dot notation to denote the time derivative,˙ ≡ ddt8where the scale factor is a monotonically increasing function depending only ontime and is defined to be 1 today.Taking the assumption of general relativity and applying it to a homogeneousand isotropic expanding universe in four dimensions results in the Freidmann equa-tionH2+ka2=8piG3ρ (1.5)where k is the curvature parameter, G is the Newtonian gravitational potential,and ρ is the density. This can be interpreted physically by the statement that theexpansion of the universe depends on the matter in the universe. The continuityequation for the conservation of energy is given byρ˙ =−3H (P+ρ) (1.6)where P is the pressure. These equations are derived in detail in section 1.2.The equation of state relates the energy density and pressure viaP = ρw (1.7)where w is a dimensionless parameter. The scale factor depends on the equation ofstate viaa(t) =(tt0) 23(1+w)(1.8)The equation of state follows the dominant component in the universe. For a non-relativistic matter dominated universe, the pressure is zero and so w = 0 implyinga ∝ t23 . For a radiation dominated universe, w = 13 implying a ∝ t12 . For darkenergy, w = −1. During dark matter production, the universe was in a radiationdominated era.ThermodynamicsThe early universe is a hot, dense soup of elementary particles. These particles arein equilibrium and share a common temperature, so the best way to describe themis statistically. The average properties of the ith particle species are completely9determined by the distribution functionfi =1e(Ei−µi)T ±1(1.9)where E is the energy, µ is the chemical potential, and T is the temperature. Theminus sign indicates the Bose-Einstein distribution used for photons and bosons,while the plus sign indicates the Fermi-Dirac distribution used for fermions. Thenumber density for each particle species i is given byni = gi∫ d3 p(2pi)2fi(p) (1.10)where gi is the degeneracy of the species (i.e. the number of internal degrees offreedom such as spin, colour, etc.) and p is the momentum. Similarly, the energydensity is given byρi = gi∫ d3 p(2pi)2fi(p)E(p) (1.11)and the pressure is given byPi = gi∫ d3 p(2pi)3fi(p)p23E(p)(1.12)The momentum of a particle in the plasma is of order T . Therefore, if the massof the particle is small compared to the temperature, then that particle is relativistic.The energy and number densities for radiation are given byρi =pi230giξρT 4ni =ζ (3)pi2giξnT 3(1.13)where ξρ is 1 for bosons and 78 for fermions, and ξn is 1 for bosons and34 forfermions. Matter is defined by having a large mass compared to the temperature.10The energy and number densities for matter are given byρi = minini = gi(miT2pi) 32e−(mi−µi)T(1.14)The matter density is therefore suppressed when the mass is larger than the tem-perature, implying it is much smaller than the radiation density. Dark matter willend up dominating because it is not in thermodynamic equilibrium.The expansion of the universe is adiabatic, so the amount of entropy per co-moving volume is constant.d(sa3)dt= 0 (1.15)where the entropy is given bys =2pi245g∗sT 3 (1.16)where g∗s is the number of relativistic degrees of freedom in the plasma. Therefore,the temperature is inversely proportional to the scale factorT ∝1a(1.17)Thermal historyAs stated previously, the universe started off as a very hot, dense, uniform plasmaof elementary particles. Below are some highlights of the various transitions theparticles go through as the universe expands and cools.Inflation In the very early universe, when T  GeV, the universe undergoes aperiod of inflation. This accelerated expansion leads to a hot, uniform plasma.QCD phase transition At T ∼GeV mesons and baryons are formed by the merg-ing of quarks and gluons.11Big bang nucleosynthesis At T . 1 MeV deuterium is formed from baryons.Matter-radiation equality At T ∼ 1 eV the universe transitions from being radia-tion dominated to being dominated by matter. The matter starts to clump into whatwill later form stars and galaxies.Recombination At T ∼ 0.1 eV atoms form. The free protons and electrons com-bine to form neutral hydrogen. Once these charged particles are in bound states,photons decouple and propagate freely. These photons are what we observe as theCMB (the surface of last scattering).Particle dark matter productionFrom the observations described in section 1.1.1, we know that our new dark matterparticle must be stable, neutral, and massive. However, just assuming the existenceof a new particle is not enough; we require a mechanism to get the observed densityin the universe. This mechanism is called freeze out, and we give an of it overviewhere.For this section, we will refer to the dark matter particle as χ . To start, χ isin equilibrium with the rest of the particles in the plasma of the early universe. Atthis time, the universe is a hot thermal bath, and all particles are relativistic. Asthe universe begins to cool and the temperature drops below the mass of χ , the χinteractions can no longer keep up with the expansion of the universe. Dark matterannihilations are essentially turned off, and the dark matter is said to have frozenout of the thermal bath, leaving behind a relic density that we observe throughgravitational interactions today.Assuming freeze out occurs when dark matter is non-relativistic, and that thedark matter remains in equilibrium during the freeze out process, the Boltzmannequation for dark matterdnχdt+3Hnχ =−〈σv〉(n2χ −n2χeq)(1.18)gives the evolution equation for the dark matter number density. The 3Hnχ termcomes from the dilution of the number density due to the expansion of the universe,12and the right hand side is the collision term. σ is the cross section and v is the rel-ative velocity of the incoming annihilation particles. The angled brackets indicatethe thermal average.Figure 1.2: Dark matter freeze out. The red dashed line represents the equi-librium number density, nχeq , while the blue line represents the numberdensity for dark matter during freeze out as a function of x≡ mχT . As thetemperature decreases, and x increases, the annihilation process for darkmatter shuts off, effectively freezing out the dark matter and locking ina relic density.When T  mχ , χ is relativistic, and the number density is given by equation1.13. Therefore nχeq ∼ T 3 is unsuppressed. Since the collision term in equation1.18 is much larger than the Hubble term, nχ is driven by the equilibrium densityvalue.When the temperature cools below dark matter mass, equation 1.14 shows thatthe equilibrium density is exponentially suppressed. Therefore, the collision termbecomes less important, and the number density will stop tracking the equilibriumvalue when the Hubble term becomes large. This is the point at which the annihi-lation process shuts off since the mean time between collisions exceeds the Hubbletime H−1.After freeze out, the dark matter number density dilutes with universe expan-sion and remains larger than the equilibrium value, as shown in figure 1.2.13It is useful to solve equation 1.18 in terms of the yieldYχ =nχs(1.19)defined in terms of the plasma entropy density s, as defined in equation 1.16. Wecan also rewrite equation 1.18 with respect to temperature. Since dark matter freezeout occurs during radiation domination when a∝ t 12 , the Hubble equation becomesH2 =(a˙a)2=(12t)2= g∗pi2901M2PlT 4 (1.20)The Boltzmann equation 1.18 can then be written asdYχdx=− xsH(mχ) 〈σv〉(Y 2χ −Y 2χeq) (1.21)where x ≡ mχT and H(mχ)can be found from equation 1.20 by taking x = 1. Theyield equation 1.21 above can be easily solved numerically, but we will consideran approximation to solve it analytically and get a better intuition for the end yieldvalue after freeze out. Since at very late times, the equilibrium value is smallcompared to Yχ , we can drop Yχeq . By integrating from freeze out, x = xf, to today,x = ∞, we can see how the yield depends on our parametersYχ(t0) ∝xf〈σv〉 (1.22)The yield is not explicitly dependent on the mass of the dark matter, and inverselyproportional to the annihilation cross sections. Therefore, the smaller the annihila-tion cross section, the larger the relic density of dark matter.1.1.3 Alternative theoriesAll of our current dark matter evidence is from its gravitational influence on visiblematter. The fact that particle dark matter has yet to be directly detected, combinedwith the notion that 95% of our universe is made of unknown ‘dark’ stuff as shownin figure 1.1, have led many physicists to consider alternative theories to dark mat-ter. Instead of adding extra, unseen matter, alternative theories focus on modifying14the existing gravity laws or changing the gravity paradigm entirely (see [29–31]for some recent reviews). In the 1970s, David Lovelock showed that the Einsteinequations are the only possible second order equations of motion for a single met-ric in four dimensions that are local [32, 33]. Therefore, modifying our currentunderstanding of general relativity requires breaking underlying assumptions. Forexample, this can be done by including higher derivatives in the equations, addingextra degrees of freedom, allowing a higher dimensional spacetime, or doing awaywith locality.In addition, there are specific observations that are not explained via conven-tional dark matter. The ‘Missing Satellites Problem’ is a discrepancy between darkmatter simulations and observations. The simulations predict a far larger amountof substructure in dark matter haloes, such as satellites, than what is observed [34].Another discrepancy between simulation and observation occurs when comparinggalactic scales. The simulations predict cusp like behaviour at small radii, withthe density behaving like a power law, while observations show a more constantdensity. This is known as the ‘Core-Cusp Problem’ [35]. Finally, the Tully-Fisherrelation is an empirical result that relates the total mass of baryons, mb, in a rotatingcircular galaxy to the rotation velocity, v, viamb =A vx (1.23)where A and x are constants [36]. The fact that this relation does not depend onthe mass of the dark matter is puzzling.There are numerous alternative theories for gravity and dark matter. We outlinea handful of popular theories below, before giving an overview of emergent gravitywhich serves as motivation for this thesis.Modified Newtonian dynamicsMOdified Newtonian Dynamics (MOND) theories reject general relativity as theproper mathematical description of gravity and elevate Newtonian physics in anew framework for