Models and Probes of the Early andDark UniverseInflation and 21-cm Radiation in CosmologybyMichael SitwellB.Sc., Queen’s University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2014c Michael Sitwell 2014AbstractThe prevailing model of modern cosmology stipulates the existence of exoticsubstances such as dark matter and dark energy and events such as inflation.However, their underlying nature is not currently known. In this thesis,we explore new models and measurement techniques that may be used tocharacterize their cosmological e↵ects and shed light on their inner workings.A model of inflation driven by a substance that may be described macro-scopically as a cosmological elastic solid is studied. The proper techniquesfor the quantization of perturbations within the elastic solid are presented.We find that a su ciently rigid elastic solid with slowly varying sound speedscan produce an inflationary period. Interestingly, we find models where theelastic solid has an equation of state significantly greater than 1 that nev-ertheless produces nearly scale-invariant scalar and tensor spectra.The remaining chapters of this thesis concern the use of 21-cm radiationas a probe of the physics of dark matter and dark energy.The e↵ects of warm dark matter on the highly-redshifted 21-cm signalis examined. If dark matter is warm instead of cold, its non-negligible ve-locities may inhibit the formation of low-mass halos, thereby delaying star-formation, which may delay the emission and absorption signals expectedin the mean 21-cm signal. The e↵ects of warm dark matter on both themean 21-cm signal, as well as on its power spectrum, are described and de-generacies between the e↵ects of warm dark matter and other astrophysicalparameters are quantified.One of the primary goals of 21-cm radiation intensity mapping is to mea-sure baryon acoustic oscillations over a wide range of redshifts to constrainthe properties of dark energy from the expansion history of the late-timeUniverse. We forecast the constraining power of the CHIME radio telescopeon the matter power spectrum and dark energy parameters. Lastly, wedevise new calibration algorithms for the gains of an interferometric radiotelescope such as CHIME.iiPrefaceThis thesis contains reprinted material originally found in the following pa-pers:1. Chapter 4: M. Sitwell, & K. Sigurdson, “Quantization of Perturbationsin an Inflating Elastic Solid,” Phys. Rev. D, vol. 89, 123509, 2014.2. Chapter 6: M. Sitwell, A. Mesinger, Y. Ma, & K. Sigurdson, “TheImprint of Warm Dark Matter on the Cosmological 21-cm Signal,”MNRAS, vol. 438, p. 2664, 2014.3. Chapter 7: J. R. Shaw, K. Sigurdson, M. Sitwell, A. Stebbins, &U. Pen, “Coaxing Cosmic 21cm Fluctuations from the Polarized Skyusing m-mode Analysis,” arXiv:1401.2095, 2014.All calculations found in Paper 1 were done by M. Sitwell, which wereperformed under the supervision of K. Sigurdson. The preparation of thispaper was done entirely by MS, with advice from KS.The work in Paper 2 made heavy use of the 21CMFAST code, which waswritten and provided by A. Mesinger. Some modifications to the code weremade by MS. All analysis done on the output of this code was performedby MS. The forecasts used in this paper were provided by AM. This paperwas written entirely by MS with the consultation of AM. Further feedbackfor this paper was given by Y. Ma and KS. Section 6.2, which does notappear in the published paper, was added to provide additional backgroundinformation.Some of the forecasting methods described in Paper 3 can be found inChaper 7 of this thesis. The majority of the research described in this paperwas conducted by J. R. Shaw and KS. The forecasts of distance measure-ments from the power spectrum, as well as the forecasts for the dark energyparameters, were performed by MS. The preparation of this paper was donealmost entirely by JRS, in collaboration with KS. Appendix E of this pa-per, which was written by MS, describes the forecasting methods covered inSections 7.6 and 7.7 of this thesis. In addition, forecasts appearing in theiiiPrefaceCFI grant proposal for CHIME [10] (specifically those shown in Figs. 3-6)were made by MS using the methods described in Chaper 7.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Physical Cosmology . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Origin of Perturbations and Inflation . . . . . . . . . . . 31.3 Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . . 31.4 21-cm Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Measuring the E↵ects of Dark Energy . . . . . . . . . . . . . 61.6 Cosmological History in Brief . . . . . . . . . . . . . . . . . . 72 The Universe: Background, Linear Perturbations, Nonlin-ear Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 The Unperturbed Universe . . . . . . . . . . . . . . . . . . . 92.1.1 The FLRW Spacetime . . . . . . . . . . . . . . . . . . 92.1.2 Distances and Times in Cosmology . . . . . . . . . . 112.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Linear Perturbation Theory . . . . . . . . . . . . . . . . . . 142.3.1 Notation and Conventions . . . . . . . . . . . . . . . 142.3.2 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . 162.3.3 Linear Einstein Equations . . . . . . . . . . . . . . . 182.3.4 Adiabatic and Entropy Modes . . . . . . . . . . . . . 182.4 Linear Perturbations in Our Universe . . . . . . . . . . . . . 19vTable of Contents2.5 Collapse into Nonlinear Structures . . . . . . . . . . . . . . . 212.5.1 Spherical Collapse . . . . . . . . . . . . . . . . . . . . 222.5.2 The Press-Schecther model . . . . . . . . . . . . . . . 232.5.3 The Excursion Set Formalism . . . . . . . . . . . . . 242.5.4 Improvements to the Mass Function . . . . . . . . . . 262.5.5 Halo Virialization . . . . . . . . . . . . . . . . . . . . 263 A Brief Tour Through Cosmological Inflation . . . . . . . . 283.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Problems with the Standard Cosmological Model . . . . . . . 283.3 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 End of Inflation and Reheating . . . . . . . . . . . . . . . . . 323.6 Generation of Perturbations . . . . . . . . . . . . . . . . . . 333.6.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . 333.6.2 Beyond the Horizon . . . . . . . . . . . . . . . . . . . 364 Inflation with an Elastic Solid . . . . . . . . . . . . . . . . . . 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Elastic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.1 Quantization of Scalar Modes . . . . . . . . . . . . . 474.4.2 Quantization of Tensor Modes . . . . . . . . . . . . . 514.5 Superhorizon Evolution . . . . . . . . . . . . . . . . . . . . . 524.6 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6.1 Inflation with Constant Sound Speeds and Equationof State . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.2 The ‘Horizon Problem’ Revisited . . . . . . . . . . . . 594.6.3 Non-Constant Sound Speeds and Equation of State . 604.6.4 Slowly Varying Sound Speeds and Equation of State . 624.7 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 654.8 End of Inflation and Reheating . . . . . . . . . . . . . . . . . 674.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 The Physics of 21-cm Radiation . . . . . . . . . . . . . . . . 755.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Properties of 21-cm Radiation . . . . . . . . . . . . . . . . . 755.2.1 The Brightness Temperature . . . . . . . . . . . . . . 755.2.2 The Spin Temperature . . . . . . . . . . . . . . . . . 77viTable of Contents5.3 History of the 21-cm Signal . . . . . . . . . . . . . . . . . . . 805.4 Radio Interferometry and Detection of 21-cm Signal . . . . . 816 The Imprint of Warm Dark Matter on the Cosmological 21-cm Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Thermal Relic . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3 E↵ect of WDM on structure formation . . . . . . . . . . . . 876.3.1 Free-streaming . . . . . . . . . . . . . . . . . . . . . . 876.3.2 Residual velocities . . . . . . . . . . . . . . . . . . . . 886.3.3 Halo Abundances . . . . . . . . . . . . . . . . . . . . 886.4 Cosmic 21-cm signal . . . . . . . . . . . . . . . . . . . . . . . 906.5 Simulation of 21-cm signal . . . . . . . . . . . . . . . . . . . 916.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 936.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 Forecasting 21-cm BAO Experiments . . . . . . . . . . . . . 1057.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Constraining Dark Energy Parameters . . . . . . . . . . . . . 1067.3 Measuring the Acoustic Scale . . . . . . . . . . . . . . . . . . 1077.3.1 The Sound Horizon . . . . . . . . . . . . . . . . . . . 1077.3.2 Baryon Acoustic Oscillations . . . . . . . . . . . . . . 1087.4 Fisher Matrix Formalism . . . . . . . . . . . . . . . . . . . . 1097.5 Measuring the 21-cm Power Spectrum . . . . . . . . . . . . . 1107.6 The ‘Wiggles Only’ Method . . . . . . . . . . . . . . . . . . 1187.6.1 Modelling the BAO Power Spectrum . . . . . . . . . 1197.6.2 Distance Uncertainties . . . . . . . . . . . . . . . . . 1217.7 Dark Energy Constraints . . . . . . . . . . . . . . . . . . . . 1237.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288 Redundant Baseline Calibration . . . . . . . . . . . . . . . . 1298.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.2 Calibration Requirements for CHIME . . . . . . . . . . . . . 1308.3 Gain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.4 Amplitude Calibration . . . . . . . . . . . . . . . . . . . . . 1328.4.1 The Logarithm Method . . . . . . . . . . . . . . . . . 1328.4.2 Identical Beams . . . . . . . . . . . . . . . . . . . . . 1338.4.3 Nonidentical Beams . . . . . . . . . . . . . . . . . . . 1358.4.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . 1378.4.5 Amplitude Calibration Results . . . . . . . . . . . . . 139viiTable of Contents8.5 Phase Calibration . . . . . . . . . . . . . . . . . . . . . . . . 1458.5.1 The Eigenvector Method . . . . . . . . . . . . . . . . 1458.5.2 Phase Degeneracies . . . . . . . . . . . . . . . . . . . 1488.5.3 Phase Calibration Results . . . . . . . . . . . . . . . 1488.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156AppendixA Supplemental Details for Elastic Solid Model of Inflation 170A.1 Equations of Motion for Scalar and Tensor Perturbations . . 170A.2 Multicomponent System with Energy-Momentum Transfer . 171A.3 Scalar Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 173viiiList of Tables2.1 Popular gauge choices for the scalar perturbations. . . . . . . 174.1 Examples of parameters for slowly varying sound speeds andequation of state . . . . . . . . . . . . . . . . . . . . . . . . . 667.1 Telescope parameters for CHIME used for BAO forecasting . 114ixList of Figures4.1 Evolution of h modes in the = B = 0 gauge . . . . . . . . . 594.2 Power spectrum of ⇣ during the decay of elastic solid to radi-ation for a superhorizon mode . . . . . . . . . . . . . . . . . . 705.1 Hyperfine levels relevant for the WF mechanism . . . . . . . 786.1 Mean collapse fraction for CDM and WDM . . . . . . . . . . 906.2 Mean spin temperatures T¯S for CDM and WDM . . . . . . . 956.3 Mean 21-cm brightness temperature T¯b . . . . . . . . . . . . 976.4 Critical points in the mean 21-cm signal . . . . . . . . . . . . 986.5 Parameter space curves ze(f⇤|CDM) = ze(mX|WDM) for var-ious critical points . . . . . . . . . . . . . . . . . . . . . . . . 996.6 Evolution of f⇤(z) in CDM required to match the mean bright-ness temperature T¯b in WDM . . . . . . . . . . . . . . . . . 1006.7 Evolution of the power spectrum of Tb for WDM . . . . . . 1026.8 Power spectrum of the brightness temperature Tb . . . . . . 1037.1 Contributions to the 21-cm power spectrum noise per mode . 1167.2 Survey volume per unit redshift over the CHIME band . . . . 1167.3 Forecasted power spectrum uncertainties . . . . . . . . . . . . 1187.4 Forecast uncertainties for DA and H . . . . . . . . . . . . . . 1227.5 Measurement uncertainties on DV . . . . . . . . . . . . . . . 1237.6 Derivatives of lnH and lnDA with respect to w0 and wa . . . 1247.7 Forecasted constraints in the w0 wa plane . . . . . . . . . . 1257.8 Relative improvement of figure of merit FOM with CHIMEover fiducial value FOM0 . . . . . . . . . . . . . . . . . . . . 1267.9 Constraints on wDE . . . . . . . . . . . . . . . . . . . . . . . 1288.1 Beam basis functions . . . . . . . . . . . . . . . . . . . . . . . 1388.2 Fiducial simulated values of gains and the beam perturbationparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.3 Calibrated gain amplitude bias and standard deviation . . . . 141xList of Figures8.4 Gain amplitude calibration as a function of the maximumbeam perturbation . . . . . . . . . . . . . . . . . . . . . . . . 1438.5 Amplitude calibration as a function of error on prior . . . . . 1448.6 Amplitude calibration as a function of beam uncertainty . . . 1468.7 Phase calibrations after each iteration . . . . . . . . . . . . . 1498.8 Phase calibration as a function of maximum beam perturbation1508.9 Phase calibrations as a function of the error on the phase prior151xiList of AbbreviationsBAO Baryon Acoustic OscillationsBBN Big Bang NucleosynthesisBOSS Baryon Oscillation Spectroscopic SurveyCDM Cold Dark MatterCHIME Canadian Hydrogen Intensity Mapping ExperimentCL Galaxy ClusterCMB Cosmic Microwave BackgroundCOBE Cosmic Background ExplorerDM Dark MatterEOR Epoch of ReionizationEPS Extended Press-SchectherFLRW Friedmann-Lemaˆıtre-Robertson-WalkerFWHM Full Width at Half MaximumFOM Figure of MeritIGM Intergalactic MediumPS Press-SchectherSDSS Sloan Digital Sky SurveySKA Square Kilometre ArraySN SupernovaSVD Singular Value DecompositionUV UltravioletWDM Warm Dark MatterWF Wouthuysen-FieldWL Weak LensingWMAP Wilkinson Microwave Anisotropy ProbexiiChapter 1Introduction1.1 Physical CosmologyPhysical cosmology, the study of the largest scales of the Universe and itsfundamental constituents, in its modern form began to take shape in theearly 20th century, with such revelations as Albert Einstein’s formulation ofthe theory of general relativity and Edwin Hubble’s observational evidencefor an expanding universe. Since then, cosmology has grown into a precisionscience, due to the remarkable measurements of the cosmic microwave back-ground (CMB), the study of galaxies and galaxy clusters, and observationsof supernovae, among many other experiments, whose successes have movedcosmology from a largely qualitative field to a quantitative one.From these theoretical and observational leaps, the standard model ofBig Bang cosmology emerged. Chiefly, it describes an expanding universethat on large scales is homogenous and isotropic. The rate of expansionis determined by basic properties of the contents of the Universe. Theseconstituents are divided into the broad categories of matter (or baryons1),radiation, dark matter and dark energy. Radiation refers to relativisticspecies, which in the standard cosmological model include photons and neu-trinos. Dark matter (DM) is a non-luminous substance that while actinggravitationally in a similar manner to normal visible matter, does not inter-act (or at least interacts very weakly) with the photon. Although the ideaof such matter dates back to the early 1930s, its exact internal structure iscurrently not known. The paradigm of dark energy emerged in the 1990sto explain the observed acceleration of the expansion of the Universe. Aswith the similarly named dark matter, its fundamental nature is currentlyunknown.2Augmenting the general descriptions given above, dark matter is oftenassumed to be cold (CDM), denoting that the dark matter should be non-1This is a misnomer, as in cosmology baryons commonly refers all types of visiblenon-relativistic matter, including leptons.2See Refs. [1, 2, 3] for a general introduction to modern cosmology, Ref. [4] for anintroduction to particle dark matter and Refs. [5, 6] for an introduction to dark energy,11.1. Physical Cosmologyrelativistic, both currently and in the early Universe. One simple and oftenemployed model of dark energy is that of a ‘cosmological constant’ ⇤, whichwhen combined with the above assumptions for dark matter form the stan-dard ⇤CDM model of the Universe. While this model has been extremelysuccessful in describing our Universe, it highlights large gaps in our currentunderstanding, most importantly the true nature of dark matter and darkenergy. The search to uncover the inner workings of dark matter and darkenergy drives a significant amount of research in cosmology, as well as inphysics as a whole.Since the discovery of the expansion of our Universe, researchers haveattempted to look further and further back in time, when densities andtemperatures were much higher then they are currently. One early successof modern cosmology was that of Big Bang nucleosynthesis (BBN), whichdescribes the production and abundances of the lightest nuclei, occurring atkeV to MeV scales [7].An essential component of modern cosmology is perturbation theory,which in the cosmological context describes small perturbations to the oth-erwise homogeneous and isotropic Universe [8, 9]. While these perturbationsremain small in the early Universe, they become highly non-linear at latertimes and provide the early structure that eventually grows into dark matterhaloes and galaxies. In this sense, these small disturbances in homogeneityand isotropy lay the seeds for the structure that we see all around us in ourUniverse. Through the use of general relativity, we can track the evolutionof these perturbations, and we can thereby extrapolate their properties toearlier and earlier times (as long as we are in a regime where general rela-tivity holds). These perturbations can be seen in the early Universe fromthe imprint left in the CMB, released approximately 380 000 years after theBig Bang, at a time known as recombination. This imprint is manifestedas small anisotropies in the otherwise isotropic signal. Anisotropies in theCMB have been measured to great precision through satellite experimentssuch as COBE3, WMAP4, and Planck5, ground-based telescopes such asACT6 and SPT7, and balloon-borne experiments such as BOOMERanG8.3http://lambda.gsfc.nasa.gov/product/cobe/4http://map.gsfc.nasa.gov5http://www.rssd.esa.int/index.php?project=planck6http://www.princeton.edu/act/7http://pole.uchicago.edu8http://www.astro.caltech.edu/ lgg/boomerang/boomerang front.htm21.2. The Origin of Perturbations and Inflation1.2 The Origin of Perturbations and InflationFrom experiments that measure the CMB or large-scale structure, we caninfer some basic properties of these perturbations when extrapolated backinto the very early Universe. For example, these very early perturbationsare nearly scale invariant, with a very slight preference for larger scales.A natural question to ask is: what is the origin of these perturbations?Since currently there are only a few observables that describe the pre-BBNUniverse, this is a di cult question to answer.Currently, the most popular answer is that these perturbations origi-nated as quantum mechanical fluctuations that were stretched to cosmicscales during a brief period of extremely rapid expansion in the very earlyUniverse, known as inflation. The popularity of inflation is due in part toits ability to solve a handful of problems that emerged in classical moderncosmology. One such problem deals with why the CMB temperature is veryisotropic over the sky, even though many of the regions where the CMB wasreleased were not in causal contact with one another according to classicalmodern