"Science, Faculty of"@en .
"Physics and Astronomy, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Sitwell, Michael"@en .
"2014-11-19T15:44:20Z"@en .
"2014"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"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. \n\nA 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.\n\nThe remaining chapters of this thesis concern the use of 21-cm radiation as a probe of the physics of dark matter and dark energy. \n\nThe 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.\n\nOne 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."@en .
"https://circle.library.ubc.ca/rest/handle/2429/51117?expand=metadata"@en .
"Models and Probes of the Early andDark UniverseInflation and 21-cm Radiation in CosmologybyMichael SitwellB.Sc., Queen\u00E2\u0080\u0099s 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\u0000 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\u00E2\u0086\u00B5ects 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\u0000ciently 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 \u00001 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\u00E2\u0086\u00B5ects 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\u00E2\u0086\u00B5ects 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\u00E2\u0086\u00B5ects 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, \u00E2\u0080\u009CQuantization of Perturbationsin an Inflating Elastic Solid,\u00E2\u0080\u009D Phys. Rev. D, vol. 89, 123509, 2014.2. Chapter 6: M. Sitwell, A. Mesinger, Y. Ma, & K. Sigurdson, \u00E2\u0080\u009CTheImprint of Warm Dark Matter on the Cosmological 21-cm Signal,\u00E2\u0080\u009DMNRAS, vol. 438, p. 2664, 2014.3. Chapter 7: J. R. Shaw, K. Sigurdson, M. Sitwell, A. Stebbins, &U. Pen, \u00E2\u0080\u009CCoaxing Cosmic 21cm Fluctuations from the Polarized Skyusing m-mode Analysis,\u00E2\u0080\u009D 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\u00E2\u0086\u00B5ects 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 \u00E2\u0080\u0098Horizon Problem\u00E2\u0080\u0099 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\u00E2\u0086\u00B5ect 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 \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 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 \u00E2\u0087\u00A3 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\u00C2\u00AFS for CDM and WDM . . . . . . . 956.3 Mean 21-cm brightness temperature \u0000T\u00C2\u00AFb . . . . . . . . . . . . 976.4 Critical points in the mean 21-cm signal . . . . . . . . . . . . 986.5 Parameter space curves ze(f\u00E2\u0087\u00A4|CDM) = ze(mX|WDM) for var-ious critical points . . . . . . . . . . . . . . . . . . . . . . . . 996.6 Evolution of f\u00E2\u0087\u00A4(z) in CDM required to match the mean bright-ness temperature \u0000T\u00C2\u00AFb in WDM . . . . . . . . . . . . . . . . . 1006.7 Evolution of the power spectrum of \u0000Tb for WDM . . . . . . 1026.8 Power spectrum of the brightness temperature \u0000Tb . . . . . . 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 \u0000 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-Lema\u00CB\u0086\u00C4\u00B1tre-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\u00E2\u0080\u0099s formulation ofthe theory of general relativity and Edwin Hubble\u00E2\u0080\u0099s 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 \u00E2\u0080\u0098cosmological constant\u00E2\u0080\u0099 \u00E2\u0087\u00A4, whichwhen combined with the above assumptions for dark matter form the stan-dard \u00E2\u0087\u00A4CDM 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\u0000cult 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 \u00E2\u0087\u00A0 150Mpc.9 As the primordial perturbation are distributed over awide range of scales and directions, these waves overlap with one another,making it di\u0000cult 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\u00E2\u0086\u00B5ect 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\u00E2\u0080\u0099s 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-\u00E2\u0086\u00B5 forest by BOSS [16, 17] andin the cross-correlation of Lyman-\u00E2\u0086\u00B5 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 \u00E2\u0087\u00A0 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\u00E2\u0086\u00B5erent 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 \u00E2\u0080\u0098tomographic\u00E2\u0080\u0099 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 \u00E2\u0080\u0098dark ages\u00E2\u0080\u0099, 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 \u00E2\u0080\u0098bubbles\u00E2\u0080\u0099 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\u0000cient 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\u0000cientlevel.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\u00E2\u0086\u00B5ects 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-\u00E2\u0086\u00B5 line, places theend of reionization around z \u00E2\u0087\u00A0 6 [21, 22]. However, due to its high cross-section, the Universe is opaque to Lyman-\u00E2\u0086\u00B5 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\u00E2\u0086\u00B5ects 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\u00E2\u0086\u00B5erent 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 \u00001. However, di\u00E2\u0086\u00B5erentmodels predict slight departures from wDE = \u00001.10 With current measure-ments consistent with wDE = \u00001, 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 = \u00001 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 \u00E2\u0087\u00A5 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\u0000 800MHz (wavelengthsof \u00E2\u0087\u00A0 37 \u0000 75 cm) in order to measure the BAO at redshifts in the rangez \u00E2\u0087\u00A1 0.8 \u0000 2.5, a time period when the e\u00E2\u0086\u00B5ects 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.\u00E2\u0080\u00A2 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).\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 0.1\u000010MeV, z \u00E2\u0087\u00A0 4 \u00E2\u0087\u00A5 108\u00004 \u00E2\u0087\u00A5 1010 Big-Bang Nucleosynthesis :Light nuclei are formed.\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 1MeV, z \u00E2\u0087\u00A0 4 \u00E2\u0087\u00A5 109 Neutrino Decoupling : Neutrinos decouplefrom other species and free-stream thereafter.\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 0.5MeV, z \u00E2\u0087\u00A0 2 \u00E2\u0087\u00A5 109 Electron-Positron Annihilation: Electronsand positrons annihilate with one another, a relatively small numberof electrons persist past the annihilation.\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 1 eV, z \u00E2\u0087\u00A0 4250 Matter-Radiation Equality : The Universe becomesmatter dominated past this point.71.6. Cosmological History in Brief\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 0.26 eV, z \u00E2\u0087\u00A0 1100 Recombination: Free electrons and protonscombine to form hydrogen, the CMB is released.\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 1.6\u00002.6meV, z \u00E2\u0087\u00A0 6\u000010 Reionization: Radiation from early as-trophysical sources ionize hydrogen in the IGM (see Chapters 5 and6).\u00E2\u0080\u00A2 T \u00E2\u0087\u00A0 0.33meV, z \u00E2\u0087\u00A0 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-Lema\u00CB\u0086\u00C4\u00B1tre-Robertson-Walker (FLRW)metric, which can be represented byds2 = g\u00C2\u00AF\u00C2\u00B5\u00E2\u008C\u00ABdx\u00C2\u00B5dx\u00E2\u008C\u00AB = dt2 \u0000 a2(t) \u00EF\u00A3\u00BF dr21\u0000Kr2 + r2d\u00E2\u008C\u00A62\u0000 , (2.1)where g\u00C2\u00AF\u00C2\u00B5\u00E2\u008C\u00AB is the background FLRW metric and d\u00E2\u008C\u00A62 = d\u00E2\u009C\u00932 + sin2 \u00E2\u009C\u0093d\u00002. Thespatial curvature constant K can assume the values 0, 1,\u00001 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\u00CB\u0099(t)/a(t), where an overdot denotes di\u00E2\u0086\u00B5erentiation withrespect to t. Often it will be more convenient to work with the conformaltime \u00E2\u008C\u0098 \u00E2\u008C\u0098 R dt/a(t) instead of the coordinate time t. In these cases, we willalso make use of the conformal Hubble rate H \u00E2\u008C\u0098 a0/a = aH, where prime 0stands for @/@\u00E2\u008C\u0098.It is easy to show that the stress-energy tensor for a perfect fluidT\u00E2\u0086\u00B5\u0000 = (\u00E2\u0087\u00A2 + P )u\u00E2\u0086\u00B5u\u0000 \u0000 P \u0000\u00E2\u0086\u00B5\u0000 (2.2)92.1. The Unperturbed Universeyields a homogenous and isotropic spacetime, where \u00E2\u0087\u00A2 is the energy density,P is the pressure scalar, and u\u00E2\u0086\u00B5 is the 4-velocity. Solving the Einsteinequations with this metric and stress-energy tensor yields the Friedmannequations H2 + Ka2 = 8\u00E2\u0087\u00A1G3 \u00E2\u0087\u00A2, (2.3a)a\u00C2\u00A8a = \u00004\u00E2\u0087\u00A13 G\u00E2\u0087\u00A2(1 + 3w). (2.3b)In the above equation, we have introduced the equation of state parameterw \u00E2\u008C\u0098 P/\u00E2\u0087\u00A2. It is important to note that, from (2.3b), if the equation of stateis larger than \u00001/3, the expansion of the spacetime will be decelerating,while a value of w smaller than \u00001/3 leads to an accelerating expansion.Eliminating \u00E2\u0087\u00A2 in the above equations yields the useful di\u00E2\u0086\u00B5erential equationH0 = \u00001 + 3w2H2. (2.4)In the case where w is constant, this di\u00E2\u0086\u00B5erential equation can easily be solvedasH =2(1 + 3w)(\u00E2\u008C\u0098 \u0000 \u00E2\u008C\u0098c) , (2.5)where \u00E2\u008C\u0098c is a constant of integration.For a noninteracting perfect fluid, the covariant conservation of thestress-energy tensor yields the energy (density) conservation equation\u00E2\u0087\u00A2\u00CB\u0099 + 3H\u00E2\u0087\u00A2(1 + w) = 0. (2.6)For constant w, this implies that \u00E2\u0087\u00A2 / a\u00003(1+w).The energy density is typically decomposed as a sum of components withdi\u00E2\u0086\u00B5erent equations of state. In the \u00E2\u0087\u00A4CDM 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 \u00E2\u0087\u00A4 with w = \u00001. The present-day energy density of each substancei is often expressed as a fraction \u00E2\u008C\u00A6i of the critical density \u00E2\u0087\u00A2cr, 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 (\u00E2\u008C\u00A6ma\u00003 + \u00E2\u008C\u00A6ra\u00004 + \u00E2\u008C\u00A6\u00E2\u0087\u00A4 + \u00E2\u008C\u00A6ka\u00002), (2.7)where the subscripts m and r denote matter and radiation, respectively, thesubscript 0 denotes the present-day value, and \u00E2\u008C\u00A6k = \u0000K/(a0H0)2. The102.1. The Unperturbed Universescale factor is normalized as a0 = 1. Current observed values for theseparameters are very roughly \u00E2\u008C\u00A6\u00E2\u0087\u00A4 \u00E2\u0087\u00A1 0.73,\u00E2\u008C\u00A6m \u00E2\u0087\u00A1 0.27,\u00E2\u008C\u00A6k \u00E2\u0087\u00A1 0,\u00E2\u008C\u00A6r \u00E2\u0087\u00A1 8 \u00E2\u0087\u00A5 10\u00005,and h \u00E2\u0087\u00A1 0.7, where h is the present-day Hubble parameter H0 expressed inunits of 100 km s\u00001Mpc\u00001 [25]. In certain situations it is convenient to usethe parameters !m = \u00E2\u008C\u00A6mh2 and !b = \u00E2\u008C\u00A6bh2.2.1.2 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 = (\u0000ob \u0000\u0000em)/\u0000em of light emitted with wavelength \u0000em and observed with wave-length \u0000ob. The redshift is commonly used in place of the scale factor a,related by a0/a = 1 + z.An important distinction is the di\u00E2\u0086\u00B5erence 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 \u00E2\u008C\u0098 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\u0000 = \u00E2\u008C\u0098(t2)\u0000 \u00E2\u008C\u0098(t1) between the times t1 and t2. Unless otherwise stated, thelater time t2 is assumed to be the present day and then \u0000 is a function of asingle time parameter. Thus, if the zero point of \u00E2\u008C\u0098 is chosen appropriately,\u00E2\u008C\u0098(t) measures the particle horizon of a massless particle at time t.By expressing the conformal time as the integral\u00E2\u008C\u0098 = Z da\u00CB\u009C\u00CB\u009Ca H\u00001(a\u00CB\u009C) (2.8)we can see that massless particles can travel roughly a comoving distanceH\u00001 in the time that the scale factor increases by a factor of e. In thislight, the Hubble radius H\u00001 (or H\u00001 in comoving coordinates) is commonlyreferred to as the \u00E2\u0080\u0098horizon\u00E2\u0080\u0099.The angular diameter distance DA is used to relate the physical size Lphof a very distant object to the angle \u00E2\u009C\u0093 that it subtends by DA = Lph/\u00E2\u009C\u0093. In112.2. Thermodynamicsgeneral, the angular diameter distance is given byDA(z) = 11 + z 1H0p\u00E2\u008C\u00A6k sinh\u00E2\u0087\u00A3p\u00E2\u008C\u00A6kH0\u0000(z)\u00E2\u008C\u0098 , (2.9)but greatly simplifies in a flat universe to DA(z) = a\u0000(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 \u00E2\u0087\u00A2, and pressure P can be expressedas [1, 2] n = g Z d3p(2\u00E2\u0087\u00A1)3 f(E), (2.10a)\u00E2\u0087\u00A2 = g Z d3p(2\u00E2\u0087\u00A1)3 f(E)E, (2.10b)P = g Z d3p(2\u00E2\u0087\u00A1)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 = \u00E2\u0087\u00A2 + P \u0000 \u00C2\u00B5nT . (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 \u0000 m, in which case for a boson the above relations simplify tonB = \u00E2\u0087\u00A3(3)\u00E2\u0087\u00A12 gT 3, \u00E2\u0087\u00A2B = \u00E2\u0087\u00A1230gT 4, PB = \u00E2\u0087\u00A1290gT 4, (2.12)where \u00E2\u0087\u00A3 is the zeta function. For a fermion, we get nF = (3/4)nB, \u00E2\u0087\u00A2F =(7/8)\u00E2\u0087\u00A2B, 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\u00E2\u0087\u00A1245gT 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 \u00E2\u008C\u00A7 m, for both bosons and fermions wehave n = g\u00E2\u009C\u0093mT2\u00E2\u0087\u00A1 \u00E2\u0097\u00863/2 e(\u00C2\u00B5\u0000T )/T , \u00E2\u0087\u00A2 = mn, P = nT. (2.14)For nonrelativistic matter we see that w \u00E2\u0087\u00A1 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\u00E2\u0086\u00B5ective degrees of freedom as\u00E2\u0087\u00A2r(T ) = \u00E2\u0087\u00A1230g(T )T 4, (2.15a)sr(T ) = 2\u00E2\u0087\u00A1245 gs(T )T 3, (2.15b)nr(T ) = \u00E2\u0087\u00A3(3)\u00E2\u0087\u00A12 gn(T )T 3, (2.15c)where T is the photon temperature, and g, gs, and gn are the number of e\u00E2\u0086\u00B5ec-tive relativistic degrees of freedom contributing towards the energy, entropy,and number densities, respectively, given byg(T ) = Xb gb\u00E2\u009C\u0093TbT \u00E2\u0097\u00864 + 78 Xf gf \u00E2\u009C\u0093TfT \u00E2\u0097\u00864 , (2.16a)gs(T ) = Xb gb\u00E2\u009C\u0093TbT \u00E2\u0097\u00863 + 78 Xf gf \u00E2\u009C\u0093TfT \u00E2\u0097\u00863 , (2.16b)gn(T ) = Xb gb\u00E2\u009C\u0093TbT \u00E2\u0097\u00863 + 34 Xf gf \u00E2\u009C\u0093TfT \u00E2\u0097\u00863 . (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\u00E2\u0086\u00B5ective degrees of freedom decrease as particles becomenonrelativistic. All relativistic particles in the plasma are in thermodynamicequilibrium with each other (so g = gs) until T \u00E2\u0087\u00A0 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 \u00E2\u008C\u00A7 MeV, after electron-positron annihilation, we will have g \u00E2\u0087\u00A1 3.36and gs = 43/11 \u00E2\u0087\u00A1 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) = \u00E2\u009C\u0093gs(T2)gs(T1)\u00E2\u0097\u00861/3 T2T1 , (2.17)where T1 and T2 are photon temperatures at two di\u00E2\u0086\u00B5erent 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(\u00E2\u008C\u0098)(\u00E2\u008C\u0098\u00E2\u0086\u00B5\u0000 \u0000 h\u00E2\u0086\u00B5\u0000)dx\u00E2\u0086\u00B5dx\u0000 , (2.18)where \u00E2\u008C\u0098\u00E2\u0086\u00B5\u0000 is the metric for Minkowski space and h\u00E2\u0086\u00B5\u0000 represents small pertur-bations about the background. It will prove useful to further parameterizethe perturbations h\u00E2\u0086\u00B5\u0000 . To do this, we first separate the time-like and spatialparts as ds2 = a2(\u00E2\u008C\u0098) \u00E2\u0087\u00A5(1 + 2\u0000)d\u00E2\u008C\u00982 \u0000 2Bidxid\u00E2\u008C\u0098 \u0000 hijdxidxj\u00E2\u0087\u00A4 . (2.19)16This section largely uses the notation of Ref. [9]. See also Ref. [26].142.3. Linear Perturbation TheoryThis introduces a scalar \u0000, 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)\u0000ij + 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 \u0000 13\u0000ijr2E (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 = \u000016h+ 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\u0000T 00 = \u0000\u00E2\u0087\u00A2, (2.21a)\u0000T i0 = (\u00E2\u0087\u00A2 + P )vi, (2.21b)\u0000T ij = \u0000(\u0000P \u0000ij + P\u00E2\u0087\u00A7ij), (2.21c)where \u0000\u00E2\u0087\u00A2 and \u0000P are the energy density and pressure perturbations, respec-tively, vi is the velocity perturbation, and \u00E2\u0087\u00A7ij is the anisotropic stress. Thevelocity perturbation vi and anisotropic stress \u00E2\u0087\u00A7ij 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 \u00E2\u0087\u00A7, 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 \u00E2\u0087\u00A7), 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\u00E2\u0087\u00A1)\u00003/2 R d3kfkeik\u00C2\u00B7x.152.3. Linear Perturbation Theorydescribed by a power spectrum Pf (k), which describes the variance of itsFourier components and is given byhfkfk\u00CB\u009Ci = (2\u00E2\u0087\u00A12/k3)Pf (k)\u0000(k+ k\u00CB\u009C). (2.22)The correlation function \u00E2\u0087\u00A0f for f is then\u00E2\u0087\u00A0f (|x\u0000 x\u00CB\u009C|) = Z dkk Pf (k)sin(k|x\u0000 x\u00CB\u009C|)k|x\u0000 x\u00CB\u009C| . (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\u00E2\u0086\u00B5erent 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\u00E2\u0086\u00B5erent coordi-nate systems in the perturbed spacetime. We can relate two such coordinatesystems x\u00E2\u0086\u00B5 and x\u00CB\u009C\u00E2\u0086\u00B5 by x\u00CB\u009C\u00E2\u0086\u00B5 = x\u00E2\u0086\u00B5 + \u00E2\u0087\u00A0\u00E2\u0086\u00B5. Switching coordinate systems in thismanner is known as a gauge transformation. The spatial part of the vectorrelating the two coordinate systems \u00E2\u0087\u00A0\u00E2\u0086\u00B5 can be decomposed as \u00E2\u0087\u00A0i = \u00E2\u0087\u00A3 ,i + \u00E2\u0087\u00A0i?,where \u00E2\u0087\u00A0? is divergenceless. When changing coordinates, the scalar metricperturbations transform as 18\u00CB\u009C\u0000 = \u0000\u0000H\u00E2\u0087\u00A00 \u0000 (\u00E2\u0087\u00A00)0, (2.24a) \u00CB\u009C = +H\u00E2\u0087\u00A00, (2.24b)B\u00CB\u009C = B \u0000 \u00E2\u0087\u00A00 + \u00E2\u0087\u00A3 0, (2.24c)E\u00CB\u009C = E + \u00E2\u0087\u00A3, (2.24d)where perturbations in the coordinate system x\u00CB\u009C\u00E2\u0086\u00B5 (x\u00E2\u0086\u00B5) are denoted with(without) a tilde. A scalar variable q defined in the perturbed spacetime thatis decomposed into a background component q\u00C2\u00AF and a perturbed component\u0000q transforms as \u0000q\u00CB\u009C = \u0000q \u0000 q\u00C2\u00AF0\u00E2\u0087\u00A00. (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 \u00E2\u0087\u00A2 and pressure P . A 4-vector wi, such as the 4-velocity, will transform asw\u00CB\u009C0 = w0 + w\u00C2\u00AF0(\u00E2\u0087\u00A00)0 \u0000 (w\u00C2\u00AF0)0\u00E2\u0087\u00A00, w\u00CB\u009Ci = wi + w\u00C2\u00AF0(\u00E2\u0087\u00A0i)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 \u0000C\u00CB\u009Cij = \u0000Cij \u0000 13\u0000ij(C\u00C2\u00AFkk )0\u00E2\u0087\u00A00, the traceless partof the tensor \u0000Cij , given by \u0000Cij \u0000 13\u0000ij\u0000Ckk , 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 \u00E2\u0087\u00A0\u00E2\u0086\u00B5 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 \u0000 = B = 0Comoving v = B = 0O\u00E2\u0086\u00B5-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\u0000 = \u0000 +H(B \u0000 E0) + (B \u0000 E0)0, (2.27a) = \u0000H(B \u0000 E0), (2.27b)\u0000\u00E2\u0087\u00A2(gi) = \u0000\u00E2\u0087\u00A2 + \u00E2\u0087\u00A20(B \u0000 E0), (2.27c)\u0000P (gi) = \u0000P + P 0(B \u0000 E0), (2.27d)v(gi) = v +B \u0000 E0. (2.27e)In the above equations, \u00E2\u0087\u00A2 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 \u0000 3H( 0 +H\u0000) = 32l2a2\u0000\u00E2\u0087\u00A2(gi), (2.28a) 0 +H\u0000 = \u0000v(gi), (2.28b) 00 +H(\u00000 + 2 0) + (H2 + 2H0)\u0000+ 13r2(\u0000\u0000 ) = 32l2a2\u0000P (gi), (2.28c) \u0000 \u0000 = 3l2a2P\u00E2\u0087\u00A7, (2.28d)where \u0000 = H2\u0000H0 = 32 l2a2(\u00E2\u0087\u00A2+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\u0000(gi)0 \u0000 (1 + w)(r2v(gi) + 3 0) + 3H \u0000P (gi)\u00E2\u0087\u00A2 \u0000 w\u0000(gi)! = 0, (2.29a)v(gi)0+H(1\u00003w)v(gi)+ w01 + wv(gi)\u0000 \u0000P (gi)\u00E2\u0087\u00A2 + P \u0000\u0000\u0000 23 w1 + wr2\u00E2\u0087\u00A7 = 0, (2.29b)where \u0000 = \u0000\u00E2\u0087\u00A2/\u00E2\u0087\u00A2 is the density contrast.The Einstein equations are very simple for the tensor perturbations andyield the sole equation(hT)i00j + 2H(hT)i0j \u0000r2(hT)ij = 6l2a2P (\u00E2\u0087\u00A7T)ij , (2.30)where (\u00E2\u0087\u00A7T)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 (\u0000\u00E2\u0087\u00A2 6= 0, \u0000s = 0) and entropy mode(s) with (\u0000\u00E2\u0087\u00A2 = 0, \u0000s 6= 0). Thepressure of a \u00E2\u0080\u0098fluid-like\u00E2\u0080\u0099 substance 19 in general is a function of both theenergy and entropy densities so that\u0000P = @P@\u00E2\u0087\u00A2 \u0000\u0000\u0000\u0000s\u0000\u00E2\u0087\u00A2 + @P@s \u0000\u0000\u0000\u0000\u00E2\u0087\u00A2\u0000s. (2.31)19By \u00E2\u0080\u0098fluid-like\u00E2\u0080\u0099 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\u00E2\u0080\u0099s 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 \u00E2\u0087\u00A1 \u0000.We will now examine some of the basics of the evolution of the matterdensity contrast \u0000 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\u00E2\u0086\u00B5ect of the radiation on the matter distribution is negligible at thispoint. Consequently, the Einstein equations in Section 2.3.3 are su\u0000cientfor 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 \u0000 evolves as [1]d2\u0000kda2 + \u00E2\u009C\u0093d lnHda + 3a\u00E2\u0097\u0086 d\u0000kda \u0000 3\u00E2\u008C\u00A6m2a5(H/H0)2 \u0000k = 0. (2.32)Note that for these late-time sub-horizon modes, the evolution of \u0000k is inde-pendent of k. We can then separate the evolution of \u0000k into two regimes: anearly scale-dependent evolution and a late scale-independent evolution. Theexpression for \u0000k at late times is most often expressed through its relationto the metric perturbation \u0000, which in the current limit from Eq. (2.28a)implies \u0000k2\u0000k = (3/2)l2a2\u00E2\u0087\u00A2m\u0000k.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) = \u0000(k, alate)\u0000(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\u0000culty that for large-scale superhorizon modes, \u0000k 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 \u0000 and \u0000. It is defined asG(a) = a \u0000(a)\u0000(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\u00E2\u008C\u00A6mH(a)H0 Z a0 da\u00CB\u009C(a\u00CB\u009CH(a\u00CB\u009C)/H0)\u00003. (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 \u0000. This is conventionally parameterized as 21P\u0000,I(k) = 50\u00E2\u0087\u00A129k3 \u00002H(k/H0)ns\u00001\u00E2\u008C\u00A62m/G(a = 1)2. (2.36)The amplitude of the power spectrum is set by the parameter \u0000H 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 \u0000 / 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 \u0000H \u00E2\u0087\u00A0 10\u00005. Withthis, we now arrive at the expression for the linear matter power spectrumfor a > alate Pm(k, a) = 2\u00E2\u0087\u00A12\u00002H knsHns+30 T 2(k)\u00E2\u009C\u0093 G(a)G(a = 1)\u00E2\u0097\u00862 . (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 \u00002R = h\u00002Ri, where \u0000R(x) = R d3x\u00CB\u009C\u0000(x\u00CB\u009C)WR(x \u0000 x\u00CB\u009C) 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)\u0000 kR cos(kR))(kR)3 . (2.38)This variance can be written in terms of the the power spectrum P (k) by\u00002R = Z d3k(2\u00E2\u0087\u00A1)3P (k)|WR(k)|2. (2.39)Its value at R = 8Mpch\u00001, denoted by \u00008, is a frequently measured param-eter (measured to by about \u00008 \u00E2\u0087\u00A0 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\u00E2\u0086\u00B5ects 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 = \u0000GMR2 = \u000043\u00E2\u0087\u00A1G\u00E2\u0087\u00A2R, (2.40)where M = (4/3)\u00E2\u0087\u00A1\u00E2\u0087\u00A2R3, until shell crossing occurs. Since in a matter-dominated universe we have \u00E2\u0087\u00A2 / a\u00003, the full nonlinear overdensity \u0000nlis given by \u0000nl = \u00E2\u009C\u0093 a(t)R(t)/R0\u00E2\u0097\u00863 \u0000 1, (2.41)where R0 is the initial size of the overdense region. By substituting \u0000nl intoEq. (2.40) and solving for the overdensity yields the following parametricsolutions R = GM(1\u0000 cos \u00E2\u008C\u00A7)C\u00001, (2.42a)t = GM(\u00E2\u008C\u00A7 \u0000 sin \u00E2\u008C\u00A7)C\u00003/2, (2.42b)\u0000nl = 9(\u00E2\u008C\u00A7 \u0000 sin \u00E2\u008C\u00A7)22(1\u0000 cos \u00E2\u008C\u00A7)3 \u0000 1, (2.42c)\u0000 = 35\u00E2\u009C\u009334(\u00E2\u008C\u00A7 \u0000 sin \u00E2\u008C\u00A7)\u00E2\u0097\u00862/3 , (2.42d)where \u00E2\u008C\u00A7 2 (0, 2\u00E2\u0087\u00A1) 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, \u0000 is the solution for the overdensity by linearizingall equations in the overdensity, while \u0000nl 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 \u0000 and \u0000nl 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 \u00E2\u008C\u00A7 = \u00E2\u0087\u00A1 and collapse is complete when \u00E2\u008C\u00A7 = 2\u00E2\u0087\u00A1.At final collapse, the linear overdensity is \u0000 = \u0000c \u00E2\u0087\u00A1 1.69, where \u0000c 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\u00E2\u0087\u00A1/3)\u00E2\u0087\u00A2R3 collapses into a halo. To this end, we smooth the densitycontrast on a scale R to yield the field \u0000M , which will have a variance \u00002M (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\u0000c d\u0000M 1p2\u00E2\u0087\u00A1\u0000M (z)exp\u00E2\u009C\u0093\u0000 \u00002M2\u00002M (z)\u00E2\u0097\u0086 = erfc\u00E2\u009C\u0093 \u0000cp2\u0000M (z)\u00E2\u0097\u0086 .(2.43)Above, a somewhat precarious factor of 2 was added whose inclusion wasoriginally justified to allow for underdense regions with \u0000M < 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\u00E2\u0086\u00B5erentiating 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 = \u0000 \u00E2\u0087\u00A2M d ln\u0000dM F (\u00E2\u008C\u00AB), (2.44)where \u00E2\u008C\u00AB(z) = \u0000c/\u0000M (z) and for the PS model F (\u00E2\u008C\u00AB) isFPS(\u00E2\u008C\u00AB) = r 2\u00E2\u0087\u00A1\u00E2\u008C\u00ABe\u0000\u00E2\u008C\u00AB2/2. (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 \u00E2\u0080\u0098cloud-in-cloud\u00E2\u0080\u0099problem.