gravity. MOND was originally proposed in 1983 by MordehaiMilgrom. Newton’s second law, F = ma, is modified such that the force due to15gravity scales likeF = ma2a0(1.24)where a0 ≈ 1.2× 10−10 m/s2 is the acceleration constant. In the limit of smalla a0, which is the case for objects with orbits far away from the galactic centre,the acceleration constant becomes significant and MOND predicts the observed flatrotation curves [37].MOND is compatible with observed galaxy rotation curves [38–40], and evenprovides an explanation for the Tully-Fisher relation [36]. However, any alternativetheory to dark matter should be able to explain all of the phenomena attributed todark matter, and MOND cannot do this. For example, it is not successful on thescale of galaxy clusters, and, in particular, cannot provide a good explanation forthe Bullet Cluster [23] without invoking dark matter.Scalar-tensor theoriesScalar-tensor theories are well studied theories of gravity that modify general rela-tivity by adding an extra scalar field. Gravity is then mediated by a rank-2 tensor,as in Einstein’s general relativity, as well as a scalar field. In addition, the Newto-nian gravitational constant is no longer constant; it depends on the scalar field. Thesimplest scalar-tensor theory is Brans-Dicke gravity [41], which is well understoodand can be solved explicitly for specific cases such as static, spherical symmetry.The action for this theory is given byS =116pi∫ (φR− ωφ∇µφ∇µφ +Lm)√−g d4x (1.25)where φ is the scalar field, R is the Ricci scalar, ω is the Brans-Dicke couplingparameter, and Lm is the matter Lagrangian. ω can be tuned to fit the data, andthere are strong constraints on this coupling from solar system data [42] and cos-mological observations [43].16f (R) theoriesIn f (R) theories, the Ricci scalar is generalized by a function f (R), which leadsto fourth order derivatives in the equations of motion [44]. Higher order theorieshave been shown to introduce instabilities [45], but f (R) theories can avoid theinstabilities since the higher orders only act on the conformal mode which is non-dynamical. Thus, these theories are interesting to study, and, in the context ofhigher order theories, relatively simple. The action is given in general byS =∫ ( f (R)16piG+Lm)√−g d4x (1.26)There are many different f (R) theory models that need to be addressed specificallywhen considering constraints. However, similarities between classes of modelsallow constraints from both solar system data [44, 46, 47] and cosmological obser-vations [48, 49].Kaluza-Klein theoriesThe Kaluza-Klein theories were originally motivated by the unification of gravityand electrodynamics [50, 51]. They are higher dimensional theories that modifygravity by considering a 4+1 dimensional spacetime [52, 53]. The fourth spatialdimension is compact, and the characteristic size of the compact spatial dimen-sion is strongly constrained by particle collider experiments [54]. At early times,space is small, and the characteristic size is important. However, at late times,space is large in comparison, and we need only consider the effective theory infour dimensions. The scalar-tensor theories mentioned above are examples of sucheffective theories. Historically, Kaluza-Klein theories were also important in thedevelopment of string theory [55]. The cosmological implications have been stud-ied extensively (for example, see [56–58]).Emergent gravityIn 2010, Erik Verlinde proposed that gravity is not a fundamental interaction, butrather an emergent phenomenon [5]. Consider the holographic principle, wherethe information in the bulk of a space can be encoded on the lower dimensional17boundary [59, 60]. Starting from the assumption that the information associatedwith space obeys this holographic principle, Verlinde showed that gravity emergesfrom thermodynamic principles in the form of an entropic force. The connectionbetween gravity and thermodynamics has been explored before [61–65], and Ver-linde was not the first to suggest a thermodynamic origin [66, 67]. However, hewas able to provide a mechanism for the origin of gravity.In this paradigm, information is the key concept. At the microscopic level,information is associated with the matter and its position in space and is mea-sured as entropy. This system, with many degrees of freedom, then moves towardsmaximizing its entropy. At the macroscopic level, this effectively manifests in anentropic force. When there are changes in information due to the change in loca-tion of matter, an entropic force emerges in the form of gravity. This force does notdepend on the dynamics at the microscopic level, and there is no field associatedwith it. In this setting, Verlinde was able to reproduce Newtonian gravitation andthen the Einstein equations through relativistic generalization.In 2016, Verlinde expanded on his original paper by considering the conse-quences of emergent gravity for cosmology [6]. In his paper, an associated positivedark energy turns the ‘stiff’ geometry of spacetime into an elastic medium, creat-ing some ‘memory’ effects. The backreaction of the medium on the matter takesthe form of an extra ‘dark’ gravitational force that appears to be due to dark matter.His main result is given by∫B(8piGa0ΣD)2dV =(d−2d−1)∮∂BΦBa0dA (1.27)whereB is an arbitrary integration region, ΣD is the surface mass density for appar-ent dark matter, d is the dimension, ∂B is the boundary, and ΦB is the Newtoniangravitational potential of the baryonic matter.In order to try to compare equation 1.27 with observation, we consider spher-ical symmetry in d = 4 dimensions. Then the surface mass density can be writtenin terms of the Newtonian gravitational potential asΣ=− 14piGΦr(1.28)18allowing us to find a relationship for the apparent dark matter mass MD in terms ofthe baryonic mass MB ∫ r0GM2D(r′)r′2dr′ =MB(r)a0r6(1.29)This equation holds for spherically symmetric mass distributions that are isolatedand in dynamic equilibrium. Equation 1.29 does not, for example, apply in the caseof the Bullet Cluster.Though this theory reproduces some of the MOND results, such as the scalingrelation in equation 1.24 and the baryonic Tully-Fisher relation in equation 1.23,the physics in getting there is very different. Verlinde is quick to point out that histheory is not a derivation of MOND and should not be interpreted as the modifica-tion of a gravitational field. These results are not new laws of gravity, but insteadgive estimates for the strength of the dark gravitational force in specific situations.Emergent gravity is also promising in ways that MOND is not. Where MOND failsto explain the observed acceleration in galaxy clusters without invoking dark mat-ter, emergent gravity is able to greatly reduce the amount of dark matter requiredby considering the proper mass density profile of galaxy clusters.In conclusion, Verlinde states “The observed phenomena that are currentlyattributed to dark matter are the consequence of the emergent nature of gravityand are caused by an elastic response due to the volume law contribution to theentanglement entropy of our universe.” In other words, the gravitational effectthat we observe and attribute to dark matter is rather a memory effect from theemergence of space and is an intrinsic property of the spacetime itself. This boldstatement, if true, would revolutionize our understanding of gravity; it requires aheavy dose of skepticism. The work presented in this thesis is aimed at building aframework in which to properly test the emergent gravity theory.1.2 The homogeneous and isotropic expanding universeIn this section, we consider the homogeneous and isotropic universe presented insection 1.1 and derive and calculate the relevant quantities for this work. Section1.2.1 is dedicated to the FRW metric, section 1.2.2 to general relativity, and section191.2.3 to the stress energy tensor. In preparing this section, we consulted [9, 68, 69].1.2.1 Friedmann-Robertson-Walker metricThe evolution of a homogeneous and isotropic universe can be represented as ho-mogeneous and isotropic spacelike hypersurfaces that are constant in time. Thereare only three types of these spacelike hyperspaces with the required homogene-ity and isotropy: a flat space with no curvature, a sphere with constant positivecurvature, or a hyperbolic space with constant negative curvature. By embeddinga three dimensional homogeneous and isotropic space in a four dimensional Eu-clidean space, the induced metric for a space of constant curvature can be writtenasd`2 = a2[11− kr2 dr2+ r2(dθ 2+ sin2 θdϕ2)], k =+1, spherical0, flat−1, hyperbolic(1.30)where a2 > 0 and k is the normalized curvature parameter. The scale factor acharacterizes the relative sizes of the hypersurfaces and has arbitrary normalization.We will consider the flat case k= 0. Now we want to generalize to a time dependentfour dimensional spacetime while maintaining the homogeneity and isotropy. Theonly way to do this is to allow the time evolution to be completely described bythe scale factor a(t). The line element is then, in both spherical and Cartesiancoordinates, given byds2 =−dt2+a2(t)[dr2+ r2 (dθ 2+ sin2 θdϕ2)]=−dt2+a2(t)δi jdxidx j= gµνdxµdxν(1.31)where gµν is known as the FRW metric. The spatial coordinates xi are the co-moving coordinates and the time coordinate t is the proper time measured by acomoving observer. Then the distance between any two comoving observers at agiven time t is proportional to the scale factor a(t) as expected. How does a(t)evolve? For the FRW universe, we can determine the form of the scale factor a(t)20via the Einstein equations as shown in the following section.1.2.2 General relativityIn general relativity, the relationship between the gravitational field, characterizedby the metric, and matter in the universe is governed by the Einstein equations.The Einstein equations are given by2Gµν ≡ Rµν − 12Rgµν +Λgµν =8piM2PlTµν (1.32)where Λ is the cosmological constant which, for the purposes of this work, will beneglected. The Ricci tensor Rµν and the Ricci scalar R are given by3Rµν = Γαµν ,α −Γαµα ,ν +ΓαβαΓβµν −ΓαβνΓβµαR≡ gµνRµν(1.33)The Γαµν are the Christoffel symbols. They come from the geodesic equation andare given byΓαµν ≡12gαβ[gµβ ,ν +gβν ,µ −gµν ,β](1.34)The nonzero components of the Christoffel symbols are given byΓ0i j = a2Hδi jΓi0 j = Γij0 = Hδij(1.35)Therefore, the nonzero components of the Ricci tensor areR00 =−3(H2+ H˙)Ri j = a2(3H2+ H˙)δi j(1.36)and the Ricci scalar isR = 6(2H2+ H˙)(1.37)2The Planck mass MPl is defined such that M2Pl ≡ G−1, where G is the Newtonian gravitationalconstant.3We use the notation ,α ≡ ddxα for derivatives with respect to specific coordinates.21The Einstein tensor is constructed from the Ricci tensor and Ricci scalar givenabove, and its nonzero components are given byG00 = 3H2Gi j =−a2(3H2+2H˙) (1.38)Since spacetime bends with matter, we need a more robust definition of thederivative. This comes in the form of the covariant derivative ∇α . Specifically, weare interested in how the covariant derivative acts on scalars, vectors, and tensors.