cosmology. Another problem is why the Universe appears to havevery little, if any, spatial curvature.The persistence of inflationary theories in modern cosmology is largelydue to the fact that they both provide solutions to these problems as wellas producing the initial set of perturbations in the Universe. On the otherhand, due to the small number of observables currently available that canplace constraints on models of inflation, if inflation did occur its exact modeldescription is not yet known. However, as inflationary models in general pro-duce propagating gravitational disturbances, known as gravitational waves,measuring these relic gravitational waves may provide crucial evidence ofinflation.1.3 Acoustic OscillationsSometime shortly after the events of the very early Universe, the primordialcosmological perturbations found themselves in a radiation-dominated uni-verse. During this extremely hot and dense era, the baryons were stronglycoupled to the photons, forming a so-called baryon-radiation fluid, where toa good approximation the baryons and photons moved as one. In overdenseareas, the strong radiation pressure of the photons pushes outwards, causingthe photons to disperse from the area. Since the baryons are strongly cou-pled to the photons at this time, they are dragged along with the photons.31.3. Acoustic OscillationsParticles rush out of overdense areas in a wave that propagates until thesound speed of these acoustic waves drops to zero, a time labeled as thedrag era, occurring after recombination.The presence of these acoustic waves are embedded in both the distribu-tion of radiation, in the form of the anisotropies in the CMB, and matter,by means of the distribution of galaxies and dark matter haloes. The im-print of these waves on the baryons is known as baryon acoustic oscillations(BAO). As the acoustic waves were only able to propagate from the begin-ning of the radiation-dominated era until the drag era, the material flowingout of overdense regions propagated a finite distance, leaving extra mat-ter a certain distance away from the location of the original overdensity.This creates a preferential scale in the distribution of matter, occurring atroughly ⇠ 150Mpc.9 As the primordial perturbation are distributed over awide range of scales and directions, these waves overlap with one another,making it di cult to see individual signs of these waves. However, as therewill be a preferential separation distance of matter at the BAO scale, theBAO signal can be observed statistically, for example as a bump at the BAOscale in the two-point correlation function (ontop of the correlation functionthat disregards the e↵ect of the baryons) or equivalently as an oscillation inthe matter power spectrum.As the BAO imprints a preferential (comoving) scale into the distributionof matter, it can be used as a statistical standard ruler for measuring theexpansion of the Universe, thereby giving BAO great importance in moderncosmology. The BAO scale corresponds to the sound horizon at the drag era.The first BAO detections were made in 2005 from galaxy surveys consistingof 10,000’s of galaxies made by the Sloan Digital Sky Survey (SDSS) [11]and also by the Two-degree-Field Galaxy Redshift Survey [12]. SubsequentBAO detections using galaxy surveys have been made by the Six-degree-Field Galaxy Survey [13], WiggleZ [14], and BOSS [15] and has recentlybeen detected at high redshifts in the Lyman-↵ forest by BOSS [16, 17] andin the cross-correlation of Lyman-↵ with quasars [18].By measuring the BAO at various redshifts, we can use the BAO asa standard ruler to track the expansion of the Universe. By using thisprocedure with redshifts up to z ⇠ 3, a detailed expansion history of thedark energy dominated Universe may be measured. This process can beused to place constraints on models of dark energy, as various models predictslightly di↵erent expansion histories.9Unless stated otherwise, all quoted distances are comoving distances chosen to coincidewith present-day physical distances.41.4. 21-cm Radiation1.4 21-cm RadiationA promising new tool for the exploration of cosmology is 21-cm radiation,the radiation emitted by the hyperfine spin-flip of neutral hydrogen (HI),which is emitted with a wavelength of about 21-cm in the rest frame ofthe hydrogen atom. The low excitation energy for this hyperfine transitiongives it some desirable properties: it is sensitive to low temperatures andhas a relatively low optical depth so can be used to probe far into the high-redshift Universe. As we can infer radial distances through the redshiftof the observed radiation, in addition to the angular distribution of theemission, 21-cm radiation can be used to construct 3D ‘tomographic’ mapsof the HI distribution in our Universe, potentially containing a plethora ofnew and valuable information. Furthermore, 21-cm radiation may provideour only glimpse into the ‘dark ages’, a time in which very few structureshave formed. However, removing bright foregrounds that may be as high asthree to four orders of magnitude larger than the 21-cm signal presents aformidable challenge.The nature of the 21-cm signal changes throughout cosmic history. The21-cm signal is measured against the CMB and may appear in either emis-sion or absorption [19]. The 21-cm signal is likely to appear in absorptionduring the dark ages and slightly afterwards, and in emission shortly beforereionization and afterwards. During reionization, regions of ionized hydro-gen (HII) form, creating non-emitting ‘bubbles’ in the intergalactic medium(IGM). These bubbles can grow to the Mpc scale and eventually overlap atthe end of reionization. After reionization, when HII regions in the IGM havecoalesced, the origin of 21-cm emission is relegated to only dense collapsedhalos that contain su cient amounts of neutral hydrogen.The post-reionization 21-cm signal may be used to map the underlyingdistribution of matter, from which the BAO signal may be extracted. Sincethe BAO scale is on the order of 150Mpc, high-resolution maps of the matterdistribution, such as those made from galaxy surveys, are not necessary tomeasure the BAO. Lower resolution maps made from the 21-cm signal maybe used to measure the BAO at many redshifts, a process potentially easierthan conducting vast galaxy surveys. The caveat to this is that for 21-cmmeasurements of the BAO scale to be successful, the very bright foregroundscomprised mainly of synchrotron radiation must be removed to a su cientlevel.21-cm radiation may also shed light onto the details of exactly how andwhen reionization took place. Much is currently unknown about how longthe epoch of reionization (EOR) lasted and exactly how ionized HII regions51.5. Measuring the E↵ects of Dark Energyformed. Observations of the Gunn-Peterson trough [20] in the spectrumof distant quasars, caused by the scattering of photons that pass throughcontinuous HI regions while redshifting through the Lyman-↵ line, places theend of reionization around z ⇠ 6 [21, 22]. However, due to its high cross-section, the Universe is opaque to Lyman-↵ emission at higher redshifts.On the other hand, the lower optical depth of 21-cm radiation makes it wellsuited as a probe of the EOR at even higher redshift eras, including when the21-cm signal may have been in absorption both before and after significantstructure formation has taken place.1.5 Measuring the E↵ects of Dark EnergyWhat many consider to be the first substantial evidence of a late-time accel-erating Universe came in 1998 with the observations of type Ia supernovae(SN). These SN can act as standard candles, as their peak brightness con-sistently hits at approximately the same point, and so can be used to tracethe expansion of the Universe. The SN observations of Riess et al. [23] andPerlmutter el al. [24] both showed evidence for a late-time acceleration, aresult that has since been supported by further observations.There exist many di↵erent models of dark energy (or models that pro-duce a similar real or perceived acceleration), such as a cosmological con-stant, scalar field models (quintessence), and modified gravity, to name afew. For a model to be consistent with observations, the late-time equationof state of the dark energy wDE must be close to 1. However, di↵erentmodels predict slight departures from wDE = 1.10 With current measure-ments consistent with wDE = 1, a driving force in dark energy research isto obtain more constraining measurements of wDE.As previously mentioned, 21-cm experiments designed to measure theBAO are well suited for this purpose, many of which have recently goneinto operation or are to be built in the near future. Many of these ex-periments are interferometric telescopes that are similar in design to EORexperiments (e.g. LOFAR11, MWA12, PAPER13). The Canadian HydrogenIntensity Mapping Experiment14 (CHIME) is one such radio telescope, be-ing built in Penticton, British Columbia. CHIME is a drift scan telescope10A cosmological constant predicts wDE = 1 exactly.11http://www.lofar.org12http://www.mwatelescope.org13http://eor.berkeley.edu14http://chime.phas.ubc.ca61.6. Cosmological History in Briefwith no moving parts that will consist of five 100m ⇥ 20m cylindrical re-flectors with 256 dual-polarization feeds running down the focal line of eachcylinder. A smaller scale pathfinder telescope of two 35m long cylinderswith 128 feeds on each cylinder, constructed in late 2013, will prototypethe full CHIME telescope. The cylinders are aligned with the North-Southdirection to provide at any one time a wide field of view of the sky in theNS direction and narrow one in the East-West direction. CHIME will becapable of mapping nearly half of the sky in the course of a day. CHIMEwill observe the sky in the frequency range of 400 800MHz (wavelengthsof ⇠ 37 75 cm) in order to measure the BAO at redshifts in the rangez ⇡ 0.8 2.5, a time period when the e↵ects of dark energy first becomesprominent. The measurement of the BAO scale in this redshift range willcomplement measurements already made at lower redshifts.1.6 Cosmological History in BriefAs cosmology studies the evolution of the Universe from its birth to thepresent day, there have been many important events that have occurred inthe history of the Universe. To conclude this introduction, I give a briefoverview of cosmic history. In the following list, time is demarcated byeither a temperature T or redshift z, where the former is more convenientat early times and the latter at later times.• T & fewMeV, z & 109 The very early Universe: Many significantevents might have occurred during this time, for example grand unifi-cation or baryogenesis. Inflation, if it occurred, would belong to thistime period (see Chapters 3 and 4).• T ⇠ 0.1 10MeV, z ⇠ 4 ⇥ 108 4 ⇥ 1010 Big-Bang Nucleosynthesis :Light nuclei are formed.• T ⇠ 1MeV, z ⇠ 4 ⇥ 109 Neutrino Decoupling : Neutrinos decouplefrom other species and free-stream thereafter.• T ⇠ 0.5MeV, z ⇠ 2 ⇥ 109 Electron-Positron Annihilation: Electronsand positrons annihilate with one another, a relatively small numberof electrons persist past the annihilation.• T ⇠ 1 eV, z ⇠ 4250 Matter-Radiation Equality : The Universe becomesmatter dominated past this point.71.6. Cosmological History in Brief• T ⇠ 0.26 eV, z ⇠ 1100 Recombination: Free electrons and protonscombine to form hydrogen, the CMB is released.• T ⇠ 1.6 2.6meV, z ⇠ 6 10 Reionization: Radiation from early as-trophysical sources ionize hydrogen in the IGM (see Chapters 5 and6).• T ⇠ 0.33meV, z ⇠ 0.4 Matter-Dark Energy Equality : The Universe isdominated by dark energy past this point (see Chapter 7).8Chapter 2The Universe: Background,Linear Perturbations,Nonlinear Structures2.1 The Unperturbed Universe2.1.1 The FLRW SpacetimeOn large scales (& 100Mpc) the Universe is very homogenous and isotropic.While this was established empirically in the late 20th century, this was acommonly used assumption well before this time. As such, much insight canbe gained from perfectly homogenous and isotropic models of the Universe,which can later be extended to allow for small amounts of inhomogeneityand anisotropy.The most general homogenous and isotropic spacetime admitted from theEinstein equations is the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)metric, which can be represented byds2 = g¯µ⌫dxµdx⌫ = dt2 a2(t) dr21 Kr2 + r2d⌦2 , (2.1)where g¯µ⌫ is the background FLRW metric and d⌦2 = d✓2 + sin2 ✓d 2. Thespatial curvature constant K can assume the values 0, 1, 1 for a spatiallyflat, open, and closed spacetime, respectively. The expansion of the space-time is controlled by the scale factor a(t) and its evolution is given by theHubble rate H(t) = a˙(t)/a(t), where an overdot denotes di↵erentiation withrespect to t. Often it will be more convenient to work with the conformaltime ⌘ ⌘ R dt/a(t) instead of the coordinate time t. In these cases, we willalso make use of the conformal Hubble rate H ⌘ a0/a = aH, where prime 0stands for @/@⌘.It is easy to show that the stress-energy tensor for a perfect fluidT↵ = (⇢ + P )u↵u P ↵ (2.2)92.1. The Unperturbed Universeyields a homogenous and isotropic spacetime, where ⇢ is the energy density,P is the pressure scalar, and u↵ is the 4-velocity. Solving the Einsteinequations with this metric and stress-energy tensor yields the Friedmannequations H2 + Ka2 = 8⇡G3 ⇢, (2.3a)a¨a = 4⇡3 G⇢(1 + 3w). (2.3b)In the above equation, we have introduced the equation of state parameterw ⌘ P/⇢. It is important to note that, from (2.3b), if the equation of stateis larger than 1/3, the expansion of the spacetime will be decelerating,while a value of w smaller than 1/3 leads to an accelerating expansion.Eliminating ⇢ in the above equations yields the useful di↵erential equationH0 = 1 + 3w2H2. (2.4)In the case where w is constant, this di↵erential equation can easily be solvedasH =2(1 + 3w)(⌘ ⌘c) , (2.5)where ⌘c is a constant of integration.For a noninteracting perfect fluid, the covariant conservation of thestress-energy tensor yields the energy (density) conservation equation⇢˙ + 3H⇢(1 + w) = 0. (2.6)For constant w, this implies that ⇢ / a 3(1+w).The energy density is typically decomposed as a sum of components withdi↵erent equations of state. In the ⇤CDM model, all substances have oneof three equations of state: nonrelativistic matter that has w = 0, radiationwhich has the relativistic equation of state w = 1/3, and the cosmologicalconstant ⇤ with w = 1. The present-day energy density of each substancei is often expressed as a fraction ⌦i of the critical density ⇢cr, where thecritical density is the density that yields a flat spacetime with a Hubble ratematching the present-day value. The Friedmann equation (2.3a) can thenbe neatly expressed asH2 = H20 (⌦ma 3 + ⌦ra 4 + ⌦⇤ + ⌦ka 2), (2.7)where the subscripts m and r denote matter and radiation, respectively, thesubscript 0 denotes the present-day value, and ⌦k = K/(a0H0)2. The102.1. The Unperturbed Universescale factor is normalized as a0 = 1. Current observed values for theseparameters are very roughly ⌦⇤ ⇡ 0.73,⌦m ⇡ 0.27,⌦k ⇡ 0,⌦r ⇡ 8 ⇥ 10 5,and h ⇡ 0.7, where h is the present-day Hubble parameter H0 expressed inunits of 100 km s 1Mpc 1 [25]. In certain situations it is convenient to usethe parameters !m = ⌦mh2 and !b = ⌦bh2.2.1.2 Distances and Times in CosmologyWhen dealing with an expanding universe, the notion of a distance can beambiguous and requires a more precise definition then when thought of inthe Newtonian sense. Furthermore, when using units where c = 1, distancesand times share the same units and in many contexts can be thought ofinterchangeably.One of the most basic of such measures is the redshift z = ( ob em)/ em of light emitted with wavelength em and observed with wave-length ob. The redshift is commonly used in place of the scale factor a,related by a0/a = 1 + z.An important distinction is the di↵erence between physical (proper) dis-tances Lph and comoving distances L, where comoving coordinates denotecoordinates that are defined such that they remain constant with respectto the motion of particular objects. In other words, for objects that arecomoving with the Hubble flow, the separation in their comoving distanceremains constant. Physical and comoving distances are related by Lph = aL.Up to an integration constant, the conformal time ⌘ measures the comovingdistance that a massless particle travels in a certain duration. More pre-cisely, a particle traveling at the speed of light travels a comoving distance = ⌘(t2) ⌘(t1) between the times t1 and t2. Unless otherwise stated, thelater time t2 is assumed to be the present day and then is a function of asingle time parameter. Thus, if the zero point of ⌘ is chosen appropriately,⌘(t) measures the particle horizon of a massless particle at time t.By expressing the conformal time as the integral⌘ = Z da˜˜a H 1(a˜) (2.8)we can see that massless particles can travel roughly a comoving distanceH 1 in the time that the scale factor increases by a factor of e. In thislight, the Hubble radius H 1 (or H 1 in comoving coordinates) is commonlyreferred to as the ‘horizon’.The angular diameter distance DA is used to relate the physical size Lphof a very distant object to the angle ✓ that it subtends by DA = Lph/✓. In112.2. Thermodynamicsgeneral, the angular diameter distance is given byDA(z) = 11 + z 1H0p⌦k sinh⇣p⌦kH0 (z)⌘ , (2.9)but greatly simplifies in a flat universe to DA(z) = a (z).2.2 ThermodynamicsMacroscopic thermodynamic quantities are ubiquitous in cosmology, as it iscommonplace to find substances in thermodynamic equilibrium.The number density n, energy density ⇢, and pressure P can be expressedas [1, 2] n = g Z d3p(2⇡)3 f(E), (2.10a)⇢ = g Z d3p(2⇡)3 f(E)E, (2.10b)P = g Z d3p(2⇡)3 f(E) p23E , (2.10c)where g is the number of degrees of freedom and f is the distribution function(Bose-Einstein or Fermi-Dirac).15 The entropy density s can be found viathe thermodynamic identity ass = ⇢ + P µnT . (2.11)In most situations, substances are either non-relativistic or ultra-relativistic.An ultra-relativistic substance with temperature T and particle mass m sat-isfies T m, in which case for a boson the above relations simplify tonB = ⇣(3)⇡2 gT 3, ⇢B = ⇡230gT 4, PB = ⇡290gT 4, (2.12)where ⇣ is the zeta function. For a fermion, we get nF = (3/4)nB, ⇢F =(7/8)⇢B, PF = (7/8)PB. From these expressions we can see that indeedw = 1/3 for an ultra-relativistic species, for both bosons and fermions. Theentropy density for a boson is thensB = 2⇡245gT 3 (2.13)15We set c = ~ = kb = 1 in this section.122.2. Thermodynamicsand sF = (7/8)sB for a fermion.In the nonrelativistic limit T ⌧ m, for both bosons and fermions wehave n = g✓mT2⇡ ◆3/2 e(µ T )/T , ⇢ = mn, P = nT. (2.14)For nonrelativistic matter we see that w ⇡ 0, as asserted in the previoussection.An important application of the above thermodynamic relations is toa plasma composed of relativistic particles. Suppose the plasma containsNb (Nf ) relativistic bosons (fermions), with each species b (f) having gb(gf ) degrees of freedom and is at equilibrium temperature Tb (Tf ). UsingEqs. (2.12) and (2.13), we can write the total energy, entropy, and num-ber densities of the relativistic plasma in terms of temperature dependente↵ective degrees of freedom as⇢r(T ) = ⇡230g(T )T 4, (2.15a)sr(T ) = 2⇡245 gs(T )T 3, (2.15b)nr(T ) = ⇣(3)⇡2 gn(T )T 3, (2.15c)where T is the photon temperature, and g, gs, and gn are the number of e↵ec-tive relativistic degrees of freedom contributing towards the energy, entropy,and number densities, respectively, given byg(T ) = Xb gb✓TbT ◆4 + 78 Xf gf ✓TfT ◆4 , (2.16a)gs(T ) = Xb gb✓TbT ◆3 + 78 Xf gf ✓TfT ◆3 , (2.16b)gn(T ) = Xb gb✓TbT ◆3 + 34 Xf gf ✓TfT ◆3 . (2.16c)Note that g = gs if all species are in thermodynamic equilibrium at temper-ature T = Tb = Tf .These expressions are useful for describing the energy and entropy den-sities in the early Universe, since the early Universe is radiation dominated.132.3. Linear Perturbation TheoryFor example, in the standard model of particle physics, at temperatures wellabove the top mass (T > 173GeV) all particles in the standard model willbe relativistic and we will have g = gs = 106.75. As the temperature drops,the values of the e↵ective degrees of freedom decrease as particles becomenonrelativistic. All relativistic particles in the plasma are in thermodynamicequilibrium with each other (so g = gs) until T ⇠ MeV when neutrinos de-couple. In addition, just after neutrino decoupling, electrons and positronsannihilate, which dumps their entropy into the photons but not into theneutrinos, since by this time they have decoupled, which results in a heatingof the photons relative to the neutrinos. Thus after electron-positron anni-hilation, not all relativistic species share the same temperature and g 6= gs.For T ⌧ MeV, after electron-positron annihilation, we will have g ⇡ 3.36and gs = 43/11 ⇡ 3.91.Throughout much of the history of the Universe, local thermal equilib-rium held and thus the total entropy density remained constant. A usefulconsequence of this is that for these times, we can use the conservation ofentropy to relate the expansion of the Universe to the temerpature bya(T1)a(T2) = ✓gs(T2)gs(T1)◆1/3 T2T1 , (2.17)where T1 and T2 are photon temperatures at two di↵erent times.2.3 Linear Perturbation Theory2.3.1 Notation and ConventionsWe now extend the results from the previous section to allow for smallperturbations that break homogeneity and isotropy. Since these deviationsremain small for a large part of cosmic history on many relevant scales,linear perturbation has become an essential tool in cosmology.16We begin by perturbing the metricds2 = a2(⌘)(⌘↵ h↵ )dx↵dx , (2.18)where ⌘↵ is the metric for Minkowski space and h↵ represents small pertur-bations about the background. It will prove useful to further parameterizethe perturbations h↵ . To do this, we first separate the time-like and spatialparts as ds2 = a2(⌘) ⇥(1 + 2 )d⌘2 2Bidxid⌘ hijdxidxj⇤ . (2.19)16This section largely uses the notation of Ref. [9]. See also Ref. [26].142.3. Linear Perturbation TheoryThis introduces a scalar , a vector Bi, and a tensor hij . The vector and ten-sor perturbations can be further decomposed into scalar, vector and tensorparts according to how each part transforms under spatial transformations.Bi decomposes as Bi = B,i +Si (where ,i = @i), comprised of a scalar Band a divergenceless vector Si. For the tensor hij , we first decompose it ashij = (1 + h/3) ij + 2Eij , where h is the trace of hij and Eij is traceless.Eij is subsequently decompose into scalar, vector, and tensor components,where the full tensor formed from each component is denoted by ESij , EVij ,and ETij , respectively. The tensors formed from the scalar component E andvector component Ei are given byESij = E,ij 13 ijr2E (2.20a)EVij = E(i,j) (2.20b)where the curved brackets in the subscript denotes symmetrization (in otherwords E(i,j) = 12(Ei,j + Ej,i)). In addition to being traceless, the tensor ETijis transverse meaning ETij,i = 0. Lastly, we make two notational changes toconform to popular conventions by denoting 2ETij by hTij and using the scalarperturbation = 16h+ 13r2E in place of h.Since perfect fluids are often used to model substances in cosmology, wenow perturb the stress-energy tensor of a perfect fluid given in Eq. (2.2) as T 00 = ⇢, (2.21a) T i0 = (⇢ + P )vi, (2.21b) T ij = ( P ij + P⇧ij), (2.21c)where ⇢ and P are the energy density and pressure perturbations, respec-tively, vi is the velocity perturbation, and ⇧ij is the anisotropic stress. Thevelocity perturbation vi and anisotropic stress ⇧ij can be decomposed intoscalar, vector, and tensor parts in the same manner as the metric perturba-tions Bi and Eij , and denote the scalar parts by v and ⇧, respectively.We will often transform from position space into Fourier space with wavevectors k.17 When doing so, we use the convention of including an extrafactor of k = |k| in the Fourier variables for the scalar component of vectors(such as B and v) and a factor of k2 for the scalar component of tensors(such as E and ⇧), so that perturbations in Fourier space all have the samedimensions. A real, homogenous, and isotropic Gaussian field f can be17Here we use the Fourier conventions f(x) = (2⇡) 3/2 R d3kfkeik·x.152.3. Linear Perturbation Theorydescribed by a power spectrum Pf (k), which describes the variance of itsFourier components and is given byhfkfk˜i = (2⇡2/k3)Pf (k) (k+ k˜). (2.22)The correlation function ⇠f for f is then⇠f (|x x˜|) = Z dkk Pf (k)sin(k|x x˜|)k|x x˜| . (2.23)2.3.2 Choosing a GaugeImplicit in decomposing our spacetime into background and perturbed space-times are the coordinate systems used on each [26]. If one first defines acoordinate system on the background spacetime, there can be many map-pings of points on the background spacetime to points on the perturbedspacetime. For functions defined on the perturbed spacetime, each choiceof mapping will yield a di↵erent value for the function for the same pointon the background spacetime. In other words, the perturbation variablesdefined in the previous section may change their values for di↵erent coordi-nate systems in the perturbed spacetime. We can relate two such coordinatesystems x↵ and x˜↵ by x˜↵ = x↵ + ⇠↵. Switching coordinate systems in thismanner is known as a gauge transformation. The spatial part of the vectorrelating the two coordinate systems ⇠↵ can be decomposed as ⇠i = ⇣ ,i + ⇠i?,where ⇠? is divergenceless. When changing coordinates, the scalar metricperturbations transform as 18˜ = H⇠0 (⇠0)0, (2.24a) ˜ = +H⇠0, (2.24b)B˜ = B ⇠0 + ⇣ 0, (2.24c)E˜ = E + ⇣, (2.24d)where perturbations in the coordinate system x˜↵ (x↵) are denoted with(without) a tilde. A scalar variable q defined in the perturbed spacetime thatis decomposed into a background component q¯ and a perturbed component q transforms as q˜ = q q¯0⇠0. (2.25)18We remind the reader that a prime represents a partial derivative with respect toconformal time.162.3. Linear Perturbation TheoryRelevant examples of this are the energy density ⇢ and pressure P . A 4-vector wi, such as the 4-velocity, will transform asw˜0 = w0 + w¯0(⇠0)0 (w¯0)0⇠0, w˜i = wi + w¯0(⇠i)0. (2.26)Vector perturbations are generally not considered, as in most situations theydecay very rapidly. Lastly, we note that since the perturbation to the spatialpart of a tensor transforms as C˜ij = Cij 13 ij(C¯kk )0⇠0, the traceless partof the tensor Cij , given by Cij 13 ij Ckk , will be unchanged by a gaugetransformation and thus the tensor perturbations are gauge-invariant.Choosing a particular coordinate system in the perturbed spacetime cor-responds to picking a gauge for the perturbation variables. One can thenmove between gauges by using the 4-vector ⇠↵ that relates the gauges andthe transformations listed above. Choosing a coordinate system for thetime and spatial variables is referred to a slicing and threading, respectively.Often it is convenient to pick a gauge where certain perturbations vanish.The usefulness of a gauge usually depends on the situation. A few popularchoices of gauge are listed in Table 2.1.Gauge ConditionConformal-Newtonian B = E = 0Synchronous = B = 0Comoving v = B = 0O↵-Diagonal = E = 0Table 2.1: Popular gauge choices for the scalar perturbations.Instead of picking a gauge to work in, in some situations it is helpful touse gauge-invariant variables. Although there are multiple ways of definingsuch variables, they are most commonly defined as = +H(B E0) + (B E0)0, (2.27a) = H(B E0), (2.27b) ⇢(gi) = ⇢ + ⇢0(B E0), (2.27c) P (gi) = P + P 0(B E0), (2.27d)v(gi) = v +B E0. (2.27e)In the above equations, ⇢ and P denote their background quantities, a con-vention which we use from here onwards unless stated otherwise.172.3. Linear Perturbation Theory2.3.3 Linear Einstein EquationsPerturbing the Einstein equations is a straightforward but somewhat lengthlyprocedure (see Ref. [9] for more details). Using the stress-energy tensor inEq. (2.21), the scalar gauge-invariant equations arer2 3H( 0 +H ) = 32l2a2 ⇢(gi), (2.28a) 0 +H = v(gi), (2.28b) 00 +H( 0 + 2 0) + (H2 + 2H0) + 13r2( ) = 32l2a2 P (gi), (2.28c) = 3l2a2P⇧, (2.28d)where = H2 H0 = 32 l2a2(⇢+P ). While the energy-momentum conserva-tion equations gained from the covariant conservation of the stress-energytensor are not independent of the Einstein equations listed above, they areoften useful and their gauge-invariant form is given by (gi)0 (1 + w)(r2v(gi) + 3 0) + 3H P (gi)⇢ w (gi)! = 0, (2.29a)v(gi)0+H(1 3w)v(gi)+ w01 + wv(gi) P (gi)⇢ + P 23 w1 + wr2⇧ = 0, (2.29b)where = ⇢/⇢ is the density contrast.The Einstein equations are very simple for the tensor perturbations andyield the sole equation(hT)i00j + 2H(hT)i0j r2(hT)ij = 6l2a2P (⇧T)ij , (2.30)where (⇧T)ij is the tensor part of the anisotropic stress tensor.2.3.4 Adiabatic and Entropy ModesAnother useful decomposition of perturbations is the separation into adia-batic and entropic parts. This divides perturbations into adiabatic mode(s)with ( ⇢ 6= 0, s = 0) and entropy mode(s) with ( ⇢ = 0, s 6= 0). Thepressure of a ‘fluid-like’ substance 19 in general is a function of both theenergy and entropy densities so that P = @P@⇢ s ⇢ + @P@s ⇢ s. (2.31)19By ‘fluid-like’ we are not necessarily referring to a perfect fluid, but to a substancewhose stress-energy tensor can be parameterized by Eq. (2.21).182.4. Linear Perturbations in Our Universe2.4 Linear Perturbations in Our UniverseTwo of