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 \u0000M \u00E2\u0087\u00A1 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\u00E2\u0086\u00B5usion 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 \u00E2\u0080\u0098up-crossing\u00E2\u0080\u0099through 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\u00E2\u0086\u00B5 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\u00CB\u009C and has density contrast\u0000. 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 \u0000M = 0and M ! 1, we start from \u0000M = \u0000 and M = M\u00CB\u009C . The e\u00E2\u0086\u00B5ect of this is tosimply make the replacement \u0000c ! \u0000c \u0000 \u0000 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 \u0000. For cases where\u0000 is small, it is useful to expand the mass function as a Taylor series, whichto linear order can be expressed asdndM (\u0000) = dndM \u00E2\u0087\u00A31 + b(M)\u0000\u00E2\u008C\u0098, (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 + \u00E2\u008C\u00AB2(M, z)\u0000 1\u0000c . (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 \u0000c 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\u00E2\u0087\u00A1 \u00E2\u008C\u00AB\u00CB\u0086(1 + \u00E2\u008C\u00AB\u00CB\u0086\u00002p)e\u0000\u00E2\u008C\u00AB\u00CB\u00862/2, (2.48)where \u00E2\u008C\u00AB\u00CB\u0086 = pa\u00E2\u008C\u00AB 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 = \u00002K. 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 = \u0000(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 \u00E2\u008C\u00A7 = 2\u00E2\u0087\u00A1). 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 \u00E2\u0087\u00A2cr will decrease by a factorof 4 during this time. Putting all of these factors together, we approximatethe ratio \u0000c = \u00E2\u0087\u00A2vir/\u00E2\u0087\u00A2\u00C2\u00AFcr of the density of the virialized halo \u00E2\u0087\u00A2vir to the criticaldensity at virialization as \u0000c = 32[1 + \u0000nl(\u00E2\u008C\u00A7 = \u00E2\u0087\u00A1)] = 18\u00E2\u0087\u00A12. This result canbe generalized to a flat universe with both matter and cosmological constant(\u00E2\u008C\u00A6m + \u00E2\u008C\u00A6\u00E2\u0087\u00A4 = 1) with the fitting formula [36]\u0000c = 18\u00E2\u0087\u00A12 + 82d\u0000 39d2, (2.49)where d \u00E2\u008C\u0098 \u00E2\u008C\u00A6zm \u0000 1 and \u00E2\u008C\u00A6zm is the matter density parameter at redshift z,which in this case is given by\u00E2\u008C\u00A6zm = \u00E2\u008C\u00A6m(1 + z)3\u00E2\u008C\u00A6m(1 + z)3 + \u00E2\u008C\u00A6\u00E2\u0087\u00A4 . (2.50)We can now write the the virialized radius asrvir = 1.49\u00E2\u009C\u0093 h0.7\u00E2\u0097\u0086\u00002/3\u00E2\u009C\u0093\u00E2\u008C\u00A6m0.3\u00E2\u0097\u0086\u00001/3\u00E2\u009C\u0093 1\u00E2\u008C\u00A6zm \u0000c18\u00E2\u0087\u00A12\u00E2\u0097\u0086\u00001/3\u00E2\u0087\u00A5\u00E2\u009C\u00931 + z10\u00E2\u0097\u0086\u00001\u00E2\u009C\u0093 M108 M\u0000\u00E2\u0097\u00861/3 kpc (2.51)and can subsequently find the corresponding circular velocity Vc = pGM/rvirand can define the virial temperature as Tvir = \u00C2\u00B5mpV 2c /2kb, where \u00C2\u00B5 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\u00E2\u0087\u00A5 107 \u00E2\u0087\u00A3 \u00C2\u00B50.6\u00E2\u008C\u0098\u00003/2\u00E2\u009C\u0093 h0.7\u00E2\u0097\u0086\u00001\u00E2\u009C\u0093\u00E2\u008C\u00A6m0.3\u00E2\u0097\u0086\u00001/2\u00E2\u0087\u00A5\u00E2\u009C\u00931\u00E2\u008C\u00A6zm \u0000c18\u00E2\u0087\u00A12\u00E2\u0097\u0086\u00001/2\u00E2\u009C\u00931 + z10 \u00E2\u0097\u0086\u00003/2\u00E2\u009C\u0093 Tvir104 K\u00E2\u0097\u00863/2 M\u0000. (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\u00E2\u0086\u00B5erent 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\u00001(t). We can then estimate this ratio byH\u000010H\u00001i = aiHia0H0 \u00E2\u0087\u00A0 1028 Timp , (3.1)where the subscript i denotes quantities evaluated at some early \u00E2\u0080\u0098initial\u00E2\u0080\u0099time 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\u00E2\u0086\u00B5erent regions that were causally disconnected fromone another at ti. Decreasing Ti doesn\u00E2\u0080\u0099t 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 \u00E2\u008C\u00A6k is currentlybounded by roughly |\u00E2\u008C\u00A6k| < 0.04. However, a fine-tuning problem ariseswith the realization that in the standard cosmological model \u00E2\u008C\u00A6k increaseswith time, such that \u00E2\u008C\u00A6k must have been initially fine-tuned to an extremelysmall value. We can compare the present-day value of \u00E2\u008C\u00A6k to that at ti bythe fraction\u00E2\u008C\u00A6k(t0)\u00E2\u008C\u00A6k(ti) = \u00E2\u009C\u0093 HiaiH0a0\u00E2\u0097\u00862 \u00E2\u0087\u00A0 1056\u00E2\u009C\u0093 Timp\u00E2\u0097\u00862 , (3.2)again assuming that the Universe is dominated by radiation at ti. Thus,\u00E2\u008C\u00A6k(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\u00001 monotoni-cally increases with \u00E2\u008C\u0098, and \u00E2\u008C\u0098 and H\u00001 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\u00E2\u0086\u00B5erence between theHubble length and particle horizon. Unlike the particle horizon, which in-creases monotonically (this is why we can use \u00E2\u008C\u0098 as a time variable), theHubble length can decrease. Examining Eq. (2.5) again, if w < \u00001/3 thenH\u00001 would decrease as \u00E2\u008C\u0098 increases, so that a su\u0000ciently long stage with suchan equation of state would create a drastic di\u00E2\u0086\u00B5erence between H\u00001 and \u00E2\u008C\u0098.During inflation, the comoving Hubble length \u00E2\u0080\u0098zooms in\u00E2\u0080\u0099 to a much smallerscale than at the start of inflation. Spacetimes with \u00001 < w < \u00001/3 havean event horizon 27, whose comoving length is of order H\u00001. A (comoving)scale k \u00E2\u0080\u0098leaves the horizon\u00E2\u0080\u0099 when k \u00E2\u0087\u00A0 H; communication on this scale ispossible before this time but not after. After its rapid decrease during infla-tion, H\u00001 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\u00001) 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\u00001.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 \u00E2\u0080\u0098initial\u00E2\u0080\u0099 time ti, we require H\u000010 \u00EF\u00A3\u00BF H\u00001i , 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 \u00001/3.303.4. A Simple Modelary paradigm ti is before inflation. This ratio is nowH\u000010H\u00001i = aiae aea0 HiH0 = e\u0000N aea0 HiH0 \u00E2\u0087\u00A0 e\u0000N1028 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 \u00E2\u008C\u0098 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\u000010 \u00EF\u00A3\u00BF H\u00001i necessitates at least N \u00E2\u0087\u00A0 64 e-folds of inflation.29It is easy to see that the requirement of H\u000010 \u00EF\u00A3\u00BF H\u00001i solves both the horizonand flatness problems.3.4 A Simple ModelFrom Eq. (2.3b), we see that having \u00001 < w < \u00001/3 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@\u00E2\u0086\u00B5'@\u00E2\u0086\u00B5'\u0000 V ('). (3.4)The stress-energy tensor T\u00C2\u00B5\u00E2\u008C\u00AB = (@L/@(@\u00C2\u00B5'))@\u00E2\u008C\u00AB'\u0000 L\u0000\u00C2\u00B5\u00E2\u008C\u00AB in this case is thenT\u00C2\u00B5\u00E2\u008C\u00AB = @\u00C2\u00B5'@\u00E2\u008C\u00AB'\u0000 \u00EF\u00A3\u00BF12@\u00E2\u0086\u00B5'@\u00E2\u0086\u00B5'\u0000 V (')\u0000 \u0000\u00C2\u00B5\u00E2\u008C\u00AB . (3.5)At this point, we decompose our field as '(x, t) = '\u00C2\u00AF(t) + \u0000'(x, t), assumingthe homogenous part '\u00C2\u00AF is much larger than \u0000', which is treated as a per-turbation. Parameterizing the stress-energy tensor as in Section 2.1.1, wecan identify the background energy density \u00E2\u0087\u00A2 and pressure P as\u00E2\u0087\u00A2 = 12a\u00002'\u00C2\u00AF02 + V, P = 12a\u00002'\u00C2\u00AF02 \u0000 V, (3.6)and note that all o\u00E2\u0086\u00B5-diagonal terms in T\u00C2\u00AF\u00C2\u00B5\u00E2\u008C\u00AB are zero. The equation of statefor the field is then w = 12a\u00002'\u00C2\u00AF02 \u0000 V12a\u00002'\u00C2\u00AF02 + V . (3.7)28See Section 4.6.2 for a more rigorous expression for Eq. (3.3).29See Section 4.6.2 for some caveats.313.5. End of Inflation and ReheatingWe can now see that w \u00E2\u0087\u00A1 \u00001 if we have '\u00C2\u00AF0 \u00E2\u008C\u00A7 V . To formalize thisapproximation, we introduce the \u00E2\u0080\u0098slow-roll\u00E2\u0080\u0099 variables\u00E2\u009C\u008F = \u0000 H\u00CB\u0099H2 , \u00E2\u008C\u00A7 = \u0000 '\u00C2\u00A8H'\u00CB\u0099 . (3.8)When \u00E2\u009C\u008F, \u00E2\u008C\u00A7 \u00E2\u008C\u00A7 1, the field ' is said to be in the slow-roll regime and inflationends when one of these conditions is violated. When \u00E2\u009C\u008F \u00E2\u008C\u00A7 1, the equation ofstate is approximately w \u00E2\u0087\u00A1 \u00001 + 23\u00E2\u009C\u008F and thus the background expands nearthe de Sitter solution. In this case, the scale factor and conformal Hubblerate approximately evolve asa / (\u0000\u00E2\u008C\u0098)\u0000(1+\u00E2\u009C\u008F), H = \u00001 + \u00E2\u009C\u008F\u00E2\u008C\u0098 , (3.9)where the conformal time is bounded by \u00001 < \u00E2\u008C\u0098 < 0.The evolution of the field ' can be found using the Klein-Gordon equa-tion, which for the background field implies'\u00C2\u00AF00 + 3H'\u00C2\u00AF0 + a2V,' = 0 (3.10)and from Eq. (2.3a), the Hubble rate in this case is given by 30H2 = 8\u00E2\u0087\u00A1G3\u00EF\u00A3\u00BF12a\u00002'\u00C2\u00AF02 + V \u0000 . (3.11)Using these equations we can find the values of the slow-roll variables for aparticular inflationary potential. For example, for power-law inflation wherethe scale factor evolves as a / tp and p > 1, we have \u00E2\u009C\u008F = \u00E2\u008C\u00A7 = p\u00001.3.5 End of Inflation and ReheatingAt some point, inflation must end and produce the standard model particlesthat comprise the present-day visible Universe. In terms of the slow-rollvariables introduced in the previous section, inflation ends when at leastone of the slow-roll variables attains a value comparable to unity. Afterthis point, standard model particles must be produced in some manner,usually through the decay of the inflating substance, and then thermalize in aprocess known as reheating. The standard hot big bang cosmological modelthen commences after the end of reheating. The thermalized temperatureattained by the produced particles is referred to as the reheat temperatureTRH, which must be at least TRH & 5\u000010MeV in order for BBN to proceedsuccessfully [3].30We have assumed a flat background here, which should be valid once inflation haslasted long enough to drive the background to a nearly flat state.323.6. Generation of Perturbations3.6 Generation of PerturbationsA key feature of inflation is that it generates the perturbations that, for ex-ample, source the anisotropies in the CMB and give the initial conditions forlarge-scale structure formation. While inflation wipes away pre-inflationaryfeatures, the accelerating background quantum-mechanically excites pertur-bations that get stretched to superhorizon scales.3.6.1 QuantizationFollowing Ref. [9], our prescription for properly quantizing the perturbationsthat arise during inflation will be to write the inflationary action in the formof a harmonic oscillator so we can use the same well-known quantizationprocedure as used for the harmonic oscillator. We can write the total actionas S = Sgr + Sm, where Sgr / R Rp\u0000gd4x is the GR action, with g ithedeterminant of the metric, and Sm is the action for the matter present inthe inflationary spacetime. As we wish to recover equations of motion forour perturbations to linear order, we expand the action to second order inthe perturbations variables. For definiteness, we continue to examine theinflationary scenario of a single scalar field.As per usual, we examine the quantization of scalar and tensor pertur-bations separately and forgo examining the vector perturbations, as theydecay very rapidly. We start with the scalar perturbations. Obtaining thesecond order scalar parts of Sgr and Sm is straightforward, yet very tedious,so we will simply quote the result for the second order terms in the totalaction \u00002S, which is [9]\u00002S = 12 Z \u00E2\u0087\u00A5u02 \u0000 u2,i \u0000m2e\u00E2\u0086\u00B5,Su2\u00E2\u0087\u00A4 d4x, (3.12)where \u00002 denotes the second order terms. The gauge-invariant variable u,commonly known as the Mukhanov-Sasaki variable, is given by u = a(\u0000'+('\u00C2\u00AF0/H) ) and will be our canonical variable for the quantization procedure.The action in Eq. (3.12) was written in a suggestive way by defining ane\u00E2\u0086\u00B5ective mass term, given by m2e\u00E2\u0086\u00B5,S = \u0000z00z , (3.13)where z = a'\u00C2\u00AF0/H, as to make the analogy with a harmonic oscillator moreapparent. In general, the e\u00E2\u0086\u00B5ective mass will be a function of time.333.6. Generation of PerturbationsTurning now to the tensor perturbations, the Ricci scalar R can be foundfor the tensor perturbations (hT)ij without much di\u0000culty, yielding\u00002Sgr = 124l2 Z a2 h(hT)i0j (hT)j0i \u0000 (hT)ij,l(hT)j,li i d4x (3.14)for the gravitation part of the action, where l = p8\u00E2\u0087\u00A1G/3 is the Plancklength. The second order tensor part of the action for a single scalar fieldvanishes. By expressing hij in terms of the individual polarization states hp,where (hT)ij(hT)ji = 2Pp(hTp )2, the total tensor part of the action can bewritten as \u00002S = 12 Z \u00E2\u0087\u00A5U 02p \u0000 U2p,i \u0000m2e\u00E2\u0086\u00B5,TU2p \u00E2\u0087\u00A4 d4x, (3.15)where we have defined the canonical variable Up = ahp/p6l2 for each polar-ization state. The e\u00E2\u0086\u00B5ective mass for the tensors is given bym2e\u00E2\u0086\u00B5,T = \u0000a00a . (3.16)We see that the actions for the scalar and tensor modes are very similar, withonly the e\u00E2\u0086\u00B5ective mass for each perturbation type being slightly di\u00E2\u0086\u00B5erentfrom one another. We continue our discussion using the notation for thescalar modes (except we will drop the subscript S on m2e\u00E2\u0086\u00B5), with the tensorcase found by a trivial change of variables.Varying the action with respect to u yields the equation of motionu00 \u0000r2u+m2e\u00E2\u0086\u00B5u = 0, (3.17)which can readily be identified as having the form the action for a harmonicoscillator with time-varying mass.The next step is to identify the conjugate momentum \u00E2\u0087\u00A1 to u, found tobe \u00E2\u0087\u00A1 = @L@u0 = u0. (3.18)We can now promote u and \u00E2\u0087\u00A1 to operators u\u00CB\u0086 and \u00E2\u0087\u00A1\u00CB\u0086 and impose the standardcommutation relations[u\u00CB\u0086(x, \u00E2\u008C\u0098), u\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = [\u00E2\u0087\u00A1\u00CB\u0086(x, \u00E2\u008C\u0098), \u00E2\u0087\u00A1\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = 0[u\u00CB\u0086(x, \u00E2\u008C\u0098), \u00E2\u0087\u00A1\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = i\u0000(x\u0000 x\u00CB\u009C) (3.19)343.6. Generation of PerturbationsWe can write u\u00CB\u0086(x, \u00E2\u008C\u0098) in terms of the creation and annihilation operators a\u00CB\u0086\u00E2\u0080\u00A0kand a\u00CB\u0086k for a mode k as 31u\u00CB\u0086(x, \u00E2\u008C\u0098) = 1p2Z d3k(2\u00E2\u0087\u00A1)3/2 ha\u00CB\u0086k\u0000\u00E2\u0087\u00A4k(\u00E2\u008C\u0098)eik\u00C2\u00B7x + a\u00CB\u0086\u00E2\u0080\u00A0k\u0000k(\u00E2\u008C\u0098)e\u0000ik\u00C2\u00B7xi . (3.20)The mode function \u0000k(\u00E2\u008C\u0098) obeys\u000000k + \u00E2\u0087\u00A5k2 +m2e\u00E2\u0086\u00B5\u00E2\u0087\u00A4\u0000k = 0 (3.21)and the commutation relations in Eq. (3.19) imply the normalization\u00000k\u0000\u00E2\u0087\u00A4k \u0000 \u0000k\u0000\u00E2\u0087\u00A40k = 2i. (3.22)With this normalization condition, the second order di\u00E2\u0086\u00B5erential equa-tion for the mode function given in Eq. (3.21) has one remaining integrationconstant, the choice of which determines the e\u00E2\u0086\u00B5ect of the creation and anni-hilation operators on a physical state, which we now explore in more detail.Before proceeding any further, we remark that when quantizing fields inframework of general relativity, the notion of a state, in particular the vac-uum state, is dependent on one\u00E2\u0080\u0099s frame of reference [40, 41, 42, 43]. Indeed,two di\u00E2\u0086\u00B5erent observers can each define a set of mode functions that definesthe vacuum state within their frame of reference, but these will not necessar-ily coincide with the vacuum state defined in the other\u00E2\u0080\u0099s frame of reference(see [44, 45] for more details).In the present context of an expanding FLRW spacetime, this can beseen by defining a vacuum state |0i for a particular annihilation operator a\u00CB\u0086kwhich yields the minimum energy eigenvalue of the HamiltonianH\u00CB\u0086 = 12Z d3k \u00E2\u0087\u00A5\u00E2\u0087\u00A1\u00CB\u0086k\u00E2\u0087\u00A1\u00CB\u0086-k + !2ku\u00CB\u0086ku\u00CB\u0086-k\u00E2\u0087\u00A4 (3.23)at a particular time, where !2k(\u00E2\u008C\u0098) = k2+m2e\u00E2\u0086\u00B5(\u00E2\u008C\u0098). The last point is significantin that it is not guaranteed that the mode function will evolve in such a waythat this state will be the state of lowest energy at a later time. Acting withthe Hamiltonian on this state yields the energy density [44]\u00E2\u0087\u00A2 = 14Z d3k(|\u00000k|2 + !2k|\u0000k|2). (3.24)At a particular time \u00E2\u008C\u0098p, for !2k(\u00E2\u008C\u0098p) > 0 the mode function\u0000k(\u00E2\u008C\u0098p) = 1p!k(\u00E2\u008C\u0098p)eik\u00E2\u008C\u0098p , \u00000k(\u00E2\u008C\u0098p) = iq!k(\u00E2\u008C\u0098p)eik\u00E2\u008C\u0098p (3.25)31Here we use the Fourier conventions f(x) = (2\u00E2\u0087\u00A1)\u00003/2 R d3kfkeik\u00C2\u00B7x.353.6. Generation of Perturbationsminimizes Eq. (3.24) at \u00E2\u008C\u0098 = \u00E2\u008C\u0098p.32 However, if at a later time we have!2k(\u00E2\u008C\u0098p) < 0 then not only is this state no longer the minimum energy state,but there is not even a clearly defined minimum energy state for such amode at this time. If the mode function evolves in this manner, then modesdefined to be in the vacuum state when well within the horizon (k \u0000 H)will be in an excited state when well outside the horizon (k \u00E2\u008C\u00A7 H).Since it is not clear how to use our quantization procedure if one cannotclearly define a vacuum state, we assume that all modes of interest are wellwithin the horizon at some \u00E2\u0080\u0098initial\u00E2\u0080\u0099 time. We choose the mode function suchthat it selects the instantaneous vacuum state when the mode is well withinthe horizon. From Eq. (3.25), the mode function that minimizes the energydensity when well within the horizon at time \u00E2\u008C\u0098i is given by\u0000k(\u00E2\u008C\u0098i) \u00E2\u0087\u00A1 1pkeik\u00E2\u008C\u0098i , \u00000k(\u00E2\u008C\u0098i) \u00E2\u0087\u00A1 ipkeik\u00E2\u008C\u0098i . (3.26)The state selected by this initial condition is known as the Bunch-Davisvacuum [46], and approximates the Minkowski vacuum when the mode iswell within the horizon.The final step for determining the mode function is to specify me\u00E2\u0086\u00B5 . Formost cases of interest, me\u00E2\u0086\u00B5 will be proportional to \u00E2\u008C\u0098\u00002. For example, fora single scalar field in the slow-roll regime characterized by the slow-rollvariables in Eq. (3.8), the scalar e\u00E2\u0086\u00B5ective massive term is me\u00E2\u0086\u00B5,S \u00E2\u0087\u00A1 \u0000(2 +6\u00E2\u009C\u008F \u0000 3\u00E2\u008C\u00A7)/\u00E2\u008C\u00982. For me\u00E2\u0086\u00B5 = (14 \u0000 \u00E2\u008C\u00AB2)/\u00E2\u008C\u00982 with \u00E2\u008C\u00AB constant, from Eq. (3.21) themode function is found to be\u0000k = r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 hC1H(1)\u00E2\u008C\u00AB (k|\u00E2\u008C\u0098|) + C2H(2)\u00E2\u008C\u00AB (k|\u00E2\u008C\u0098|)i , (3.27)whereH(1)\u00E2\u008C\u00AB andH(2)\u00E2\u008C\u00AB are the Hankel functions of the first and second kind andC1 and C2 are integration constants. The Bunch-Davis vacuum is selectedby choosing C1 = 0, C2 = 1, so that the mode function is\u0000k = r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 H(2)\u00E2\u008C\u00AB (k|\u00E2\u008C\u0098|). (3.28)3.6.2 Beyond the HorizonNow that we have the mode function, we can form the two-point correlationfunction h0|u\u00CB\u0086(x, \u00E2\u008C\u0098)u\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)|0i with the Bunch-Davis vacuum state, which can32Any phase of \u0000k will minimize the energy density. We set the phase as eik\u00E2\u008C\u0098 for laterconvenience.363.6. Generation of Perturbationsbe written in terms of the mode function ash0|u\u00CB\u0086(x, \u00E2\u008C\u0098)u\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)|0i = Z dk k24\u00E2\u0087\u00A12 |\u0000k(\u00E2\u008C\u0098)|2 sin(k|x\u0000 x\u00CB\u009C|)k|x\u0000 x\u00CB\u009C| . (3.29)The power spectrum Pu(k) = k34\u00E2\u0087\u00A12 |\u0000k|2 for u is thenPu(k) = k38\u00E2\u0087\u00A1 |\u00E2\u008C\u0098|\u0000\u0000H(2)\u00E2\u008C\u00AB (k|\u00E2\u008C\u0098|)\u0000\u00002. (3.30)In general, we are interested in modes that are on superhorizon scalesat the end of inflation. Although the details of the end of inflation andreheating may be very complicated, we can circumvent the need to knowthese details by introducing a few new perturbation variables. We definethe gauge-invariant variable R asR \u00E2\u008C\u0098 Hv + , (3.31)which is related to the scalar canonical variable by u = zR. This canbe interpreted as the curvature perturbation in a comoving gauge or thevelocity perturbation on uniform curvature hypersurfaces. Now introducethe gauge-invariant variable \u00E2\u0087\u00A3, defined by\u00E2\u0087\u00A3 = +H\u0000\u00E2\u0087\u00A2\u00E2\u0087\u00A20 , (3.32)which can be interpreted as the either the curvature perturbation on uniformdensity hypersurfaces or the density perturbation on uniform curvature hy-persurfaces. From the Einstein equation Eq. (2.28a), R and \u00E2\u0087\u00A3 can be relatedby3\u0000(R\u0000 \u00E2\u0087\u00A3) = r2 . (3.33)In many situations, we can neglect the e\u00E2\u0086\u00B5ect of spatial derivatives when amode is on superhorizon scales, in which case Rk \u00E2\u0087\u00A1 \u00E2\u0087\u00A3k for a superhorizonmode k.33 By using the Einstein equation Eq. (2.28b) and momentum con-servation equation in Eq. (2.29b) in the comoving gauge, we can readilyarrive at the equationR0H=11 + w \u0000Pnad\u00E2\u0087\u00A2 + 3@P@\u00E2\u0087\u00A2 (R\u0000 \u00E2\u0087\u00A3) + 23 w1 + wr2\u00E2\u0087\u00A7 (3.34)for the evolution of R. We can now see that in the absence of nonadiabaticpressure and anisotropic stress, if we have Rk \u00E2\u0087\u00A1 \u00E2\u0087\u00A3k on superhorizon scales,33See Section 4.8 for a case where this is not a valid approximation.373.6. Generation of Perturbationsthen Rk (and therefore \u00E2\u0087\u00A3k) will be approximately constant on superhorizonscales. Therefore, as long as these conditions hold, by keeping track ofthe superhorizon values of R (or equivalently \u00E2\u0087\u00A3) during inflation, we do notrequire the details of reheating to track a mode into the radiation dominatedera.In this light, the power spectrum PR for R for modes that are on super-horizon scales at the end of inflation is of particular interest. PR is mostoften described by its value at the pivot scale kp = 0.002Mpc\u00001 and by itsspectral index ns = 1 + d lnPR/d ln k evaluated at some scale (usually kp).A scale-invariant scalar spectrum corresponds to ns = 1. When a mode ison superhorizon scales, we can use the small argument approximation forthe Hankel function so that \u0000k / k\u0000\u00E2\u008C\u00AB and ns = 4\u0000 2\u00E2\u008C\u00AB. Using the slow-rollvariables, the scalar spectral index is expressed as ns = 1\u0000 4\u00E2\u009C\u008F+2\u00E2\u008C\u00A7 . We cansee that slow-roll inflation produces a nearly scale-invariant scalar spectrum(consistent with observations), which arises because the conditions when amode leaves the horizon are approximately the same for all modes of interest.By use of Eq. (2.30), it can easily be shown that for the same conditionsas described above, the tensor perturbations (hT)ij remain approximatelyconstant on superhorizon scales. The power spectrum PT for hT can bedefined in the same manner as done for the scalar perturbations, with ashape characterized by the spectral index nT = d lnPT/d ln k. For inflationdriven by a single scalar field in slow-roll, the tensor spectral index is givenby nT = \u00002\u00E2\u009C\u008F. Of prime interest is the ratio between scalar and tensor modesr = PT/PR.We have finally arrived at the primordial spectrum for the scalar andtensor perturbations (at least for the simple model described in Section3.4). With this information, we can continue to follow the perturbationsinto later times using the tools described in Chapter 2.38Chapter 4Inflation with an ElasticSolid4.1 IntroductionThe inflationary paradigm, the existence of a brief period of acceleratedexpansion in the early Universe, provides an explanation for the observedhomogeneity, isotropy and flatness of the Universe [37, 38, 39]. On largescales it successfully accounts for the distribution of fluctuations seen in thecosmic microwave background (CMB) and the large-scale structure of theUniverse. Inflation is often modelled in terms of a scalar field slowly evolv-ing in its potential. Yet, the physical model of inflation is not known andeven within the context of scalar fields many models are compatible withcurrent observations. It is worthwhile exploring whether or not other phys-ical frameworks, more general than a scalar field, can successfully accountfor a period of inflation.In this chapter, we build a model of inflation that describes the substancethat drives inflation by a continuous medium that can be characterized byits macroscopic properties. The simplest model of a continuous medium ingeneral relativity is a perfect fluid. To drive an accelerating expansion, themedium must have an equation of state w \u00E2\u008C\u0098 P/\u00E2\u0087\u00A2 < \u00001/3, where \u00E2\u0087\u00A2 and P arethe energy density and pressure of the fluid, respectively. However, a perfectfluid with constant w has a sound speed for longitudinal (density) wavesof cs = pw, so demanding that the fluid drive an accelerated expansionformally results in an imaginary sound speed and an instability to smallperturbations.One generalization of a perfect fluid is a relativistic elastic solid. Elasticsolids have a rigidity, and so can support both longitudinal and transversewaves. An elastic solid (both relativistic and nonrelativistic) can be char-acterized by a bulk modulus \u00EF\u00A3\u00BF that depends on the equation of state w,and shear modulus \u00C2\u00B5 that determines how rigid the solid is [47]. As in thenonrelativistic case, the longitudinal sound speed cs depends upon both \u00EF\u00A3\u00BF394.1. Introductionand \u00C2\u00B5, while the transverse sound speed cv only depends upon \u00C2\u00B5. In therelativistic case, a su\u0000ciently rigid elastic solid can result in a real longi-tudinal sound speed cs, even in cases where w is negative enough to driveacceleration.In this chapter we describe a model of a homogeneous and isotropicelastic solid coupled to general relativity. This model has previously beenconsidered as a potential model of dark energy [48, 49] and recently similar-ities between a relativistic elastic solid and massive gravity have been noted[50].In this work, we discuss in detail how an elastic solid can drive an in-flationary epoch in the early Universe.34 Linear perturbations in an elasticsolid satisfy the equations of motion found in Ref. [49], which uses the frame-work for describing a macroscopic relativistic medium developed in Ref. [54].We develop the quadratic action for a generic elastic medium, quantize thelinear modes that are excited during inflation, and determine the spectraof scalar and tensor modes produced by an inflationary stage driven by anelastic solid.A novel feature of this model is that, in contrast to what typically occurswhen the Universe is dominated by a single substance, the anisotropic stressof the solid causes modes to evolve on superhorizon scales. As such, the finalspectrum of superhorizon modes is sensitive to the manner in which inflationends. We show here that the case where the sound speeds and equation ofstate are perfectly constant results in a blue-tilted scalar power spectrum,but if these quantities vary slowly in time then a red-tilted scalar power34 The notion that a relativistic elastic solid could drive inflation was first discussedin Ref. [51] and more recently in Refs. [52, 53]. The present work includes an in-depthtreatment of the quantization of the scalar and tensor linear perturbations and theirsuperhorizon evolution. Furthermore, new states of the elastic solid are found in whichits equation of state is far from the fiducial value of \u00001 that nonetheless produces nearlyscale-invariant spectra for the linear perturbations. In comparison to the aforementionedreferences, our work uses a di\u00E2\u0086\u00B5erent approach and treats the problem starting directly fromthe quadratic action for an elastic solid and includes an extended treatment of superhorizonevolution and reheating. The work of Ref. [52] uses an e\u00E2\u0086\u00B5ective field theory approachthat involves the presence of three scalar fields, which if certain symmetries are appliedgives the same physical behaviour as an elastic solid. Although the e\u00E2\u0086\u00B5ective field theoryapproach used in Ref. [52] is distinct from the analysis given here, the same equations ofmotion are reached for both models for cases where w and cs evolve slowly near \u00001 and 0,respectively. Specifically, the equations of motion in Ref. [52] can be recovered using thosefound in this chapter by setting cs0 = \u00E2\u009C\u008F0 = 0 and making the notational substitutionscs,1 ! cL,c, \u00E2\u009C\u008F1 ! \u00E2\u009C\u008Fc, \u00E2\u008C\u00A7s ! sc, and \u00E2\u008C\u00A7\u00E2\u009C\u008F ! \u00E2\u008C\u0098c. In addition, Ref. [52] includes a discussion ofnon-Gaussianities not included in this chapter. Ref. [53] has also recently examined theimplications of anisotropic superhorizon evolution in an inflating elastic solid.404.2. Einstein Equationsspectrum is possible. Interestingly, we find here that in models with slowlyevolving material properties a scalar spectral index near ns . 