∇αφ = φ,α∇αAβ = Aβ,α +ΓβαγAγ∇αAβ = Aβ ,α −ΓγβαAγ∇αhβγ = hβγ,α +Γβασhσγ +Γγασhβσ∇αhβγ = hβγ ,α −Γσγαhσβ −Γσαβhγσ(1.39)1.2.3 Stress energy tensorIn the Einstein equations as shown in equation 1.32, all of the matter is included viathe stress energy tensor Tµν . The equations that govern the matter are determinedby the conservation of the stress energy tensor, also called the continuity equation.∇µT µν ≡ T µν,µ +ΓµαµTαν +ΓναµT µα = 0 (1.40)On large enough scales, matter can be approximated as a perfect fluid, and thegeneral form of the stress energy tensor for a perfect fluid is given byTµν = (ρ+P)uµuν +Pgµν (1.41)22where ρ is the density, P is the pressure, and uµ is the four-velocity. Therefore, inthe comoving frame where uµ = (1,0,0,0), the stress energy tensor is given byT µν =−ρ 0 0 00 P 0 00 0 P 00 0 0 P (1.42)From the Einstein equations we can derive the Friedmann equations which aregiven byH2 =(a˙a)2=8pi3M2PlρH2+ H˙ =a¨a=− 4pi3M2Pl(ρ+3P)(1.43)The continuity equation 1.40 can be used to determine the evolution of ρ andP. From the time-time component, we get∂ρ∂ t+3a˙a(ρ+P) = 0 =⇒ a−3 ∂∂ t(ρa3)+3a˙aP = 0 (1.44)Therefore, since matter has no pressure, ρm ∝ a−3. This makes sense as we expectthe matter density to scale like inverse volume since the universe is expanding.Since P = ρ3 for radiation, ρr ∝ a−4. Recall the equation of state for a perfectfluid from equation 1.7. Therefore, in general, the density evolves according toρ ∝ a−3(1+w), where w = 0 for matter, w = 13 for radiation, and w = −1 for darkenergy.Matter is coupled to gravity through the Einstein-Hilbert actionS =∫ ( 12κR+L)√−g d4x (1.45)By varying this action, we can derive the stress energy tensor viaTµν =− 2√−gδ (√−gL )δgµν= gµνL −2 δLδgµν (1.46)This form of the stress energy tensor is more easily accessible when given a model23with a specific Lagrangian density.For example, consider a single scalar field φ with potential V (φ). The La-grangian density for this scalar field is given byL =−12∇µφ∇µφ −V (φ) (1.47)Then the stress energy tensor is derived from this Lagrangian according to equation1.46, and is given byTµν = ∇µφ∇νφ −gµν(12gαβ∇αφ∇βφ +V (φ))(1.48)For a homogeneous universe, the scalar field φ only depends on the time coordinatet and not on any of the spatial coordinates; the individual nonzero stress energytensor components are given byT 00 =−(12φ˙ 2+V (φ))=−ρT ij =(12φ˙ 2−V (φ))δ ij = Pδij(1.49)1.3 Cosmological perturbation theoryUp until this point we have assumed that our universe is described by the homoge-neous and isotropic FRW universe. Let us now consider the fact that our universeis not perfectly homogeneous and isotropic. We will assume that any inhomo-geneities can be represented by small perturbations around the homogeneous back-ground solutions and present an overview of cosmological perturbation theory inthe following sections. In section 1.3.1 we consider the metric perturbations, whilein section 1.3.2 we consider the matter perturbations. In preparing this section, weconsulted [9, 68, 69].241.3.1 Metric perturbationsConsider the following perturbed metricgµν(t,x) = g¯µν(t)+δgµν(t,x) (1.50)where g¯µν is the unperturbed FRW metric given in equation 1.31, and δgµν isa small metric perturbation around the FRW background.4 This metric is givenexplicitly and in full generality byds2 =−(1+2Φ)dt2+2aBidxidt+a2[(1−2Ψ)δi j +Ei j]dxidx j (1.51)whereBi ≡ ∂iB−Si, ∂ iSi = 0Ei j ≡ 2∂i jE +2∂(iFj)+δgi j, ∂ iFi = 0, ∂ ihi j = 0(1.52)For the purposes of this work, we will neglect the vector perturbations Si and Fi,as well as the tensor perturbation hi j, and focus solely on the scalar perturbationsΦ, Ψ, B, and E. In general, these scalar perturbations all depend on the time andspatial coordinates. In matrix form, the metric is given bygµν =(− [1+2Φ] a∂iBa∂iB a2 [(1−2Ψ)δi j +2∂i∂ jE])(1.53)and the matrix form of the inverse metric, to first order, is given bygµν =(− [1−2Φ] −a−1∂iB−a−1∂iB a−2 [(1+2Ψ)δi j−2∂i∂ jE])(1.54)Gauge transformationsNow that the spacetime is not homogeneous and isotropic, we need to determinehow the scalar perturbations transform under a change of coordinates. Consider4In general, we denote the background fields and metric using overbars, as in f¯ , and the corre-sponding perturbations using δ , as in δ f .25the following gauge transformationt→ t+αxi→ xi+∂ iβ(1.55)Using the invariance of the line element ds2, the metric transforms asg˜αβ (x˜)dx˜αdxµdx˜βdxν= gµν(x) (1.56)Therefore, the scalars transform according toΦ→Φ− α˙B→ B+ αa−aβ˙E→ E−βΨ→Ψ+Hα(1.57)At first glance, it appears that the scalar metric perturbations are defined by fourfunctions. However, with coordinate transformations, we can use the two functionsα and β to manipulate and eliminate the scalar perturbations at will and so there arereally only two independent functions that describe the scalar metric perturbations.Since we are able to remove perturbations through a coordinate transformation(gauge choice), we want to define a new set of variables that is gauge invariant andcan be used to extract physical results. The Bardeen variables [70] are an exampleof such a set of perturbations and are given byΦB ≡Φ− ddt[a2(E˙− Ba)]ΨB ≡Ψ+a2H(E˙− Ba) (1.58)Since these perturbations are impossible to remove through a gauge transformation,they are considered the true physical perturbations.26Newtonian gaugeAt this point we are free to choose a gauge that is convenient, and we chooseto work in the so-called Newtonian gauge. The Newtonian gauge is defined byE = 0 = B, giving ΦB = Φ and ΨB =Ψ. In this gauge, Φ is the Newtonian grav-itational potential. Starting from a generic spacetime, this gauge is easily realizedin choosing a coordinate transformation from equation 1.55 by t → t + a2E˙− aBand xi→ xi+∂ iE.In the Newtonian gauge, the metric simplifies to a diagonal metric and is givenbyds2 =−(1+2Φ)dt2+a2 (1−2Ψ)δi jdxidx j (1.59)In the following calculations, it is convenient to deconstruct the above metric intoits background and perturbation parts by gµν = g¯µν +δgµν and similarly with theinverse metric gµν = g¯µν +δgµν . The matrix forms of these metrics are given byg¯µν =(−1 00 a2δi j)δgµν =(−2Φ 00 −2a2Ψδi j)g¯µν =(−1 00 a−2δ i j)δgµν =(2Φ 00 2a−2Ψδ i j) (1.60)The perturbed Christoffel symbols to first order are given byΓµαβ =[12g¯µν(g¯αν ,β + g¯βν ,α − g¯αβ ,ν)]+[12g¯µν(δgαν ,β +δgβν ,α −δgαβ ,ν)+12δgµν(g¯αν ,β + g¯βν ,α − g¯αβ ,ν)]= Γ¯µαβ +δΓµαβ(1.61)The nonzero components of the background Christoffel symbols are given by equa-tion 1.35, and the nonzero components of the first order Christoffel symbol pertur-27bations are given byδΓ000 = Φ˙δΓ0i0 = ∂iΦδΓ0i j =−a2[2H (Φ+Ψ)+ Ψ˙]δi jδΓi00 = a−2∂ iΦδΓij0 =−Ψ˙δ ijδΓijk = ∂iΨδ jk−∂ jΨδ ik−∂kΨδ ij(1.62)The perturbed Ricci tensor to first order can be calculated from the perturbedChristoffel symbols and is given byRµν =[Γ¯αµν ,α − Γ¯αµα ,ν + Γ¯αβα Γ¯βµν − Γ¯αβν Γ¯βµα]+[δΓαµν ,α −δΓαµα ,ν + Γ¯αβαδΓβµν +δΓαβα Γ¯βµν − Γ¯αβνδΓβµα −δΓαβν Γ¯βµα]= R¯µν +δRµν(1.63)The nonzero components of the background Ricci tensor are given by equation1.36, and the nonzero components of the first order Ricci tensor perturbations aregiven byδR00 = a−2∇2Φ+3[Ψ¨+H(Φ˙+2Ψ˙)]δRi0 = 2(H∂iΦ+∂iΨ˙)δRi j ={∇2Ψ−a2 [6H2 (Φ+Ψ)+2H˙ (Φ+Ψ)+H (Φ˙+6Ψ˙)+ Ψ¨]}δi j+∂i∂ j (Ψ−Φ)(1.64)The perturbed Ricci scalar to first order is given byR =[g¯µν R¯µν]+[g¯µνδRµν +δgµν R¯µν]= R¯+δR(1.65)where the background Ricci scalar is given in equation 1.37 and the first orderperturbation of the Ricci scalar is given byδR = 2a−2∇2 (2Ψ−Φ)−6[4H2Φ+2H˙Φ+H (Φ˙+4Ψ˙)+ Ψ¨] (1.66)28Now that we have found the perturbed metric, Ricci tensor, and Ricci scalar, wecan construct the perturbed Einstein tensor viaGµν =[R¯µν − 12 g¯µν R¯]+[δRµν − 12(g¯µνδR+δgµν R¯)]= G¯µν +δGµν(1.67)The nonzero components of the background Einstein tensor are given in equation1.38, and the components of the first order perturbation in the Einstein tensor aregiven byδG00 = 2a−2∇2Ψ−6HΨ˙δGi0 = 2(H∂iΦ+∂iΨ˙)δGi j ={∇2 (Φ−Ψ)+2a2 [(3H2+2H˙)(Φ+Ψ)+2H (Φ˙+ Ψ˙)+ Ψ¨]}δi j+∂i∂ j (Ψ−Φ)(1.68)Conformal Newtonian gaugeOn occasion, it is convenient to work in the conformal FRW universe. The confor-mal time is given byη ≡∫ t0dta(t)(1.69)This is also referred to as the comoving horizon, since it is the total comovingdistance light could have travelled since t = 0. This is an important distance incosmology, since it defines causally connected regions. The metric in the confor-mal Newtonian gauge is given byds2 = a2 (η)[−(1+2Φ)dt2+(1−2Ψ)δi jdxidx j]= gµνdxµdxν (1.70)The Hubble parameter becomes5H ≡ a′ (η)a(η)(1.71)5For different sections we use the prime notation to indicate different things. In this section, weuse it to indicate the derivative with respect to the conformal time, i.e. ′ ≡ ddη .29The Christoffel symbols are defined in equation 1.61 to first order. The nonzerobackground components are given byΓ¯000 =HΓ¯0i j =H δi jΓ¯i0 j =H δij = Γ¯ij0(1.72)and the nonzero perturbation components are given byδΓ000 =Φ′δΓ00i = ∂iΦ= δΓ0i0δΓ0i j =−[2H (Φ+Ψ)−Ψ′]δi jδΓi00 = ∂iΦδΓi0 j =−Ψ′δ ij = δΓij0δΓijk = δ jk∂iΨ−δ ij∂kΨ−δ ik∂ jΨ(1.73)The Ricci tensor is defined in equation 1.63 to first order. The nonzero backgroundcomponents are given byR¯00 =−3H ′R¯i j =(2H 2+H ′)δi j(1.74)and the nonzero perturbation components are given byδR00 = ∇2Φ+3H(Φ′+Ψ′)+3Ψ′′δR0i = 2(H ∂iΦ+∂iΨ′)= δRi0δRi j =[−Ψ′′+∇2Ψ−2(H 2+H ′)(Φ+Ψ)−H (Φ′+5Ψ′)]δi j +∂i∂ j (Ψ−Φ)(1.75)The Ricci scalar is defined in equation 1.65 to first order. The background andperturbation components are given byR¯ = 6a−2(H 2+H ′)δR =−2a−2[6(H 2+H ′)Φ+∇2Φ−2∇2Ψ+3H (Φ′+3Ψ′)+3Ψ′′] (1.76)30Putting everything together, we can write down the nonzero components of theEinstein tensor from the definition to first order in equation 1.67. The backgroundcomponents are given byG¯00 = 3H 2G¯i j =−(H 2+2H ′)δi j(1.77)and the perturbation components are given byδG00 = 2(∇2Ψ−3H Ψ′)δG0i = 2(H ∂iΦ+∂iΨ′)= δGi0δGi j =[2Ψ′′+∇2(Φ−Ψ)+2(H 2+2H ′)(Φ+Ψ)+2H (Φ′+2Ψ′)]δi j+∂i∂ j (Ψ−Φ)(1.78)1.3.2 Matter perturbationsFor this section, we go back to working in an arbitrary gauge. The perturbed den-sity and pressure functions areρ(t,x) = ρ¯(t)+δρ(t,x)P(t,x) = P¯(t)+δP(t,x)(1.79)and the general form of the stress energy tensor is given byTµν = (ρ+P)uµuν +Pgµν +Σµν (1.80)which has the same form as the background stress energy tensor given in equa-tion 1.41 with the addition of an anisotropic stress term Σµν . The perturbed four-velocity is given by uµ = u¯µ +δuµ , where u¯µ = (−1,0,0,0) and δuµ = (−Φ,avi),where vi is the coordinate velocity.The perturbed stress energy tensor is given in terms of the density, pressure,31and four-velocity byTµν =[(ρ¯+ P¯)u¯µ u¯ν + P¯g¯µν]+[(δρ+δP) u¯µ u¯ν +(ρ¯+ P¯)(δuµ u¯ν + u¯µδuν)+(δPg¯µν + P¯δgµν)+Σµν]= T¯µν +δTµν(1.81)The background stress energy tensor is given in equation 1.42, and the componentsof the first order perturbation to the stress energy tensor are given byδT 00 =−δρδT 0i = (ρ¯+ P¯)avi = δq,iδT i0 =−(ρ¯+ P¯)a−1(vi−Bi)δT ij = δPδij +Σij(1.82)From the above δT ij component and the corresponding Einstein tensor component,we have the following∂i∂ j (Ψ−Φ) = 8piMPlΣi j (1.83)Therefore, if the anisotropic stress term is zero, then Ψ=Φ.Just like the metric tensor, the stress energy tensor transforms according toT µν (x) =∂xµ∂ x˜α∂ x˜β∂xνT˜αβ (x˜) (1.84)Using this transformation of the stress energy tensor, we find that the various firstorder matter perturbations transform according toδρ → δρ− ˙¯ραδP→ δP− ˙¯Pαvi→ vi+∂iαδq→ δq+(ρ¯+ P¯)αδφ → δφ − φ˙α(1.85)32Next, consider the conservation equation for the perturbed stress energy tensor.∇µTµν = gµα∇αTµν= g¯µα(T¯µν ,α − Γ¯λµα T¯λν − Γ¯λαν T¯µλ)+[g¯µα(δTµν ,α −δΓλµα T¯λν −δΓλαν T¯µλ − Γ¯λµαδTλν − Γ¯λανδTµλ)+δgµα(T¯µν ,α − Γ¯λµα T¯λν − Γ¯λαν T¯µλ)]= 0(1.86)The background conservation equation is the same as was given in equation 1.40,and setting the second set of square brackets above to zero gives the perturbedconservation of stress energy tensor equation.Let us revisit our example of a free scalar field, and consider perturbations inthe form of φ (t,x) = φ¯(t)+δφ (t,x) and the potential in the form of V = V¯ +δV .Then the perturbed stress energy tensor, as derived from the matter Lagrangian inequation 1.47, is given byTµν =[∇µ φ¯∇ν φ¯ − g¯µν(12g¯αβ∇α φ¯∇β φ¯ +V¯)]+[(∇µ φ¯∇νδφ +∇µδφ∇ν φ¯)− 12g¯µν(g¯αβ(∇α φ¯∇βδφ +∇αδφ∇β φ¯)+δgαβ∇α φ¯ I∇β φ¯ J)−12δgµν g¯αβ∇α φ¯∇β φ¯ − g¯µνδV −δgµνV¯]= T¯µν +δTµν(1.87)For the Newtonian gauge in FRW, the components of the background stress energytensor are given in equation 1.49, and the first order perturbations of the stressenergy tensor components are given byδT00 = 2V¯Φ+δV + ˙¯φδ˙φδTi0 = ˙¯φ∂iδφδTi j = a2[2V¯Ψ−δV − ˙¯φ 2 (Φ+Ψ)+ ˙¯φδ˙φ]δi j(1.88)33Since there is no anisotropic stress, we can take Ψ=Φ.Now that we have expressions for the Einstein tensor perturbations and thestress energy tensor perturbations, we are able to find the perturbed Einstein equa-tions in the Newtonian gauge in terms of the potential.2a2∇2Φ−6HΦ˙= 8piM2Pl[δV +2V¯Φ+ ˙¯φδ˙φ]4piM2Pl˙¯φ∂iδφ = H∂iΦ+∂iΦ˙4piM2Pl[δV −2V¯Φ+2Φ ˙¯φ 2− ˙¯φδ˙φ]+2Φ(3H2+2H˙)+4Φ˙H + Φ¨= 0(1.89)34Chapter 2The gravity-like YukawainteractionThe Yukawa interaction was originally proposed by Hideki Yukawa to describe theinteraction between nucleons [71]. It has a coupling constant g that is analogousto the electron charge e. In the Standard Model of particle physics, the Yukawainteraction is well studied as it couples the Higgs scalar field to leptons and quarks.In cosmology, the Yukawa interaction has also been studied as a ‘fifth force’ [72]and there are strong current limits on the strength of this Yukawa coupling fromsolar system and other data [73–75].In this chapter, we establish that the Yukawa interaction is gravity-like in cer-tain limits. We show that the Yukawa interaction is analogous to gravity from botha particle perspective in section 2.1 and a general relativity perspective in section2.2. In section 2.3, we explore how the addition of the Yukawa force affects spher-ical collapse.2.1 From a particle perspectiveThe goal of this section is to compute the Yukawa potential and compare it togravity. We are interested in the scattering of two massive fermions, ψψ → ψψ ,through the Yukawa coupling, as shown by the Feynman diagram in figure 2.1. Wefirst compute the scattering amplitude in the nonrelativistic limit using the Feyn-35man rules for Yukawa theory, as given in appendix A. By comparing to the Bornapproximation, we derive the Yukawa potential. In this section only, we adopt thenegative metric signature, as is the convention in particle physics. In the prepara-tion of this section, we consulted [76, 77].Figure 2.1: Tree level Feynman diagram for two fermions ψ scatteringthrough a scalar Yukawa interaction φ with coupling g. The particleson the left are the incoming fermions, with momenta p and k, and spinss and r. The particles on the right are the outgoing fermions, with mo-menta p′ and k′, and spins s′ and r′. The momentum for the scalarmediator is labelled q.In Minkowski space, the Lagrangian density for a fermion field, ψ , coupled toa scalar field, φ , through a Yukawa interaction is given byLYukawa =LDirac+LKlein-Gordon−gψ¯ψφ (2.1)where fermions are described by the Dirac equation, and scalars are described bythe Klein-Gordon equation. Their respective Lagrangian densities areLDirac = ψ¯(i/∂ −mψ)ψLKlein-Gordon =12(∂µφ)2− 12m2φφ2(2.2)36and gψ¯ψφ is the Yukawa interaction.Consider a two particle scattering event. The tree level Feynman diagram isgiven by figure 2.1. Assuming the particles are distinguishable, this is the onlyrelevant diagram. From the Feynman diagram, we can calculate the scattering am-plitude in momentum space using the Feynman rules. We will assign a momentumlabel of p to the top pair of fermions, and a momentum label of k to the bottompair. Outgoing fermions will be distinguished with a prime. The momentum of thescalar mediator is given by q. Similarly, we label the spin with s and s′ for the toppair, and r and r′ for the bottom pair. From the propagator, we have a factor ofiq2−m2φ + iε(2.3)The two vertices give(−ig)2 (2.4)The external legs give u¯s′(p′)us (p) and u¯r′ (k′)ur (k). By conservation of momen-tum, q = p′− p = k− k′ wherep =(mψ ,p)p′ =(mψ ,p′)k =(mψ ,k)k′ =(mψ ,k′)(2.5)Thus q = p′−p. Therefore, the scattering amplitude is given byiM = i(−ig)2 u¯s′ (p′)us (p) i(p′− p)2−m2φu¯r′ (k′)ur (k) (2.6)Since we are interested in the nonrelativistic regime, we only need to keep the firstorder for the 3-momenta. Therefore(p′− p)2 =−∣∣p′−p∣∣2+O (p4) (2.7)The u(p) arise when writing a Dirac field as a linear combination of plane37waves according toψ(x) = u(p)e−ip·x (2.8)In the rest frame, u(p) is given explicitly byus (p) =√m(ξ sξ s)(2.9)where ξ is an arbitrary two component spinor normalized such that ξ †ξ = 1.1u¯s(p) is defined viau¯s(p) = us†(p)γ0 (2.10)where γ0 is the Dirac matrixγ0 =1 0 0 00 1 0 00 0 −1 00 0 0 −1 (2.11)In the nonrelativistic limit, we can use the definition of us (p) given by equation 2.9.Combining this with the definition for u¯s(p) from equation 2.10, we can calculateu¯s′ (p′)us (p) = 2mψξ s′†ξ s = 2mψδ ss′u¯r′ (k′)ur (k) = 2mψξ r′†ξ r = 2mψδ rr′ (2.12)Therefore, spin is conserved, and the scattering amplitude from equation 2.6 be-comesiM =ig2|p′−p|2−m2φ4m2ψδss′δ rr′(2.13)Now we can compare our scattering amplitude to the Born approximation inquantum mechanics [78] which is valid for weak scattering.〈p′∣∣iT ∣∣p〉=−iV˜ (q)(2pi)δ (Ep′−Ep) (2.14)By comparing the scattering amplitude from equation 2.13 with the Born approxi-1The † notation is used to indicate the conjugate transpose.38mation from equation 2.14, we getV˜ (q) =−g2|q|2+m2φ(2.15)where q is the momentum of the scalar mediator φ and we neglect the factor of4m2ψ since it is just due to our normalization convention. Since our calculation wasdone in momentum space, we take the Fourier transform of equation 2.15 to getback to position space. This results in the usual Yukawa potential given byV (r) =− g24pi1re−mφ r (2.16)There are a couple of important things to note about this potential. First, it isnegative, so the force is attractive. Second, the range of this potential is given by1mφ. Therefore, if we let the scalar field φ be massless, then the range is infinite andV ∝ 1r . Recall that the Newtonian gravitational potential is negative and Φ ∝1r .Thus, the Yukawa potential connects to the Newtonian gravitational potential inthis way. Therefore, coupling fermions with a scalar field, we end up with a longrange, scalar mediated force that looks like gravity.2.2 From the Einstein-Hilbert actionIn this section, we examine the Yukawa interaction from a general relativity per-spective, assuming scalar field matter. First, we consider a free massive scalar fieldψ minimally coupled to gravity. We then consider the massive scalar field coupledto a massless scalar Yukawa field φ as well. Using perturbation theory, we showthat the Yukawa coupling is analogous to gravity in the limit of small Φ and εφ .For this section, we work in the conformal Newtonian gauge, where the metricis given by equation 1.70. Assuming no anisotropic stress, we can set Ψ=Φ, andthe metric becomesds2 = a2 (η)[−(1+2Φ)dt2+(1−2Φ)δi jdxidx j]= gµνdxµdxν (2.17)39where the determinant is given byg = a8(1−2Φ)3(1+2Φ) (2.18)Matter is coupled to gravity through the Einstein-Hilbert action given in equa-tion 1.45 and reprinted below for convenienceS =∫ ( 12κR+L)√−g d4xThe gravity coupling is given by√−g L . Given equation 2.18, we can calculate√−g to first order as √−g = a4 (1−2Φ) (2.19)Before adding the Yukawa interaction, let us consider a free massive scalarfield ψ . The Lagrangian density is given byL =−12∇µψ∇µψ− 12m2ψ2 (2.20)If we consider the coupling of gravity to the mass term, we end up with√−g12m2ψ2 = a4 [1−2Φ] 12m2ψ2 (2.21)We now consider a massive scalar field ψ coupled to a massless scalar field φthrough the Yukawa interaction. The Lagrangian density is given byL =−12∇µψ∇µψ− 12∇µφ∇µφ − 12m2ψ2+gφψ2 (2.22)where g is the strength of the Yukawa coupling and m is the mass of ψ . To bet-ter explore how the Yukawa interaction relates to gravity in general relativity, wereparametrize g with ε ≡ gm2 . The Lagrangian density is then given byL =−12∇µψ∇µψ− 12∇µφ∇µφ − 12m2 (1−2εφ)ψ2 (2.23)40If we consider the same gravity coupling as before, we see that√−g12m2 (1−2εφ)ψ2 = a4 [1−2(Φ+ εφ)] 12m2ψ2 (2.24)Comparing equations 2.21 and 2.24, we see that εφ is analogous to the gravitationalpotentialΦ in the limit of smallΦ and εφ . Therefore, the Yukawa field φ acts like agravitational field. We have now shown that the Yukawa interaction is gravity-likefrom both a particle physics perspective and a general relativity perspective.2.3 Spherical collapseIn cosmology, the evolution of density perturbations is important as the growthof these perturbations is thought to give rise to the structure we see today. Nowthat we have established that the Yukawa interaction is gravity-like, we apply it tothe spherical collapse model. In this section we consider a matter overdensity inEinstein-de Sitter cosmology, a flat, matter dominated cosmology. In the prepara-tion for this section, we consulted [79–81]In the spherical top hat model we consider, the universe contains only matter asa collisionless fluid, and there is one spherical overdensity perturbation. Definingthe density contrast asδ ≡ ρ− ρ¯ρ¯=δρρ¯(2.25)we can calculate the mass enclosed in a radius rM =4pi3r3ρ¯(1+δ ) (2.26)where, in general, r, ρ¯ , and δ all depend on time, but the mass M must be constantdue to energy conservation.The spherical nature of our model allows us to apply Birkoff’s theorem, whichrelates general relativity to Newton’s gravity, and gives the follow equation of mo-tiond2rdt2=−∇V (r) (2.27)41In the case of simple Newtonian gravity,VN(r) =−GMr (2.28)However, in our calculation, we will include the Yukawa potential from equation2.16 with infinite range, given byVY(r) =− g24pir(2.29)Therefore, the total potential is given byV (r) =VN(r)+VY(r) =−(G+g24piM)M1r=− G˜Mr(2.30)This potential looks exactly like the Newtonian gravitational potential in equation2.28 with modified G. Since G, g, and M are all constant, we can solve our equa-tions with a constant, modified Newtonian gravitational constantG˜ = G+g24piM(2.31)Our equation of motion to solve is then given byd2rdt2=− G˜Mr2(2.32)Integrating once, equation 2.32 becomes12(drdt)2− G˜Mr= E (2.33)where E is the (constant) energy per unit mass of our system. In order for oursystem to be gravitationally bound, we must have E < 0. To solve equation 2.33,42we rewrite the variables in parametric form.r =G˜M2|E|(1− cosθ)t =G˜M(2|E|) 32(θ − sinθ)(2.34)where θ ∈ [0,2pi]. From equation 2.34 above, we can see that when θ = 0 thent = 0 and r = 0. Then r will expand until rmax when θ = pi , after which r willcollapse back to r = 0 when θ = 2pi . The time at which r = rmax is called theturnaround time. rmax and tTA are given byrmax =G˜M|E|tTA =G˜Mpi(2|E|) 32(2.35)The time at which r collapses back to 0 is called the collapse time and is given bytcollapse = 2tTA =2G˜Mpi(2|E|) 32(2.36)We now explore the evolution of the density contrast. Since the spherical col-lapse universe is totally matter dominated, the scale factor evolves according toa ∝ t23 so the Hubble parameter from equation 1.4 is given byH(t) =23t(2.37)Since we are in a flat universe, the corresponding Friedman equation 1.5 for thebackground density givesH2 =8piG˜3ρ¯ =⇒ ρ¯ = 16piG˜ t2(2.38)Therefore, we can calculate the mass from equation 2.26 in terms of the initial43conditionsM =4pir303ρ¯0(1+δ0) =r30H202G˜(1+δ0) (2.39)Consider the initial velocity in terms of the comoving radius xv0 =dr0dt=d(ax0)dt=dadtx0+adx0dt(2.40)Assuming dx0dt = 0, the initial velocity is just given by the initial Hubble parameterv0 = H0r0 (2.41)Then the initial kinetic energy per unit mass is given byK0 =12v20 =12H20 r20 (2.42)and the initial potential energy per unit mass is given byU0 =− G˜Mr0 =−12H20 r20(1+δ0) (2.43)Therefore, the total energy per unit mass is given byE0 = K0+U0 =−12H20 r20δ0 (2.44)Since E0 < 0 is required for gravitational collapse, this implies that gravitationalcollapse will occur whenever δ0 > 0. In other words, in the spherical collapsemodel, collapse will occur for any overdensity. Now consider the energy at turnaroundtime. The kinetic energy is zero, so we only need to consider the potential energy.ETA =UTA =− G˜Mrmax =−H20 r302rmax(1+δ0) (2.45)Plugging this value for energy into equation 2.35, the maximum radius and turnaround44time becomermax =G˜M|E| = rmaxtTA =pi√G˜M(rmax2) 32(2.46)From our definition of G˜ given in equation 2.31, it follows that G˜ > G. There-fore, by adding the Yukawa potential to the spherical collapse model, both theturnaround time and the collapse time are smaller than for gravity alone. The max-imum radius rmax, however, does not depend on G˜ and therefore is the same as thecase for gravity alone.Using conservation of energy with equations 2.44 and 2.45, we findE0 = ETA =⇒ −12H20 r20δ0 =−12H20 r30rmax(1+δ0) (2.47)Thus1+δ0δ0=rmaxr0=⇒ 1δ0' rmaxr0(2.48)So, the smaller the density perturbation, the larger the maximum radius and thelonger the turnaround time. Therefore, when adding the Yukawa interaction, thedensity perturbation effectively looks larger, which is why the turnaround and col-lapse times are shorter than for the case of gravity alone.Comparing the mean top hat density in parametric formρ =3M4pir3=3M2piδ 30(1+δ0)31(1− cosθ)3 (2.49)to the mean background density from equation 2.38 in parametric formρ¯ =M3piδ 30(1+δ0)31(θ − sinθ)2 (2.50)results in the following evolution for the density contrast in terms of θδ =ρρ¯−1 = 9(θ − sinθ)22(1− cosθ)3 −1 (2.51)45This equation does not depend on G˜, so the density contrast will evolve in exactlythe same way for both the addition of the Yukawa force and for gravity on its own.The only difference is the turnaround time. These results are encouraging since weexpect the Yukawa force to look exactly like gravity in this limit. By adding theYukawa interaction, we have effectively just scaled time. The value of the densitycontrast at turnaround, when θ = pi is δTA' 4.55, while the value at collapse, whenθ = 2pi is δcollapse = ∞, which corresponds to a black hole.Finally, we compare our evolution with the linear perturbation theory. In lineartheory, θ  1. To find the evolution of the density contrast, we can Taylor expandsinθ and cosθ around θ = 0. This results inδlinear =320(6pi)23(ttTA) 23(2.52)Recall that a ∝ t23 for matter. Therefore, for the linear theory, the density contrastscales like the scale factor, δ ∼ a. The value of the linear density contrast is 1.062at turnaround and 1.686 at collapse. Again, neither of these numbers depend on G˜,and so they are the same for both Yukawa and regular gravity. The linear densitycontrast at collapse is known as the critical overdensity for collapse, δc. In numer-ical simulations, if the density contrast is greater than δc, we consider the systemto have collapsed.46Chapter 3Coupling matter to the YukawaforceIn chapter 2 we established that the Yukawa interaction is gravity-like in certainlimits. In this chapter, we explore the equations of motion more thoroughly byconsidering the Yukawa interaction coupled to a massive scalar field. In section3.1, we derive the equations of motion, and in 3.2 we plot the numerical solutionsand analyze the resulting plots. In this chapter we work in the Newtonian gauge,where the metric is given by equation The equations of motionConsider a massive scalar field, ψ , coupled to as massless scalar field, φ , througha Yukawa coupling, as explored in section 2.2. The Lagrangian density is given byequation 2.23 and reprinted here for convenience.L =−12∇µψ∇µψ− 12∇µφ∇µφ − 12m2(1−2εφ)ψ2To find the equations of motion for the ψ and φ fields, we use the Euler-Lagrangeequation, which arises from setting the functional derivative of the Lagrangian den-47sity with respect to the field to zero.∂L∂θ−∇µ(∂L∂∇µθ)= 0 (3.1)This leads to the equations of motion for ψ and φ .ψ−m2 (1−2εφ)ψ = 0φ + εm2ψ2 = 0(3.2)where ≡ gµν∇µ∇ν . Then the background equations of motion are given by¨¯ψ+3H ˙¯ψ−m2 (1−2εφ¯) ψ¯ = 0¨¯φ +3H ˙¯φ + εm2ψ¯2 = 0(3.3)and the equations of motion for the perturbations to first order are given by¨δψ−a−2∇2δψ+3H ˙δψ+m2 (1−2εφ¯)(δψ+2ψ¯Φ)−2εm2ψ¯δφ − (Φ˙+3Ψ˙) ˙¯ψ = 0δ¨φ −a−2∇2δφ +3Hδ˙φ −2εm2ψ¯ (δψ+ ψ¯Φ)− (Φ˙+3Ψ˙) ˙¯φ = 0(3.4)where ∇2 ≡ δ i j∂i∂ j.The stress energy tensor can be calculated from the Lagrangian density accord-ing to equation 1.46, which givesTµν = ∂µψ∂νψ+∂µφ∂νφ− 12gµν[gαβ(∂αψ∂βψ+∂αφ∂βφ)+m2 (1−2εφ)ψ2](3.5)Then the nonzero components of the background stress energy tensor are given byT¯00 =12[˙¯ψ2+ ˙¯φ 2+m2(1−2εφ¯) ψ¯2]T¯i j =12a2[˙¯ψ2+ ˙¯φ 2−m2 (1−2εφ¯) ψ¯2]δi j (3.6)and the nonzero components of the stress energy tensor perturbation to first order48are given byδT00 = ˙¯ψ ˙δψ+ ˙¯φδ˙φ +m2(1−2εφ¯) ψ¯ (δψ+ ψ¯Φ)− εm2ψ¯2δφδT0i = ˙¯ψ∂iδψ+ ˙¯φ∂iδφ = δTi0δTi j = a2[˙¯ψ ˙δψ+ ˙¯φδ˙φ +m2(1−2εφ¯) ψ¯ (ψ¯Ψ−δψ)+ εm2ψ¯2δφ −(˙¯ψ2+ ˙¯φ 2)(Φ+Ψ)]δi j (3.7)Since there are no anisotropic stress terms, we can set Ψ=Φ and write everythingin terms of the Newtonian gravitational potential Φ. Using equation 1.32 we cannow write down the Einstein equations. The background equations are given by3H2 =4piM2Pl[˙¯ψ2+ ˙¯φ 2+m2(1−2εφ¯) ψ¯2]H˙ =− 4piM2Pl[˙¯ψ2+ ˙¯φ 2] (3.8)and the first order perturbation equations are given bya−2∇2Φ−3HΦ˙= 4piM2Pl[˙¯ψ ˙δψ+ ˙¯φδ˙φ +m2(1−2εφ¯) ψ¯ (δψ+ ψ¯Φ)− εm2ψ¯2δφ]H∂iΦ+∂iΦ˙=4piM2Pl[˙¯ψ∂iδψ+ ˙¯φ∂iδφ]Φ¨+4HΦ˙+(3H2+2H˙)Φ=4piM2Pl[˙¯ψ ˙δψ+ ˙¯φδ˙φ +m2(1−2εφ¯) ψ¯ (ψ¯Φ−δψ)+ εm2ψ¯2δφ −2(˙¯ψ2+ ˙¯φ 2)Φ](3.9)Since some of the equations are redundant, we will come up with a minimalset of equations to solve numerically. The others are good consistency checks. Theset of background equations we will solve is given by¨¯ψ+3H ˙¯ψ+m2(1−2εφ¯) ψ¯ = 0¨¯φ +3H ˙¯φ − εm2ψ¯2 = 03H2 =4piM2Pl[˙¯ψ2+ ˙¯φ 2+m2(1−2εφ¯) ψ¯2] (3.10)49and the set of first order perturbation equations is given by¨δψ−a−2∇2δψ+3H ˙δψ+m2 (1−2εφ¯)(δψ+2ψ¯Φ)−2εm2ψ¯δφ −4 ˙¯ψΦ˙= 0δ¨φ −a−2∇2δφ +3Hδ˙φ −2εm2ψ¯ (δψ+ ψ¯Φ)−4 ˙¯φΦ˙= 0Φ¨−a−2∇2Φ+7HΦ˙=− 8piM2Pl[m2(1−2εφ¯) ψ¯ (δψ+ φ¯Φ)− εm2ψ¯2δφ](3.11)Since H can be written in terms of a according to equation 1.4, we have six un-knowns, ψ¯ , φ¯ , a, δψ , δφ , and Φ, and six equations.In order to solve these equations numerically, we consider the perturbationfields in terms of their Fourier modes and consider the equations in Fourier space.1δψk (t)≡∫e−i(k·x)δψ (t,x)d3xδφk (t)≡∫e−i(k·x)δφ (t,x)d3xΦk (t)≡∫e−i(k·x)Φ(t,x)d3x(3.12)where the wavenumbers k are the same to linear order. Therefore, ∇2θ = −k2θ .In Fourier space, all of the equations are now only equations of time and easier tosolve.In cosmology, it is often more useful to determine how the equations evolvewith respect to the scale factor a. This will also result in one less backgroundequation to solve. The first and second order time derivatives can be rewritten interms of the derivatives of the