the most fruitful pursuits in modern cosmology have been the studyof linear perturbations in the matter and in the radiation permeating ourUniverse, which are manifested in large-scale structure and anisoptropies inthe CMB, respectively. In this section, we briefly describe the evolution ofthese perturbations.Species capable of free-streaming can be described by the use of a set ofmultipole moments. For example, this decomposition can be done with thetemperature field of the photons [26]. The evolution of the multipole mo-ments can be found from Boltzmann equations (see Ref. [1, 26] for details).In the context of a fluid, the density, velocity, and anisotropy perturbationsare associated with the first three moments of such a decomposition. Beforerecombination, the baryons and photons were tightly-coupled by Comptonscattering, which suppresses higher moments of the photon’s temperaturefield and thus the baryons and photons can be well described by a fluid.At early times, the quadrupole of the radiation is small due to the tight-coupling between the photons and baryons. After decoupling, radiation isa subdominant component in the Universe and so the quadrupole remainssmall. As such, we can safely neglect the quadrupole in many cases. Animportant consequence is that in such cases we have ⇡ .We will now examine some of the basics of the evolution of the matterdensity contrast in the Newtonian gauge. At times past recombination, wemust revert back to using the full set of multipole moments to describe thephoton distribution. However, as these times are far into matter domination,the e↵ect of the radiation on the matter distribution is negligible at thispoint. Consequently, the Einstein equations in Section 2.3.3 are su cientfor describing the matter perturbations during these times.At late times, most modes of interest are inside the horizon. The Einsteinequations in Section 2.3.3 imply that for these modes, the matter densitycontrast evolves as [1]d2 kda2 + ✓d lnHda + 3a◆ d kda 3⌦m2a5(H/H0)2 k = 0. (2.32)Note that for these late-time sub-horizon modes, the evolution of k is inde-pendent of k. We can then separate the evolution of k into two regimes: anearly scale-dependent evolution and a late scale-independent evolution. Theexpression for k at late times is most often expressed through its relationto the metric perturbation , which in the current limit from Eq. (2.28a)implies k2 k = (3/2)l2a2⇢m k.192.4. Linear Perturbations in Our UniverseA transfer function T (k) is used to describe the early scale-independentevolution and is defined as T (k) = (k, alate) (kLS, alate) , (2.33)where alate is some late time well into the scale-independent regime. Thetransfer function is normalized so that it equals unity for some large-scalemode kLS. It can be shown from the Einstein equations without too muchdi culty that for large-scale superhorizon modes, k decreases by a factorof 9/10 from its primordial value [9]. Although the transfer function can befound analytically in small and large scale limits, expressions for the transferfunction valid for both small and large scales are typically expressed as afitting formula found numerically. Two of the most popular fitting formulasfor the transfer function are that of Bardeen, Bond, Kaiser, and Szalay [27]and Eisenstein and Hu [28].The growth function G(a) parameterizes the late scale-independent evo-lution of and . It is defined asG(a) = a (a) (alate) , (2.34)for a > alate.20 The growth function can be found by solving Eq. (2.32) andonly retaining the growing mode. With appropriate initial conditions, thegrowth function is found to beG(a) = 52⌦mH(a)H0 Z a0 da˜(a˜H(a˜)/H0) 3. (2.35)The primary descriptive statistic of the matter density field is its two-point correlation function, or as more commonly used, its Fourier transform,the matter power spectrum. The last remaining piece before we write downthe linear matter power spectrum is specifying the primordial power spec-trum for . This is conventionally parameterized as 21P ,I(k) = 50⇡29k3 2H(k/H0)ns 1⌦2m/G(a = 1)2. (2.36)The amplitude of the power spectrum is set by the parameter H and itsscale dependence is specified by the scalar spectral index ns. Most often20The extra factor of a is added to the definition of the growth function so that / G(a).21The form of this parameterization is chosen to simplify the expression for the matterpower spectrum evaluated at the present.202.5. Collapse into Nonlinear Structuresns is taken to be independent of k and from observational constraints isslightly less than unity, while the amplitude is roughly H ⇠ 10 5. Withthis, we now arrive at the expression for the linear matter power spectrumfor a > alate Pm(k, a) = 2⇡2 2H knsHns+30 T 2(k)✓ G(a)G(a = 1)◆2 . (2.37)A related statistic often employed in cosmology is the expectation valueof the variance of the linear overdensity within a sphere of radius R, sym-bolized by 2R = h 2Ri, where R(x) = R d3x˜ (x˜)WR(x x˜) is the linearoverdensity smoothed on the scale R with the top hat window function WR,which in Fourier space is given byWR(k) = 3(sin(kR) kR cos(kR))(kR)3 . (2.38)This variance can be written in terms of the the power spectrum P (k) by 2R = Z d3k(2⇡)3P (k)|WR(k)|2. (2.39)Its value at R = 8Mpch 1, denoted by 8, is a frequently measured param-eter (measured to by about 8 ⇠ 0.8 [25, 29]) and is often used to normalizethe linear power spectrum.2.5 Collapse into Nonlinear StructuresSo far we have examined our Universe approximated as homogeneous andisotropic and then considered linear perturbations about the homogeneousand isotropic background. On large scales, one can go far with this model.On the other hand, on smaller scales the behaviour of the perturbationsis highly nonlinear, as evident from the galaxies, stars, planets, and otherastrophysical structures present in our Universe. In this section we give ashort review of some simple but powerful models for describing the collapseof linear perturbations into nonlinear structures. In particular, we examinecollapse into dark matter halos, which subsequently act as the breedingground for galaxies. In this section, we assume that the dark matter is coldand will examine some of the e↵ects of relaxing this assumption in Chapter 6.212.5. Collapse into Nonlinear Structures2.5.1 Spherical CollapseBefore we are able to predict quantities like the abundances of collapsedstructures, we must be able to track a perturbation from the linear to thenonlinear regime. To accomplish this, we aim to find the value of the over-density predicted in linear theory when the full nonlinear perturbation hascollapsed.To start, we consider an isolated, spherical, and uniform overdensity ofcold, pressureless matter. In this simple model, particles move in sphericalshells without crossing one another until far into its collapse, after which themotion of the particles will be chaotic, eventually relaxing into a virializedstate [5, 30]. We focus our attention on times when the Universe is matterdominated. As we are considering a region smaller than the horizon size,Newtonian dynamics should be reasonably accurate, so that a shell of mattera distance R away from the centre of the overdensity moves according tod2Rdt2 = GMR2 = 43⇡G⇢R, (2.40)where M = (4/3)⇡⇢R3, until shell crossing occurs. Since in a matter-dominated universe we have ⇢ / a 3, the full nonlinear overdensity nlis given by nl = ✓ a(t)R(t)/R0◆3 1, (2.41)where R0 is the initial size of the overdense region. By substituting nl intoEq. (2.40) and solving for the overdensity yields the following parametricsolutions R = GM(1 cos ⌧)C 1, (2.42a)t = GM(⌧ sin ⌧)C 3/2, (2.42b) nl = 9(⌧ sin ⌧)22(1 cos ⌧)3 1, (2.42c) = 35✓34(⌧ sin ⌧)◆2/3 , (2.42d)where ⌧ 2 (0, 2⇡) is a parametric variable, C is an integration constant,and we have used the fact that a / t2/3 in a matter-dominated universe.22In the above equations, is the solution for the overdensity by linearizingall equations in the overdensity, while nl is the solution without any such22Similar expressions exist for the evolution of an underdense region.222.5. Collapse into Nonlinear Structuresapproximations. From Eq. (2.42a), we can see that initially the size of theoverdense region expands with the background until it reaches a turnaroundpoint where the overdense region starts to collapse. This simplified modelof collapse should be useful until late in the collapse, when significant shellcrossing occurs.We can now use the evolution equations for and nl given in Eqs. (2.42)to map the evolution of the linear perturbations to the full nonlinear be-haviour (at least in this simplified case). From Eq. (2.42a), we can see thatthe turn around occurs when ⌧ = ⇡ and collapse is complete when ⌧ = 2⇡.At final collapse, the linear overdensity is = c ⇡ 1.69, where c is re-ferred to as the (linear) critical collapse threshold. Conveniently, we havefound that in a matter dominated universe, the critical collapse threshold forspherical collapse is a constant. For the same model but in a universe withboth matter and dark energy (with constant equation of state), a similaranalysis can be done, but the collapse threshold now evolves with time [31].The critical collapse threshold plays a central role in the Press-Schecthermodel, which will be the focus of the next section.2.5.2 The Press-Schecther modelThe Press-Schecther (PS) model [32] is a simple but powerful tool thatpredicts the abundance of dark matter halos, which has been relatively suc-cessful matching predictions to observations and simulations.23 The PSmodel considers a Gaussian random (linear) density field that is consideredcollapsed into a halo when it reaches a critical collapse threshold.More precisely, we would like to know when a region of size R and massM = (4⇡/3)⇢R3 collapses into a halo. To this end, we smooth the densitycontrast on a scale R to yield the field M , which will have a variance 2M (z),given by Eq. (2.39).24 Since the field is Gaussian, the fraction of collapsedregions (known as the collapse fraction) with mass M or above is simplyfcoll(z) = 2Z 1 c d M 1p2⇡ M (z)exp✓ 2M2 2M (z)◆ = erfc✓ cp2 M (z)◆ .