1, compatiblewith current observational constraints, can be found for w relatively far fromthe nominal inflationary value w ' \u00001.While we do not specify a particular microphysical model for or forma-tion mechanism of the elastic solid, we note that relativistic elastic solidshave been used to model a variety of physical systems including networksof topological defects [55, 56]. A frustrated network of topological defects[57, 58] could potentially form a system with a negative equation of statethat is su\u0000ciently stable to provide the \u00E2\u0087\u00A0 60 e-folds of inflation needed.These systems lack a preferred length scale and as such may undergo vaststretching without fracturing [48].This chapter is organized as follows: In Sections 4.2 and 4.3 we reviewthe relevant Einstein equations and linearized perturbation equations for arelativistic elastic solid. In Section 4.4 we derive the action for the scalar andtensor linear perturbations of a relativistic elastic solid. The superhorizonevolution in this model is discussed in Section 4.5 and its application to aperiod of inflation in the early Universe is discussed in Sections 4.6 to 4.8.4.2 Einstein EquationsWe consider the flat Friedmann-Robertson-Walker (FRW) metric 35ds2 = a2(\u00E2\u008C\u0098)(\u00E2\u008C\u0098\u00E2\u0086\u00B5\u0000 \u0000 h\u00E2\u0086\u00B5\u0000)dx\u00E2\u0086\u00B5dx\u0000 , (4.1)where \u00E2\u008C\u0098 is the conformal time, \u00E2\u008C\u0098\u00E2\u0086\u00B5\u0000 is the metric for Minkowski space andthe tensor h\u00E2\u0086\u00B5\u0000 represents small perturbations to \u00E2\u008C\u0098\u00E2\u0086\u00B5\u0000 . The energy density \u00E2\u0087\u00A2and pressure P of the background in a flat universe can be expressed as\u00E2\u0087\u00A2 = H2l2a2 , P = \u00002H0 +H23l2a2 , (4.2)where a prime 0 represents a derivative with respect to the conformal time \u00E2\u008C\u0098,H = a0/a = aH, H is the Hubble parameter, and l = p8\u00E2\u0087\u00A1G/3 is the Plancklength. We assume that we can parameterize the pressure by P = w\u00E2\u0087\u00A2, wherew is referred to as the equation of state, so that with Eq. (4.2) we can formthe di\u00E2\u0086\u00B5erential equation for HH0 = \u00001 + 3w2H2. (4.3)35We use signature (+,-,-,-) and units in which c = ~ = kb = 1. Our notation largelyfollows Ref. [9].414.2. Einstein EquationsUsing the Friedmann equation\u00E2\u0087\u00A20 = \u00003H\u00E2\u0087\u00A2(1 + w), (4.4)the relationship between dP/d\u00E2\u0087\u00A2 and w is found to bedPd\u00E2\u0087\u00A2 = w \u0000 w03H(1 + w) . (4.5)We parameterize the metric in Eq. (4.1) asds2 = a2(\u00E2\u008C\u0098) \u00E2\u0087\u00A5(1 + 2\u0000)d\u00E2\u008C\u00982 \u0000 2B,i dxid\u00E2\u008C\u0098\u0000((1 + h/3)\u0000ij + 2Eij)dxidxj\u00E2\u0087\u00A4 , (4.6)where h is the trace of the spatial part of the metric perturbation and Eijis traceless. If we decompose the tensor Eij into scalar, vector, and tensorparts, then the scalar component of the spatial part of h\u00E2\u0086\u00B5\u0000 ishSij = h3 \u0000ij + 2(@i@j \u0000 13\u0000ijr2)E = \u00002 \u0000ij + 2E,ij , (4.7)where \u00E2\u008C\u0098 \u000016h + 13r2E is the curvature perturbation and we denote thetensor part of 2Eij by the conventional notation hTij , which in addition tobeing traceless is transverse (hTij,i = 0).The stress-energy tensor is parameterized in the standard form as\u0000T 00 = \u0000\u00E2\u0087\u00A2, (4.8a)\u0000T i0 = (\u00E2\u0087\u00A2 + P )vi, (4.8b)\u0000T ij = \u0000(\u0000P \u0000ij + P\u00E2\u0087\u00A7ij), (4.8c)where \u0000\u00E2\u0087\u00A2 and \u0000P are the energy density and pressure perturbations, respec-tively, vi is the velocity perturbation, and \u00E2\u0087\u00A7ij is the anisotropic stress.The gauge-invariant Einstein equations for the scalar perturbations arer2 \u0000 3H( 0 +H\u0000) = 32l2a2\u0000\u00E2\u0087\u00A2(gi), (4.9a) 0 +H\u0000 = \u0000v(gi), (4.9b) 00 +H(\u00000 + 2 0) + (H2 + 2H0)\u0000+ 13r2(\u0000\u0000 ) = 32l2a2\u0000P (gi), (4.9c) \u0000 \u0000 = 3l2a2P\u00E2\u0087\u00A7, (4.9d)424.3. Elastic Solidwhere \u0000 = H2\u0000H0 = 32 l2a2(\u00E2\u0087\u00A2+P ), and the energy-momentum conservationequations are\u0000(gi)0 \u0000 (1 + w)(r2v(gi) + 3 0) + 3H \u0000P (gi)\u00E2\u0087\u00A2 \u0000 w\u0000(gi)! = 0, (4.10a)v(gi)0 +H(1\u0000 3w)v(gi) + w01 + wv(gi)\u0000\u0000P (gi)\u00E2\u0087\u00A2 + P \u0000 \u0000\u0000 23 w1 + wr2\u00E2\u0087\u00A7 = 0, (4.10b)where the gauge-invariant perturbation variables are defined as\u0000 = \u0000 +H(B \u0000 E0) + (B \u0000 E0)0, (4.11a) = \u0000H(B \u0000 E0), (4.11b)\u0000\u00E2\u0087\u00A2(gi) = \u0000\u00E2\u0087\u00A2 + \u00E2\u0087\u00A20(B \u0000 E0), (4.11c)\u0000P (gi) = \u0000P + P 0(B \u0000 E0), (4.11d)v(gi) = v +B \u0000 E0, (4.11e)noting that \u00E2\u0087\u00A7 is already a gauge-invariant quantity.The sole Einstein equation for the tensor perturbations is(hT)i00j + 2H(hT)i0j \u0000r2(hT)ij = 6l2a2P (\u00E2\u0087\u00A7T)ij . (4.12)4.3 Elastic SolidIn this section, we briefly summarize the formalism in Ref. [54] for a contin-uous relativistic medium and the findings of Ref. [49] for the linear pertur-bations in an isotropic relativistic elastic solid.As shown in Ref. [54], the behaviour of a continuous relativistic mediumcan be described by the use of two di\u00E2\u0086\u00B5erent manifolds: a three-dimensionalmanifold F used to characterize the internal state of the medium and a four-dimensional spacetime manifoldM used to describe its relativistic evolution.A projection P : M! F is used to project timelike lines inM onto points inthe material space on F . This can be interpreted as projecting the worldlineof a \u00E2\u0080\u0098particle\u00E2\u0080\u0099 of the medium onto a single point in the material space. Theinternal properties of the medium are characterized through tensors definedon F that are then mapped onto M via the inverse image P\u00001. We use434.3. Elastic Soliduppercase Latin letters A,B, . . . and lowercase Greek letters \u00C2\u00B5, \u00E2\u008C\u00AB, . . . to labelthe indices of tensors defined on F and M, respectively.As the four-demensional projection tensor \u0000\u00C2\u00B5\u00E2\u008C\u00AB = g\u00C2\u00B5\u00E2\u008C\u00AB\u0000u\u00C2\u00B5u\u00E2\u008C\u00AB can be usedto find the distance between adjacent particles in their local rest frame,\u0000\u00C2\u00B5\u00E2\u008C\u00AB and its material space counterpart \u0000AB characterize the strain of themedium. Working in material space, we assume that the energy density \u00E2\u0087\u00A2and pressure tensor PAB can be expressed in terms of the strain tensor \u0000ABand are related by @(p|\u0000|\u00E2\u0087\u00A2) = \u0000p|\u0000|2PAB@\u0000AB, (4.13a)@(p|\u0000|PAB) = \u0000p|\u0000|2EABCD@\u0000CD, (4.13b)in close analogy to the classical case, where |\u0000| is the determinant of \u0000AB.The elasticity tensor EABCD has been introduced in Eq. (4.13b) to relatestress and strain tensors, as in the classical case.As shown in Ref. [49], specifying the pressure and elasticity tensors issu\u0000cient for describing the behaviour of linear perturbations in a relativisticelastic solid. By relating derivatives in F and M, the spacetime pressureand elasticity tensors for an isotropic elastic solid areP\u00C2\u00B5\u00E2\u008C\u00AB = P\u0000\u00C2\u00B5\u00E2\u008C\u00AB , (4.14a)E\u00C2\u00B5\u00E2\u008C\u00AB\u00E2\u0087\u00A2\u0000 = \u00E2\u008C\u0083\u00C2\u00B5\u00E2\u008C\u00AB\u00E2\u0087\u00A2\u0000 + \u00E2\u009C\u0093(\u00E2\u0087\u00A2 + P )dPd\u00E2\u0087\u00A2 \u0000 P\u00E2\u0097\u0086 \u0000\u00C2\u00B5\u00E2\u008C\u00AB\u0000\u00E2\u0087\u00A2\u0000 + 2P\u0000\u00C2\u00B5(\u00E2\u0087\u00A2\u0000\u0000)\u00E2\u008C\u00AB , (4.14b)where P is the pressure scalar and \u00E2\u008C\u0083\u00C2\u00B5\u00E2\u008C\u00AB\u00E2\u0087\u00A2\u0000 is the shear tensor given by\u00E2\u008C\u0083\u00C2\u00B5\u00E2\u008C\u00AB\u00E2\u0087\u00A2\u0000 = 2\u00C2\u00B5\u00E2\u009C\u0093\u0000\u00C2\u00B5(\u00E2\u0087\u00A2\u0000\u0000)\u00E2\u008C\u00AB \u0000 13\u0000\u00C2\u00B5\u00E2\u008C\u00AB\u0000\u00E2\u0087\u00A2\u0000\u00E2\u0097\u0086 , (4.15)with \u00C2\u00B5 being the shear modulus. For a perfectly elastic medium, the stress-energy tensor is related to the pressure tensor byT\u00C2\u00B5\u00E2\u008C\u00AB = \u00E2\u0087\u00A2u\u00C2\u00B5u\u00E2\u008C\u00AB + P\u00C2\u00B5\u00E2\u008C\u00AB , (4.16)where u\u00C2\u00B5 are flow vectors tangent to worldlines.An elastic solid has a resistance to compressive and shearing motionsand thus can support both longitudinal and transverse waves, which travelat speeds cs and cv, respectively. In both the relativistic and nonrelativisticcases, cs is dependent upon both the bulk modulus \u00EF\u00A3\u00BF and the shear modulus444.3. Elastic Solid\u00C2\u00B5, while cv is dependent only upon \u00C2\u00B5. In the nonrelativistic case, the soundspeeds and bulk modulus are given by [47]c2s = \u00EF\u00A3\u00BF + 43\u00C2\u00B5\u00E2\u0087\u00A2 , c2v = \u00C2\u00B5\u00E2\u0087\u00A2 , \u00EF\u00A3\u00BF = \u00E2\u0087\u00A2dPd\u00E2\u0087\u00A2 , (4.17)where the energy density \u00E2\u0087\u00A2 is dominated by the mass contribution in thenonrelativistic limit. The sound speeds for the relativistic case can be foundby making the substitution \u00E2\u0087\u00A2 ! \u00E2\u0087\u00A2 + P so that [59]c2s = dPd\u00E2\u0087\u00A2 + 43c2v, c2v = \u00C2\u00B5\u00E2\u0087\u00A2 + P , (4.18)and the bulk modulus is now given by \u00EF\u00A3\u00BF = (\u00E2\u0087\u00A2 + P )dP/d\u00E2\u0087\u00A2. We can see thateven in the case where dP/d\u00E2\u0087\u00A2 is negative, a real value for longitudinal soundspeed cs can be obtained if the rigidity is su\u0000ciently large.We now examine the perturbations in the elastic solid. Perturbationsin a continuous medium can be described by a shift vector \u00E2\u0087\u00A0\u00E2\u0086\u00B5 = \u00E2\u0087\u00A0\u00E2\u0086\u00B5(x\u00000 ), sothat if a particle in a medium is at position x\u00000 when no perturbations arepresent, then the particle would be at position x\u00E2\u0086\u00B5(x\u00000 ) = \u00E2\u0087\u00A0\u00E2\u0086\u00B5(x\u00000 ) + x\u00E2\u0086\u00B50 whenperturbations are present. By use of the above equations, it can be shownthat the linear perturbations that arise in the stress-energy tensor can bewritten in terms of the shift vector \u00E2\u0087\u00A0\u00E2\u0086\u00B5 as [49]\u0000T 00 = \u0000(\u00E2\u0087\u00A2 + P )\u00E2\u009C\u0093\u00E2\u0087\u00A0k,k +h2\u00E2\u0097\u0086 , (4.19a)\u0000T i0 = (\u00E2\u0087\u00A2 + P )\u00E2\u0087\u00A0i0, (4.19b)\u0000T ij = dPd\u00E2\u0087\u00A2 (\u00E2\u0087\u00A2 + P )\u00E2\u009C\u0093\u00E2\u0087\u00A0k,k +h2\u00E2\u0097\u0086 \u0000ij+ \u00C2\u00B5 \u00EF\u00A3\u00BF2\u00E2\u0087\u00A0(i,j) +hij \u0000 23\u0000ij \u00E2\u009C\u0093\u00E2\u0087\u00A0k,k +h2\u00E2\u0097\u0086\u0000 . (4.19c)By comparing these equations to the standard parameterizations given inEq. (4.8), we can make the following identifications:\u0000\u00E2\u0087\u00A2 = \u0000(\u00E2\u0087\u00A2 + P )\u00E2\u009C\u0093\u00E2\u0087\u00A0k,k +h2\u00E2\u0097\u0086 , (4.20a)vi = \u00E2\u0087\u00A0i0, (4.20b)\u0000P = dPd\u00E2\u0087\u00A2 \u0000\u00E2\u0087\u00A2, (4.20c)\u00E2\u0087\u00A7ij = \u0000 \u00C2\u00B5P \u00E2\u009C\u00932\u00E2\u0087\u00A0(i,j) +hij \u0000 23\u0000ij \u00E2\u009C\u0093\u00E2\u0087\u00A0k,k +h2\u00E2\u0097\u0086\u00E2\u0097\u0086 . (4.20d)454.4. ActionWe note that Eq. (4.20c) implies that entropy perturbations are not presentin the solid in the sense that the pressure perturbation is fully specified bythe energy density and not the entropy. Taking the scalar parts of theseequations yields \u0000 = \u0000(1 + w)(r2\u00E2\u0087\u00A0S \u0000 3 +r2E), (4.21a)v = \u00E2\u0087\u00A0S0, (4.21b)r2\u00E2\u0087\u00A7 = 2c2v(1 + w\u00001) \u00E2\u0087\u00A5\u0000r2\u00E2\u0087\u00A0S \u0000r2E\u00E2\u0087\u00A4 , (4.21c)where \u00E2\u0087\u00A0S is the scalar part of the shift vector. Using Eq. (4.21a), we canrewrite the anisotropic stress asr2\u00E2\u0087\u00A7 = 2c2v(1 + w\u00001) \u00EF\u00A3\u00BF \u00001 + w \u0000 3 \u0000= \u00006c2v(1 + w\u00001)\u00E2\u0087\u00A3, (4.22)where we have identified the gauge-invariant variable \u00E2\u0087\u00A3 as\u00E2\u0087\u00A3 \u00E2\u008C\u0098 +H\u0000\u00E2\u0087\u00A2\u00E2\u0087\u00A20 , (4.23)which can be interpreted as the curvature perturbation on uniform densityhypersurfaces or as the density perturbation on uniform curvature hyper-surfaces.The tensor perturbations are simple in comparison. Eq. (4.20) impliesthat the tensor part of the anisotropic stress is simply(\u00E2\u0087\u00A7T )ij = \u0000 \u00C2\u00B5P (hT)ij . (4.24)Having characterized the general properties of our material, we can nowbegin to examine how perturbations are excited in an elastic solid.4.4 ActionTo quantize the linear perturbations in the elastic solid, we start with itsaction and perturb it to second order in the perturbation variables to yieldlinear equations of motion. We decompose the action as S = Sm+Sgr, whereSm and Sgr are the matter and gravitational parts of the action, respectively.The gravitational part of the action is given bySgr = \u0000 16l2 Z Rp\u0000gd4x, (4.25)464.4. Actionwhere g \u00E2\u008C\u0098 det(g\u00C2\u00B5\u00E2\u008C\u00AB) and R is the Ricci scalar. The matter part of the actionfor a continuous medium is given by [60]Sm = \u0000 Z \u00E2\u0087\u00A2totp\u0000gd4x. (4.26)In the above equation, \u00E2\u0087\u00A2tot is the total energy density, which we decomposeas \u00E2\u0087\u00A2tot = \u00E2\u0087\u00A2totf + \u00E2\u0087\u00A2tote , where \u00E2\u0087\u00A2totf is the energy density corresponding to aperfect fluid and \u00E2\u0087\u00A2tote is the additional energy density arising from shearstresses in the elastic solid. The perfect fluid part of the action taken tosecond order in the perturbation variables can be expressed as [9]\u00002Sf = \u0000 Z \u00EF\u00A3\u00BF\u00E2\u0087\u00A2\u00002p\u0000gp\u0000g0 + (\u00E2\u0087\u00A2 + P )\u00E2\u009C\u0093\u00001nn0 \u00001p\u0000gp\u0000g0 + \u00002nn0 \u00E2\u0097\u0086+12dPd\u00E2\u0087\u00A2 (\u00E2\u0087\u00A2 + P )\u00E2\u009C\u0093\u00001nn0 \u00E2\u0097\u00862#p\u0000g0d4x, (4.27)where here the subscript 0 indicates the background value, \u00001 and \u00002 denotethe terms in a variable containing first and second order perturbations, re-spectively, and n is the number density. For a relativistic isotropic elasticsolid, \u00E2\u0087\u00A2tote is given by [48] \u00E2\u0087\u00A2tote = P 24\u00C2\u00B5\u00E2\u0087\u00A7ij\u00E2\u0087\u00A7ij , (4.28)so that the action for the elastic part perturbed to second order is\u00002Se = \u0000 Z a4P 24\u00C2\u00B5 \u00E2\u0087\u00A7ij\u00E2\u0087\u00A7ijd4x. (4.29)4.4.1 Quantization of Scalar ModesUsing the expressions for the action as described above, the scalar part ofthe perfect fluid action, including the gravitational part, perturbed to secondorder can be found to be [9]\u00002Sf + \u00002Sgr = 16l2 Z a2 \"\u00006 02 + 2H\u0000 0 + H2 \u0000 \u00003dPd\u00E2\u0087\u00A2 !\u00002!\u00004( 0 +H\u0000)r2(B \u0000 E0)\u0000 2 ,i (2\u0000,i\u0000 ,i ) + 2\u0000(v,i +B,i )(v,i +B,i )\u00002\u0000 dPd\u00E2\u0087\u00A2 3 \u0000r2E \u0000r2\u00E2\u0087\u00A0S + \u0000dPd\u00E2\u0087\u00A2 !235 d4x.474.4. ActionWe now wish to put the action in canonical form. We will work in thecomoving gauge where v = B = 0. The main reasons for this choice ofgauge are that the action above is simplified greatly in this gauge and thatthe gauge-invariant variable R, defined byR \u00E2\u008C\u0098 Hv + , (4.30)in the comoving gauge is simply related to the metric perturbation byR = and so represents the curvature perturbation in this gauge. If weare able to form an expression solely in terms of in this gauge, the gauge-invariant expression can then be trivially found by substituting R for .In the comoving gauge, the Einstein equations (4.9a) and (4.9b) and themomentum conservation equation (4.10b) arer2( +HE0)\u0000 3H( 0 +H\u0000)\u0000 32H2\u0000 = 0, (4.31a) 0 +H\u0000 = 0, (4.31b)dPd\u00E2\u0087\u00A21 + w \u0000 + \u0000 + 23 w1 + wr2\u00E2\u0087\u00A7 = 0. (4.31c)Using Eqs. (4.31b) and (4.31c), the fluid part of the action in the comovinggauge becomes\u00002Sf + \u00002Sgr = 13l2 Z \"3(1 + w)dPd\u00E2\u0087\u00A2 02 \u0000 3(1 + w) 2,i\u000043w21 + wH2dPd\u00E2\u0087\u00A2 (r2\u00E2\u0087\u00A7)2# d4x, (4.32)where a total derivative term has been dropped.We now turn our attention to the additional part of the action for anelastic solid given in Eq. (4.29). For the scalar part of the anisotropic stresstensor, we have (\u00E2\u0087\u00A7S)ij(\u00E2\u0087\u00A7S)ij = 23(r2\u00E2\u0087\u00A7)2 + (total derivative term), so thescalar part of the elastic part of the action is\u00002Se = \u0000 16l2 Z w2a2H2c2v(1 + w)(r2\u00E2\u0087\u00A7)2d4x, (4.33)where we have used the sounds speeds in Eq. (4.18). We can then write the484.4. Actiontotal action for the scalar perturbations as\u00002S = 16l2 Z a2 \"3(1 + w)dPd\u00E2\u0087\u00A2 02 \u0000 3(1 + w) 2,i\u0000c2sw2H2c2v(1 + w)dPd\u00E2\u0087\u00A2 (r2\u00E2\u0087\u00A7)2# d4x. (4.34)We can express r2\u00E2\u0087\u00A7 as a function of by using Eqs. (4.22), (4.31b), and(4.31c), which yieldsr2\u00E2\u0087\u00A7 = 2c2vc2s 1 + ww \u00EF\u00A3\u00BF 0H \u0000 3dPd\u00E2\u0087\u00A2 \u0000 . (4.35)Using this expression in the action above gives\u00002S = 13l2 Z z2 \u00E2\u0087\u00A5R02 \u0000 c2sR2,i \u0000 4c2v\u0000R2\u00004c2vH\u00E2\u009C\u0093(c2v)0c2v \u0000 (c2s )0c2s \u00E2\u0097\u0086R2\u0000 d4x, (4.36)where we have cast the action into a gauge-invariant form and a total deriva-tive term has been dropped. We have defined z byz \u00E2\u008C\u0098 ap\u0000csH = acsr32(1 + w). (4.37)We can now define the canonical variable u asu \u00E2\u008C\u0098r 23l2 zR, (4.38)so that the action becomes\u00002S = 12 Z \u00E2\u0087\u00A5u02 \u0000 c2su2,i \u0000m2e\u00E2\u0086\u00B5,S(\u00E2\u008C\u0098)u2\u00E2\u0087\u00A4 d4x, (4.39)where another total derivative term has been dropped and the e\u00E2\u0086\u00B5ective massis m2e\u00E2\u0086\u00B5,S(\u00E2\u008C\u0098) \u00E2\u008C\u0098 \u0000z00z + 4c2v\u0000 + 4c2vH\u00E2\u009C\u0093(c2v)0c2v \u0000 (c2s )0c2s \u00E2\u0097\u0086 . (4.40)Varying the action with respect to u leads to the equation of motionu00 \u0000 c2sr2u+m2e\u00E2\u0086\u00B5,S(\u00E2\u008C\u0098)u = 0. (4.41)494.4. ActionIt is clear that the action in Eq. (4.39) has the same form as the action fora harmonic oscillator with time-dependent mass, so we may use the samequantization procedure as is used to quantize a harmonic oscillator. Theconjugate momentum \u00E2\u0087\u00A1 to u is\u00E2\u0087\u00A1 = @L@u0 = u0. (4.42)We now promote u and \u00E2\u0087\u00A1 to operators u\u00CB\u0086 and \u00E2\u0087\u00A1\u00CB\u0086 and impose the commutationrelations[u\u00CB\u0086(x, \u00E2\u008C\u0098), u\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = [\u00E2\u0087\u00A1\u00CB\u0086(x, \u00E2\u008C\u0098), \u00E2\u0087\u00A1\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = 0,[u\u00CB\u0086(x, \u00E2\u008C\u0098), \u00E2\u0087\u00A1\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)] = i\u0000(x\u0000 x\u00CB\u009C). (4.43)Using the Fourier conventionsf(x) = Z d3k(2\u00E2\u0087\u00A1)3/2 fkeik\u00C2\u00B7x, (4.44)we can write u\u00CB\u0086k in terms of the creation and annihilation operators a\u00CB\u0086\u00E2\u0080\u00A0k anda\u00CB\u0086k as u\u00CB\u0086k(\u00E2\u008C\u0098) = (a\u00CB\u0086k\u0000\u00E2\u0087\u00A4k(\u00E2\u008C\u0098) + a\u00CB\u0086\u00E2\u0080\u00A0\u0000k\u0000k(\u00E2\u008C\u0098))/p2 so thatu\u00CB\u0086(x, \u00E2\u008C\u0098) = 1p2Z d3k(2\u00E2\u0087\u00A1)3/2 ha\u00CB\u0086k\u0000\u00E2\u0087\u00A4k(\u00E2\u008C\u0098)eik\u00C2\u00B7x + a\u00CB\u0086\u00E2\u0080\u00A0k\u0000k(\u00E2\u008C\u0098)e\u0000ik\u00C2\u00B7xi , (4.45)and the mode function \u0000k(\u00E2\u008C\u0098) obeys\u000000k + \u00E2\u0087\u00A5c2sk2 +m2e\u00E2\u0086\u00B5,S\u00E2\u0087\u00A4\u0000k = 0. (4.46)The commutation relations in Eq. (4.43) imply the normalization\u00000k\u0000\u00E2\u0087\u00A4k \u0000 \u0000k\u0000\u00E2\u0087\u00A40k = 2i. (4.47)Once we solve for the mode function \u0000k from the di\u00E2\u0086\u00B5erential equation inEq. (4.46), subject to the the normalization condition above, we can calcu-late the power spectrum for the scalar perturbations PR(k) = |Rk|2k3/2\u00E2\u0087\u00A12.We will see in Section 4.5 that the perturbations associated with a sin-gle scalar mode with wavevector k is anisotropic, due to the presence ofanisotropic stress. However, each mode has the same evolution (the solu-tion for \u0000k in Eq. (4.46) is the same for each k with the same magnitudek). As discussed in Section 4.5, we will assume that expectation values areisotropic, so that hukuk\u00CB\u009Ci = |uk|2\u0000(k + k\u00CB\u009C). As a result, the integrand ofthe integral over d3k in the two-point correlation function for the operator504.4. Actionu\u00CB\u0086(x, \u00E2\u008C\u0098) with the vacuum state will depend only on k, so we can triviallyintegrate over the solid angle d\u00E2\u008C\u00A6k, so thath0|u\u00CB\u0086(x, \u00E2\u008C\u0098)u\u00CB\u0086(x\u00CB\u009C, \u00E2\u008C\u0098)|0i = Z dk k24\u00E2\u0087\u00A12 |\u0000k(\u00E2\u008C\u0098)|2 sin(k|x\u0000 x\u00CB\u009C|)k|x\u0000 x\u00CB\u009C| . (4.48)We can now identify the power spectrum for u as Pu(k) = k34\u00E2\u0087\u00A12 |\u0000k|2 andusing Eq. (4.38) the power spectrum for R will bePR(k) = 3l2k38\u00E2\u0087\u00A12z2 |\u0000k|2. (4.49)4.4.2 Quantization of Tensor ModesWe now turn our attention to the tensor modes. Using the tensor part ofthe metric in Eq. (4.6) to calculate the tensor part of the Ricci scalar R, thegravitational part of the action can be found to be\u00002Sgr = 124l2 Z a2 h(hT)i0j (hT)j0i \u0000 (hT)ij,l(hT)j,li i d4x. (4.50)The only contribution to the matter part of the action is from the tensorpart of the anisotropic stress, given by Eq. (4.24), which using the elasticpart of the action in Eq. (4.29) is\u00002Se = \u0000 Z a4\u00C2\u00B54P (hT)ij(hT)jid4x. (4.51)With the transverse sound speed given in Eq. (4.18), the total action be-comes\u00002S = 124l2 Z a2 h(hT)i0j (hT)j0i \u0000 (hT)ij,l(hT)j,li\u00004c2v\u0000(hT)ij(hT)jii d4x. (4.52)It is convenient to express (hT)ij in terms of the individual polarization stateshTp , where (hT)ij(hT)ji = 2Pp(hTp )2, so that the action for each polarizationstate is \u00002S = 112l2 Z a2 \u00E2\u0087\u00A5(hTp )02 \u0000 (hTp )2,i \u0000 4c2v\u0000(hTp )2\u00E2\u0087\u00A4 d4x. (4.53)We can define the canonical variable Up for the tensor perturbations asUp = ahTpp6l2 , (4.54)514.5. Superhorizon Evolutionso the action becomes\u00002S = 12 Z \u00E2\u0087\u00A5U 02p \u0000 U2p,i \u0000m2e\u00E2\u0086\u00B5,TU2p \u00E2\u0087\u00A4 d4x, (4.55)where a total derivative has been dropped and the e\u00E2\u0086\u00B5ective mass for thetensor modes is given by m2e\u00E2\u0086\u00B5,T = \u0000a00a + 4c2v\u0000. (4.56)As with the scalar perturbations, the action for the tensor perturbations hasthe same form as the action of a harmonic oscillator with time-dependentmass. We can therefore use the same quantization procedure as was usedwith the scalar perturbations. We promote the canonical variable Up to anoperator and write it in terms of creation and annihilation operators asU\u00CB\u0086p(x, \u00E2\u008C\u0098) = 1p2Z d3k(2\u00E2\u0087\u00A1)3/2\u00EF\u00A3\u00BFa\u00CB\u0086kX\u00E2\u0087\u00A4k(\u00E2\u008C\u0098)eik\u00C2\u00B7x + a\u00CB\u0086\u00E2\u0080\u00A0kXk(\u00E2\u008C\u0098)e\u0000ik\u00C2\u00B7x\u0000, (4.57)and so the equation of motion for the mode function Xk isX 00k + \u00E2\u0087\u00A5k2 +m2e\u00E2\u0086\u00B5,T\u00E2\u0087\u00A4Xk = 0. (4.58)The commutation relations analogous to Eq. (4.43) for the tensor case yieldsthe normalization