scale factor.ddt= a˙dda= aHddad2dt2= a˙2d2da2+ a¨dda= a2H2d2da2+aH2[1− 4piM2Pla2((dψ¯da)2+(dφ¯da)2)] dda(3.13)where we have substituted H˙ according to the background Einstein equations 3.8.We now have an explicit form for the Hubble parameter from the Friedmann1We adopt the notation δψ ≡ δψk, δφ ≡ δφk, Φ≡Φk for the remainder of this section.50equation in terms of the scale factor.2H2 =4pi3M2Plm2(1−2εφ¯)ψ¯2[1− 4pi3M2Pla2(ψ¯ ′2+ φ¯ ′2)]−1(3.14)The equations of motion for the background equations 3.10 written in terms of thescale factor area2H2ψ¯ ′′+[4aH2− 4piM2Pla3H2(ψ¯ ′2+ φ¯ ′2)]ψ¯ ′+m2(1−2εφ¯)ψ¯ = 0a2H2φ¯ ′′+[4aH2− 4piM2Pla3H2(ψ¯ ′2+ φ¯ ′2)]φ¯ ′− εm2ψ¯2 = 0(3.15)where the Hubble parameter H can be replaced according to equation 3.14.Similarly, the equations of motion for the first order perturbation equations 3.11can be rewritten in terms of the scale factor.a2H2δψ ′′+k2a2δψ+[4aH2− 4piM2Pla3H2(ψ¯ ′2+ φ¯ ′2)]δψ ′−4a2H2ψ¯ ′Φ′+m2(1−2εφ¯)(δψ+2ψ¯Φ)−2εm2ψ¯δφ = 0a2H2δφ ′′+k2a2δφ +[4aH2− 4piM2Pla3H2(ψ¯ ′2+ φ¯ ′2)]δφ ′−4a2H2φ¯ ′Φ′−2εm2ψ¯(δψ+ ψ¯Φ) = 0a2H2Φ′′+k2a2Φ+[4aH2− 4piM2Pla3H2(ψ¯ ′2+ φ¯ ′2)]Φ′ =− 8piM2Pl[m2(1−2εφ¯)ψ¯(δψ+ ψ¯Φ)− εm2ψ¯2δφ] (3.16)In order to better understand the upcoming plots, we can also rewrite theseequations in terms of dimensionless variables. a and Φ are already dimensionless,2In this section, the prime notation indicates derivatives with respect to a, i.e. ′ ≡ dda .51and the remaining dimensionful quantities are rewritten as followsk˜ ≡ kH0H˜ ≡ HH0ψ˜ ≡ ψMPlφ˜ ≡ φMPlm˜≡ mMPlε˜ ≡MPlε(3.17)Using the new form of the Hubble parameter in equation 3.14, we can calculate H0in terms of the initial conditions ψ¯0 ≡ ψ¯(0) and φ¯0 ≡ φ¯(0).H20 =4pi3M2Plm2(1−2εφ¯0)ψ¯20[1− 4pi3M2Pla2(ψ¯ ′20 + φ¯′20)]−1(3.18)Now we are free to choose ψ¯ ′0 = 0 = φ¯ ′0. Thus, in terms of the dimensionlessvariables, H0 becomesH20 =4pi3M2Plm˜2(1−2ε˜ ˜¯φ0)˜¯ψ20 (3.19)Therefore, in terms of the dimensionless variables, the Hubble parameter isH˜2 =(1−2ε˜ ˜¯φ)˜¯ψ2(1−2ε˜ ˜¯φ0)˜¯ψ20[1− 4pi3a2(˜¯ψ ′2+ ˜¯φ ′2)]−1(3.20)where we have effectively reparametrized the dimensionless mass asM2PlH20m˜ =34pi1(1−2ε˜ ˜¯φ0)˜¯ψ20(3.21)However, keep in mind that the parameter ε also depends on the mass of the ψfield.52Combining everything, we obtain a new set of equations to solve. The set ofbackground equations can be solved independently and are given by3a2ψ¯ ′′+4aψ¯ ′−4pia3 (ψ¯ ′2+ φ¯ ′2) ψ¯ ′+ 34pi[1− 4pi3a2(ψ¯ ′2+ φ¯ ′2)] 1ψ¯= 0a2φ¯ ′′+4aφ¯ ′−4pia3 (ψ¯ ′2+ φ¯ ′2) φ¯ ′− 34pi[1− 4pi3a2(ψ¯ ′2+ φ¯ ′2)] ε(1−2εφ¯) = 0(3.22)Notice that ψ¯ only depends on ε implicitly through φ¯ . Then, using the solutions toequations 3.22, we can solve the set of first order perturbation equations given bya2δψ ′′+k2a2H2δψ+4aδψ ′−4pia3 (ψ¯ ′2+ φ¯ ′2)δψ ′−4a2ψ¯ ′Φ′+34pi[1− 4pi3a2(ψ¯ ′2+ φ¯ ′2)][δψ+2ψ¯Φψ¯2− 2εδφ(1−2εφ¯) ψ¯]= 0a2δφ ′′+k2a2H2δφ +4aδφ ′−4pia3 (ψ¯ ′2+ φ¯ ′2)δφ ′−4a2φ¯ ′Φ′− 32pi[1− 4pi3a2(ψ¯ ′2+ φ¯ ′2)] ε (δψ+ ψ¯Φ)(1−2εφ¯) ψ¯ = 0a2Φ′′+k2a2H2Φ+8aΦ′−4pia3 (ψ¯ ′2+ φ¯ ′2)Φ′= 6[1− 4pi3a2(ψ¯ ′2+ φ¯ ′2)][δψ+ ψ¯Φψ¯− εδφ(1−2εφ¯)](3.23)where H2 can be substituted via equation 3.203.2 AnalysisIn this section, we solve the equations of motion given in equations 3.22 and 3.23numerically in Mathematica using NDSolve. For the initial conditions, we chooseψ¯0 = 1 = φ¯0.Before considering the first order equations, let us explore the behaviour ofthe background equations when the Yukawa coupling strength is small and largecompared to the mass of the scalar field ψ . The solutions to equations 3.22 are3For easier readability, we suppress the tilde notation for the remainder of this section.53ε = 0.1ε = 100������������������ψ[� ��]��� ��� ��� ��� ��� ���������������������ϕ[� ��]Figure 3.1: Background solutions for a massive scalar field ψ coupled to amassless scalar Yukawa field φ for small coupling constant ε = 0.1 andlarge coupling constant ε = 100. The blue curves show the solutionswhen the coupling is small when compared to the mass of ψ , while theorange curves show the solutions when the coupling is large when com-pared to the mass of ψ . The top plot is the background solution forthe massive scalar field ψ . The oscillations are more strongly dampedfor stronger coupling strength. The bottom plot is the massless scalarYukawa field φ . φ is increasing for small coupling strength and decreas-ing for large coupling strength. See figure 3.2 for the plot of φ¯ ′.54shown in figure 3.1. For the massive scalar field ψ , the strength of the Yukawacoupling affects the strength of the damping. The stronger the coupling strength,the stronger the damping. For the massless scalar Yukawa field φ , a large or smallcoupling dramatically changes the behaviour of the background solution. For asmall coupling, the background field is continuously increasing, while for a largecoupling, the background field is continuously decreasing. Looking at the deriva-tive of this field in figure 3.2, we can see that the derivative is always positive for asmall coupling, and always negative for a large coupling. It asymptotes to 0 in bothcases, implying the background field should reach a constant. We did attempt tosolve the equations when ε ∼ 1, however we were unable to succeed as the systemwas too stiff. In figure 3.3, we have the results for H2, obtained by plugging in theε = 0.1ε = 100��-� ����� ����� ����� �-���-���-���-�����ϕ′ [���]Figure 3.2: Background solution for the derivative of the massless scalarYukawa field, φ ′. For small coupling strength, the derivative is posi-tive and asymptotes to zero. Therefore, the Yukawa field should con-tinuously increase until it reaches a maximum. Similarly, for large cou-pling strength, the derivative is negative and asymptotes to zero, and theYukawa field should continuously decrease until it reaches a minimum.See figure 3.1 for the plot of φ .solutions for the background fields. For a large coupling, H2 < 0. Therefore, tak-ing the large coupling doesn’t make sense. This is not problematic because we aremore interested in the smaller coupling. Recall from section 2.2 that we assumed55a small ε so as to get an analogy to gravity to first order in ε .ε = 0.1ε = 100��-� ����� ����� ����� �-�-�-�-����Figure 3.3: Background solutions for the Hubble parameter. For large cou-pling, H2 < 0. Therefore, the corresponding solutions are meaningless,and we can focus on a small coupling strength.The background plots are reprinted in figure 3.4 using a logarithmic scale forconvenience. We have also included a gray dashed line at a = ε . Notice that ψbehaves like an overdamped field when a < ε and an underdamped field whena > ε . The Hubble parameter also behaves as it would for an overdamped fieldwhen a < ε and an underdamped field when a > ε . We will see that it also tracksthe background density, as expected from the Friedmann equation 1.43.We now consider the first order plots. For all first order plots, we plot just oneFourier mode where k = 1 in order to analyze behaviour. Additional plots withmore modes are provided in appendix B. The first order plots are shown in figure3.5. All three scalar perturbations are oscillating rapidly. The perturbation to themassive scalar field δψ and the perturbation to the scalar Yukawa field δφ havevery similar shapes, with the amplitude of the oscillations growing with the scalefactor. However, δψ is two orders of magnitude larger than δφ . The amplitudeof the oscillations for the Newtonian gravitational potential Φ decay very rapidlyat the beginning. They start to grow again at about the same time the scalar fieldperturbations start to grow, but level off around a = ε and exhibit some bouncy56������������������ψ[� ��]��������������������ϕ[� ��]��-� ����� ����� ����� ����������������������������[��]Figure 3.4: Background solutions for ψ , φ , and H, where ψ is a massivescalar field coupled to a massless scalar Yukawa field φ , and H is theHubble parameter. a = ε is shown by the grey, dashed line. The topplot, ψ¯ , is overdamped when a < ε and underdamped when a > ε , asexpected from equation 3.22. The middle plot shows φ¯ . The bottomplot, H, tracks the density, as shown in figure 3.6. This is also expectedgiven the Friedmann equation 1.43.57-��× ��-��-��× ��-�����× ��-����× ��-��δψ[���]-��× ��-��-��× ��-�����× ��-����× ��-��δϕ[���]��-� ����� ����� ����� �-���× ��-��-��× ��-��-��× ��-�����× ��-����× ��-�����× ��-��ΦFigure 3.5: First order perturbation solutions δψ , δφ , and Φ, where ψ is amassive scalar field coupled to a massless scalar Yukawa field φ , andΦ is the Newtonian gravitational potential. a = ε is shown by the grey,dashed line. The top plot, δψ , and the middle plot, δφ , are oscillatingrapidly and the amplitude of the oscillations is growing. The bottomplot,Φ, is also rapidly oscillating. The amplitude decays, before startingto grow at about the same time as that for the other scalar perturbations.However, it levels off around a= ε and exhibits some bouncy behaviour.58behaviour.��-�����������������ρ[ρ �]��-� ����� ����� ����� �-���-���-���-���-�������[ρ �]Figure 3.6: Background solutions for the normalized total energy density andpressure. a= ε is shown by the grey, dashed line. In the top energy den-sity plot, ρ¯ behaves as for an overdamped field that is slow to decreasedue to the Hubble friction. For a< ε , it behaves as for an underdampedfield, with the oscillations showing up as wiggles. In the bottom pres-sure plot, P starts out negative, before oscillating close to 0 at a> ε .We now consider the density and pressure. The dimensionful background den-sity and pressure with respect to time, t, can be calculated from the background59��-� ����� ����� ����� �-���-���������������� ���� ���� �-���-���-��������� AverageFigure 3.7: Background solution for the equation of state. a = ε is shownby the grey, dashed line. The top plot is the background solution forthe equation of stat, starting out at w¯∼−1 for a< ε , before oscillatingabout 0 when a> ε . The bottom plot is the moving average for the back-ground equation of state, showing that the equation of state averages to0 for a> ε .60stress energy tensor components given in equation 3.6.ρ¯ =12[˙¯ψ2+ ˙¯φ 2+m2(1−2εφ¯) ψ¯2]P¯ =12[˙¯ψ2+ ˙¯φ 2−m2 (1−2εφ¯) ψ¯2] (3.24)The first order perturbation equations can be calculated from the first order stressenergy equations 3.7.