(2.43)Above, a somewhat precarious factor of 2 was added whose inclusion wasoriginally justified to allow for underdense regions with M < 0 to be incor-23This agreement improves significantly with small modifications to the formulation,such as accounting for non-spherical collapse, as considered by Ref. [33].24In general, we will write the smoothing scale in terms of the mass M within a regionof size R instead of R itself.232.5. Collapse into Nonlinear Structuresporated into larger halos. This factor of 2 was later more rigorously justifiedand we will return to the matter in Section 2.5.3.By di↵erentiating the collapse fraction, we find that the mass function,the number density dn of halos with mass M between M and M + dM , isgiven by dndM = ⇢M d ln dM F (⌫), (2.44)where ⌫(z) = c/ M (z) and for the PS model F (⌫) isFPS(⌫) = r 2⇡⌫e ⌫2/2. (2.45)2.5.3 The Excursion Set FormalismThe Press-Schecther model of collapse provides a starting point for a pow-erful analysis tool known as the excursion set or extended Press-Schecther(EPS) formalism [34, 35]. By reframing the standard Press-Schecther model,the EPS formalism adds many useful extensions to the standard PS model,as well as providing new insights into our simple model of collapse.To motivate the excursion set formalism, we first discuss a conceptualdrawback of the standard PS theory. This drawback concerns how to prop-erly form halo statistics to account for the situation when there is a smallerregion below the collapse threshold (when the density field is smoothed ona smaller scale), which is contained in a larger region that is above the col-lapse threshold (when smoothed on a larger scale). One would expect thatthe smaller region would be amalgamated with the matter in the larger re-gion into a collapsed structure [35]. This is known as the ‘cloud-in-cloud’problem.In light of the cloud-in-cloud problem, we slightly alter the objectiveof the standard PS method: We would like to find the largest smoothingscale where the smoothed density field exceeds the critical threshold. Thisis accomplished by starting at a very large smoothing scale, where M ⇡ 0,so that the probability of collapse at this scale is negligible, and decreasethe smoothing scale until we find the first point where the smoothed densityfield exceeds the critical threshold. The largest scale that exceeds the criticalthreshold is marked as a collapsed halo and any smaller scales inside thisregion that surpasses the critical threshold is considered part of the largerhalo.As the smoothing scale decreases, more Fourier modes become relevantfor the collapse and the probability of collapse increases. At this point, we242.5. Collapse into Nonlinear Structuresmay do this process numerically with a particular realization of the densityfield. Alternatively, we may proceed to calculate halo statistics analyticallyusing the probability distributions given in the problem, a description ofwhich follows. We can imagine adding Fourier modes to the density fieldas the smoothing scale decreases, which, since we are considering a Gaus-sian random field where the modes are independent of one another, has thesame statistics as a di↵usion process. This amounts to the density contrasttaking a random walk as the scale decreases, starting from a value of zeroat large scales. The goal is to find the probability of the first ‘up-crossing’through the critical threshold at a particular scale. If we choose the win-dow function used in the smoothing to be a spherical top hat in k-space,each step in the random walk will be independent of one another, yieldinga simple analytic solution. However, in Section 2.5.1 we calculated the col-lapse threshold assuming that our overdensity had the profile of a sphericaltop hat in real space. Thus using both the critical threshold as previouslyderived and the spherical top-hat smoothing window function in k-space isnot fully consistent with one another. Fortunately, predictions using theaforementioned method match well to simulations and observations and us-ing more self-consistent approaches seem to yield little improvement whilemaking the analysis more cumbersome. In this light, we continue with themethod stated above with less trepidation. With uncorrelated steps in ourrandom walk, the expression for the collapse fraction can be found by cal-culating the fraction of random walks trajectories that remain below thecritical threshold for all modes with k less than the the cut-o↵ scale, set byM in our k-space window function. The resulting expression for the collapsefraction coincides with that of Eq. (2.43), including the addition of the factorof 2.The excursion set formalism allows us to tackle many more problems,such as how halos accrete mass and merge over time, the length of time forformation, and many other similar questions.25 We may now also calculatethe spatial biasing of halos. Until now, we have only examined global quan-tities, but now we wish to determine halo statistics in a particular region ofspace with a finite size that encompasses a mass M˜ and has density contrast . This local collapse fraction, known as the biased collapse fraction, canbe calculated in a similar manner as described above for the global collapsefraction, expect that instead of starting the random walk process at M = 0and M ! 1, we start from M = and M = M˜ . The e↵ect of this is tosimply make the replacement c ! c in Eq. (2.43). We can now find25For a comprehensive review of these subjects, see Ref. [30].252.5. Collapse into Nonlinear Structuresthe biased mass function dn/dM , which is a function of . For cases where is small, it is useful to expand the mass function as a Taylor series, whichto linear order can be expressed asdndM ( ) = dndM ⇣1 + b(M) ⌘, (2.46)where b is referred to as the halo bias. For the PS mass function, the halobias bPS is given by bPS(M) = 1 + ⌫2(M, z) 1 c . (2.47)2.5.4 Improvements to the Mass FunctionThe EPS formalism provides a simple but powerful analysis tool for examin-ing the basic properties of halos. However, as formulated above, the PS massfunction underestimates the number of high-mass halos and overestimatesthe number of low-mass halos, as compared to numerical simulations. Oneof the most successful extensions to the PS model is to allow for ellipsoidalcollapse. The mass function of Sheth and Tormen [33] allows for such devia-tions from spherical collapse. Conveniently, the resulting mass function canstill be expressed in terms of the critical collapse threshold c for sphericalcollapse and has a similar form as in the PS model. The Sheth-Tormen massfunction can be expressed using Eq. (2.44), but where F is now given byFST = Ar 2⇡ ⌫ˆ(1 + ⌫ˆ 2p)e ⌫ˆ2/2, (2.48)where ⌫ˆ = pa⌫ and A, a, and p are fitting parameters.2.5.5 Halo VirializationThe ESP formalism has proved to be a valuable tool for statistically describ-ing the collapse of matter into halos. However, if we would like to examinebasic characteristics of the halo after significant shell-crossing has occured,we require new tools that can accommodate for the chaotic behaviour of thematter as it enters its final stages of collapse and its subsequent relaxation.In this section we describe how basic halo properties can be estimated byusing the virial theorem.For a self-gravitating system, the virial theorem relates the time-averagedkinetic and potential energy, K and U , respectively, by U = 2K. As-suming conservation of energy within the system, the energy of the system262.5. Collapse into Nonlinear Structuresat turnaround, given simply by U at this time, will equal the energy ofthe relaxed system Uvir + Kvir = Uvir/2. Since for a spherical overdensityU = (3/5)GM/r, the (physical) virial radius rvir will be half the value ofthe radius at turnaround and the volume at virialization will decrease by afactor of 8 compared to that at turnaround. We can approximate the timeof virialization as occurring when the overdensity would collapse completelyaccording to Eqs. (2.42) (at ⌧ = 2⇡). Since a / t2/3 in a matter-dominateduniverse, from Eq. (2.42b) we see that a expands by a factor of 22/3 betweenturnaround and virilization, and consequently ⇢cr will decrease by a factorof 4 during this time. Putting all of these factors together, we approximatethe ratio c = ⇢vir/⇢¯cr of the density of the virialized halo ⇢vir to the criticaldensity at virialization as c = 32[1 + nl(⌧ = ⇡)] = 18⇡2. This result canbe generalized to a flat universe with both matter and cosmological constant(⌦m + ⌦⇤ = 1) with the fitting formula [36] c = 18⇡2 + 82d 39d2, (2.49)where d ⌘ ⌦zm 1 and ⌦zm is the matter density parameter at redshift z,which in this case is given by⌦zm = ⌦m(1 + z)3⌦m(1 + z)3 + ⌦⇤ . (2.50)We can now write the the virialized radius asrvir = 1.49✓ h0.7◆ 2/3✓⌦m0.3◆ 1/3✓ 1⌦zm c18⇡2◆ 1/3⇥✓1 + z10◆ 1✓ M108 M ◆1/3 kpc (2.51)and can subsequently find the corresponding circular velocity Vc = pGM/rvirand can define the virial temperature as Tvir = µmpV 2c /2kb, where µ is themean molecular weight and mp is the proton mass. The halo mass as afunction of its virial temperature can then be written asM = 9.37⇥ 107 ⇣ µ0.6⌘ 3/2✓ h0.7◆ 1✓⌦m0.3◆ 1/2⇥✓1⌦zm c18⇡2◆ 1/2✓1 + z10 ◆ 3/2✓ Tvir104 K◆3/2 M . (2.52)27Chapter 3A Brief Tour ThroughCosmological Inflation3.1 IntroductionFrom the start of modern cosmology in the early 20th century through the1960s, a standard cosmological model emerged that described the expansionof our Universe and its basic constituents. However, beginning in the 1970s,puzzling