conditionX 0kX\u00E2\u0087\u00A4k \u0000XkX\u00E2\u0087\u00A40k = 2i. (4.59)Accounting for both polarization states, the two-point function for thetensor perturbations with the vacuum state is thenh0|(h\u00CB\u0086T )ij(x, \u00E2\u008C\u0098)(h\u00CB\u0086T )ji (x\u00CB\u009C, \u00E2\u008C\u0098)|0i = Z dk 6l2\u00E2\u0087\u00A12a2k2|Xk(\u00E2\u008C\u0098)|2\u00E2\u0087\u00A5 sin(k|x\u0000 x\u00CB\u009C|)k|x\u0000 x\u00CB\u009C| , (4.60)with which we can identify the tensor power spectrum PT asPT(k) = 6l2k3\u00E2\u0087\u00A12a2 |Xk|2. (4.61)4.5 Superhorizon EvolutionAn interesting phenomenon in this model is that both \u00E2\u0087\u00A3 and R evolve onsuperhorizon scales, even when the elastic solid is the only substance present524.5. Superhorizon Evolutionin the Universe. Typically, this type of superhorizon evolution only arisesin the presence of a nonadiabatic pressure \u0000Pnad = \u0000P \u0000 (dP/d\u00E2\u0087\u00A2)\u0000\u00E2\u0087\u00A2. FromSection 4.3, we saw that \u0000P/\u0000\u00E2\u0087\u00A2 = dP/d\u00E2\u0087\u00A2 for an elastic solid, so the nona-diabatic pressure vanishes. However, the addition of the anisotropic stressin the elastic solid adds another type of stress to the system, which causessuperhorizon evolution in a similar manner to cases when nonadiabatic pres-sures are present.In the standard case when only adiabatic and isotropic pressures arepresent, both \u00E2\u0087\u00A3 and R remain approximately constant on superhorizon scalesbecause once smoothed on a scale much larger than the horizon, each patchof the Universe smaller than the smoothing scale evolves approximately likea separate unperturbed FRW universe. This idea is known as the \u00E2\u0080\u0098separateuniverse approach\u00E2\u0080\u0099 [61]. The locally defined expansion \u00E2\u009C\u0093\u00CB\u009C(x, t) with respectto coordinate time t is given by\u00E2\u009C\u0093\u00CB\u009C(x, t) = 3H \u0000 3 \u00CB\u0099(x, t) +r2\u0000(x, t), (4.62)where an overdot denotes a derivative with respect to coordinate time and\u0000 = E\u00CB\u0099 \u0000B is the (local) shear. Considering a flat slicing ( = 0), the localexpansion will be equal to the background value if we can safely neglect thee\u00E2\u0086\u00B5ects of the shear on large scales. In this case, since the (total) energydensity evolves according to the local energy conservation equation, whichto linear order is\u00E2\u0087\u00A2\u00CB\u0099tot(x, t) = \u0000[\u00E2\u009C\u0093\u00CB\u009C(x, t) +r2v(x, t)][\u00E2\u0087\u00A2tot(x, t) + P tot(x, t)]. (4.63)Then after smoothing on superhorizon scales, the local energy density ateach location will (approximately) follow the same unperturbed FRW evo-lution. Therefore, the di\u00E2\u0086\u00B5erence between the energy density perturbationsat di\u00E2\u0086\u00B5erent locations will be kept approximately constant in time and since\u00E2\u0087\u00A3 is proportional to the energy density perturbation in a flat slicing, \u00E2\u0087\u00A3 willbe approximately constant on superhorizon scales.36As was shown in Ref. [62], when the anisotropic stress is neglected, theshear is in fact negligible on large scales. However, the anisotropic stress actsas a source term for the shear (see Eq. (31) of Ref. [63]), causing the shearto be non-negligible on superhorizon scales in the case of an elastic solid. Ifthe shear cannot be neglected, then di\u00E2\u0086\u00B5erent locations in the Universe aftersmoothing on superhorizon scales will not evolve as an unperturbed FRW36A similar argument can be made for R, which is proportional to v in a spatially flatslicing, by using the local conservation of momentum.534.5. Superhorizon Evolutionuniverse owing to the fact that a FRW spacetime is shear free and in generalthe local expansion will be position dependent in a flat slicing.Working in the gauge where = B = 0 so that the shear is \u0000 = E\u00CB\u0099,the trace-free part of the spatial components of the Einstein equations inFourier space is \u0000\u00CB\u0099k + 3H\u0000k \u0000 k2a2\u0000k = 3wH2\u00E2\u0087\u00A7k. (4.64)As we can see, the shear is indeed sourced by the anisotropic stress and willevolve on superhorizon scales unless the anisotropic stress is negligible onthese scales. It is easily seen from Eq. (4.22) that for an elastic solid theanisotropic stress is significant (i.e. comparable to the energy density andpressure perturbations) on superhorizon scales since in a spatially flat slicing\u00E2\u0087\u00A7k = \u00002(c2v/w)\u0000k. Therefore, we will have |\u00E2\u0087\u00A7k| \u00E2\u0087\u00A0 |\u0000k| in this slicing for asu\u0000ciently rigid solid and the shear will evolve as\u0000\u00CB\u0099k + 3H\u0000k \u0000 k2a2\u0000k = \u00006c2vH2\u0000k, (4.65)so that the shear is sourced by the (non-negligible) density perturbations onsuperhorizon scales when viewed in this gauge.If we consider a mode with wavevector k = (0, 0, k) then the scalarpart of the anisotropic stress tensor in Fourier space, given by (\u00E2\u0087\u00A7Sk)ij =(\u0000kikj/k2 + \u0000ij/3)\u00E2\u0087\u00A7k, is (\u00E2\u0087\u00A7Sk)ij = diag(13 , 13 ,\u000023)\u00E2\u0087\u00A7k. From Eq. (4.8), thescalar perturbation to the spatial part of the stress tensor in a spatiallyflat slicing will be (\u0000TSk )ij = \u0000diag(c2s \u0000 2c2v, c2s \u0000 2c2v, c2s )\u0000\u00E2\u0087\u00A2k, which is in-herently anisotropic, having a di\u00E2\u0086\u00B5erent pressure in directions parallel andperpendicular to the direction of propagation.Instead of considering perturbations about a FRW spacetime, we nowexamine the behaviour of an unperturbed Bianchi spacetime, which has thedefining properties of being homogeneous and in general anisotropic. Weconcentrate on the Bianchi type I spacetime that has the metricds2 = dt2 \u0000 ax(t)2dx2 \u0000 ay(t)2dy2 \u0000 az(t)2dz2, (4.66)where ax, ay, and az are directional scale factors. The properties of nearlyisotropic Bianchi spacetimes were detailed in Ref. [64], which treated theirdeparture from isotropy as a linear perturbation. After smoothing on su-perhorizon scales, the metric perturbation Eij in Eq. (4.6) from perturbingabout a flat FRW spacetime is simply the symmetric trace-free tensor char-acterizing the anisotropy of a nearly isotropic Bianchi I spacetime to linear544.6. Inflationorder.37 For example, consider the mode k = (0, 0, k). After smoothing, Eijwill be approximately uniform in a local patch of the Universe. In a spatiallyflat gauge where h = 2r2E, if we denote the average value of h in this patchas h\u00C2\u00AF, then a scalar mode with this wavevector will evolve approximately asa Bianchi I spacetime with ax = ay = a and az = a+ h\u00C2\u00AF. A tensor mode with\u00E2\u0080\u0098plus\u00E2\u0080\u0099 polarization, with an average value of h\u00C2\u00AF+ in this patch, would evolvewith directional scale factors ax = a+ h\u00C2\u00AF+, ay = a\u0000 h\u00C2\u00AF+, and az = a.For a single mode, we can absorb the shear on superhorizon scales intothe background spacetime by perturbing about a Bianchi I spacetime insteadof a flat FRW (which is the isotropic special case of Bianchi I). In this case,after smoothing on superhorizon scales, perturbations would again evolve ac-cording to an unperturbed metric, but in general would be of type Bianchi I,not FRW. In the standard inflationary scenario, the shear is negligible onsuperhorizon scales, so Eij is approximately constant on these scales andcan be removed from the metric by a simple coordinate redefinition, leavingthe isotropic special case of our spacetime.Although a single mode is formally anisotropic, in the case of inflation,modes are excited on a wide range of scales in all directions. We assume thatinitial perturbations are drawn from an isotropic Gaussian distribution andthat expectation values will be isotropic and therefore continue to examinethe perturbations in the metric in Eq. (4.1) in which perturbations are takenabout an isotropic FRW spacetime.384.6 InflationWe now apply the results of the previous sections to the case where inflationis driven by an elastic solid. We divide the analysis into two parts: thesimple case with constant sound speeds and equation of state and the casewhere they are varying in time. We then consider the more specialized casewhere the sound speeds and equation of state slowly vary with time.37See Eq. (59) of Ref. [64]. Also note that on superhorizon scales, by making thesubstitution \u0000 ! E\u00CB\u0099, Eq. (4.64) is approximately Eq. (39) in Ref. [64].38Although we take the expectation values over a single realization to be isotropic,in principle, a residual net anisotropy might persist. The persistence of anisotropic ge-ometries in this context was recently studied in Ref. [53]. We set aside here questionspertaining to the precise size and impact of sustained anisotropies and only assume thatcorrections to the evolution of linear perturbations in an isotropic background appear athigher order.554.6. Inflation4.6.1 Inflation with Constant Sound Speeds and Equationof StateWith constant equation of state, H can easily be solved from Eq. (4.3) withan appropriate integration constant asH =2(1 + 3w)\u00E2\u008C\u0098 . (4.67)From Eq. (4.40), we see that in this case the e\u00E2\u0086\u00B5ective mass for the scalarmodes becomes m2e\u00E2\u0086\u00B5,S(\u00E2\u008C\u0098) = \u00002\u0000 6w \u0000 24c2v(1 + w)(1 + 3w)2\u00E2\u008C\u00982 . (4.68)The general solution for the mode function \u0000k(\u00E2\u008C\u0098) can now easily be foundfrom Eq. (4.46) as\u0000k = r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 hC1H(1)\u00E2\u008C\u00AB (csk|\u00E2\u008C\u0098|) + C2H(2)\u00E2\u008C\u00AB (csk|\u00E2\u008C\u0098|)i , (4.69)where H(1)\u00E2\u008C\u00AB and H(2)\u00E2\u008C\u00AB are the Hankel functions of the first and second kind,C1 and C2 are integration constants and the index \u00E2\u008C\u00AB is\u00E2\u008C\u00AB = 12s1 + 4\u00E2\u009C\u00932\u0000 6w \u0000 24c2v(1 + w)(1 + 3w)2 \u00E2\u0097\u0086. (4.70)When a mode is well within the horizon with csk|\u00E2\u008C\u0098|\u0000 1, the mode functioncan be approximated by\u0000k \u00E2\u0087\u00A1 1pcsk \u00E2\u0087\u00A3C1eicsk\u00E2\u008C\u0098 + C2e\u0000icsk\u00E2\u008C\u0098\u00E2\u008C\u0098 , (4.71)which is the solution for Minkowski space. We initialize the mode by assum-ing that it is in its lowest energy state when it is well within the horizon,with mode function \u0000k \u00E2\u0087\u00A1 1pcskeicsk\u00E2\u008C\u0098. (4.72)With these constants of integration, the mode function becomes\u0000k = r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 H(2)\u00E2\u008C\u00AB (csk|\u00E2\u008C\u0098|). (4.73)564.6. InflationWhen the mode is far outside the horizon with csk|\u00E2\u008C\u0098|\u00E2\u008C\u00A7 1, the mode functioncan be approximated as\u0000k \u00E2\u0087\u00A1r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 i\u0000(\u00E2\u008C\u00AB)\u00E2\u0087\u00A1 \u00E2\u009C\u0093csk|\u00E2\u008C\u0098|2 \u00E2\u0097\u0086\u0000\u00E2\u008C\u00AB , (4.74)where \u0000(\u00E2\u008C\u00AB) is the gamma function. With the evolution of the mode function\u0000k for modes well outside the horizon, we find the power spectrum for R tobePR(k) \u00E2\u0087\u00A1 c2(1\u0000\u00E2\u008C\u00AB)s \u00002(\u00E2\u008C\u00AB)4\u00E2\u008C\u00AB8\u00E2\u0087\u00A13(1 + w) l2k3\u00002\u00E2\u008C\u00AB |\u00E2\u008C\u0098|1\u00002\u00E2\u008C\u00ABa2 . (4.75)Since for constant equation of state the scale factor evolves as a / |\u00E2\u008C\u0098| 21+3w , wedo indeed see that Rk evolves with time when the mode is on superhorizonscales if cv is nonzero. Using Eq. (4.67), we can write the power spectrumin terms of the Hubble parameter asPR(k) \u00E2\u0087\u00A1 c2(1\u0000\u00E2\u008C\u00AB)s \u00002(\u00E2\u008C\u00AB)4\u00E2\u0087\u00A13(1 + w)|1 + 3w|1\u00002\u00E2\u008C\u00AB l2(k/a)3\u00002\u00E2\u008C\u00ABH\u00001+2\u00E2\u008C\u00AB . (4.76)Although modes evolve on superhorizon scales, all modes well outsidethe horizon share the same time evolution. In other words, the presenceof a superhorizon evolution will not a\u00E2\u0086\u00B5ect the relative scale dependence ofmodes on superhorizon scales. Thus, we can calculate quantities like thescalar spectral index ns = 1 + dlnPR/dlnk and its running using the samemethods that are used in the case where the superhorizon evolution is small.For constant sound speeds and equation of state, the scalar spectral indexis ns = 4 \u0000 2\u00E2\u008C\u00AB. The necessary restrictions of \u00001 < w < \u00001/3, 0 \u00EF\u00A3\u00BF c2v \u00EF\u00A3\u00BF 1,and 0 < c2s \u00EF\u00A3\u00BF 1 imply that ns is bound from below by one. Thus, the scalarpower spectrum for this case can only have a blue tilt, which has been ruledout to a high degree of likelihood [25]. If w is near \u00001, we can see fromEq. (4.68) that the e\u00E2\u0086\u00B5ect of the shear stress on m2e\u00E2\u0086\u00B5,S will be small and anearly scale-invariant spectrum will be produced, as is the case in manymodels of inflation.It is interesting to note that cs near zero (w near \u000043c2v) also produces anearly scale-invariant two-point spectrum. In this case, the w dependenceof the \u0000z00/z and 4c2v\u0000 terms in m2e\u00E2\u0086\u00B5,S cancels with one another so that anearly scale-invariant spectrum can be produced for values of w far from\u00001 (but still bounded by \u00001 < w < \u00001/3). This result is not possible instandard inflationary models, since the 4c2v\u0000 term is absent in these cases,so the w dependence of m2e\u00E2\u0086\u00B5,S remains important.574.6. InflationAs discussed in Ref. [66], inflationary scenarios that produce a nearlyscale-invariant two-point function with a background in the far-from-de Sit-ter regime generically do not have nearly scale-invariant higher-point cor-relations, provided the perturbations are adiabatic in the sense of Ref. [67]\u00E2\u0080\u0094 which requires the anisotropic stress to be negligible on large scales.However, the elastic solid model we describe here requires non-negligibleanisotropic stress on large scales for a consistent description of linear per-turbations. We leave the interesting question of higher-point correlationfunctions in far-from-de Sitter accelerating elastic solid models for futurework.Additionally, when w is far from \u00001, one must take care that an ad-equate number of e-folds of inflation can occur as the energy density andhorizon size may evolve significantly during inflation. This can alter theminimum number of e-folds required to solve the \u00E2\u0080\u0098horizon problem\u00E2\u0080\u0099. Anupper bound on the number of e-folds of inflation will be set by puttingbounds on the energy density, set on the lower end by the reheat temper-ature and on the higher end by a high-energy limit (see Section 4.6.2 forfurther details). Requiring inflation to start below the Planck scale and endwith temperatures above \u00E2\u0087\u00A010s of MeV puts an upper bound on the equationof state of w . \u00002/5 when a nearly scale-invariant spectrum is achieved.However, as will be discussed in Section 4.8, a power spectrum amplitudecompatible with current observations requires either very small values of csor super-Planckian densities for values of w extremely far from \u00001.Returning to the discussion of Section 4.5, for constant sound speeds andequation of state with cv 6= 0, the superhorizon modes of h in the = B = 0gauge (as well as E since h = 2r2E in this gauge) evolve ash0k / Akk\u0000\u00E2\u008C\u00AB |\u00E2\u008C\u0098|\u0000 5+3w2+6w\u0000\u00E2\u008C\u00AB , (4.77)where the factor Ak determines the initial amplitude of h0k for a particularmode. If we scale the wavevector of the mode as k! \u00E2\u0086\u00B5k for some constant\u00E2\u0086\u00B5, the same late-time evolution of the superhorizon mode can remain un-changed by simultaneously scaling Ak ! \u00E2\u0086\u00B5\u00E2\u008C\u00ABAk, an example of which canbe seen in Fig. 4.1. As such, we cannot determine which particular modesa superhorizon sized anisotropy originated from.584.6. Inflation-1 -0.1 -0.010.70.80.91.01.1csk\u00C3\u00A9h\u00C2\u00BBhk\u00C2\u00BB\u00C3\u00AA\u00C2\u00BBhk\u00C3\u00A9\u00C2\u00BBh.c.Figure 4.1: Evolution of h modes in the = B = 0 gauge for w = \u00000.9and c2v = 0.8 (note that a plot of Ek would look identical as hk = \u00002Ekin this gauge). The solid and dashed lines show the evolution for modeswith |k| = k\u00CB\u009C and |k| = 2k\u00CB\u009C, respectively, and the subscript h.c. denoteshorizon crossing. Initial amplitudes of the perturbations are chosen so thatthe modes coincide when both are on superhorizon scales.4.6.2 The \u00E2\u0080\u0098Horizon Problem\u00E2\u0080\u0099 RevisitedAn interesting feature of this model of inflation is that it allows the possi-bility of far from de Sitter backgrounds that nevertheless produce a nearlyscale-invariant two-point correlation function. In such a case, the horizonmay change significantly during inflation, thus altering the \u00E2\u0080\u0098horizon prob-lem\u00E2\u0080\u0099, in which we expect to be able to fit the present-day horizon size intothe horizon at the beginning of inflation expanded to today. Labelling quan-tities evaluated at the beginning of inflation, reheating, and the present-dayby the subscripts i, RH, and 0, respectively, we requireH\u000010 \u00EF\u00A3\u00BF a0ai H\u00001i=a0aRH aRHai H\u00001i\u00E2\u0087\u00A1\u00E2\u009C\u0093gsRHgs0 \u00E2\u0097\u00861/3 TRHT0 eNH\u00001i , (4.78)where N is the number of e-folds of inflation and gs is the e\u00E2\u0086\u00B5ective numberof relativistic degrees of freedom contributing to the entropy density.594.6. InflationIf the equation of state during inflation is w, then assuming w is approxi-mately constant, the horizon size changes as / a3(1+w)/2 during inflation, sothat H\u00001i \u00E2\u0087\u00A1 e\u00003(1+w)N/2H\u00001RH. The minimum number of e-folds of inflationto solve the \u00E2\u0080\u0098horizon problem\u00E2\u0080\u0099 then becomesN \u0000 11\u0000 \u00E2\u009C\u008F \u00EF\u00A3\u00BFln\u00E2\u009C\u0093 T0H0\u00E2\u0097\u0086+ ln\u00E2\u009C\u0093HRHTRH \u00E2\u0097\u0086+ 13ln\u00E2\u009C\u0093 gs0gsRH\u00E2\u0097\u0086\u0000\u00E2\u0087\u00A111\u0000 \u00E2\u009C\u008F \u00EF\u00A3\u00BF54 + ln\u00E2\u009C\u0093 TRH1013 GeV\u00E2\u0097\u0086+ 13ln\u00E2\u009C\u0093106.75gsRH \u00E2\u0097\u0086\u0000 , (4.79)where \u00E2\u009C\u008F = 3(1 + w)/2 and we have taken gs0 = 43/11. Thus, the mini-mum number of e-folds required when w is far from \u00001 (\u00E2\u009C\u008F is large) may besubstantially larger than that for the standard w ' \u00001 case.If \u00E2\u0087\u00A4 is a high-energy cut-o\u00E2\u0086\u00B5 scale that is an upper bound for the initialenergy density of inflation (presumably the Planck scale), then for \u00E2\u009C\u008F 6= 0, Nis bounded from above byN . 12\u00E2\u009C\u008F \u00EF\u00A3\u00BF172 + 4 \u00EF\u00A3\u00BFln\u00E2\u009C\u0093 \u00E2\u0087\u00A4mp\u00E2\u0097\u0086\u0000 ln\u00E2\u009C\u0093 TRHGeV\u00E2\u0097\u0086\u0000\u0000 ln\u00E2\u0087\u00A3 gRH106.75\u00E2\u008C\u0098\u0000 , (4.80)where mp = p8\u00E2\u0087\u00A1/3 l\u00001 is the Planck mass and gRH is the e\u00E2\u0086\u00B5ective numberof relativistic degrees of freedom contributing to the energy density. If thebound from Eq. (4.79) provides the strongest constraint on the minimumnumber of e-folds of inflation, which will likely be the case if w is far from\u00001, then the maximum reheat temperature such that N is appropriatelybounded islog10\u00E2\u009C\u0093 TRHGeV\u00E2\u0097\u0086 \u00E2\u0087\u00A1 11\u0000 \u00E2\u009C\u008F/2\u00E2\u009C\u009319\u0000 24\u00E2\u009C\u008F+0.44(1\u0000 \u00E2\u009C\u008F)ln\u00E2\u009C\u0093 \u00E2\u0087\u00A4mp\u00E2\u0097\u0086+ (0.18\u00E2\u009C\u008F\u0000 0.11)ln\u00E2\u0087\u00A3 gRH106.75\u00E2\u008C\u0098\u00E2\u0097\u0086 , (4.81)where we have assumed gRH = gsRH.4.6.3 Non-Constant Sound Speeds and Equation of StateSince having the sound speeds and equation of state perfectly constant can-not result in a red-tilted scalar spectrum, we would like to examine if addinga time dependence can result in a red-tilted scalar spectrum. It will proveuseful to introduce an alternative time variable q, defined byq(\u00E2\u008C\u0098) \u00E2\u008C\u0098 \u0000 Z \u00E2\u008C\u0098 cs(\u00E2\u008C\u0098\u00CB\u009C)d\u00E2\u008C\u0098\u00CB\u009C, (4.82)604.6. Inflationand a new field yk \u00E2\u008C\u0098 pcs\u0000k, so that the equation of motion for the modefunction in Eq. (4.46) becomesyk,qq + \u00E2\u0087\u00A5k2 + m\u00CB\u009C2e\u00E2\u0086\u00B5,S\u00E2\u0087\u00A4 yk = 0, (4.83)where ,q = d/dq and m\u00CB\u009C2e\u00E2\u0086\u00B5,S \u00E2\u008C\u0098 m2e\u00E2\u0086\u00B5,Sc2s \u0000 (pcs),qqpcs . (4.84)The advantage of this change of variables is that the squared sound speed c2sdoes not appear in front of the k2 term in Eq. (4.83), as it does in Eq. (4.46),so that the same methods used for solving for the equation of motion of \u0000k inthe case where cs is constant can be used to solve for the new mode functionyk as a function of the new time variable q.In the previous section, the mode function was easily solved becausem2e\u00E2\u0086\u00B5,S was inversely proportional to \u00E2\u008C\u00982. Accordingly, we reparameterizem\u00CB\u009C2e\u00E2\u0086\u00B5,S by BS \u00E2\u008C\u0098 \u0000q2m\u00CB\u009C2e\u00E2\u0086\u00B5,S , so that the equation of motion for yk becomesyk,qq + \u00EF\u00A3\u00BFk2 \u0000 BSq2 \u0000 yk = 0. (4.85)To obtain a solution where the running of ns is small, we consider solutionswhere BS is nearly constant, in which case Eq. (4.85) would have solutionyk(q) \u00E2\u0087\u00A1r\u00E2\u0087\u00A1q2 hC1H(1)\u0000S (kq) + C2H(2)\u0000S (kq)i , (4.86)where the index \u0000S is \u0000S \u00E2\u008C\u0098 12p1 + 4BS. (4.87)As with the previous case, we choose the integration constants C1 and C2to select the lowest energy state when kq \u0000 1. This coincides with theasymptotic solution in Eq. (4.72) if the change of cs with time is small atsome early time when all modes of interest are well within the horizon.Writing the normalization condition in Eq. (4.47) in terms of the modefunction yk and the time variable q gives(yk),q y\u00E2\u0087\u00A4k \u0000 yk(yk)\u00E2\u0087\u00A4,q = \u00002i. (4.88)With these choices, the mode function yk evolves asyk(q) = r\u00E2\u0087\u00A1q2 H(1)\u0000S (kq). (4.89)614.6. InflationWhen dealing with the time variable q, k|q| \u00E2\u0087\u00A0 1 does not necessarily implycsk|\u00E2\u008C\u0098| \u00E2\u0087\u00A0 1. However, in the case when cs varies slowly in time, as is consid-ered in Section 4.6.4, then k|q| \u00E2\u0087\u00A0 1 when csk|\u00E2\u008C\u0098| \u00E2\u0087\u00A0 1, in which case there willbe no confusion about what is meant by a mode crossing the horizon.Analogously with the previous section, on superhorizon scales, the modefunction yk is approximately equal toyk(q) \u00E2\u0087\u00A1 \u0000r\u00E2\u0087\u00A1q2 i\u0000(\u0000S)\u00E2\u0087\u00A1 \u00E2\u009C\u0093kq2 \u00E2\u0097\u0086\u0000\u0000S , (4.90)at which time the power spectrum for R becomesPR(k) \u00E2\u0087\u00A1 cs\u00002(\u0000S)4\u0000S8\u00E2\u0087\u00A13(1 + w) l2k3\u00002\u0000Sq1\u00002\u0000Sa2 , (4.91)which implies that the scalar spectral index ns is now given byns = 4\u0000 2\u0000S. (4.92)It is trivial to check that in the case that the sound speeds and equationof state are constant, the above equations simplify to those given in theprevious section. However, if we can find time-varying sound speeds and/orequation of state such that BS is approximately constant, the bounds on theindex \u0000S may be extended to include red-tilted scalar spectra.4.6.4 Slowly Varying Sound Speeds and Equation of StateFrom the above considerations, we wish to find a parameterization of thesound speeds and equation of state that allows for a small variation in timein such a way that results in BS being approximately constant and allowsthe scalar spectral index to be less than one.We will use the variable \u00E2\u009C\u008F, which coincides with the slow-roll variablefrom standard inflationary scenarios, defined by\u00E2\u009C\u008F \u00E2\u008C\u0098 \u0000 H\u00CB\u0099H2 = 1\u0000 H0H2 , (4.93)so that w = \u00001 + 23\u00E2\u009C\u008F. In this context, \u00E2\u009C\u008F simply parameterizes the departureof w from -1. We parameterize \u00E2\u009C\u008F as \u00E2\u009C\u008F(\u00E2\u008C\u0098) = \u00E2\u009C\u008F0+f\u00E2\u009C\u008F(\u00E2\u008C\u0098) for some slowly varyingfunction f\u00E2\u009C\u008F(\u00E2\u008C\u0098). We write the time dependence of f\u00E2\u009C\u008F asdlnf\u00E2\u009C\u008Fdln\u00E2\u008C\u0098 = \u0000\u00E2\u008C\u00A7\u00E2\u009C\u008F. (4.94)624.6. InflationIf we assume that |\u00E2\u008C\u00A7\u00E2\u009C\u008F|\u00E2\u008C\u00A7 1, then \u00E2\u009C\u008F(\u00E2\u008C\u0098) will be given by\u00E2\u009C\u008F(\u00E2\u008C\u0098) \u00E2\u0087\u00A1 \u00E2\u009C\u008F0 + \u00E2\u009C\u008F1(\u00E2\u008C\u0098/\u00E2\u008C\u0098\u00E2\u0087\u00A4)\u0000\u00E2\u008C\u00A7\u00E2\u009C\u008F (4.95)for some reference time \u00E2\u008C\u0098\u00E2\u0087\u00A4. With this time dependence, w slowly varies near\u00001 + 23\u00E2\u009C\u008F0 for some time, but at some later time, it will evolve at a morerapid pace. We will choose the reference time \u00E2\u008C\u0098\u00E2\u0087\u00A4 to be the end of inflationto ensure that the time dependence of w is small during inflation. We alsoallow cs to vary in time and use a parameterization analogous to the oneused for \u00E2\u009C\u008F, so that cs(\u00E2\u008C\u0098) \u00E2\u0087\u00A1 cs0 + cs1(\u00E2\u008C\u0098/\u00E2\u008C\u0098\u00E2\u0087\u00A4)\u0000\u00E2\u008C\u00A7s , (4.96)where |\u00E2\u008C\u00A7s|\u00E2\u008C\u00A7 1. With this parameterization, our new time variable q isq(\u00E2\u008C\u0098) = \u0000\u00E2\u008C\u0098 \u00EF\u00A3\u00BFcs0 + cs1(\u00E2\u008C\u0098/\u00E2\u008C\u0098\u00E2\u0087\u00A4)\u0000\u00E2\u008C\u00A7s1\u0000 \u00E2\u008C\u00A7s \u0000 . (4.97)The requirement that BS be approximately constant for \u00E2\u008C\u0098 \u00EF\u00A3\u00BF \u00E2\u008C\u0098\u00E2\u0087\u00A4 is metso long as both \u00E2\u008C\u00A7\u00E2\u009C\u008F and \u00E2\u008C\u00A7s are su\u0000ciently small. Note that solutions where wdeparts significantly from \u00001 are valid since \u00E2\u009C\u008F0 is not required to be small.If we desire w to stay close to \u00001, we would add the restrictions |\u00E2\u009C\u008F0| \u00E2\u008C\u00A7 1and |\u00E2\u009C\u008F1| \u00E2\u008C\u00A7 1. To obtain a slightly red-tilted scalar spectrum, we will wantthe sound speeds and equation of state to evolve near constant values thatresult in a nearly scale-invariant (blue-tilted) scalar spectrum. Therefore,from Section 4.6.1, we will want to consider solutions where at least one of\u00E2\u009C\u008F0 and cs0 are small.For the rest of this chapter, we restrict ourselves to cases where |\u00E2\u009C\u008F1|\u00E2\u008C\u00A7 1,in which case Eq. (4.3) can be used to solve for H and subsequently thescale factor a, which to linear order in \u00E2\u008C\u00A7\u00E2\u009C\u008F and \u00E2\u009C\u008F1 isH \u00E2\u0087\u00A1 \u00001\u0000 \u00E2\u009C\u008F0 + \u00E2\u009C\u008F1\u00E2\u008C\u0098(1\u0000 \u00E2\u009C\u008F0)2 , a \u00E2\u0087\u00A1 a\u00E2\u0087\u00A4(\u00E2\u008C\u0098/\u00E2\u008C\u0098\u00E2\u0087\u00A4)\u0000 1\u0000\u00E2\u009C\u008F0+\u00E2\u009C\u008F1(1\u0000\u00E2\u009C\u008F0)2 , (4.98)where a\u00E2\u0087\u00A4 is the value of the scale factor at \u00E2\u008C\u0098 = \u00E2\u008C\u0098\u00E2\u0087\u00A4. In the case where\u00E2\u009C\u008F0 = cs0 = 0, BS to first order in our small parameters is found to beBS \u00E2\u0087\u00A1 2 + 152 \u00E2\u008C\u00A7s + 32\u00E2\u008C\u00A7\u00E2\u009C\u008F \u0000 3c2s1\u00E2\u009C\u008F1, (4.99)and the scalar spectral index ns becomesns \u00E2\u0087\u00A1 1\u0000 5\u00E2\u008C\u00A7s \u0000 \u00E2\u008C\u00A7\u00E2\u009C\u008F + 2c2s1\u00E2\u009C\u008F1, (4.100)from which we see can yield a red-tilted scalar spectrum (see the first threerows of Table 4.1 for examples). Note that although cs ! 