δρ = ˙¯ψ ˙δψ+ ˙¯φδ˙φ −(˙¯ψ2+ ˙¯φ 2)Φ+m2[(1−2εφ¯) ψ¯δψ− εψ¯2δφ]δP = ˙¯ψ ˙δψ+ ˙¯φδ˙φ −(˙¯ψ2+ ˙¯φ 2)Φ−m2 [(1−2εφ¯) ψ¯δψ− εψ¯2δφ] (3.25)We can change variables from time to scale factor using equation 3.13 as before.The background density and pressure from equation 3.24 are rewritten asρ¯ =12[a2H2(ψ¯ ′2+ φ¯ ′2)+m2(1−2εφ¯) ψ¯2]P¯ =12[a2H2(ψ¯ ′2+ φ¯ ′2)−m2 (1−2εφ¯) ψ¯2] (3.26)and the first order density and pressure perturbations from equation 3.25 are rewrit-ten asδρ = a2H2[ψ¯ ′δψ ′+ φ¯ ′δφ ′− (ψ¯ ′2+ φ¯ ′2)Φ]+m2 [(1−2εφ¯) ψ¯δψ− εψ¯2δφ]δP = a2H2[ψ¯ ′δψ ′+ φ¯ ′δφ ′− (ψ¯ ′2+ φ¯ ′2)Φ]−m2 [(1−2εφ¯) ψ¯δψ− εψ¯2δφ](3.27)Using equations 3.26, we find the background equation of state to bew¯ =P¯ρ¯=a2H2(ψ¯ ′2+ φ¯ ′2)+m2(1−2εφ¯) ψ¯2a2H2(ψ¯ ′2+ φ¯ ′2)−m2 (1−2εφ¯) ψ¯2 (3.28)and the perturbation equation to beδw =δPρ¯− w¯δ (3.29)where δ ≡ δρρ¯ is the density contrast.61The background solutions to the density and pressure are shown in figure 3.6,plotted in terms of the initial value for the background density from equation 3.26given byρ¯0 =12m2(1−2εφ¯0)ψ¯20 =−P0 (3.30)The background equation of state and moving average are show in in figure 3.7.These background plots look very much like the plot for an axion-like particle[82]. For a < ε , the energy density behaves as for an overdamped field slow todecrease due to the Hubble friction. For a < ε , it behaves as for an underdampedfield, with the oscillations showing up as wiggles. From this shape, it is apparentthat the energy density is dominated by the massive scalar field ψ . The pressurestarts negative. This is evident in the background equation of state, as shown infigure 3.7, where w¯ ∼−1 for a < ε , which is reflective of something that behaveslike dark energy. However, as a > ε , the pressure goes close to 0, as reflective ofmatter. This is also evident in the equation of state plot, where w¯ starts oscillatingrapidly between −1 and 1 about 0. The moving average in figure 3.7 shows thatthis should average out to 0.Figure 3.8 shows the density perturbation with respect to the initial backgrounddensity, the pressure perturbation with respect to the initial background pressure,and the equation of state perturbation. Both the density and pressure perturbationsare rapidly oscillating with decreasing amplitude. This is to be expected since theuniverse is expanding. The equation of state perturbation is also rapidly oscillating.The oscillation amplitude starts decreasing, but it increases with the scale factorafter a> ε .Finally, we consider the density contrast δ , as shown in figure 3.9. From theplot, δ oscillates rapidly with the amplitude decreasing initially before increasing.Recall from section 2.3, that the density contrast grows like the scale factor, δ ∼ a,at linear order in an Einstein-de Sitter universe. In order to determine if the densitycontrast exhibits this behaviour here, we plot(δa)2in figure 3.9, where we havesquared δa so we can plot it logarithmically. If δ ∼ a, we expect the magnitude ofthe oscillations in this plot to be constant. From the plot, we observe that after aninitial time, the density contrast does appear to be growing like the scale factor fora period before and after a = ε . In the vicinity of a = ε , the oscillation amplitude62-��× ��-�-��× ��-����× ��-���× ��-�δρ[ρ �]-��× ��-�-��× ��-����× ��-���× ��-�δ�[ρ�]��-� ����� ����� ����� �-��× ��-�-��× ��-����× ��-���× ��-�δFigure 3.8: First order perturbation solutions for the density, pressure, andequation of state. a = ε is shown by the grey, dashed line. The top plotis the solution for the density perturbation, δρ , and the middle plot isthe solution for the pressure perturbation, δP, with respect to the initialbackground density ρ¯0. Both of these plots are oscillating rapidly withshrinking amplitude. The bottom plot is the solution to the perturbedequation of state, δw. The oscillation amplitude starts decreasing, but itincreases with the scale factor after a> ε .63-��× ��-�-��× ��-����× ��-���× ��-�δ��-� ����� ����� ����� ���-����-����-����-����-����-�δ ��Figure 3.9: a = ε is shown by the grey, dashed line. The top plot is the den-sity contrast δ . It is oscillating rapidly and growing. We expect theamplitude to grow like a . Therefore, the bottom plot is the square ofthe density contrast over the scale factor a. After an initial time,(δa)2is flat for a period before and after a = ε . In the vicinity of a = ε , theoscillation amplitude appears to grow.64appears to grow. From figure 3.7, we know that we are in a matter dominateduniverse when a > ε . Further investigation into the behaviour around a = ε isrequired.65Chapter 4Toward a slow forceNow that we have explored the equations of motion for the gravity-like Yukawainteraction, we are finally ready to consider a slow force. In section 4.1, we givean overview of the concepts and motivation related to a slow force. In section 4.2,we look into deriving an equation of motion mathematically for the case of a singleslow massless scalar field.4.1 Concepts and motivationWe define a slow force to be a force that propagates at a speed slower than thespeed of light. As described in section 1.1.3, there is an entropic gravity theorywherein dark matter is a memory effect due to the emergence of gravity. This isthe effect we hope to emulate with a slow, gravity-like force.Consider two massive point particles in Newtonian gravity alone. Their po-sitions are well-determined by Keplerian orbits, as shown in the top diagram offigure 4.1, where the red particle is influenced by the force of gravity from thecurrent position of the blue particle. However, if we introduce a slow, gravity-likeforce, the red particle will not only react to the force of gravity from the currentposition of the blue particle, but also to the gravity-like effect of the slow forcebased on a past position of the blue particle, as shown in the bottom diagram offigure 4.1.In general relativity, we need to take a more careful approach when thinking66Test particleCurrent position Force of gravityTest particleForce of gravitySlow forcePast positionCurrent positionFigure 4.1: The top diagram shows two point particles in Newtonian grav-ity. The bottom diagram shows the effect of adding a slow, gravity-likeforce. The red particle reacts to both the force of gravity from the cur-rent position of the blue particle, as well as the gravity-like force from apast position of the blue particle.about a slow, gravity-like force due to the retarded effect of gravity. To illustratea slow force in general relativity, consider the light cone shown in figure 4.2. Ina light cone, the horizontal axis represent space, and the vertical axis representstime. The origin is the present space and time. The black lines at 45◦ representthe regular light cone. Since gravity propagates at the speed of light, we can usethe black light cone to illustrate gravity. A particle in the present is reacting to thegravitational effects from everything within the past light cone, where the dashedblack line represent space at a particular time. Now consider adding a slow, gravity-like force. Since the slow force propagates at a speed that is less than the speedof light, it is represented by the narrower slow force cone. Therefore, a particle inthe present is also reacting to the gravity-like effect of everything within the pastslow force cone. In particular, the blue dashed line represents the same space asthe black dashed line, but at an earlier time. Therefore, our particle is reacting tothe universe at different times in the past.Lastly, we consider the effects of a slow, gravity-like force on the baryon acous-tic oscillations (BAO), as illustrated in figure 4.3. As discussed in section 1.1.1, the67SpaceTimeLight coneSlow forceconeFigure 4.2: Light cone diagram in Minkowski space. The black lines at 45◦represent the regular light cone, while the blue lines represent the nar-rower slow force cone. A particle at the present reacts to the gravi-tational effects of everything within the past light cone as well as thegravity-like effect of everything within the slow force cone. The dashedline represents space at different times. Therefore, the particle is react-ing to the universe at different times in the past.BAO observations are intimately connected to dark matter. In the standard cosmol-ogy regime, matter and photons form a hot plasma in the early universe. Considera Gaussian overdensity in the plasma with adiabatic initial conditions. Matter ispushed outward by pressure forces and inward by gravitational forces. These op-posing forces of gravity and pressure create oscillations analogous to sound in air.Baryonic matter is pushed outward by the pressure from photons, but since darkmatter does not interact with photons, it disperses more slowly. This can be rep-resented as the blue dashed line in figure 4.3. Now, if we were to consider a slow,gravity-like force instead of dark matter, the baryonic matter would experience thegravity-like effect of the past position of baryonic matter. This can also be repre-sented by the blue dashed line in 4.3. Effectively, you get a contribution that looks68Baryonic matterPressureGravityFigure 4.3: Baryon acoustic oscillations. Opposing pressure and gravita-tional forces create oscillations in the baryonic matter analogous tosound waves in air. With the addition of a slow, gravity-like force, thebaryonic matter experiences the gravity-like effect from the past posi-tion of the baryonic matter represented by the blue dashed line. This isanalogous to the effect of dark matter on the BAO.like a bunch of matter at the centre, which is exactly what dark matter looks like.These thought experiments further motivate us to study a slow, gravity-likeforce. In particular, the BAO considerations show that a slow, gravity-like forcemight be plausible as a replacement for dark matter. However, it also shows thatthis theory is highly constrained by the past position of baryonic matter.4.2 A slow, massless scalar fieldWe require the Yukawa field to be massless in order to ensure the resulting forcehas infinite range. In this section, we illustrate mathematically how to get a slow,massless scalar field. Consider the Lagrangian density for a free, massless scalarfieldL =−12∇µφ∇µφ (4.1)69The equation of motion is given byφ = 0 (4.2)In Minkowski space, this is just the regular wave equation given by− φ¨ +∇2φ = 0 (4.3)Notice that the coefficients in front of the time and spatial derivatives are both 1,corresponding to the speed of light. In order to achieve a ‘slow’ massless scalarfield, we need the coefficient in front of the spatial derivative to be less than 1.A natural thing to try, is to add an extra field coupled to our scalar field φ . Wearbitrarily add derivatives of a