questions arose that made this standard model seem incongruentwith observations. Among others, one such question was why the CMB wasso isotropic in spite of the fact that, according to the prevailing cosmologicalmodel of the time, many CMB photons coming from di↵erent directionswould have originated from locations what were not yet in causal contactwith one another. Alan Guth proposed the theory of inflation in 1980 asa solution to these problems [37], later developed by Linde [38], Albrechtand Steinhardt [39], among others. It was later realized that inflation alsoprovided a mechanism that seeds the perturbations in our Universe, whichlater imprinted themselves as anisotropies in the CMB and sourced large-scale structures. In this chapter, we give a brief introduction to inflationarytheory.3.2 Problems with the Standard CosmologicalModelThe problems alluded to in the previous section all in some way deal with theinitial conditions set in the early Universe. Here we outline these problems.The Horizon ProblemWe have already briefly touched on the horizon problem, one view of whichasks why the Universe is isotropic to such a high degree. We can formulatethis problem more precisely by comparing the comoving size of the present-283.3. The Basicsday observable Universe to that of a causal patch at some early time. Inthe standard cosmological model, where the total equation of state w liesbetween 0 and 1/3 throughout, the comoving particle horizon at time t is ofthe order of H 1(t). We can then estimate this ratio byH 10H 1i = aiHia0H0 ⇠ 1028 Timp , (3.1)where the subscript i denotes quantities evaluated at some early ‘initial’time ti and have assumed that the Universe is radiation dominated at thistime with temperature Ti. If ti is near the Planck scale, then the comovinglength scale of the present-day observable Universe was about 1028 timesbigger than that of causal regions at ti, so that the present-day observableUniverse encloses 1084 di↵erent regions that were causally disconnected fromone another at ti. Decreasing Ti doesn’t help the situation much; for Ti atthe GeV scale, the present-day horizon volume would still encompass around1030 causally disconnected regions. With this in mind, it is unusual that ourUniverse would be so isotropic as well as homogenous on large scales if theregions that comprise the present-day horizon volume were not in causalcontact with one another at some time in the distant past.The Flatness ProblemThe present-day Universe is very flat, in the sense that ⌦k is currentlybounded by roughly |⌦k| < 0.04. However, a fine-tuning problem ariseswith the realization that in the standard cosmological model ⌦k increaseswith time, such that ⌦k must have been initially fine-tuned to an extremelysmall value. We can compare the present-day value of ⌦k to that at ti bythe fraction⌦k(t0)⌦k(ti) = ✓ HiaiH0a0◆2 ⇠ 1056✓ Timp◆2 , (3.2)again assuming that the Universe is dominated by radiation at ti. Thus,⌦k(ti) must be fine-tuned to an extremely small value at the Planck scale,when one might expect it to be of order unity at this time.3.3 The BasicsAll of the aforementioned problems in some way deal with the horizon sizein the very early Universe. The problems stem from the fact that in the293.3. The Basicsstandard cosmological model, the particle horizon26 is of the same orderof magnitude as the Hubble length, which can be thought of as the lengthscale over which particles can communicate with one another at a certaintime (within the time that the scale factor grows by a factor of e). FromEq. (2.5), we can see that if w is bounded by 0 and 1/3 throughout, as in thestandard cosmological model, the comoving Hubble length H 1 monotoni-cally increases with ⌘, and ⌘ and H 1 are of the same order of magnitude.In other words, in the standard model, when a scale enters the horizon, itis the first time there can be causal contact on this scale.Inflation resolves these issues by creating a large di↵erence between theHubble length and particle horizon. Unlike the particle horizon, which in-creases monotonically (this is why we can use ⌘ as a time variable), theHubble length can decrease. Examining Eq. (2.5) again, if w < 1/3 thenH 1 would decrease as ⌘ increases, so that a su ciently long stage with suchan equation of state would create a drastic di↵erence between H 1 and ⌘.During inflation, the comoving Hubble length ‘zooms in’ to a much smallerscale than at the start of inflation. Spacetimes with 1 < w < 1/3 havean event horizon 27, whose comoving length is of order H 1. A (comoving)scale k ‘leaves the horizon’ when k ⇠ H; communication on this scale ispossible before this time but not after. After its rapid decrease during infla-tion, H 1 begins to grow again as the subsequent evolution of the Universeproceeds as described in the standard cosmological model. With inflation,when a scale reenters the horizon well after inflation, the particle horizon ismany orders of magnitude larger than the Hubble length and so althoughcommunication can only commence on this scale once it enters the horizon,communication could have taken place on this scale well before this time(i.e. before it left the horizon during inflation). Standard inflationary mod-els assume that the inflationary spacetime is nearly de Sitter (w is close to 1) so that H is nearly constant during inflation. The parallel view in termsof physical scales has rapidly growing physical scales during inflation passthrough a nearly constant Hubble length H 1.We can now revisit the problems discussed in Section 3.2 with an in-flationary period assumed to have occurred in the very early Universe. Asbefore, to fit the present-day observable Universe into a causal region ofspace during an ‘initial’ time ti, we require H 10 H 1i , but in the inflation-26The particle horizon is the maximum distance from which particles could have trav-elled to an observer at a particular time over the entire history of the Universe until theobservation time.27Technically speaking, there is only a true event horizon if the equation of state wcontinues to stay below 1/3.303.4. A Simple Modelary paradigm ti is before inflation. This ratio is nowH 10H 1i = aiae aea0 HiH0 = e N aea0 HiH0 ⇠ e N1028 Temp , (3.3)where ae is the scale factor at the end of inflation and we have parameterizedthe duration of inflation by the number of e-folds N ⌘ ln(ae/ai).28 In thelast step in Eq. (3.3), we have assumed that H is approximately constantthroughout inflation. If inflation occurs near the Planck scale, then therequirement ofH 10 H 1i necessitates at least N ⇠ 64 e-folds of inflation.29It is easy to see that the requirement of H 10 H 1i solves both the horizonand flatness problems.3.4 A Simple ModelFrom Eq. (2.3b), we see that having 1 < w < 1/3 results in an accel-erating background. The next step is to find what substances are capableof driving an accelerating expansion. Here we examine the simple case of asingle scalar field ' with potential V (') [2, 3, 9]. Its Lagrangian is given byL =12@↵'@↵'
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Models and probes of the early and dark Universe : inflation and 21-cm radiation in cosmology Sitwell, Michael 2014
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Title | Models and probes of the early and dark Universe : inflation and 21-cm radiation in cosmology |
Creator |
Sitwell, Michael |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | The prevailing model of modern cosmology stipulates the existence of exotic substances such as dark matter and dark energy and events such as inflation. However, their underlying nature is not currently known. In this thesis, we explore new models and measurement techniques that may be used to characterize their cosmological effects and shed light on their inner workings. A model of inflation driven by a substance that may be described macroscopically as a cosmological elastic solid is studied. The proper techniques for the quantization of perturbations within the elastic solid are presented. We find that a sufficiently rigid elastic solid with slowly varying sound speeds can produce an inflationary period. Interestingly, we find models where the elastic solid has an equation of state significantly greater than -1 that nevertheless produces nearly scale-invariant scalar and tensor spectra. The remaining chapters of this thesis concern the use of 21-cm radiation as a probe of the physics of dark matter and dark energy. The effects of warm dark matter on the highly-redshifted 21-cm signal is examined. If dark matter is warm instead of cold, its non-negligible velocities may inhibit the formation of low-mass halos, thereby delaying star-formation, which may delay the emission and absorption signals expected in the mean 21-cm signal. The effects of warm dark matter on both the mean 21-cm signal, as well as on its power spectrum, are described and degeneracies between the effects of warm dark matter and other astrophysical parameters are quantified. One of the primary goals of 21-cm radiation intensity mapping is to measure baryon acoustic oscillations over a wide range of redshifts to constrain the properties of dark energy from the expansion history of the late-time Universe. We forecast the constraining power of the CHIME radio telescope on the matter power spectrum and dark energy parameters. Lastly, we devise new calibration algorithms for the gains of an interferometric radio telescope such as CHIME. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-11-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
IsShownAt | 10.14288/1.0167036 |
URI | http://hdl.handle.net/2429/51117 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2015-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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