0 if cs0 = 0634.6. Inflationin the limit where \u00E2\u008C\u0098 ! \u00001, cs at the beginning of inflation will not besignificantly di\u00E2\u0086\u00B5erent from its value at the end of inflation (cs(\u00E2\u008C\u0098\u00E2\u0087\u00A4) = cs1) aslong as |\u00E2\u008C\u00A7s| \u00E2\u008C\u00A7 1 and the number of e-folds of inflation is modest (i.e. 100\u00E2\u0080\u0099sof e-folds).As previously stated, we can still estimate the running of ns by conven-tional means where relevant quantities are evaluated at horizon crossing.As horizon crossing occurs when csk|\u00E2\u008C\u0098| \u00E2\u0087\u00A0 1, at horizon crossing dlnk \u00E2\u0087\u00A0\u0000(\u00E2\u008C\u0098\u00001 + c0s/cs)\u00001d\u00E2\u008C\u0098, so thatdnsdlnk \u00E2\u0087\u00A0 \u00E2\u009C\u00931\u00E2\u008C\u0098 + c0scs\u00E2\u0097\u0086\u00001 dd\u00E2\u008C\u0098p1 + 4BS\u0000\u0000\u0000\u0000csk|\u00E2\u008C\u0098|\u00E2\u0087\u00A01. (4.101)For \u00E2\u009C\u008F0 = cs0 = 0, the running to second order in our small parameters isdnsdlnk \u00E2\u0087\u00A0 2c2s1\u00E2\u009C\u008F1(\u00E2\u008C\u00A7\u00E2\u009C\u008F + 2\u00E2\u008C\u00A7s), (4.102)from which we see that the running of ns vanishes at first order.As in the constant sound speeds and equation of state case, we find thatwe are not restricted to very small values of \u00E2\u009C\u008F0 and cs0, although for brevitywe will not explicitly write out ns and its running and instead illustratethrough numerical examples. In fact, we can formally find solutions withw varying slowly near values up to \u00001/3, corresponding to values of \u00E2\u009C\u008F0 justbelow 1, that result in a slightly red-tilted scalar spectrum with a smallrunning of its spectral index, although, as previously mentioned, achievingthe necessary number of e-folds of inflation becomes increasingly challengingfor values of w very far from \u00001.As an example, choosing \u00E2\u009C\u008F0 = 1/4 so that w varies close to \u00005/6 andtaking the other parameters to have the values listed in the fourth rowof Table 4.1, we obtain a scalar spectral index ns \u00E2\u0087\u00A1 0.96 and runningdns/dlnk \u00E2\u0087\u00A0 \u000010\u00005. Another example where w varies near \u00002/3 is givenin the fifth row of Table 4.1. Note that the reheat temperature used inthis example is significantly lower compared to the other examples listed inthe table to allow for the required number of e-folds of inflation. Choosingcs0 to be nonzero instead of \u00E2\u009C\u008F0, with cs0 = 0.15 and the values in the lastrow of Table 4.1 yields a scalar spectral index of ns \u00E2\u0087\u00A1 0.96 and runningdns/dlnk \u00E2\u0087\u00A0 \u000010\u00003.We conclude this section by writing the power spectrum for R at the end644.7. Gravitational Wavesof inflation for superhorizon modes, which from Eq. (4.91), isPR(k) \u00E2\u0087\u00A1 3\u00002(\u0000S)(cs0 + cs1)8\u00E2\u0087\u00A13(\u00E2\u009C\u008F0 + \u00E2\u009C\u008F1)\u00E2\u0087\u00A5\u00EF\u00A3\u00BFcs0(1\u0000 \u00E2\u009C\u008F0 + \u00E2\u009C\u008F1) + cs1(1 + \u00E2\u009C\u008F1 + \u00E2\u008C\u00A7s \u0000 \u00E2\u009C\u008F0(1 + \u00E2\u008C\u00A7s))2(1\u0000 \u00E2\u009C\u008F0)2 \u00001\u00002\u0000S\u00E2\u0087\u00A5 l2(k/a\u00E2\u0087\u00A4)3\u00002\u0000SH\u00001+2\u0000S\u00E2\u0087\u00A4 , (4.103)where the subscript \u00E2\u0087\u00A4 denotes evaluation at \u00E2\u008C\u0098 = \u00E2\u008C\u0098\u00E2\u0087\u00A4. Since modes can evolveon superhorizon scales, PR(k) at the end of reheating may be di\u00E2\u0086\u00B5erent fromthe expression given above. This concern will be addressed in Section 4.8.4.7 Gravitational WavesTo find the amplitude of gravitational waves produced during inflation, weneed solve for the equation of motion of the mode function Xk in Eq. (4.58),which can be solved in an analogous manner to \u0000k in the scalar case. Thistask will be relatively easy compared to the scalar perturbations, since thetensor perturbations travel with a sound speed equal to unity so there is noneed to switch time variables in order to solve the di\u00E2\u0086\u00B5erential equation forXk. Analogous to the scalar case, for the tensor modes we can defineBT \u00E2\u008C\u0098 \u0000\u00E2\u008C\u00982m2e\u00E2\u0086\u00B5,T. (4.104)Using the parameterizations of the sound speeds and equation of state inSection 4.6.4, BT to first order in \u00E2\u008C\u00A7s, \u00E2\u008C\u00A7\u00E2\u009C\u008F, and \u00E2\u009C\u008F1 with \u00E2\u009C\u008F0 = cs0 = 0 isBT \u00E2\u0087\u00A1 2\u0000 3c2s1\u00E2\u009C\u008F1. (4.105)Since BT is constant to linear order in our small parameters, the tensor modefunction in Eq. (4.58) has solutionXk \u00E2\u0087\u00A1r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 hC1H(1)\u0000T (k|\u00E2\u008C\u0098|) + C2H(2)\u0000T (k|\u00E2\u008C\u0098|)i , (4.106)where C1 and C2 are new constants of integration and the index \u0000T is\u0000T = 12p1 + 4BT. (4.107)The choice of C1 = 0 and C2 = 1 satisfies the normalization condition ofEq. (4.59) and selects the lowest energy state when the mode is well within654.7.GravitationalWavescs0 cs1 \u00E2\u009C\u008F0 \u00E2\u009C\u008F1 \u00E2\u008C\u00A7s \u00E2\u008C\u00A7\u00E2\u009C\u008F TRH(GeV) ns 109A\u00E2\u0087\u00A3 r(kp) nT dns/dlnk0 0.01 0 0.01 0.005 0.015 1.55\u00E2\u0087\u00A5 1013 0.96 2.43 \u00E2\u0087\u00A0 10\u000012 \u00003.1\u00E2\u0087\u00A5 10\u00004 \u00E2\u0087\u00A0 \u000010\u000050 0.8 0 0.01 0.0086 0.01 3.69\u00E2\u0087\u00A5 1015 0.96 2.43 0.002 0.013 \u00E2\u0087\u00A0 10\u000040 0.3 0 0.001 0.005 0.015 6.06\u00E2\u0087\u00A5 1014 0.96 2.43 \u00E2\u0087\u00A0 10\u00006 1.5\u00E2\u0087\u00A5 10\u00004 \u00E2\u0087\u00A0 10\u000060 0.001 0.25 0.01 0.0078 0.0078 3.05\u00E2\u0087\u00A5 1012 0.96 2.43 \u00E2\u0087\u00A0 10\u000015 \u00000.0002 \u00E2\u0087\u00A0 \u000010\u000050 10\u00008 0.5 0.01 0.008 0.001 3.84\u00E2\u0087\u00A5 106 0.96 2.43 \u00E2\u0087\u00A0 10\u000040 \u00004.1\u00E2\u0087\u00A5 10\u00005 \u00E2\u0087\u00A0 \u000010\u000070.15 0.05 0 0.01 0.024 0.01 1.79\u00E2\u0087\u00A5 1014 0.96 2.43 \u00E2\u0087\u00A0 10\u00006 6.1\u00E2\u0087\u00A5 10\u00004 \u00E2\u0087\u00A0 \u000010\u00003Table 4.1: Examples of choice of parameters for slowly varying sound speeds and equation of state. For caseswhere cs0 = \u00E2\u009C\u008F0 = 0, ns, nT, and r are calculated directly from Eqs. (4.100), (4.110), and (4.131), respectively. Inexamples where either \u00E2\u009C\u008F0 6= 0 or cs0 6= 0, ns and nT are calculated by first computing BS or BT, as defined inSections 4.6.3 and 4.7, respectively. The power spectrum for \u00E2\u0087\u00A3 has been parameterized as P\u00E2\u0087\u00A3(k) = A\u00E2\u0087\u00A3(k/kp)ns\u00001where kp = 0.002Mpc\u00001. In calculating A\u00E2\u0087\u00A3 and r, the model of a rapid decay into radiation was assumed to endinflation and reheat the Universe. For cases with cs0 = \u00E2\u009C\u008F0 = 0, A\u00E2\u0087\u00A3 and r are calculated using Eqs. (4.126) and(4.131), respectively. For examples with either \u00E2\u009C\u008F0 6= 0 or cs0 6= 0, A\u00E2\u0087\u00A3 and r are calculated using the power spectragiven in Eqs. (4.103) and (4.111), respectively, along with the relevant equations found in Section 4.8 to relate R,hT just before the end of inflation to \u00E2\u0087\u00A3, hT following the end of inflation. All examples use gRH = 106.75.664.8. End of Inflation and Reheatingthe horizon. When modes are well outside the horizon, the mode functioncan be approximated asXk \u00E2\u0087\u00A1r\u00E2\u0087\u00A1|\u00E2\u008C\u0098|2 i\u0000(\u0000T)\u00E2\u0087\u00A1 \u00E2\u009C\u0093k|\u00E2\u008C\u0098|2 \u00E2\u0097\u0086\u0000\u0000T . (4.108)Using this approximation for the mode function with Eq. (4.61) yields thepower spectrum for gravitational waves on superhorizon scalesPT(k) \u00E2\u0087\u00A1 6\u00E2\u0087\u00A5 22\u0000T\u00001l2\u00002(\u0000T)\u00E2\u0087\u00A13 k3\u00002\u0000T |\u00E2\u008C\u0098|1\u00002\u0000Ta2 . (4.109)When \u00E2\u009C\u008F0 = cs0 = 0, the tensor spectral index nT = dlnPT/dlnk isnT \u00E2\u0087\u00A1 2c2s1\u00E2\u009C\u008F1. (4.110)At the end of inflation, the tensor power spectrum can be written asPT(k) \u00E2\u0087\u00A1 6\u00002(\u0000T)\u00E2\u0087\u00A13 \u00E2\u009C\u00931\u0000 \u00E2\u009C\u008F0 + \u00E2\u009C\u008F12(1\u0000 \u00E2\u009C\u008F0)2 \u00E2\u0097\u00861\u00002\u0000T \u00E2\u0087\u00A5 l2(k/a\u00E2\u0087\u00A4)3\u00002\u0000TH\u00001+2\u0000T\u00E2\u0087\u00A4 . (4.111)4.8 End of Inflation and ReheatingUnlike in cases where the superhorizon evolution of modes is small, thedetails of the end of inflation will a\u00E2\u0086\u00B5ect the amplitude of modes on super-horizon scales, although the time evolution of all superhorizon modes willbe the same. There are many possibilities for ending inflation and reheatingwithin this model. One possibility is once all modes of interest are on super-horizon scales having w increase rapidly so that it surpasses \u00001/3 and theUniverse stops inflating. At some later point, the solid can lose its rigidity,at which point the superhorizon evolution will be small and the details ofreheating will not a\u00E2\u0086\u00B5ect modes on superhorizon scales. Another possibility,which we will examine in more detail, is that inflation ends with the decayof the elastic solid.In general, \u00E2\u0087\u00A3k and Rk will not be equal during inflation on superhori-zon scales. We can easily find the relationship between \u00E2\u0087\u00A3 and R by usingEqs. (4.22) and (4.35), which is\u00E2\u0087\u00A3 = dP/d\u00E2\u0087\u00A2c2s R\u0000 13c2sHR0, (4.112)from which we see that in general, even on superhorizon scales, \u00E2\u0087\u00A3k 6= Rk.After inflation ends and the rigidity vanishes, the superhorizon evolution674.8. End of Inflation and Reheatingwill be small and \u00E2\u0087\u00A3k \u00E2\u0087\u00A1 Rk, but the change in \u00E2\u0087\u00A3k and Rk must be trackedthrough the transition that ends inflation.We now consider the case where the elastic solid rapidly decays into aperfect fluid to end inflation. Following Ref. [65], we can write the totalstress-energy tensor as T\u00C2\u00B5\u00E2\u008C\u00AB = X\u00E2\u0086\u00B5 T\u00C2\u00B5\u00E2\u008C\u00AB(\u00E2\u0086\u00B5), (4.113)where T\u00C2\u00B5\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) is the stress-energy tensor of component \u00E2\u0086\u00B5 = {e, f}, with e and fdenoting the elastic solid and perfect fluid decay product, respectively. Forthis analysis, it will be more convenient to use the coordinate time t insteadof the conformal time. While the local energy-momentum transfer 4-vectorQ\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) for each species can be nonzero, so thatr\u00C2\u00B5T\u00C2\u00B5\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) = Q\u00E2\u008C\u00AB(\u00E2\u0086\u00B5), (4.114)we must haveP\u00E2\u0086\u00B5 Q\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) = 0 so that the total stress-energy tensor is covari-antly conserved.For the scalar perturbations, we can define a \u00E2\u0087\u00A3\u00E2\u0086\u00B5 variable for each sub-stance \u00E2\u0086\u00B5, defined by \u00E2\u0087\u00A3\u00E2\u0086\u00B5 \u00E2\u008C\u0098 +H \u0000\u00E2\u0087\u00A2\u00E2\u0086\u00B5\u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5 (4.115)that is related to the total \u00E2\u0087\u00A3 by\u00E2\u0087\u00A3 = X\u00E2\u0086\u00B5 \u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5\u00CB\u0099\u00E2\u0087\u00A2 \u00E2\u0087\u00A3\u00E2\u0086\u00B5. (4.116)Similarly, the variable R\u00E2\u0086\u00B5 for each substance, defined byR\u00E2\u0086\u00B5 = Hv\u00E2\u0086\u00B5 + , (4.117)is related to the total R byR = X\u00E2\u0086\u00B5 \u00E2\u0087\u00A2\u00E2\u0086\u00B5 + P\u00E2\u0086\u00B5\u00E2\u0087\u00A2 + P R\u00E2\u0086\u00B5. (4.118)The total energy density \u00E2\u0087\u00A2 and pressure P are given by \u00E2\u0087\u00A2 = P\u00E2\u0086\u00B5 \u00E2\u0087\u00A2\u00E2\u0086\u00B5 andP = P\u00E2\u0086\u00B5 P\u00E2\u0086\u00B5, while the entropy perturbation between substances \u00E2\u0086\u00B5 and \u0000 isS\u00E2\u0086\u00B5\u0000 \u00E2\u008C\u0098 3(\u00E2\u0087\u00A3\u00E2\u0086\u00B5 \u0000 \u00E2\u0087\u00A3\u0000) (4.119)and its relative velocity perturbation is given byv\u00E2\u0086\u00B5\u0000 \u00E2\u008C\u0098 v\u00E2\u0086\u00B5 \u0000 v\u0000 = R\u00E2\u0086\u00B5 \u0000R\u0000H. (4.120)684.8. End of Inflation and ReheatingFor the decay of the elastic solid to a perfect fluid, the energy-momentumtransfer is given by Q\u00E2\u008C\u00ABe = \u0000Q\u00E2\u008C\u00ABf = \u0000\u0000g\u00E2\u008C\u00AB\u0000u\u0000\u00E2\u0087\u00A2e(1 + we), (4.121)where u\u0000 is the total velocity 4-vector of the elastic solid and the perfectfluid and \u0000 is the decay rate of the elastic solid into the fluid (not to beconfused with the gamma function used previously).In the current case, where a single substance is decaying into anothersingle substance, we expect that entropy perturbations will not be generatedin the decay. In general, the evolution of the entropy perturbation betweenthe elastic solid and the fluid is given by(S\u00CB\u0099ef )k = \"Q\u00CB\u0099e\u00E2\u0087\u00A2\u00CB\u0099f + Qe2\u00E2\u0087\u00A2 \u00E2\u009C\u0093 \u00E2\u0087\u00A2\u00CB\u0099f\u00E2\u0087\u00A2\u00CB\u0099e \u0000 \u00E2\u0087\u00A2\u00CB\u0099e\u00E2\u0087\u00A2\u00CB\u0099f \u00E2\u0097\u0086# (Sef )k+k2a2H \u00EF\u00A3\u00BF\u00E2\u009C\u00931\u0000 Qf\u00E2\u0087\u00A2\u00CB\u0099f \u00E2\u0097\u0086 (Rf )k \u0000 \u00E2\u009C\u00931\u0000 Qe\u00E2\u0087\u00A2\u00CB\u0099e \u00E2\u0097\u0086 (Re)k\u0000 , (4.122)where Q\u00E2\u0086\u00B5 = Q0(\u00E2\u0086\u00B5) is the background value of the time component of theenergy-momentum-transfer 4-vector. We refer to Appendix A.2 for the ex-plicit form of the background and perturbation equations used to derive thisrelation. We can see that if on superhorizon scales the entropy perturbationvanishes at some time, the entropy perturbation will stay approximatelyconstant past this time. If the decay is rapid (\u0000\u0000 H), then any preexistingentropy perturbations will quickly be driven to zero at the very beginning ofthe decay. Therefore, for times of interest, we set the entropy perturbationto zero. With no entropy perturbation, \u00E2\u0087\u00A3e = \u00E2\u0087\u00A3f and \u00E2\u0087\u00A3 will be continuousacross the decay, when it changes from \u00E2\u0087\u00A3 = \u00E2\u0087\u00A3e before the decay to \u00E2\u0087\u00A3 = \u00E2\u0087\u00A3fafter the decay. On the other hand, we do not expect the relative velocityperturbation to be zero during the decay, so in general R will change rapidlyduring the decay as it goes from R = Re before the decay to R = Rf \u00E2\u0087\u00A1 \u00E2\u0087\u00A3f onsuperhorizon scales after the decay.39 In this light, we will follow \u00E2\u0087\u00A3 instead ofR from the end of inflation into radiation domination. The point where theelastic solid decays presumably occurs when a macroscopic quantity such asthe energy density or pressure reaches a critical value. Although, due to theinhomogeneities in these fields, the decay many not occur at the same valueof the scale factor at every location, since these perturbations are small andthe superhorizon evolution is not drastic when scale-invariant spectra are39The assertions that \u00E2\u0087\u00A3 is continuous and R is discontinuous through the decay wereverified numerically.694.8. End of Inflation and Reheatingproduced, the approximation of the decay occurring uniformly at a = a\u00E2\u0087\u00A4should be a reasonable assumption.To find the postinflationary scalar power spectrum, we match \u00E2\u0087\u00A3 andits first derivative at the time of the decay and will assume that the decayproduct is radiation. During radiation domination, on large scales \u00E2\u0087\u00A3k evolvesas \u00E2\u0087\u00A3 00k + 2H\u00E2\u0087\u00A3 0k \u00E2\u0087\u00A1 0, (4.123)which has solution \u00E2\u0087\u00A3k(a \u0000 a\u00E2\u0087\u00A4) \u00E2\u0087\u00A1 \u00E2\u0087\u00A3k\u00E2\u0087\u00A4 + \u00E2\u0087\u00A3 0k\u00E2\u0087\u00A4H\u00E2\u0087\u00A4\u00E2\u0087\u00A31\u0000a\u00E2\u0087\u00A4a \u00E2\u008C\u0098 , (4.124)where integration constants have been chosen so that \u00E2\u0087\u00A3k and \u00E2\u0087\u00A3 0k are contin-uous over the decay. From Eq. (4.124), we see that \u00E2\u0087\u00A3k has both a constantand decaying mode during radiation domination. Within a few e-folds afterthe decay, the decaying mode becomes negligible, as seen in Fig. 4.2.0.1 1 10 1002.432.442.452.46a\u00C3\u00AAa*109PzFigure 4.2: The power spectrum of \u00E2\u0087\u00A3 during the decay of the elastic solidto radiation for a superhorizon mode. During inflation, sound speed andequation of state parameters are chosen to be those listed in the third rowof Table 4.1.Using Eq. (4.112) to calculate \u00E2\u0087\u00A3 from R before the decay, for the casewhere \u00E2\u009C\u008F0 = cs0 = 0, we find that the relationship between P\u00E2\u0087\u00A3(k, a \u0000 a\u00E2\u0087\u00A4) andPR(k, a\u00E2\u0087\u00A4) right before the decay, given in Eq. (4.103), isP\u00E2\u0087\u00A3(k, a \u0000 a\u00E2\u0087\u00A4)PR(k, a !\u0000a\u00E2\u0087\u00A4) = \u0000\u0000\u0000\u00003\u0000 4(3 + 2c2s1)\u00E2\u009C\u008F1 + 4\u00E2\u008C\u00A7s + 2\u00E2\u008C\u00A7\u00E2\u009C\u008F3c4s1 \u0000\u0000\u0000\u0000. (4.125)704.8. End of Inflation and ReheatingAs cs1 < 1 (but not necessarily cs1 \u00E2\u008C\u00A7 1), typically \u00E2\u0087\u00A3k will be larger than Rk,in which case if \u00E2\u0087\u00A3k is in the linear regime, so will Rk.We can express the power spectrum of \u00E2\u0087\u00A3 in terms of the pivot scalekp = 0.002Mpc\u00001 as P\u00E2\u0087\u00A3(k) = A\u00E2\u0087\u00A3(k/kp)ns\u00001. Assuming rapid reheating, forthe cs0 = \u00E2\u009C\u008F0 = 0 case A\u00E2\u0087\u00A3 can be written asA\u00E2\u0087\u00A3 = 10\u000022\u0000\u0000\u0000\u0000\u0000cns\u00006s1\u00E2\u009C\u008F1 \u0000\u0000\u0000\u0000 \u00E2\u0087\u00A3 gRH106.75\u00E2\u008C\u0098 7\u0000ns6 \u00E2\u009C\u0093 TRH1013 GeV\u00E2\u0097\u00865\u0000ns , (4.126)where TRH is the reheat temperature and gRH is the e\u00E2\u0086\u00B5ective number of rela-tivistic degrees of freedom contributing to the energy density at reheating.40In the above expression, \u0000 is a constant that is given in detail in AppendixA.3. For nearly scale-invariant scalar spectra, \u0000 is of order unity. When thesound speeds and equation of state are constant, A\u00E2\u0087\u00A3 is given by the sameexpression except with the substitutions cs1 ! cs and \u00E2\u009C\u008F1 ! \u00E2\u009C\u008F.Tracking the e\u00E2\u0086\u00B5ect of a rapid decay to radiation on the tensor modes isstraightforward, since as seen in Eq. (4.12) the tensor modes will only be af-fected by the change in anisotropic stress. Accordingly, we match the tensorperturbations and their first derivatives across the decay as the anisotropicstress vanishes. From Appendix A.1, the equation of motion for the tensorperturbations is given by 41(hTk )00 + 2H(hTk )0 + (k2 + 4c2v\u0000)hTk = 0. (4.127)Since there is negligible rigidity in the radiation fluid, the transverse soundspeed will vanish after the decay. During radiation domination, superhorizontensor modes have the same approximate evolution equation as the scalarmodes in Eq. (4.123); therefore, matching the tensor modes and their firstderivatives across the decay will have the same form as the scalar modesolution in Eq. (4.124), so thathTk (a \u0000 a\u00E2\u0087\u00A4) \u00E2\u0087\u00A1 hTk\u00E2\u0087\u00A4 + (hTk\u00E2\u0087\u00A4)0H\u00E2\u0087\u00A4\u00E2\u0087\u00A31\u0000a\u00E2\u0087\u00A4a \u00E2\u008C\u0098 . (4.128)Using the above equation with Eq. (4.109), the postinflationary tensor powerspectrum of superhorizon modes is related to its value at the end of inflation40In Eq. (4.126), we have assumed that all relativistic species are in thermodynamicequilibrium at TRH so that gRH = gsRH, where gs is the e\u00E2\u0086\u00B5ective number of relativisticdegrees of freedom contributing to the entropy density, and we make this assumptionthroughout this chapter.41In this section, hT will label a component of the tensor hTij .714.8. End of Inflation and ReheatingbyPT(k, a \u0000 a\u00E2\u0087\u00A4)PT(k, a\u00E2\u0087\u00A4) = (1\u0000 2\u0000T)2(\u00E2\u009C\u008F0 \u0000 1)44(1\u0000 \u00E2\u009C\u008F0 + \u00E2\u009C\u008F1)2 , (4.129)where the tensor power spectrum at the end of inflation in given in Eq. (4.111).The tensor-to-scalar ratio r = PT/P\u00E2\u0087\u00A3 after the decaying modes men-tioned above are negligible for the \u00E2\u009C\u008F0 = cs0 = 0 case can now be found tobe r \u00E2\u0087\u00A1 16c5s1\u00E2\u009C\u008F1(a\u00E2\u0087\u00A4H\u00E2\u0087\u00A4/k)ns\u00001\u0000nT . (4.130)The tensor-to-scalar ratio is mildly dependent on the (physical) wavenumberand Hubble rate at the end of inflation, since in general the scalar and tensormodes have di\u00E2\u0086\u00B5erent tilts. At the pivot scale kp, the tensor-to-scalar ratiois r(kp) \u00E2\u0087\u00A1 1.94\u00E2\u0087\u00A5 1022.9(ns\u00000.96\u0000nT)c5s1\u00E2\u009C\u008F1\u00E2\u009C\u0093\u00E2\u0087\u00A3 gRH106.75\u00E2\u008C\u00981/6 TRH1013 GeV\u00E2\u0097\u0086ns\u00001\u0000nT . (4.131)For this case, the tensor-to-scalar ratio is suppressed by the c5s1 term, sofor small values of cs1 the tensor-to-scalar ratio will be highly suppressed.For example, with the parameter values in the first row of Table 4.1 withcs1 = 0.01, the tensor-to-scalar ratio is r \u00E2\u0087\u00A0 10\u000012. However, if cs1 assumesa higher value, then this suppression is more moderate, illustrated by thevalues in the second row of Table 4.1, which with cs1 = 0.8 yields a tensor-to-scalar ratio of r = 0.002.As in Section 4.6.4, we do not write out an explicit expression for thescalar amplitude or tensor-to-scalar ratio for cases when either cs0 or \u00E2\u009C\u008F0 arenonzero and will soon illustrate with numerical examples instead. But beforedoing this, we can gain some insight by examining the case when the soundspeeds and equation of state are constant. As previously mentioned, a nearlyscale-invariant scalar spectrum can be produced when w is far from \u00001 ifcs is su\u0000ciently small. However, in this case there are added considerationsas the energy density changes significantly during the course of inflation. Ifinflation lasts just long enough to solve the \u00E2\u0080\u0098horizon problem\u00E2\u0080\u0099 and \u00E2\u0087\u00A4 is theenergy scale at the beginning of inflation, A\u00E2\u0087\u00A3 will be given byA\u00E2\u0087\u00A3 \u00E2\u0087\u00A1 10\u0000109\u0000ns(13.1+10.4\u00E2\u009C\u008F)+30\u00E2\u009C\u008F2\u0000\u00E2\u009C\u008F \u0000\u0000\u0000\u0000\u0000cns\u00006s \u00E2\u009C\u008F \u0000\u0000\u0000\u0000\u00E2\u0087\u00A5\u00E2\u0087\u00A3 gRH106.75\u00E2\u008C\u0098ns\u00001+18\u00E2\u009C\u008F\u00004ns\u00E2\u009C\u008F6(2\u0000\u00E2\u009C\u008F) \u00E2\u009C\u0093 \u00E2\u0087\u00A4mp\u00E2\u0097\u0086 5\u0000ns1+\u00E2\u009C\u008F/(2(1\u0000\u00E2\u009C\u008F)) (4.132)724.9. Conclusionfor cs, \u00E2\u009C\u008F constant, where mp is the Planck mass. For solutions with ns \u00E2\u0087\u00A0 1,if \u00E2\u009C\u008F is raised to higher values, A\u00E2\u0087\u00A3 may drop significantly. If \u00E2\u0087\u00A4 is bounded bythe Planck scale, for large values of \u00E2\u009C\u008F (w far from \u00001), to keep A\u00E2\u0087\u00A3 \u00E2\u0087\u00A0 10\u00009,cs may have to be fine-tuned to a very small value. Alternatively, thisfine-tuning may be averted if one is comfortable having inflation start atsuper-Planckian scales.This issue is demonstrated for the slow-varying sound speeds case inthe fifth row in Table 4.1, in which w varies close to \u00002/3. To attain thesame scalar amplitude as was used in the other examples in Table 4.1 andhave inflation start at or below the Planck scale and last for a su\u0000cientlylong duration, cs had to assume the extremely small value of \u00E2\u0087\u00A0 10\u00008. Inthe fourth row of Table 4.1, w ' \u00005/6, so while the departure from \u00001 isstill significant, we find cs can assume much larger values near 10\u00003 \u00E2\u0080\u0094 andsmaller values of w may have correspondingly larger values of cs.However, even if we do not tune cs to be very small, it is theoreticallyinteresting that we can achieve a scale-invariant spectrum far from w ' \u00001even if the amplitude of perturbations are not large enough to match ob-servations. For instance, the parameter values w = \u00002/3 and cs = 1/10produce a slightly blue-tilted spectrum for both scalar and tensor perturba-tions (with ns \u00E2\u0087\u00A1 1.04 and nT \u00E2\u0087\u00A1 0.04). Understanding the physical originof this near scale invariance is an extremely interesting question that mightgive new insight into the physics of horizons.Lastly, we compute the value of the tensor-to-scalar ratio for our exam-ples where either cs0 or \u00E2\u009C\u008F0 are nonzero. Using the values in the fourth row ofTable 4.1, where w varies slowly near \u00005/6, gives a tensor-to-scalar ratio atthe pivot scale of r \u00E2\u0087\u00A0 10\u000015. We again see that the tensor-to-scalar ratio ishighly suppressed by cs1. With the values listed in the last row of Table 4.1with cs0 = 0.15 yields r \u00E2\u0087\u00A1 10\u00006 and the suppression of r is more moderate.4.9 ConclusionBy having a su\u0000ciently rigid structure, a relativistic elastic solid is capableof driving an inflationary stage in the early Universe. In the case of constantsound speeds cs and cv and equation of state w, a blue-tilted scalar powerspectrum is produced. Allowing the sound speeds and equation of state tovary slowly in time can result in a red-tilted scalar power spectrum withsmall running. When cs is small, the tensor-to-scalar ratio will be highlysuppressed, but can attain larger values for higher values of cs.An interesting feature of this model is that perturbations evolve on su-734.9. Conclusionperhorizon scales, even in the absence of nonadiabatic pressure. The su-perhorizon evolution results from the shear stresses