new scalar field, χ , coupled to derivative of φ . Sincewe want our new Lagrangian density to change the ∇2φ term in our equation ofmotion, there are two possible terms to add:(∇µχ∇µφ)2 and ∇µχ∇µφ∇νχ∇νφ .The resulting Lagrangian density isL =−12∇µφ∇µφ +λ2M4Pl(∇µχ∇µφ)2+κ2M2Pl∇µχ∇µχ∇νφ∇νφ (4.4)By working through the equations of motion, we find it convenient to add a thirdscalar field ζ coupled in such a way to χ as to force χ = 0. Choosing κ = −λ2leads to nice cancellations. Therefore, the Lagrangian for a slow massless scalarfield is given byL =−12∇µφ∇µφ +λ2M4Pl(∇µχ∇µφ)2− λ4M4Pl∇µχ∇µχ∇νφ∇νφ +12∇µχ∇µζ(4.5)The equations of motion for these fields are given byχ = 0−(1+λ2M4Pl∇µχ∇µχ)φ + λM4Pl∇µχ∇νχ∇µ∇νφ = 0ζ + 2λM2Pl(∇µχ∇µφφ +∇µφ∇νφ∇µ∇νχ)= 0(4.6)70To illustrate how this gets the desired ‘slow’ effect, consider a χ field that is depen-dent only on time. Then, in Minkowski space, the equations of motions are givenbyχ¨ = 0−φ¨ + (1−b)(1+b)∇2φ = 0 where b =λ2M4Plχ˙2−ζ¨ +∇2ζ − 2λM4Plχ˙ φ˙(−φ¨ +∇2φ)= 0(4.7)where b is constant since χ¨ = 0. Thus, b is a positive number for positive λ . Forb< 1, the coefficient in front of the spatial derivative of φ is less than 1. Therefore,if we were to add the Lagrangian density from equation 4.5 to the Lagrangiandensity for the Yukawa interaction, we should get a slow, gravity-like force.To have a deeper understand of the Lagrangian density, we derive the Hamil-tonian density. To do this, we consider the general form of the Lagrangian densitygiven in equation 4.5 without assuming Minkowski space or χ(t). The Hamiltoniandensity is given byH = piφ∇0φ +piχ∇0χ+piζ∇0ζ −L (4.8)where the pi are the conjugate fields. In general, the conjugate fields are given bypiθ =∂H∂∇0θ(4.9)Therefore, assuming a diagonal metric, the conjugate fields given the Lagrangiandensity in equation 4.5 arepiφ =−∇0φ + λM4Pl[(∇µχ∇µφ)∇0χ− 12(∇µχ∇µχ)∇0φ]piχ =λM4Pl[(∇µχ∇µφ)∇0φ − 12(∇µφ∇µφ)∇0χ]+12∇0ζpiζ =12∇0χ(4.10)Plugging in the conjugate fields from equations 4.10 and the Lagrangian densityfrom equation 4.5 into the Hamiltonian density in equation 4.8, the Hamiltonian71density becomesH =12(∇µφ∇µφ −2∇0φ∇0φ)− λ2M4Pl∇νχ∇νφ(∇µχ∇µφ −4∇0χ∇0φ)+λ8M4Pl∇νχ∇νχ(∇µφ∇µφ −4∇0φ∇0φ)+λ8M4Pl∇νφ∇νφ(∇µχ∇µχ−4∇0χ∇0χ)− 12(∇µχ∇µζ −∇0χ∇0ζ)(4.11)In Minkowski space, assuming χ depends only on time, the Hamiltonian density isH =12(1+3b)φ˙ 2+12(1+b)∇φ ·∇φ where b≡ λ2M4Plχ˙2 (4.12)where b is the same as in equation 4.7. Thus, as long as λ is positive, the Hamilto-nian density is positive and therefore bounded from below.72Chapter 5DiscussionIn this thesis, we explored the Yukawa interaction as a precursor to developing aproper framework in which to study an emergent gravity theory as an alternative todark matter.5.1 The gravity-like Yukawa interactionIn chapter 2, we establish that the Yukawa interaction is gravity-like. From a parti-cle physics perspective, we show that the Yukawa interaction leads to a gravity-likepotential in the nonrelativistic limit. By assuming the Yukawa field is massless, wecan extend the range of the force to infinity, like gravity. Again, from a generalrelativity perspective, we show the Yukawa interaction is analogous to the New-tonian gravitational potential. Finally, we consider the case of spherical collapse,and show that the addition of an extra, gravity-like force is analogous to increasingthe strength of gravity. The only difference in the evolution of the density contrastis the time it takes for an overdensity to collapse. Thus, the addition of an extra,gravity-like force has the same result as scaling time. An overdensity in a universewith an extra, gravity-like force looks exactly the same as a larger overdensity in auniverse with just regular gravity.In chapter 3, we take a deeper look at the equations of motion for a masslessYukawa field coupled to a massive scalar field. We find that the strength of theYukawa coupling can have drastic effects on the evolution of the fields, and show73that solutions have distinct behaviour before and after the scale factor is of thesame magnitude as the coupling. The background plots look similar to the plotsfor axion-like particles, including an equation of state that looks like dark energywhen the scale factor is less than the Yukawa coupling, and matter when it is bigger.5.2 Toward a slow forceIn section 4.1, we gave an overview of the concept of a slow, gravity-like force. Ina universe with an additional slow, gravity-like force, a massive particle effectivelyfeels gravity-like effects from the universe at different times in the past. For thecase of the baryon acoustic oscillations, the baryonic matter is influenced by thegravity-like effects from the past position of itself. However, in order for this tobe the case, the speed at which the slow force propagates must be less than thespeed of the acoustic oscillations. Therefore, the BAO could be used to set anupper limit on the speed of the slow force. In some sense, we have just relegateddark matter to be the scalar mediator of the Yukawa coupling. However, since thistheory is highly constrained by the past position of baryonic matter, it limits thefree parameters that can be used, increasing its predictive power.We derived the Lagrangian density for a slow, massless scalar field in Minkowskispace in section 4.2. Thus, it is possible to have both a slow and massless scalarfield. This is important in the case of the Yukawa interaction because, in orderfor the resulting force to have infinite range, the Yukawa field must be massless.Therefore, using the addition of a slow, gravity-like Yukawa interaction as a possi-ble framework in which to study emergent gravity has shown promising potentialand requires further consideration.5.3 Future workNow that we have shown that this line of inquiry is indeed worth pursuing, thereare a number of future directions in which we could take this project. Dark matterhas a rich phenomenology, and any alternative theory must adequately explain thephenomena attributed to dark matter. The overall objective of future work is toelaborate on the additional slow force idea and develop it to the level of testability,using data to either constrain or rule it out. This will hopefully shed light onto the74plausibility of this particular class of theories alternative to dark matter. The firststep is to do the calculations for the spherical collapse model with the addition of aslow, gravity-like force. There are a number of mathematical details that still needto be figured out, such as the proper Lagrangian density for a slow force in the FRWuniverse. The next step would be to explain the CMB features currently explainedvia conventional dark matter. Should this theory demonstrate success with theCMB data, it would prove an interesting pursuit and require further attention bylooking into other phenomena currently explained via the standard dark mattermodel. 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External legs:incoming scalar: = 1outgoing scalar: = 1incoming fermion: = us(p)outgoing fermion: = u¯2(p)incoming antifermion: = v¯s(k)outgoing antifermion: = vs(k)4. Impose momentum conservation at each vertex.5. Integrate over each undetermined loop.6. Figure out the overall sign for each diagram.For the solid fermion lines, the arrow indicates particle number flow. The mo-mentum for fermions follows the arrow, whereas the momentum for antifermions isopposite and indicated by an additional arrow. For the dashed scalar lines, momen-tum is always ingoing for initial state scalars and outgoing for final state scalars.The momentum direction is irrelevant for internal scalars and can be chosen forconvenience.By going backwards along number flow paths, the scattering amplitude is found88by multiplying the appropriate factors for each propagator, vertex, and externalleg. Momentum conservation is imposed by adding a Dirac delta function for themomenta associated with each vertex. Any remaining undetermined momenta dueto loops are integrated over. Finally, the sign must be determined for each diagram.For our diagram in figure 2.1, there is only one diagram without any loops.89Appendix BAdditional plotsThe following supplemental plots show different Fourier modes for the various firstorder perturbation solutions.90k=1k=0.5k=0.1-��× ��-��-��× ��-�����× ��-����× ��-��δψ[���]-��× ��-��-��× ��-�����× ��-����× ��-��δϕ[���]��-� ����� ����� ����� �-���× ��-��-��× ��-��-��× ��-�����× ��-����× ��-�����× ��-��ΦFigure B.1: First order perturbation solutions δψ , δφ , and Φ, where ψ is amassive scalar field coupled to a massless scalar Yukawa field φ , andΦ is the Newtonian gravitational potential. a = ε is shown by the grey,dashed line. The top plot, δψ , and the middle plot, δφ , are oscillatingrapidly and the amplitude of the oscillations is growing. The bottomplot, Φ, is also rapidly oscillating. The amplitude decays, before start-ing to grow at about the same time as that for the other scalar pertur-bations. However, it levels off around a = ε and exhibits some bouncybehaviour. The oscillation amplitude and period are larger for smallerk. The smaller k are also noisier.91k=1k=0.5k=0.1-��× ��-�-��× ��-����× ��-���× ��-�δρ[ρ �]-��× ��-�-��× ��-����× ��-���× ��-�δ�[ρ�]��-� ����� ����� ����� �-��× ��-�-��× ��-����× ��-���× ��-�δFigure B.2: First order perturbation solutions for the density, pressure, andequation of state. a = ε is shown by the grey, dashed line. The top plotis the solution for the density perturbation, δρ , and the middle plot isthe solution for the pressure perturbation, δP, with respect to the initialbackground density ρ¯0. Both of these plots are oscillating rapidly withshrinking amplitude, and appear similar for the various Fourier modes.The bottom plot is the solution to the perturbed equation of state, δw.The oscillation amplitude starts decreasing, but it increases with thescale factor after a > ε . The late time oscillation amplitudes appearsmaller for smaller k.92k=1k=0.5k=0.1-��× ��-�-��× ��-����× ��-���× ��-�δ��-� ����� ����� ����� ���-����-����-����-����-����-�δ ��Figure B.3: a = ε is shown by the grey, dashed line. The top plot is the den-sity contrast δ . It is oscillating rapidly and growing. We expect theamplitude to grow like a . Smaller k corresponds to smaller oscillationamplitude. Therefore, the bottom plot is the square of the density con-trast over the scale factor a. After an initial time, which appears to belonger for smaller k,(δa)2flattens out.93


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