in the solid, where thepropagation of a single perturbative mode causes an anisotropic pressure.Because of this anisotropy, when smoothed on a superhorizon scale, di\u00E2\u0086\u00B5erentlocations in the Universe will not share the same FRW evolution, as theydo when both shearing stresses and nonadiabatic pressures are absent. Asa result, the perturbations do not \u00E2\u0080\u0098freeze-out\u00E2\u0080\u0099 soon after horizon crossingand consequently, the details of the end of inflation can impact both scalarand tensor power spectra for modes that are on superhorizon scales wheninflation ends. The case of a rapid decay of the elastic solid into radiationwas explored as a specific example.Finally and intriguingly, we find this model allows for w to vary slowlynear values that are significantly di\u00E2\u0086\u00B5erent from \u00001 and can find cases wherethis produces nearly scale-independent scalar and tensor power spectra de-spite being far from the de Sitter regime. This is surprising and unexpectedand it would be interesting to determine the underlying physical reason forthis phenomena.74Chapter 5The Physics of 21-cmRadiation5.1 IntroductionThe remaining chapters of this thesis will deal with the cosmic 21-cm signalemitted by neutral hydrogen. The nature of the cosmic 21-cm signal changesdrastically throughout the evolution of the Universe, most notably duringthe reionization of hydrogen at redshifts z \u00E2\u0087\u00A0 6\u000010. In this chapter, wereview some of the basic properties of the 21-cm signal and give a roughdescription of its evolution. In addition, we will conclude this chapter witha brief introduction to measurement techniques used with interferometricradio telescopes.5.2 Properties of 21-cm Radiation5.2.1 The Brightness TemperatureAs the first step for describing the basic properties of the 21-cm signal, wewrite the equation for radiative transfer [68]dI\u00E2\u008C\u00ABds = h\u00E2\u008C\u00AB4\u00E2\u0087\u00A1\u0000(\u00E2\u008C\u00AB)[n1A10 \u0000 (n0B01 \u0000 n1B10)I\u00E2\u008C\u00AB ], (5.1)where I\u00E2\u008C\u00AB is the specific intensity, \u0000(\u00E2\u008C\u00AB) is the line profile (normalized byR \u0000(\u00E2\u008C\u00AB)d\u00E2\u008C\u00AB = 1), ds is a proper length element, and n0 and n1 are the numberdensities for the unexcited and excited hyperfine states with degeneraciesg0 and g1, respectively (in the present case g0 = 1 and g1 = 3). A10, B10,and B01 are the Einstein coe\u0000cients for spontaneous emission, stimulatedemission, and absorption, respectively. The Einstein coe\u0000cients are relatedto one another by B10/A10 = c2/2h\u00E2\u008C\u00AB3 and B10/B01 = g0/g1, where A10 =2.85 \u00E2\u0087\u00A5 10\u000015 s\u00001. The di\u00E2\u0086\u00B5erential equation in Eq. (5.1) is easily solved andcan be expressed in the Rayleigh-Jeans limit (so that I\u00E2\u008C\u00AB can be written in755.2. Properties of 21-cm Radiationterms of a brightness temperature Tb(\u00E2\u008C\u00AB) = c2I\u00E2\u008C\u00AB/2kb\u00E2\u008C\u00AB2) asT 0b(\u00E2\u008C\u00AB) = TS(1\u0000 e\u0000\u00E2\u008C\u00A7\u00E2\u008C\u00AB ) + T 0R(\u00E2\u008C\u00AB)e\u0000\u00E2\u008C\u00A7\u00E2\u008C\u00AB , (5.2)where T 0R is the background radiation brightness, \u00E2\u008C\u00A7\u00E2\u008C\u00AB = R ds\u00E2\u0086\u00B5\u00E2\u008C\u00AB is the opticaldepth, and \u00E2\u0086\u00B5\u00E2\u008C\u00AB is the absorption coe\u0000cient given by\u00E2\u0086\u00B5\u00E2\u008C\u00AB = h\u00E2\u008C\u00AB4\u00E2\u0087\u00A1\u0000(\u00E2\u008C\u00AB)(n0B01 \u0000 n1B10). (5.3)The excitation temperature TS for the 21-cm transition, known as the spintemperature, specifies the relative number density of excited to unexcitedstates by n1n0 = g1g0 e\u0000T\u00E2\u0087\u00A4/TS , (5.4)where the energy di\u00E2\u0086\u00B5erence for the hyperfine transition is E01 = 5.9\u00E2\u0087\u00A510\u00006 eVwith equivalent temperature T\u00E2\u0087\u00A4 = E10/kB = 68mK [19, 69]. In all situationsthat we will consider we will have TS \u0000 T\u00E2\u0087\u00A4 so that n1/n0 \u00E2\u0087\u00A1 3 and thusstimulated emission will be an important process.We can now write the optical depth as\u00E2\u008C\u00A7\u00E2\u008C\u00AB = Z ds\u000010\u0000(\u00E2\u008C\u00AB)n0(1\u0000 e\u0000T\u00E2\u0087\u00A4/TS), (5.5)where \u000001 = 3c2A10/8\u00E2\u0087\u00A1\u00E2\u008C\u00AB2. The integral in Eq. (5.5) can be evaluated usingds = (c/aH)da, so the brightness temperature of the 21-cm signal measuredagainst the cosmic microwave background (CMB) at redshift z is given by[19] \u0000Tb(z) = TS \u0000 T\u00001 + z (1\u0000 e\u0000\u00E2\u008C\u00A7\u00E2\u008C\u00AB0 )\u00E2\u0087\u00A1 27xHI(1 + \u0000)\u00E2\u009C\u00931\u0000 T\u0000TS\u00E2\u0097\u0086\u00E2\u009C\u00931 + z10 0.15\u00E2\u008C\u00A6mh2\u00E2\u0097\u00861/2\u00E2\u0087\u00A5\u00E2\u009C\u0093\u00E2\u008C\u00A6bh20.023\u00E2\u0097\u0086\u00E2\u009C\u0093 HH + dvk/drk\u00E2\u0097\u0086 mK, (5.6)where \u00E2\u008C\u00A7\u00E2\u008C\u00AB0 is the optical depth at the 21-cm frequency \u00E2\u008C\u00AB0, T\u0000 is the CMBtemperature, and dvk/drk is the comoving velocity gradient along the line ofsight. We have expressed \u0000Tb(z) in terms of the neutral hydrogen fractionxHI = nHI/nH, where nH = n\u00C2\u00AFH(1 + \u0000) and \u0000 is the overdensity.An important feature of the 21-cm brightness temperature is that itsaturates in emission when TS \u0000 T\u0000 , a state which is expected from the765.2. Properties of 21-cm Radiationtime of reionization onwards. In this case we can safely drop the T\u0000/TSterm in Eq. (5.6), which makes \u0000Tb independent of the spin temperature.This simplifies the situation greatly as the spin temperature is often di\u0000cultto calculate.5.2.2 The Spin TemperatureFor the pre-reionization 21-cm signal, we require the value of the spin tem-perature. In cosmological contexts, there are three main sources that cana\u00E2\u0086\u00B5ect the spin temperature: the CMB temperature from the absorption of orstimulated emission from CMB photons, the kinetic temperature of the sur-rounding gas via collisions, and ultraviolet (UV) fields via the Wouthuysen-Field (WF) mechanism [70, 71]. In the absence of the latter two e\u00E2\u0086\u00B5ects, thespin temperature reaches thermal equilibrium with the CMB temperatureon a timescale much shorter than those relevant for cosmology. In this case,TS \u00E2\u0087\u00A1 T\u0000 and no 21-cm signal can be observed. To observe the 21-cm signal,the spin temperature must depart from the CMB temperature by means ofcollisions or the WF mechanism.The manner in which the spin temperature couples to the kinetic temper-ature of the surrounding gas by collisional excitations is a familiar processin physics, while the WF mechanism is somewhat more obscure. The WFmechanism describes the process where hyperfine states are mixed by wayof the absorption and subsequent emission of a Lyman-\u00E2\u0086\u00B5 photon. Selectionrules allow transitions between 1S and 2P hyperfine levels where the electronwhen returned to the 1S state is in a di\u00E2\u0086\u00B5erent hyperfine level then it wasbefore the absorption of the Lyman-\u00E2\u0086\u00B5 photon.42 This process is illustratedin Fig 5.1.42Similar transitions are possible with higher Lyman levels, although the e\u00E2\u0086\u00B5ect of suchtransitions is negligible compared to the Lyman-\u00E2\u0086\u00B5 transition. However, transitions tohigher Lyman levels may be important from cascades through the 2P levels [19].775.2. Properties of 21-cm RadiationFigure 5.1: Hyperfine levels relevant for the WF mechanism. Transitionsusing solid lines change the 1S hyperfine levels while the dashed lines donot. Figure from Ref. [72].As the time scales of the aforementioned processes are all much shorterthan cosmological time scales, we can safely assume equilibrium and thushave the balanced equationn0(C01 + P01 +B01ICMB) = n1(C10 + P10 +A10 +B10ICMB), (5.7)where C01, C10 and P01, P10 are the excitation and de-excitation rates viacollisions and the WF mechanism, respectively, and ICMB is the specificintensity of the CMB. The ratio between excitation and de-excitation ratesby means of collisions is given by the kinetic temperature of the surroundinggas TK as C01C10 = g1g0 e\u0000T\u00E2\u0087\u00A4/TK \u00E2\u0087\u00A1 3\u00E2\u009C\u00931\u0000 T\u00E2\u0087\u00A4TK\u00E2\u0097\u0086 , (5.8)where again we have assumed that we will have TK \u0000 T\u00E2\u0087\u00A4 for all situationsunder consideration. It will prove to be convenient to keep track of theratio between excitation and de-excitation rates via the WF mechanism inan analogous manner by use of the colour temperature T\u00E2\u0086\u00B5 defined byP01P10 = 3\u00E2\u009C\u00931\u0000 T\u00E2\u0087\u00A4T\u00E2\u0086\u00B5\u00E2\u0097\u0086 . (5.9)With these definitions, Eq. (5.7) in the Rayleigh-Jeans limit becomesT\u00001S = T\u00001\u0000 + x\u00E2\u0086\u00B5T\u00001\u00E2\u0086\u00B5 + xcT\u00001K1 + x\u00E2\u0086\u00B5 + xc , (5.10)785.2. Properties of 21-cm Radiationwhere xc and x\u00E2\u0086\u00B5 are the collisional and WF coupling coe\u0000cients, respec-tively, given by xc = C10A10 T\u00E2\u0087\u00A4T\u0000 x\u00E2\u0086\u00B5 = P10A10 T\u00E2\u0087\u00A4T\u0000 . (5.11)When examining collisional coupling, the important collisions are that of HIwith other hydrogen atoms, and free electrons and protons (see Ref. [19] foran in-depth analysis).In order to calculate the WF coupling coe\u0000cient x\u00E2\u0086\u00B5, we must first findthe WF de-excitation rate P10, which will depend on the scatter rate ofLyman-\u00E2\u0086\u00B5 photons. By careful examination of the Lyman-\u00E2\u0086\u00B5 transition be-tween di\u00E2\u0086\u00B5erent hyperfine levels, one finds that if the background radiationfield is constant over the di\u00E2\u0086\u00B5erent hyperfine levels, then P10 is related to thetotal scattering rate of Lyman-\u00E2\u0086\u00B5 photons P\u00E2\u0086\u00B5 by P10 = (4/27)P\u00E2\u0086\u00B5 [70]. P\u00E2\u0086\u00B5 inturn is related to the angle-averaged specific intensity J\u00E2\u008C\u00AB of the backgroundradiation at frequency \u00E2\u008C\u00AB by P\u00E2\u0086\u00B5 = 4\u00E2\u0087\u00A1 Z d\u00E2\u008C\u00AB\u0000\u00E2\u008C\u00ABJ\u00E2\u008C\u00AB , (5.12)where \u0000\u00E2\u008C\u00AB is the local absorption cross section at frequency \u00E2\u008C\u00AB. We can writex\u00E2\u0086\u00B5 as a function of J\u00E2\u008C\u00AB evaluated at the Lyman-\u00E2\u0086\u00B5 line centre (denoted by J\u00E2\u0086\u00B5)as x\u00E2\u0086\u00B5 = S\u00E2\u0086\u00B5J\u00E2\u0086\u00B5Jc\u00E2\u008C\u00AB , (5.13)where Jc\u00E2\u008C\u00AB = 5.825 \u00E2\u0087\u00A5 10\u000012(1 + z) cm\u00002s\u00001Hz\u00001sr\u00001 and S\u00E2\u0086\u00B5 is a correctionterm to account for the variation in the background radiation field near theLyman-\u00E2\u0086\u00B5 line centre (see Ref. [73] for a calculation of S\u00E2\u0086\u00B5).The colour temperature T\u00E2\u0086\u00B5 determines the relative rate of excitations tode-exciations of the 1S hyperfine levels via the absorption and re-emissionof Lyman-\u00E2\u0086\u00B5 photons. As such, the colour temperature will depend on therelative occupation number n\u00E2\u008C\u00AB of photons with frequency \u00E2\u008C\u00AB in the vicinity ofthe Lyman-\u00E2\u0086\u00B5 frequency, as each transition will require absorption or emissionof Lyman-\u00E2\u0086\u00B5 photons with slightly di\u00E2\u0086\u00B5erent frequencies. Since this di\u00E2\u0086\u00B5erencein energy is relatively small, we can approximate the ratio P01/P10 asP01P10 \u00E2\u0087\u00A1 g1g0 \u00E2\u009C\u00931 + \u00E2\u008C\u00AB0d lnn\u00E2\u008C\u00ABd\u00E2\u008C\u00AB \u00E2\u0097\u0086 , (5.14)and by comparing to Eq. (5.9) we can write the colour temperature asT\u00E2\u0086\u00B5 \u00E2\u0087\u00A1 hkb \u00E2\u009C\u0093\u0000d lnn\u00E2\u008C\u00ABd\u00E2\u008C\u00AB \u00E2\u0097\u0086\u00001 . (5.15)795.3. History of the 21-cm SignalIf the scattering rate of Lyman-\u00E2\u0086\u00B5 photons is very high, as is the case forthe high-redshift Universe, through the exchange of energy through atomicrecoils the photon spectrum near the Lyman-\u00E2\u0086\u00B5 frequency will be given ap-proximately by a blackbody spectrum of temperature TK, in which caseEq. (5.15) implies T\u00E2\u0086\u00B5 \u00E2\u0087\u00A0 TK [74]. A more precise expression for T\u00E2\u0086\u00B5 as afunction of TK can be found in Ref. [73].5.3 History of the 21-cm SignalCurrently, the exact history of the pre-reionization 21-cm signal is not pre-cisely known. However, we anticipate the presence of a few likely generalevents in the 21-cm signal\u00E2\u0080\u0099s evolution. In this section we give a brief de-scription of some likely generic features in the evolution of the 21-cm signal.After recombination, although most electrons are found within atoms,a small fraction (\u00E2\u0087\u00A0 O(10\u00003)) of residual free electrons remain and coupleto the CMB through Compton scattering. This coupling remains strongwell after recombination until the residual free electrons become extremelydi\u00E2\u0086\u00B5use. While strongly coupled to the CMB temperature, the gas has akinetic temperature TK \u00E2\u0087\u00A1 T\u0000 . At these early times, the gas is dense enoughso that collisional coupling is strong, so at this time we have TS \u00E2\u0087\u00A1 TK \u00E2\u0087\u00A1T\u0000 and thus no 21-cm signal can be observed at this point. Numericalcalculations using the RECFAST43 code [75] predict that the decouplingof the kinetic temperature of the gas from the CMB temperature occursaround z \u00E2\u0087\u00A0 150, presumed to be well into the dark ages before significantastrophysical structure formation occurs.After decoupling from the CMB, the gas cools adiabatically with a cool-ing rate which is faster than that of the CMB. Therefore, an absorptionsignal may be present after decoupling. As the gas continues to cool, thecollisional coupling becomes less e\u0000cient, driving TS back up to the CMBtemperature. At this point, the 21-cm signal may disappear again and re-mains at zero unless the spin temperature deviates from the CMB temper-ature again due to the presence of astrophysical sources. Photons emittedfrom astrophysical sources may a\u00E2\u0086\u00B5ect the spin temperature through the WFprocess as well as may heat the the gas, raising the spin temperature. A sig-nal in emission is generally predicted, followed by reionization which drivesthe 21-cm signal emitted from the intergalactic medium (IGM) to zero onceagain. After reionization is complete, 21-cm emissions are confined to over-dense regions.43http://www.astro.ubc.ca/people/scott/recfast.html805.4. Radio Interferometry and Detection of 21-cm Signal5.4 Radio Interferometry and Detection of 21-cmSignalNow that we have a basic understanding of the 21-cm signal and a generalidea of its evolution throughout cosmic history, we now focus our atten-tion on the detection of the 21-cm signal by means of radio interferometrictelescopes.Interferometric telescopes consist of a array of antennas, each of whichmeasure the electromagnetic field at a particular location. We label thesignal from the i-th feed as Fi. The correlation between feeds i and j,known as the visibility, is given byVij = hF \u00E2\u0087\u00A4i Fji = 1p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j Z d2n\u00CB\u0086A\u00E2\u0087\u00A4i (n\u00CB\u0086)Aj(n\u00CB\u0086)e2\u00E2\u0087\u00A1in\u00CB\u0086\u00C2\u00B7uijT (n\u00CB\u0086), (5.16)where T (n\u00CB\u0086) is the brightness temperature of the sky (related to the inten-sity I by T = (\u00002/2kb)I) coming from direction n\u00CB\u0086 and uij is the spatialseparation between the two feeds in units of wavelength, which we refer toas a baseline. Ai(n\u00CB\u0086) is the antenna response which has a solid angle of\u00E2\u008C\u00A6i = R d2n\u00CB\u0086|Ai(n\u00CB\u0086)|2. The term n\u00CB\u0086 \u00C2\u00B7 uij describes the lag in the arrival of thesignal at one feed compared to the other and the exponential term describesthe interference between the two signals. Although we have not made anyimplicit mention of polarization at this point, the feed response Fi, the beamAi(n\u00CB\u0086), and the sky intensity T (n\u00CB\u0086) all implicitly refer to either a particularpolarization or combination of polarizations. For example, the sky signalmay be decomposed into separate components for each of the Stoke\u00E2\u0080\u0099s pa-rameters. However, such details are not necessary for the purposes of thissection.It is apparent from Eq. (5.16) that the visibilities measure Fourier modes(or more appropriately spherical harmonics) on the sky modulated by thebeam response and that a map of the sky can be attained by sampling thevisibility in the u plane and then Fourier transforming. We can think of eachvisibility as being sensitive to a limited number of modes on the sky. As such,the angular resolution of the telescope is set by the largest baseline. We canestimate the angular resolution of an array at wavelength \u0000 by \u0000\u00E2\u009C\u0093 \u00E2\u0087\u00A0 \u0000/L,where L is the length of the longest baseline. This angle corresponds to thecomoving distance DA(1+z)\u0000/L in the direction perpendicular to the line ofsight. For an array with a longest baseline of L = 100m, \u0000\u00E2\u009C\u0093 runs between\u00E2\u0087\u00A0 0.2\u0000\u00000.5\u0000 in the frequency range 400\u0000800 MHz, which corresponds to815.4. Radio Interferometry and Detection of 21-cm Signalcomoving distances roughly between \u00E2\u0087\u00A0 10\u000050Mpc.44 Such a resolutionshould be adequate for measuring the baryon acoustic oscillations (BAO),which has a comoving length scale of \u00E2\u0087\u00A0 150Mpc.The power p measured by the autocorrelation of a particular feed can berepresented by p = g2kb\u0000\u00E2\u008C\u00ABTsys, where g is the gain of the feed and \u0000\u00E2\u008C\u00AB isthe size of the frequency bin. The system temperature Tsys is a sum over allsources of power, including both sources on the sky as well as instrumentalnoise.By forming the four-point function and with use of Wick\u00E2\u0080\u0099s theorem, thevariance of the output of the correlator forming the visibilities can be found.If the noise between feeds i and j are uncorrelated, this variance is given by2Tsys,iTsys,j. By averaging over time, the variance can be reduced by a factorof N = 2tint\u0000\u00E2\u008C\u00AB, representing the number of independent measurementspossible within the integration time tint. The variance on a visibility thenbecomes \u00002ij = Tsys,iTsys,jtint\u0000\u00E2\u008C\u00AB . (5.17)This highlights the key factors in reducing uncertainty in measurements ofradio signals: Having long integration times and a system that adds as littleas possible additional noise to the system temperature can both improve theprecision of our instrument.44Assuming a cosmology with \u00E2\u008C\u00A6m = 0.27, \u00E2\u008C\u00A6\u00E2\u0087\u00A4 = 0.73, and h = 0.7.82Chapter 6The Imprint of Warm DarkMatter on the Cosmological21-cm Signal6.1 IntroductionHierarchical structure formation45 within the \u00E2\u0087\u00A4CDM model has been ex-ceptionally accurate in describing the large-scale Universe within the range\u00E2\u0087\u00A0 10Mpc\u00001Gpc, as demonstrated from studies of the cosmic microwavebackground (CMB) and the clustering of galaxies. However, for over adecade concerns have been raised over whether the standard assumptionof cold dark matter (CDM) provides an adequate fit to data on smaller,sub-Mpc scales. These include predictions from N -body simulations thatyield an overabundance of galactic satellites around our galaxy and in thefield [76, 77, 78], as well as in voids [79], and produce overly-dense galacticcentres with \u00E2\u0080\u0098cuspy\u00E2\u0080\u0099 density profiles [80, 81, 82] and are inconsistent withobservations of the kinetic properties of bright Milky Way satellites [83, 84].One possible explanation lies with baryonic feedback processes [85, 86,87, 88, 89], although accurately modelling these mechanisms is often chal-lenging and di\u0000culties may persist in matching to observations.Another possible explanation is to change the properties of dark matterso it is warm (WDM).46 This may alleviate these small-scale problems dueto the higher velocities of the dark matter. In this case, structures aresmoothed on scales below the dark matter\u00E2\u0080\u0099s free-streaming length. Non-relativistic residual velocities can delay halo collapse and star formation.These e\u00E2\u0086\u00B5ects may reduce the number of sub-halos and low-mass galaxiesthat are formed as well as flatten out galactic centres.45Hierarchical structure formation describes the general formation procedure of smallerobjects forming first and then merging to form larger structures.46Other possible alterations to the standard CDM model that may resolve these small-scale problems include self-interacting dark matter [90, 91, 92] and atomic dark matter orother models with acoustic damping of dark matter fluctuations [93, 94].836.1. IntroductionThe two most popular WDM candidates in the literature motivated byparticle physics have been the sterile neutrino [95, 96, 97] and the gravitino[98, 99]. While WDM may be produced in a number of di\u00E2\u0086\u00B5erent ways, itis most often described as a thermal relic that decouples while relativistic,but is non-relativistic by matter-radiation equality as to preserve structurebeyond the Mpc scale. In this case, the WDM would have a particle massmXof the order of a keV. Although for our purposes the free-streaming scaleof the dark matter is a more fundamental quantity, we use the standardconvention of discussing the WDM mass of a thermal relic instead. Wecaution that for other WDM production mechanisms the correspondencebetween free-streaming length and mass will be di\u00E2\u0086\u00B5erent. We also remarkthat the results presented in this chapter can be applicable to models otherthan WDM that have similar cut-o\u00E2\u0086\u00B5 scales in their power spectrum (see,e.g. Ref. [93]).As WDM suppresses growth of small structures, which form first in thehierarchical structure formation of CDM, early star formation is delayedin WDM models. Detection of signals emitted from high-redshift objectseither directly, such as from gamma-ray bursts (GRBs) [100] or stronglylensed galaxies [101], or indirectly through the redshift of reionization [102],can place constraints on mX. Recently, Ref. [103] using GRB cataloguesplaced a constraint of mX > 1.6 \u0000 1.8 keV at 95% CL. Requiring WDMmodels to be able to reproduce both the stellar mass function and Tully-Fisher relation places a lower bound of mX \u0000 0.75 keV [104]. The Lyman-\u00E2\u0086\u00B5forest can probe scales down to \u00E2\u0087\u00A0 1Mpc and can provide strict limits onmX [105, 106, 107, 108], with the most recent and stringent constraint ofmX > 3.3 keV at 2\u0000 [109]. Although it has been claimed that the lessdense galactic cores formed in WDM models may provide a better fit tothe kinematic data of bright Milky Way satellites [110], there is an ongoingdebate as to whether WDM with a mass above current lower bounds cancreate a large enough galactic core as needed to solve the \u00E2\u0080\u0098cusp-core\u00E2\u0080\u0099 problem[111, 112]; though see Ref. [113].Highly-redshifted 21-cm radiation emitted from the hyperfine spin-flipof neutral hydrogen is a promising new tool to probe the high-redshift Uni-verse [19, 114, 115, 116, 117]. If WDM is present in su\u0000cient quantitiesto significantly delay structure formation, it could potentially leave a tracewithin the 21-cm radiation signal. Light emitted by the first astrophysicalsources can couple the spin temperature of neutral hydrogen to the kinetictemperature of the IGM through the Wouthuysen-Field (WF) mechanism[70, 71], as well as heat and ionize the IGM. Thus, a delay in the appearanceof these early sources can alter the 21-cm signal and delay milestones in the846.2. Thermal Relicsignal. In this chapter, we will examine the e\u00E2\u0086\u00B5ects of WDM on the pre-reionization 21-cm signal. This era may be especially useful for examiningWDM since WDM inhibits the formation of low-mass halos that form firstin CDM models and thus di\u00E2\u0086\u00B5erences between the halo populations in CDMand WDM increase with redshift. As astrophysics is very poorly known athigh-redshifts (z \u0000 6), we will focus on characterizing degeneracies betweenthe unknown astrophysics and the presence of WDM.The outline of this chapter is as follows: In Section 6.3, we review thee\u00E2\u0086\u00B5ects of the free-streaming of the WDM on the linear power spectrumand its residual velocities on halo collapse. The basic properties of the21-cm signal are outlined in Section 6.4 and its simulation is described inSection 6.5, with the simulation results discussed in Section 6.6. Throughoutthis chapter, we assume cosmological parameter values of \u00E2\u008C\u00A6\u00E2\u0087\u00A4 = 0.73, \u00E2\u008C\u00A6m =0.27, \u00E2\u008C\u00A6b = 0.046, h = 0.7, \u00008 = 0.82, ns = 0.96. We quote all quantities incomoving units, unless stated otherwise.6.2 Thermal RelicAs previously mentioned, WDM is most often described as a thermal relic[4, 118, 119]. Here we derive some basic results that relate fundamentalproperties of the WDM to its free-streaming length.We begin by examining the distribution function for a WDM thermalrelic. Since the physical momentum p scales as |p| / a\u00001 for both relativisticand nonrelativistic noninteracting particles, after decoupling the comovingmomentum q = ap of the WDM remains constant with time (assumingany interactions are negligible at this time). Consequently, after decouplingthe distribution function f of the WDM written as a function of the co-moving momentum is constant in time. Therefore, the distribution functionas a function of physical momentum evolves as f(p, t) = fi((a(t)/ai)p),where the subscript i denotes evaluation at some initial time. Since theWDM decouples while relativistic, the initial distribution function fi forthe WDM is a function of p/Ti and at any later time (both when theWDM is relativistic or nonrelativistic) the distribution function will begiven by f(p, t) = fi((a(t)/ai)p/Ti). We can define an e\u00E2\u0086\u00B5ective temperatureTX(t) = (ai/a(t))Ti for the WDM that specifies its distribution functionwhile both relativistic and nonrelativistic as f(p, t) = fi(p/TX(t)), whichwill coincide with its physical temperature while relativistic.Assuming all species to be in thermal equilibrium at early enough times,we can use the conservation of entropy given in Eq. (2.17) with TX / a\u00001856.2. Thermal Relicto relate TX to the photon temperature TTXT = \u00E2\u009C\u0093gs0gs\u00E2\u0087\u00A4\u00E2\u0097\u00861/3 , (6.1)where gs0 and gs\u00E2\u0087\u00A4 are the number of relativistic degrees of freedom contribut-ing to the entropy at present and decoupling, respectively. As the WDMdecouples while relativistic, this calculation is identical to the decoupling ofneutrinos, where we recover the well-known result for the fraction betweenthe neutrino and photon temperatures by setting gs\u00E2\u0087\u00A4 to its value at neutrinodecoupling (gs\u00E2\u0087\u00A4 = 10.75) [1]. Since gs0 \u00EF\u00A3\u00BF gs\u00E2\u0087\u00A4, TX will be lower than thephoton temperature soon after the WDM decouples, as the WDM missesout on the entropy release from the annihilation of other species after itsdecoupling.The present day ratio between the WDM and photon number densitiescan be found from Eq. (2.15c) asnX0n\u00000 = \u00E2\u009C\u0093TXT \u00E2\u0097\u00863 gnXgn\u0000 = gs0gs\u00E2\u0087\u00A4 gnX2 , (6.2)where gn\u0000 = 2 and gnX are the number of relativistic degrees of freedomcontributing to the number density of the photons and WDM, respectively.Most often the WDM is assumed to be a spin-12 particle, in which casegnX = 3/2. Since at the present day the WDM is nonrelativistic, its energydensity can be found by \u00E2\u0087\u00A2X = mXnX, which results in the present-dayfractional energy density of the WDM of\u00E2\u008C\u00A6Xh2 \u00E2\u0087\u00A1 115gs\u00E2\u0087\u00A4 gnX1.5 mXkeV , (6.3)where we have used gs0 = 43/11 [3]. Therefore, the required value of gs\u00E2\u0087\u00A4 toobtain the observed present-day dark matter density isgs\u00E2\u0087\u00A4 \u00E2\u0087\u00A1 767\u00E2\u009C\u0093\u00E2\u008C\u00A6Xh20.15 \u00E2\u0097\u0086\u00001 gnX1.5 mXkeV . (6.4)On first observation, we see that a keV scale WDM thermal relic wouldhave had to decouple at a time when gs\u00E2\u0087\u00A4 is much larger than that in thestandard model while all species are relativistic, requiring physics beyondthe standard model to add an array of new particles. However, as was notedin Ref. [120], other scenarios, such as a production of entropy after WDMdecoupling, have the net e\u00E2\u0086\u00B5ect of increasing gs\u00E2\u0087\u00A4, relaxing the requirement ofhaving to add variety of new particles.866.3. E\u00E2\u0086\u00B5ect of WDM on structure formationWe now return to the distribution function of the WDM. At late timeswhen the WDM is nonrelativistic, we can use pX = mXv to write its distri-bution function as a function v/v0 where v is the velocity of the WDM andv0 = TX/mX, which using Eqs. (6.1) and (6.4) is given byv0(z) = 0.0121(1 + z)\u00E2\u009C\u0093\u00E2\u008C\u00A6Xh20.15 \u00E2\u0097\u00861/3 \u00E2\u0087\u00A3gnX1.5 \u00E2\u008C\u0098\u00001/3 \u00E2\u0087\u00A3mXkeV\u00E2\u008C\u0098\u00004/3 km s\u00001. (6.5)For the remainder of this chapter, we assume our WDM thermal relic tobe a fermion, although adapting the results for a boson is straightforward.With this, the distribution function of the WDM is f(v) = [exp(v/v0)+1]\u00001with a root-mean-squared velocity vrms = 3.597v0.6.3 E\u00E2\u0086\u00B5ect of WDM on structure formation6.3.1 Free-streamingThe free-streaming of WDM particles smears out perturbations on smallscales, as WDM particles stream out of over-dense regions and into under-dense regions. Perturbations are suppressed on scales below that corre-sponding to the WDM particle horizon.The e\u00E2\u0086\u00B5ect of free-streaming on the spectrum of linear perturbations canbe included by use of a transfer function TX(k) that dampens small-scalefluctuations as compared to those in CDM. This transfer function can befound by fitting the results of a Boltzmann code that utilizes the WDMvelocity found in Section 6.2, which we take asTX(k) = (1 + (\u00E2\u009C\u008FkR0c)2\u00E2\u008C\u00AB)\u0000\u00E2\u008C\u0098/\u00E2\u008C\u00AB , (6.6)where \u00E2\u009C\u008F = 0.361, \u00E2\u008C\u0098 = 5, and \u00E2\u008C\u00AB = 1.2 [120]. R0c is the comoving cuto\u00E2\u0086\u00B5 scale,at which the power in k = 1/R0c is reduced by half compared to that inCDM, and is given byR0c = 0.201\u00E2\u009C\u0093\u00E2\u008C\u00A6Xh20.15 \u00E2\u0097\u00860.15 \u00E2\u0087\u00A3gnX1.5 \u00E2\u008C\u0098\u00000.29 \u00E2\u0087\u00A3mXkeV\u00E2\u008C\u0098\u00001.15 Mpc, (6.7)where gnX is the number of e\u00E2\u0086\u00B5ective degrees of freedom contributing tonumber density, with bosons contributing unity to gnX and fermions con-tributing 3/4. We will use the standard assumption that the WDM is aspin-12 fermion, so that gnX = 3/2. \u00E2\u008C\u00A6X is the energy density parametercontributed by the WDM, which we set to \u00E2\u008C\u00A6X = \u00E2\u008C\u00A6m \u0000 \u00E2\u008C\u00A6b as we will only876.3. E\u00E2\u0086\u00B5ect of WDM on structure formationbe considering models where WDM constitutes the whole of the dark mat-ter. The transfer function in Eq. (6.6) serves to suppress small-scale linearperturbations in the power spectrum, which we generate using the transferfunction of Ref. [28].6.3.2 Residual velocitiesIn addition, the residual velocity dispersion of the WDM delays the growthof non-linear perturbations and consequently collapse into virialized halos.This can be thought of as an \u00E2\u0080\u0098e\u00E2\u0086\u00B5ective pressure\u00E2\u0080\u0099. Ref. [120] modelled thecollapse in WDM by studying collapse in an analogous system comprised ofa monoatomic adiabatic gas at temperature T / v2rms. The gas temperatureevolves as T / 1/a2 so its root-mean-square velocity evolves as vrms / 1/a,as the case with WDM as seen in Eq. (6.5). The initial temperature ofthe gas is set such that it shares the same vrms with the WDM as found inSection 6.2.Using the gas analogue in a spherically symmetric hydrodynamics sim-ulation, Ref. [120] computed the linear collapse threshold \u0000c(M, z), findingthat the collapse threshold rises sharply near the Jeans mass MJ, the massin which a gas cloud\u00E2\u0080\u0099s internal pressure can no longer support it againstgravitational collapse, for the analogue gas. From Eq. (6.5), the Jeans massof the gas analogue is proportional to MJ / T 3/2/\u00E2\u0087\u00A21/2 / (\u00E2\u008C\u00A6Xh2)1/2g\u00001nXm\u00004X .The results of Ref. [103] showed that using the extended Press-Schechter(EPS) formalism to compute the collapse fraction with a sharp minimummass cuto\u00E2\u0086\u00B5 at MJ and the collapse threshold for spherical collapse in CDM(\u0000c \u00E2\u0087\u00A1 1.69) is in good agreement with the full random-walk procedure withthe WDM modified collapse threshold as used in Ref. [120]. To achieve thisclose agreement, a factor of 60 was added to the expression for MJ originallyfound in Ref. [120], so that MJ is given byMJ \u00E2\u0087\u00A1 1.5\u00E2\u0087\u00A5 1010\u00E2\u009C\u0093\u00E2\u008C\u00A6Xh20.15 \u00E2\u0097\u00861/2 \u00E2\u0087\u00A3gnX1.5 \u00E2\u008C\u0098\u00001 \u00E2\u0087\u00A3mXkeV\u00E2\u008C\u0098\u00004 M\u0000. (6.8)As using the sharp cuto\u00E2\u0086\u00B5 at MJ is much less computationally intensive andeasily integrable within the EPS formalism, we employ this method insteadof the full random-walk procedure.6.3.3 Halo AbundancesThe production rate of photons that are capable of heating or ionizing theIGM, or coupling the spin temperature to the colour temperature via the886.3. E\u00E2\u0086\u00B5ect of WDM on structure formationWF mechanism, is modelled as being proportional to the collapse fractionfcoll(z,Mmin) of halos with su\u0000cient mass (\u0000 Mmin) to host star-forminggalaxies. To compute the mean collapse fraction, we use the Sheth-Tormenmass function found in Eq. (2.48), giving the comoving number density ofhalos with mass between M and M + dM asdnSTdM = \u0000Ar 2\u00E2\u0087\u00A1 \u00E2\u0087\u00A2\u00C2\u00AFmM dln\u0000dM \u00E2\u008C\u00AB\u00CB\u0086(1 + \u00E2\u008C\u00AB\u00CB\u0086\u00002p)e\u0000\u00E2\u008C\u00AB\u00CB\u00862/2, (6.9)where \u00E2\u008C\u00AB\u00CB\u0086 = pa\u0000c(M, z)/\u0000(M), \u00E2\u0087\u00A2\u00C2\u00AFm is the mean matter energy density, \u0000(M)is the rms of density fluctuations smoothed on a scale that encompasses amass M . A, a, and p are fit parameters taken as A = 0.353, a = 0.73, andp = 0.175 [121]. The mean collapse fraction is computed asfcoll(> Mmin, z) = 1\u00E2\u0087\u00A2m Z 1Mmin MdnSTdM dM, (6.10)where Mmin = max(MJ,Msf) and Msf is the minimum halo mass where star-formation can occur. MJ is assigned a value of zero in the case of CDM. Itwill be convenient to express Msf in terms of the corresponding virializedhalo temperature Tvir as (see Eq. (2.52))Msf = 9.37\u00E2\u0087\u00A5 107 \u00E2\u0087\u00A3 \u00C2\u00B50.6\u00E2\u008C\u0098\u00003/2\u00E2\u009C\u0093 h0.7\u00E2\u0097\u0086\u00001\u00E2\u009C\u0093\u00E2\u008C\u00A6m0.3\u00E2\u0097\u0086\u00001/2\u00E2\u0087\u00A5\u00E2\u009C\u00931\u00E2\u008C\u00A6zm \u0000c18\u00E2\u0087\u00A12\u00E2\u0097\u0086\u00001/2\u00E2\u009C\u00931 + z10 \u00E2\u0097\u0086\u00003/2\u00E2\u009C\u0093 Tvir104 K\u00E2\u0097\u00863/2 M\u0000. (6.11)The mean collapse fraction in CDM and WDM models can be seen inFig. 6.1. At high redshifts, small halos begin to collapse in CDM, while noor few such halos collapse in WDM, resulting in a large relative di\u00E2\u0086\u00B5erencebetween the collapse fractions in these models. However, this di\u00E2\u0086\u00B5erencebecomes smaller with lower redshifts as objects on scales larger than thatinhibited by WDM start to collapse in both models. At late times, in theCDM scenario the mass within halos of sizes suppressed by WDM onlyrepresents a small fraction of the total mass within all collapsed structures,so the relative di\u00E2\u0086\u00B5erence between the mean collapse fraction in CDM andWDM models is small at those times. Therefore, while structure formationis delayed in WDM models, the mean collapse fraction raises more rapidlyas compared to CDM.896.4. Cosmic 21-cm signal0 5 10 15 20 25 30 35z110\u0000210\u0000410\u0000610\u0000810\u00001010\u000012f coll(>M min,z)Figure 6.1: Mean collapse fraction for CDM (solid) and WDM (dashed)models. The WDM curves in ascending order are for mX = 2, 3, 4 keV. Thecollapse fraction is calculated using Eq. (6.10) with Msf set by Tvir = 104 K.6.4 Cosmic 21-cm signalThe brightness temperature of the 21-cm signal measured against the CMBat redshift z is given by\u0000Tb(z) = TS \u0000 T\u00001 + z (1\u0000 e\u0000\u00E2\u008C\u00A7\u00E2\u008C\u00AB0 )\u00E2\u0087\u00A1 27xHI(1 + \u0000)\u00E2\u009C\u00931\u0000 T\u0000TS\u00E2\u0097\u0086\u00E2\u009C\u00931 + z10 0.15\u00E2\u008C\u00A6mh2\u00E2\u0097\u00861/2\u00E2\u0087\u00A5\u00E2\u009C\u0093\u00E2\u008C\u00A6bh20.023\u00E2\u0097\u0086\u00E2\u009C\u0093 HH + dvk/drk\u00E2\u0097\u0086 mK, (6.12)where \u00E2\u008C\u00A7\u00E2\u008C\u00AB0 is the optical depth at the 21-cm frequency \u00E2\u008C\u00AB0, TS and T\u0000 arethe spin and CMB temperatures, respectively, xHI is the neutral fraction ofhydrogen, \u0000 is the overdensity, H is the Hubble parameter and dvk/drk isthe comoving velocity gradient along the line of sight. The spin temperaturecan be represented by T\u00001S = T\u00001\u0000 + x\u00E2\u0086\u00B5T\u00001\u00E2\u0086\u00B5 + xcT\u00001K1 + x\u00E2\u0086\u00B5 + xc , (6.13)906.5. Simulation of 21-cm signalwhere TK and T\u00E2\u0086\u00B5 are the kinetic and colour temperatures, respectively, andxc and x\u00E2\u0086\u00B5 are the collisional and WF coupling coe\u0000cients, respectively.The earliest possible measurable cosmic 21-cm signal would be emittedduring the \u00E2\u0080\u0098dark ages\u00E2\u0080\u0099 before significant star formation occurs. At theseearly times, the gas is dense enough so that collisional coupling is strongand TS \u00E2\u0087\u00A1 TK. Before z \u00E2\u0087\u00A0 150, residual free electrons strongly couple thegas kinetic temperature to the CMB through Compton scattering, so TS \u00E2\u0087\u00A1TK \u00E2\u0087\u00A1 T\u0000 and no 21-cm signal can be observed at this time. After this point,any remaining free electrons are so defuse that the gas is decoupled from theCMB and cools adiabatically as TK / (1+ z)2. Since the CMB temperaturedecreases at the slower pace of T\u0000 / (1 + z), a 21-cm signal in absorptionmay be observed (at least in principle) at this time [122, 123, 124, 125].As the gas continues to cool, the collisional coupling becomes less e\u0000cient,driving TS back up to the CMB temperature. As this scenario is relativelyuna\u00E2\u0086\u00B5ected by structure formation, we do not expect the presence of WDM tosignificantly a\u00E2\u0086\u00B5ect this era of the 21-cm signal and will restrict our attentionto later times with redshifts below z \u00E2\u0087\u00A0 35.47It will be important to keep in mind that the kinetic temperature of thegas will be lower than the CMB temperature when WF coupling first be-comes e\u00E2\u0086\u00B5ective. As the Lyman-\u00E2\u0086\u00B5 background grows, the increasing strengthof the WF coupling will drive TS from a value near the CMB temperature tothe lower kinetic temperature of the gas, thus producing another absorptionsignal. As WDM delays structure formation, the production of significantUV and X-ray backgrounds will be delayed, which in turn modifies the WFcoupling, X-ray heating, and reionization. We therefore focus our attentionto the astrophysical epochs in the 21-cm signal.6.5 Simulation of 21-cm signalThe 21-cm signal is simulated using the publicly available 21CMFAST code.This is a semi-numerical simulation that generates density, velocity, ioniza-tion and spin temperature fields in a 3D box with length size \u00E2\u0087\u00A0Gpc. In thissection we briefly summarize the code. See Refs. [128, 129] and referenceswithin for further details.An initial linear density field is generated as a Gaussian random field de-scribed by a power spectrum. The initial linear density field is then evolvedusing the Zeldovich approximation.47On the other hand, these early epochs may be a\u00E2\u0086\u00B5ected by dark matter decay orannihilation [126, 127].916.5. Simulation of 21-cm signalSince we will be examining high-redshift eras, it will be necessary to com-pute the spin temperature and consequently the colour and kinetic tempera-tures and their associated coupling coe\u0000cients. The WF coupling coe\u0000cientx\u00E2\u0086\u00B5 is given by x\u00E2\u0086\u00B5 = S\u00E2\u0086\u00B5J\u00E2\u0086\u00B5Jc\u00E2\u008C\u00AB , (6.14)where J\u00E2\u0086\u00B5 is the angle-averaged Lyman-\u00E2\u0086\u00B5 background flux, S\u00E2\u0086\u00B5 is a quantumcorrection term and Jc\u00E2\u008C\u00AB = 5.825 \u00E2\u0087\u00A5 10\u000012(1 + z) cm\u00002s\u00001Hz\u00001sr\u00001. S\u00E2\u0086\u00B5 andthe colour temperature T\u00E2\u0086\u00B5 are computed according to Ref. [73]. The kinetictemperature TK is calculated by solving the set of (local) coupled di\u00E2\u0086\u00B5erentialequations for TK and the ionized fraction xe in the neutral IGM, given bydxe(x, z)dz = dtdz \u0000\u00E2\u0087\u00A4ion \u0000 \u00E2\u0086\u00B5ACx2enbfH\u0000 , (6.15a)dTK(x, z)dz = 23kb(1 + xe) dtdz Xp \u00E2\u009C\u008Fp + 2TK3nb dnbdz \u0000 TK1 + xe dxedz , (6.15b)where \u00E2\u0087\u00A4ion is the ionization rate per baryon, \u00E2\u0086\u00B5A is the case-A recombinationcoe\u0000cient, C is the clumping factor, nb is the total baryon number density,fH is the hydrogen number fraction, and \u00E2\u009C\u008Fp is the heating rate for processp. The heating processes considered are X-ray heating \u00E2\u009C\u008FX and Comptonheating \u00E2\u009C\u008Fcomp.It is necessary to estimate the emission rate of photons at a particularfrequency to compute \u00E2\u009C\u008FX, \u00E2\u0087\u00A4ion, and J\u00E2\u0086\u00B5. The primary sources of X-ray pho-tons are expected to be high-mass X-ray binaries and the inverse-Comptonscattering o\u00E2\u0086\u00B5 of relativistic electrons accelerated in supernovae [19]. TheUV background is expected to be sourced from the collisional excitation ofneutral hydrogen by electrons ionized by X-rays as well as by direct stellaremission [129]. We will make the conventional assumption that the emissionrate of photons over the frequencies of interest can be approximated as beingproportional to the star-formation rate, which is a reasonable assumptiongiven the local correlation between star formation rate and X-ray luminosity[130]. The star formation rate in turn is approximated and readily computedby using the growth of the collapse fraction. The comoving emissivity e atfrequency \u00E2\u008C\u00AB is then e(\u00E2\u008C\u00AB) = f\u00E2\u0087\u00A4\u00E2\u0087\u00A2bN(\u00E2\u008C\u00AB)dfcolldt , (6.16)where f\u00E2\u0087\u00A4 is the fraction of baryons that are incorporated into stars, \u00E2\u0087\u00A2b =\u00E2\u0087\u00A2\u00C2\u00AFb(1 + \u0000nl) is the total baryon density including the non-linear overdensity\u0000nl, and N(\u00E2\u008C\u00AB) is the number of photons with frequency \u00E2\u008C\u00AB per solar mass in926.6. Simulation Resultsstars. The local collapse fraction is computed using the hybrid prescriptionof Ref. [132], where the biased EPS method is used to compute relative localhalo abundances whose mean is then normalized to fit the mean collapsefraction given by the Sheth-Tormen mass function in Eq. (6.10).Ionization fields are generated by assuming that a region is ionized ifit contains more ionizing photons than neutral hydrogen atoms (multipliedby 1 + n\u00C2\u00AFrec, where n\u00C2\u00AFrec is the mean number of recombinations per baryon).The excursion-set formalism is used with the condition that \u00E2\u0087\u00A3fcoll(x, z, R) \u00001 \u0000 xe(x, z, R) for a cell centred at location x to be fully ionized, wherefcoll(x, z, R) is the collapse fraction smoothed on scale R, \u00E2\u0087\u00A3 is the ion-ization e\u0000ciency, and 1 \u0000 xe(x, z, R) is the remaining fraction of neutralhydrogen within R. This criterion is evaluated at deceasing scales R andif the cell is not marked as fully ionized as the scale of the pixel length isreached, the cell\u00E2\u0080\u0099s ionization fraction is marked as \u00E2\u0087\u00A3fcoll(x, z, Rcell)+xe(x, z).Lastly, we note that the ionization e\u0000ciency can be decomposed as \u00E2\u0087\u00A3 =AHef\u00E2\u0087\u00A4fescNion/(1 + n\u00C2\u00AFrec), where fesc is the fraction of ionizing photons thatescape their host galaxy, Nion is the number of ionizing photons per baryoninside stars and AHe is a correction factor due to the presence of Helium.6.6 Simulation ResultsAs much is unknown about astrophysical properties during high-redshifteras, we will examine possible degeneracies in the 21-cm signal betweenWDM and astrophysical quantities. As a first step, we will compare thedelayed WDM 21-cm signal with that in CDM with a reduced photon-production e\u0000ciency. Specifically, we decrease the e\u0000ciency uniformly overfrequency by decreasing f\u00E2\u0087\u00A4, but note that f\u00E2\u0087\u00A4 is degenerate with other pa-rameters used to calculate photon production e\u0000ciencies.The box used in our simulation runs was 750 Mpc on a side and wascomprised of 3003 cells. The 21-cm signal was simulated in the redshiftrange z = 5.6 to 35. We set the minimum halo virial temperature thatsupports star formation to be Tvir = 104 K as to approximate the minimumtemperature need to e\u0000ciently cool the halo gas through atomic cooling,neglecting possible feedback processes.48Our fiducial model uses a f\u00E2\u0087\u00A4 value of f\u00E2\u0087\u00A4fid = 10%. We set the number48Although the very first stars were likely formed within smaller halos with Tvir on theorder of 103 K that were molecularly cooled, star formation in such halos can easily bedisrupted by feedback processes [133, 134] and we therefore neglect radiation from sourceslocated in such halos.936.6. Simulation Resultsof X-rays per solar mass in stars to NX(\u00E2\u008C\u00AB) = 2.2 \u00E2\u0087\u00A5 1056 M\u00001\u0000 \u00E2\u0087\u00A5 (\u00E2\u008C\u00AB/7 \u00E2\u0087\u00A51016 Hz)\u00001.5, to roughly match X-ray luminosities at low redshifts [129] andthe number of UV photons per solar mass in stars to NUV = 2.5\u00E2\u0087\u00A5 1060 M\u0000to approximately coincide with that of Pop II stars [131].49 The fiducialionization e\u0000ciency is taken as \u00E2\u0087\u00A3 = 31.5.Examples of the mean spin and kinetic temperatures for CDM and WDMmodels are plotted in Fig. 6.2. As expected, for WDM TS stays near T\u0000 fora longer time and the lowest point in the absorption trough, where the X-ray heating rate first surpasses the adiabatic cooling rate, occurs later. Asmentioned in Section 6.3.3, although the mean collapse fraction is lower inWDM models, it grows more rapidly, which is reflected in the heating ofthe gas. In addition, Fig. 6.2 shows curves for CDM with the lower f\u00E2\u0087\u00A4 valueof f\u00E2\u0087\u00A4/f\u00E2\u0087\u00A4fid = 0.1, which in our model happens to delay star formation suchthat the minimum value of T\u00C2\u00AFS occurs roughly at the same time as in theWDM example used. In this case, the X-ray heating rate increases at amuch slower rate after the minimum in T\u00C2\u00AFS as compared to the two othercases shown, since lowering f\u00E2\u0087\u00A4 reduces the photon production e\u0000ciency instars of all masses. In both non-fiducial cases shown, T\u00C2\u00AFS and thus \u0000T\u00C2\u00AFb reacha lower value in their absorption troughs since the gas undergoes furthercooling in the extra time needed for the X-ray heating to become e\u0000cient.The evolution of the mean brightness temperatures for WDM modelswith mX = 2, 3, 4 keV are shown in Fig. 6.3. 50 It is readily seen thathaving WDM with a particle mass of a few keV can substantially changethe mean 21-cm brightness temperature evolution. While lowering f\u00E2\u0087\u00A4 withinCDM models can delay the strong absorption signal, the resulting absorptiontrough is much wider than in WDM. For the same delay in the minimum of\u0000T\u00C2\u00AFb, the delay in reionization is greater for CDM than for WDM. Althoughreionization may be greatly delayed, well past z = 6, in models with lowvalues of f\u00E2\u0087\u00A4, our primary focus is on the pre-reionization 21-cm signal. Wecaution against automatically discarding these models, as the star-formatione\u0000ciency may diverge from earlier values by reionization.Examining the gradient of the global signal in Fig. 6.3b, we see thesuppressing f\u00E2\u0087\u00A4 in CDM models only shifts the mean signal to lower redshifts.49Note that since f\u00E2\u0087\u00A4 is degenerate with a frequency-independent value of N , only theratio of NX/NUV is relevant.50We caution the reader that WDM models with mX = 2, 3 keV are disfavoured by recentLyman-\u00E2\u0086\u00B5 observations [109]. However, Lyman-\u00E2\u0086\u00B5 forest constraints are still susceptibleto astrophysical (thermal and ionization history) and observational (sky and continuumsubtraction) degeneracies. Therefore, it is still useful to confirm these constraints usingthe redshifted 21-cm signal.946.6. Simulation Results10 15 20 25 30z101102103T\u00C2\u00AF(K)TSTKTgFigure 6.2: Mean spin temperatures T\u00C2\u00AFS for CDM and WDM models. Thedotted curves show T\u00C2\u00AFS for our fiducial CDM model (blue), WDM with mX =3keV (red), and CDM with f\u00E2\u0087\u00A4/f\u00E2\u0087\u00A4fid = 0.1 (green). In addition, the meankinetic temperature T\u00C2\u00AFK of each model is plotted with a dashed curve in thesame colour used for T\u00C2\u00AFS. The grey solid line is the CMB temperature.956.6. Simulation ResultsOn the other hand, decreasing mX in WDM models increases the gradientsof the mean signal. In CDM models, @\u0000T\u00C2\u00AFb/@z attains values near 33mK(\u000045mK) near its maximum (minimum) regardless of its f\u00E2\u0087\u00A4 value. This canincrease significantly in WDM models, for example to \u00E2\u0087\u00A0 64mK (\u00E2\u0087\u00A0 \u000077mK)at its maximum (minimum) for WDM with mX = 2keV.The e\u00E2\u0086\u00B5ect of WDM on the global 21-cm signal can be tracked throughdi\u00E2\u0086\u00B5erent \u00E2\u0080\u0098critical points\u00E2\u0080\u0099 in the signal\u00E2\u0080\u0099s evolution. We choose these pointsto be the redshift zmin at which \u0000T\u00C2\u00AFb reaches its minimum value, the redshiftzh when the kinetic temperature of the gas is heated above the CMB tem-perature, and the redshift of reionization zr taken to be the redshift wherethe mean ionized fraction is x\u00C2\u00AFi(zr) = 0.5. These points are plotted for bothCDM and WDM in Fig. 6.4. The solid curves track the e\u00E2\u0086\u00B5ect of lowering f\u00E2\u0087\u00A4on the redshifts of the critical points in CDM models (the values of f\u00E2\u0087\u00A4 canbe read from the upper horizontal axis). The dashed curves show the e\u00E2\u0086\u00B5ectof WDM on these redshifts, where the value of mX for each model can beread from the lower horizontal axis.We begin to explore possible degeneracies between CDM and WDMcosmologies by finding the value of f\u00E2\u0087\u00A4 required in CDM that would have aparticular critical point occur at the same redshift as it would in WDM witha particular value of mX. In other words, for a particular event that occursat redshift ze, we would like to find the curve that satisfies ze(f\u00E2\u0087\u00A4|CDM) =ze(mX|WDM). These curves for zmin, zh, and zr can be seen in Fig. 6.5.We can see that if one uses the milestone zr to distinguish between CDMand WDM with mX = 2, 3, 4 keV then f\u00E2\u0087\u00A4 has to be known within a factorof 3.0, 1.8, and 1.4, respectively. Using zmin instead, f\u00E2\u0087\u00A4 only has to beknown within a factor of 50, 13, and 4.8 for mX = 2, 3, 4 keV, respectively,since the impact of WDM is larger at higher redshifts. Near mX = 15 keV,using zmin to distinguish WDM from CDM requires f\u00E2\u0087\u00A4 to be known within afactor of 1.1 and drops to 1.01 by mX \u00E2\u0087\u00A0 20 keV (although the astrophysicalmotivations for WDM as mentioned in the introduction loses much of itsappeal past a few keV).As the value of mX is lowered, the curves in Fig. 6.5 diverge from oneanother, as the more rapid growth of structure in WDM changes the rel-ative timing of the milestones. Therefore, if f\u00E2\u0087\u00A4 is approximately constantthroughout the epochs under consideration, adjusting the value of f\u00E2\u0087\u00A4 inCDM so that a particular critical point occurs at the same redshift as itdoes in WDM will misalign other critical points and thus cannot reproducethe whole history of \u0000T\u00C2\u00AFb in WDM models.However, we can mimic the WDM mean brightness temperature evolu-tion with CDM if we allow f\u00E2\u0087\u00A4 to vary in time. To illustrate this, Fig. 6.6966.6. Simulation Resultsz\u0000200\u0000150\u0000100\u000050050dT\u00C2\u00AF b(mK)10 15 20 25 30z\u0000250\u0000200\u0000150\u0000100\u000050050dT\u00C2\u00AF b(mK)150 100 75 50n (MHz)(a)z\u00004004080\u00E2\u0088\u0082dT\u00C2\u00AFb/\u00E2\u0088\u0082z(mK)10 15 20 25 30z\u000080\u00004004080\u00E2\u0088\u0082dT\u00C2\u00AFb/\u00E2\u0088\u0082z(mK) 150 100 75 50n (MHz)(b)Figure 6.3: Mean 21-cm brightness temperature \u0000T\u00C2\u00AFb (a) and its derivativewith respect to redshift (b). In all plots, the solid curve is the fiducial CDMmodel. The upper plots show the results of WDM runs where the dashed,dotted-dashed, and dotted curves are for mX = 2, 3, 4 keV, respectively. Thelower plots show CDM runs where the dashed, dotted-dashed, and dottedcurves are for CDM models with f\u00E2\u0087\u00A4/f\u00E2\u0087\u00A4fid = 0.03, 0.1, 0.5, respectively.976.6. Simulation Results5 10 15 20mX (keV)68101214161820z0.01 0.1 1.0f\u00E2\u0087\u00A4/ f\u00E2\u0087\u00A4fid (CDM)CDMWDMFigure 6.4: \u00E2\u0080\u0098Critical points\u00E2\u0080\u0099 in the mean 21-cm signal. Redshifts of criticalpoints for CDM (solid curves) and WDM (dashed curves) models. For CDMcurves, the redshifts of the critical points are plotted as a function of f\u00E2\u0087\u00A4,which can be read from the top horizontal axis. For WDM curves, thecritical point redshifts are plotted as a function of mX, the values of whichcan be read from the lower horizontal axis. In descending order from theright, the curves are the redshifts zmin (blue), zh (green), and zr (red) foreach model.986.6. Simulation Results5 10 15 20mX(keV)0.11.0f \u00E2\u0087\u00A4/f\u00E2\u0087\u00A4fid(CDM)zminzhzrFigure 6.5: Parameter space curves ze(f\u00E2\u0087\u00A4|CDM) = ze(mX|WDM) for variouscritical points ze 2 {zmin, zh, zr}. The orange (green) hatched region showsmodels disfavoured by observations of GRBs (the Lyman-\u00E2\u0086\u00B5 forest) fromRef. [103] (Ref. [109]).shows the form of f\u00E2\u0087\u00A4(z) needed to reproduce the mean 21-cm signal for WDMwith mX = 2, 4 keV. At high redshifts (z & 15, 25 for mX = 2, 4 keV), f\u00E2\u0087\u00A4 ismore than an order of magnitude smaller than its value at the end of reion-ization to compensate for the delay of structure formation in WDM. Whenmore massive halos start to collapse (near z = 10, 20 for mX = 2, 4 keV),f\u00E2\u0087\u00A4 rises quickly by roughly an order of magnitude to mimic the more rapidchange of the collapse fraction in WDM and finally levels o\u00E2\u0086\u00B5 during reioniza-tion. While this evolution of f\u00E2\u0087\u00A4 may be possible, it seems contrived withoutan underlying model of such evolution.Even in cases where f\u00E2\u0087\u00A4 evolves in such a way as to mimic the meanbrightness temperature in WDM, one can di\u00E2\u0086\u00B5erentiate between WDM andCDM by examining the spectrum of perturbations in the 21-cm signal atcertain points in its evolution. Perturbations in the UV and X-ray fieldsadd power to the 21-cm power spectrum \u0000221 on large scales. Since the biasof sources in WDM can be greater than that in CDM [135], more power isadded on large scales in WDM than in CDM. This e\u00E2\u0086\u00B5ect is most easily seenat times when inhomogeneities in x\u00E2\u0086\u00B5 or TK are at their maximum. Fig. 6.7shows the evolution of the power spectrum for the modes k = 0.08Mpc\u00001and k = 0.18Mpc\u00001, showing a three peak structure, where the peaks from996.6. Simulation Results10 15 20 25 30 35z0.00.10.20.30.40.50.60.70.8f \u00E2\u0087\u00A4/f \u00E2\u0087\u00A4fidFigure 6.6: Evolution of f\u00E2\u0087\u00A4(z) in CDM required to match the mean bright-ness temperature \u0000T\u00C2\u00AFb in WDM with mX = 2keV (dashed) and mX = 4keV(solid). All other parameters are set to their values in the fiducial CDMmodel.high to low redshift are associated with inhomogeneities in x\u00E2\u0086\u00B5, TK, and xHI,respectively. When inhomogeneities in TK are at their maximum, the powerat k = 0.08, 0.18Mpc\u00001 can be boosted in WDM by as much as a factor of2.4, 2.0 (1.3, 1.1) for mX = 2keV (mX = 4keV). When inhomogeneous inx\u00E2\u0086\u00B5 are near their height, the power at k = 0.08Mpc\u00001 can be increased bya factor of 1.5 (1.2) for WDM with mX = 2keV (mX = 4keV).Current and next generation interferometric radio telescopes may beused to detect the boost in power associated with WDM models. The dot-ted curves in Fig. 6.7 show forecasts for the 1\u0000 power spectrum thermal noiselevels for 2000 hours of observation time, computed by Ref. [115], for theMurchison Widefield Array (MWA), the Square Kilometre Array (SKA), andfor the proposed Hydrogen Epoch of Reionization Array (HERA). This esti-mate is quite conservative in that it ignores the contribution of foreground-contaminated modes [136]. From these forecasts, we can see that the MWAmay be able to at least marginally detect the boost in power for the mX =2keV model at the reionization and X-ray heating peaks. In addition, theseestimates indicate that next generation instruments will be able to easilymeasure the excess of power at these scales for mX = 2, 4 keV models over1006.7. Conclusionsa wide range of redshifts.The 21-cm power spectrum during a redshift near the time when TKis at its most inhomogeneous state is plotted in Fig. 6.8 for WDM withmX = 2, 4 keV and their CDM counterparts. One can see that the boost inpower in WDM may continue to k values lower than those used in Fig. 6.7.In particular, the power near k = 0.01Mpc\u00001 in WDM models with mX =2keV (mX = 4keV) may be larger by a factor of 3 (1.3) as compared to inCDM models at these times.Finally, we mention that for simplicity we have chosen to vary only oneastrophysical property. By allowing other astrophysical parameters to varyas a function of redshift, most notably Mmin, it might be possible to producea 21-cm power spectrum degenerate with WDM throughout the redshiftsunder investigation and we leave this question for future work.6.7 ConclusionsIn warm dark matter models, the abundance of small halos is suppressed,which can leave a strong imprint at high redshifts. Since structure forma-tion is delayed but more rapid in WDM, the mean 21-cm signal will followsuit, resulting in a delayed, deeper and more narrow absorption trough.These e\u00E2\u0086\u00B5ects can easily be seen in the global 21-cm signal for WDM withfree-streaming lengths above current observational bounds for thermal relicmasses as high as mX \u00E2\u0087\u00A0 10\u000020 keV (R0c \u00E2\u0087\u00A0 6\u000013 kpc).Suppressing the photon-production e\u0000ciency of astrophysical sourcescan delay the 21-cm signal as well. As such, to discriminate between WDMand CDM models by measuring the redshift of reionization, the photon-production e\u0000ciency must be known to within a factor of 3.0, 1.8, and 1.4for WDM with mX = 2, 3, 4 keV (R0c \u00E2\u0087\u00A1 86, 54, 39 kpc), respectively. Sincethe impact of WDM is larger at higher redshifts, if milestones in the mean21-cm signal that occur at higher redshift are used to di\u00E2\u0086\u00B5erentiate WDMand CDM models, the precision to which this e\u0000ciency must be known de-creases. For example, if measuring the redshift of the minimum of the mean21-cm signal (during the astrophysical epoch of the signal) the e\u0000ciencymust only be known within a factor of 50, 13, and 4.8 for mX = 2, 3, 4 keV,respectively.If the star-formation remains approximately constant over the range ofredshifts under consideration, degeneracy between CDM and WDM modelsmay be broken by examining the gradient of the mean 21-cm signal, whichis larger in WDM due to its more rapid pace of structure formation. In1016.7. Conclusions8 10 12 14 16 18 20100101102103(dT\u00C2\u00AF b)2 \u00002 21(mK2 )k = 0.08Mpc\u00001SKAMWAHERA8 10 12 14 16 18 20z10\u0000210\u00001100101102WDM\u0000CDM(mK2 )SKAMWA HERA8 10 12 14 16 18 20k = 0.18Mpc\u00001SKAMWAHERA8 10 12 14 16 18 20zSKAMWAHERA(a)10 15 20 25100101102103(dT\u00C2\u00AF b)2 \u00002 21(mK2 )k = 0.08Mpc\u00001SKAMWAHERA10 15 20 25z10\u0000210\u00001100101102WDM\u0000CDM(mK2 )SKAMWAHERA10 15 20 25k = 0.18Mpc\u00001SKAMWAHERA10 15 20 25zSKAMWA HERA(b)Figure 6.7: Evolution of the angle-averaged power spectrum of \u0000Tb for WDMwith (a) mX = 2keV and (b) mX = 4keV. The top panels show power spec-tra at k = 0.08, 0.18Mpc\u00001 for WDM (dashed) and the CDM model (solid).CDM models have f\u00E2\u0087\u00A4(z) chosen to reproduce the global 21-cm signal foundfor the respective WDM model. The bottom panels show the di\u00E2\u0086\u00B5erencein the power spectrum between WDM and CDM models. Dotted curvesshow forecasts for the 1\u0000 \u0000 power spectrum thermal noise as computed inRef. [115] with 2000h of observation time. The dotted green, blue, and redcurves are the forecasts for the MWA, SKA, and HERA, respectively.1026.7. Conclusions101102103(dT\u00C2\u00AFb)2\u00002 21(k)(mK2 )z = 12.5mX = 2keV10\u00002 10\u00001 100k (Mpc\u00001)101102103z = 15mX = 4keVFigure 6.8: Power spectrum of the brightness temperature \u0000Tb. The toppanel shows the power spectrum at z = 12.5 for WDM with mX = 2keV(dashed) and CDM (solid). In the CDM model, f\u00E2\u0087\u00A4(z) evolves as shownin Fig. 6.6 such that it reproduces the global signal in the WDM model.Similarly, the bottom panel shows the power spectrum at z = 15 for WDMwith mX = 4keV (dashed) and CDM (solid) with f\u00E2\u0087\u00A4(z) chosen to matchthe global signal in this WDM model. The power spectrum of each modelis plotted at a redshift near where the X-ray background is at its mostinhomogeneous state in its respective model.1036.7. Conclusionsaddition, the spectrum of perturbations in the 21-cm signal may as well beused to break this degeneracy, as the 21-cm power spectrum in WDM hasan excess of power on large scales owing to the stronger biasing of sourcesin WDM. This is true even if the photon-production e\u0000ciency evolves withredshift in such a way as to reproduce with CDM the global 21-cm signal inWDM models. For WDM with mX = 2keV (mX = 4keV), the power in the21-cm signal at k = 0.08, 0.18Mpc\u00001 can be increased by a factor as highas 2.4, 2.0 (1.3, 1.1) as compared to that in CDM. Power spectrum measure-ments made by current interferometric telescopes, such as the MWA, shouldbe able to discriminate between CDM and WDM models with mX . 3 keV,while next generation telescopes will easily be able di\u00E2\u0086\u00B5erentiate betweenCDM and all relevant WDM models.In this work, we assume that atomically-cooled halos drive the 21-cmsignal. If instead smaller, molecularly-cooled halos, whose production issuppressed in WDM, play a significant role in producing the 21-cm signalin CDM, then the e\u00E2\u0086\u00B5ects di\u00E2\u0086\u00B5erentiating WDM from CDM described abovewould be even more pronounced. On the other hand, if star-formation wasnot e\u0000cient in halos with Tvir = 104 K, the di\u00E2\u0086\u00B5erences between CDM andWDM in the 21-cm signal would be diminished.104Chapter 7Forecasting 21-cm BAOExperiments7.1 IntroductionIn the search for the underlying nature of dark energy, precise measurementsof the expansion of the Universe are essential for constraining models of darkenergy [5, 6, 137]. One such class of experiments are designed to measurethe baryon acoustic oscillations (BAO) at di\u00E2\u0086\u00B5erent redshifts, from which anexpansion history can be inferred. With many new experiments designed tomeasure the BAO on the horizon, forecasting their ability to measure theBAO and constrain dark energy parameters plays an important role for theirdesign. These forecasts can be used to optimize the design and operationof the experiment and estimate the impact of noise and foregrounds on themeasurement of the BAO and ultimately on the dark energy equation ofstate.This chapter describes the development of software used to make suchforecasts and some of the forecasts made for the CHIME51 telescope. Theseforecasts estimate the ability of an experiment to measure the matter powerspectrum, projecting these uncertainties onto measurements of the BAO,expansion parameters, and finally onto the dark energy equation of state.For this analysis, we use the standard parameterization of the dark en-ergy equation of state [137]wDE(z) = w0 + wa[1\u0000 a(z)] = w0 + wa z1 + z , \u00E2\u0080\u0098 (7.1)and thus our ultimate goal is to determine the precision in which the param-eters w0 and wa can be measured. With this parameterization, the Hubble51http://chime.phas.ubc.ca1057.2. Constraining Dark Energy Parametersrate is given byH2(z) = H20\u00EF\u00A3\u00BF\u00E2\u008C\u00A6m(1 + z)3 + \u00E2\u008C\u00A6k(1 + z)2+ \u00E2\u008C\u00A6\u00E2\u0087\u00A4(1 + z)3(1+w0+wa) exp(\u00003waz/(1 + z))\u0000. (7.2)We begin this chapter by describing the physics of the BAO and its po-tential for constraining wDE followed by a brief discussion of other methodsthat can be used to constrain dark energy parameters, which can be used inconjunction with BAO experiments to provide more stringent constraints.The remainder of this chapter describes forecasting methods for BAO ex-periments and their resulting forecasts, where emphasis is given to cylindertransit telescopes such as CHIME.7.2 Constraining Dark Energy ParametersEach experiment designed to constrain the dark energy equation of state willhave di\u00E2\u0086\u00B5erent systematic errors, may cover a di\u00E2\u0086\u00B5erent redshift range, andwill produce a di\u00E2\u0086\u00B5erent contour in the w0\u0000wa plane. As such, the constraintson dark energy parameters improves greatly when the results from di\u00E2\u0086\u00B5erentobservational methods are combined. In particular, the report from the DarkEnergy Task Force (DETF) [137] endorses pursuing multiple techniques formeasuring the dark energy equation of state that includes measurementsof BAO, type Ia supernovae (SN), galaxies clusters (CL), and weak lensing(WL).The observables of a dark energy experiment are in some way a\u00E2\u0086\u00B5ectedby either H(z) directly or a quantity dependent on it (i.e. DA(z), G(z),etc. . .), which in turn is dependent on the dark energy parameters. As willbe discussed in more detail in Section 7.3.2, the BAO can be measured indirections both parallel and perpendicular to the line of sight, so has thepotential of measuring both H(z) and DA(z) separately. SN Ia can actas standard candles as their absolute luminosity at peak brightness occursnearly at the same point for every SN Ia. As standard candles, SN Iacan be used to measure the luminosity distance DL = (1 + z)2DA. SN Iaprovided some of the first definitive observational evidence for a late periodof acceleration and remains an important tool for constraining the darkenergy equation of state. CL abundances dN/dMd\u00E2\u008C\u00A6dz observed in a regionof solid angle d\u00E2\u008C\u00A6 and in a redshift bin dz can be compared to the massfunction dn/dM calculated assuming a particular dark energy model. The1067.3. Measuring the Acoustic Scalemass function may be calculated analytically using the methods describedin Section 2.5 or more precisely using N -body simulations. Dark energye\u00E2\u0086\u00B5ects the mass function through both the growth function G(z) via therms of density fluctuations \u00002R(z) = (G(z)/G(z = 0))2\u00002R(z = 0) as well asthrough the combinationD2A(z)/H(z) needed to convert between a comovingand physical volume. WL measures the statistical distortion of the imagesof galaxies that pass by large masses. The level of distortion depends onthe growth of the density fluctuations that distort the image, hence on thegrowth function G(z) as well as on the distances between the lens, source,and observer and so is sensitive to expansion history as well.One of the ultimate goals of these experiments is to place constraintson wDE and its time evolution. In this light, the DEFT figure of merit(FOM), defined as the reciprocal of the area of the 95% confidence contourin the w0\u0000wa plane (marginalizing over all over parameters), can be used toevaluate the e\u00E2\u0086\u00B5ectiveness of an experiment to constrain wDE, where a largerFOM indicates more constraining power.7.3 Measuring the Acoustic Scale7.3.1 The Sound HorizonOne of the most basic quantities characterizing the acoustic oscillations inthe photon-baryon fluid is the sound horizon. The sound speed cs of thefluid is given by c2s = \u0000P\u0000\u00E2\u0087\u00A2 \u00E2\u0087\u00A1 \u0000\u00E2\u0087\u00A2\u0000/3\u0000\u00E2\u0087\u00A2\u0000 + \u0000\u00E2\u0087\u00A2b = 13(1 +Rb) , (7.3a)Rb = \u0000\u00E2\u0087\u00A2b\u0000\u00E2\u0087\u00A2\u0000 = \u00E2\u0087\u00A2\u00CB\u0099b\u00E2\u0087\u00A2\u00CB\u0099\u0000 = 34 \u00E2\u0087\u00A2b\u00E2\u0087\u00A2\u0000 = 34 \u00E2\u008C\u00A6b\u00E2\u008C\u00A6\u0000 a. (7.3b)The comoving sound horizon rs is thenrs(z) = Z \u00E2\u008C\u0098(z)0 d\u00E2\u008C\u0098\u00CB\u009Ccs(\u00E2\u008C\u0098\u00CB\u009C) = H\u000010 Z a(z)0 da\u00CB\u009Ca\u00CB\u009C2E(a\u00CB\u009C)p3(1 +Rb(a\u00CB\u009C)) , (7.4)where E(a) = H(a)/H0. We will be evaluating the sound horizon duringmatter domination when E(a) = p\u00E2\u008C\u00A6mpa+ aeq/a2, where aeq is the scalefactor at matter-radiation equality. With this, the comoving sound horizonis thenrs(z) = 43H\u000010 s \u00E2\u008C\u00A6\u0000\u00E2\u008C\u00A6m\u00E2\u008C\u00A6b ln p1 +Rb(z) +pRb(z) +Req1 +pReq ! , (7.5)1077.3. Measuring the Acoustic Scalewhere Req \u00E2\u008C\u0098 Rb(aeq).We are interested in the final comoving sound horizon when decouplingoccurs. We define the decoupling time of a species as when its opticaldepth drops to unity. The optical depth for the photons \u00E2\u008C\u00A7 can be foundby integrating \u00E2\u008C\u00A7\u00CB\u0099 = ne\u0000Ta, where ne is the number density of free electronsand \u0000T is the Thomson cross section. The optical depth for the baryons\u00E2\u008C\u00A7d can be found from \u00E2\u008C\u00A7\u00CB\u0099d = \u00E2\u008C\u00A7\u00CB\u0099/Rb, where the factor of Rb accounts for thedi\u00E2\u0086\u00B5erence in population between the baryons and photons. Since there aremore photons than baryons at this time (Rb < 1), the photons decouple at aredshift z\u00E2\u0087\u00A4 slightly before the decoupling of the baryons at redshift zd. Theredshifts z\u00E2\u0087\u00A4 and zd can found analytically using fitting formulas [28, 138],which for zd is zd = 1291!0.251m1 + 0.659!0.828m (1 + b1!b2b ),b1 = 0.313!\u00000.419m (1 + 0.607!0.674m ),b2 = 0.238!0.223m . (7.6)The redshifts of decoupling and the sound horizon at these times can beinferred by the use of CMB data. The values inferred by Planck [29] arez\u00E2\u0087\u00A4 = 1090.37\u00C2\u00B1 0.65, rs(z\u00E2\u0087\u00A4) = 144.75\u00C2\u00B1 0.66Mpc, (7.7a)zd = 1059.29\u00C2\u00B1 0.65, rs(zd) = 147.53\u00C2\u00B1 0.64Mpc. (7.7b)7.3.2 Baryon Acoustic OscillationsUsing the BAO as a standard ruler, by measuring its size at a variety ofredshifts we can reconstruct an expansion history. The BAO manifests itselfas a bump in the correlation function and so will appear as an oscillation inthe power spectrum. The BAO may potentially be measured in both radialand perpendicular directions, which appear as an angular separation ' andredshift separation \u0000z that are related to the expansion parameters by'(z) = rd(1 + z)DA(z) , (7.8a)\u0000z(z) = rdH(z)/c, (7.8b)where rd = rs(zd). The first generation of BAO detections did not havesu\u0000cient data to accurately measure the parallel and perpendicular BAOscales separately and instead used the spherically averaged measure['(z)2\u0000z(z)]1/3 = rd[c(1 + z)2DA(z)2/H(z)]1/3 . (7.9)1087.4. Fisher Matrix FormalismWith new experiments such as CHIME that will be able to rapidly maplarge areas of the sky over a wide range of redshifts, DA and H may bemeasured separately, increasing the constraining power of our telescope.7.4 Fisher Matrix FormalismCreating a detailed model of the full likelihood function for a forecast of anexperiment is often a di\u0000cult task. For the purposes of forecasting, it is oftenmore useful to introduce some assumptions to simplify this task [1, 5]. In thisvein, we assume that we have a fiducial model that is su\u0000ciently accuratesuch that it is reasonable to find the maximum likelihood by expanding thelikelihood function L about the fiducial model, which yieldslnL(\u00E2\u009C\u0093) \u00E2\u0087\u00A1 lnL(\u00E2\u009C\u0093\u00CB\u009C) + @ lnL(\u00E2\u009C\u0093)@\u00E2\u009C\u0093i \u0000\u0000\u0000\u0000\u00E2\u009C\u0093=\u00E2\u009C\u0093\u00CB\u009C(\u00E2\u009C\u0093i \u0000 \u00E2\u009C\u0093\u00CB\u009Ci)+12@2 lnL(\u00E2\u009C\u0093)@\u00E2\u009C\u0093i@\u00E2\u009C\u0093j \u0000\u0000\u0000\u0000\u00E2\u009C\u0093=\u00E2\u009C\u0093\u00CB\u009C(\u00E2\u009C\u0093i \u0000 \u00E2\u009C\u0093\u00CB\u009Ci)(\u00E2\u009C\u0093j \u0000 \u00E2\u009C\u0093\u00CB\u009Cj), (7.10)where the likelihood is a function of our model parameters \u00E2\u009C\u0093 that have values\u00E2\u009C\u0093 = \u00E2\u009C\u0093\u00CB\u009C in the fiducial mode and assume that \u00E2\u009C\u0093 \u0000 \u00E2\u009C\u0093\u00CB\u009C is small. If \u00E2\u009C\u0093 is near themaximum likelihood values, after taking the expectation value of the aboveexpression, the first derivative term should vanish (or at least be small).The Fisher matrix F then can be approximated asFij = \u0000\u00E2\u008C\u00A7@2 lnL(\u00E2\u009C\u0093)@\u00E2\u009C\u0093i@\u00E2\u009C\u0093j \u0000 \u00E2\u0087\u00A1 \u0000@2 lnL(\u00E2\u009C\u0093)@\u00E2\u009C\u0093i@\u00E2\u009C\u0093j \u0000\u0000\u0000\u0000\u00E2\u009C\u0093=\u00E2\u009C\u0093\u00CB\u009C. (7.11)Assuming that the parameters \u00E2\u009C\u0093 are normally distributed, then Eq. (7.11)implies that the Fisher matrix is approximately equal to the inverse of thecovariance matrix C\u00E2\u009C\u0093 for the parameters \u00E2\u009C\u0093. The likelihood function is thenapproximatelyL \u00E2\u0087\u00A1 1q(2\u00E2\u0087\u00A1)n|F\u00001|exp\u00E2\u009C\u0093\u000012(\u00E2\u009C\u0093 \u0000 \u00E2\u009C\u0093\u00CB\u009C)TF(\u00E2\u009C\u0093 \u0000 \u00E2\u009C\u0093\u00CB\u009C)\u00E2\u0097\u0086 , (7.12)where |F\u00001| is the determinant of the inverse of the n \u00E2\u0087\u00A5 n fisher matrix F.Even if the parameters \u00E2\u009C\u0093 not not have Gaussian uncertainties, the Fishermatrix is a useful measure since as long as \u00E2\u009C\u0093 are unbiased estimators of thetrue values with covariance matrix C\u00E2\u009C\u0093, the Crame\u00C2\u00B4r-Rao inequality impliesthat C\u00E2\u009C\u0093 \u0000 F\u00001. We proceed assuming Gaussian statistics and that the1097.5. Measuring the 21-cm Power Spectrumchosen fiducial model has parameter values close to the maximum likelihoodvalues.We now summarize frequently used operations on the Fisher matrix.If we have the Fisher matrix F\u00E2\u009C\u0093 for parameters \u00E2\u009C\u0093, the Fisher matrix F\u0000for parameters \u0000 can be calculated by use of the Jacobian Jij = @\u00E2\u009C\u0093i/@\u0000jevaluated in the fiducial model. Since \u00E2\u009C\u0093i\u0000 \u00E2\u009C\u0093\u00CB\u009Ci = Jij(\u0000j\u0000 \u0000\u00CB\u009Cj), from Eq. (7.12)the Fisher matrix transforms asF\u0000 = JTF\u00E2\u009C\u0093J. (7.13)To maximize the likelihood with respect to a parameter, we simply removethe row and column of the Fisher matrix corresponding to that variable.Marginalizing over a variable amounts to removing the row and columncorresponding to that variable from the covariance matrix. If we order ourFisher matrix asF =\u00E2\u009C\u0093A BBT M\u00E2\u0097\u0086 , (7.14)where M is the submatrix that has both rows and columns correspondingto variables that we wish to marginalize over, then the marginalized Fishermatrix FM is given byFM = A\u0000BMBT . (7.15)Since adding a Gaussian prior amounts to summing the inverse of covariancematrices, a prior can be added by simply adding Fisher matrices, assumingthat they use the same fiducial model and have their rows and columnsordered in the same manner.7.5 Measuring the 21-cm Power SpectrumIn this section we outline our method for forecasting the uncertainties inmeasuring the 21-cm power spectrum for a cylinder transit telescope. Thismethod, based on the analysis in Refs. [139, 140], provides a straightforwardand computationally cheap procedure for estimating power spectrum uncer-tainties. More complex and computationally expensive forecasting meth-ods, which also include the e\u00E2\u0086\u00B5ects of foreground subtraction, can be foundin Refs. [141, 142]. For our forecast, we use the cosmological parameters\u00E2\u008C\u00A6m = 0.266,\u00E2\u008C\u00A6b = 0.0449,\u00E2\u008C\u00A6k = 0, ns = 0.963, h = 0.71,\u00008 = 0.8, consistentwith WMAP7.The visibilities measured by a telescope may be processed in several waysto produce maps and power spectra. In the flat-sky approximation, each1107.5. Measuring the 21-cm Power Spectrumvisibility measures a small number of Fourier modes on the sky. Extendingthis notion to a wide field of view telescope, a visibility measures a finite setof spherical harmonics on the sky. Alternatively, the process of beamformingcombines the measured visibilities to form localized beams on the sky, whichwe will employ for our forecasting. In this process, the localized beams areformed from the spatial Fourier transform of the feed responses or visibilities.These beams can be characterized by the point spread function (or dirty orsynthetic beam)PSF(p) = |A(p)|2 Z d2\u0000rS(\u0000r)e2\u00E2\u0087\u00A1ip\u00C2\u00B7\u0000r, (7.16)where \u0000r is the baseline vector, p is the wave vector of the radiation, A(p)is the primary beam response, and S(\u0000r) is a sampling function equal tounity for each baseline measured and zero otherwise. We have assumed herethat all primary beams are identical. An image of the sky, known as thedirty image, can be found by convolution of the point spread function andthe true sky intensity.Each cylinder in our telescope will have Nf equally spaced (dual polar-ization) feeds along its focal line and will consist of Ncyl cylinders. To startwith, consider the response of a single cylinder that lies in the e\u00CB\u0086z = 0 planewith its focal line along the e\u00CB\u0086y axis, so that the feed locations are given byrm = (0,mdf , 0) for m = 0, . . . , Nf \u0000 1. By sampling p at discrete locations,the Fourier transform in Eq. (7.16) along the cylinder can be expressed asa discrete Fourier transform, where py is sampled at (py)n = n/Nfdf forn = \u0000Nf/2 + 1, . . . , Nf/2. If \u00E2\u009C\u0093 is the angle between zenith and p projectedinto the e\u00CB\u0086y \u0000 e\u00CB\u0086z plane then we sample sin \u00E2\u009C\u0093 at even intervals withsin \u00E2\u009C\u0093n = (n\u0000 12)\u0000Nfdf n = \u0000Nf2 + 1, . . . , Nf2 . (7.17)The resolution of the telescope is set by the longest baseline in the arrayL = Nfdf , assumed to be large enough for small angle approximations tobe valid. Our feed and cylinder spacing will be such that L is the longestbaseline in both North-South and East-West directions. The resolution ofthe synthetic beams is \u0000\u00E2\u009C\u0093 \u00E2\u0087\u00A1 \u0000 /L, which by Eq. (7.17) is also the angularsampling rate. Fourier transforming between the cylinders segments eachsynthetic beam into multiple beams in the azimuth direction to improve theresolution in that direction.The foreground emissions, predominately from synchrotron radiationfrom our Galaxy as well as from extragalactic sources, will dominate over1117.5. Measuring the 21-cm Power Spectrumthe cosmological 21-cm signal. However, these foregrounds have very smoothspectra, unlike the 21-cm signal, and thus there are a variety of techniquesthat rely on this di\u00E2\u0086\u00B5erence in spectra to separated the foregrounds from thedesired signal [141, 142, 148, 149, 150]. In the following we assume thatforegrounds may be completely cleaned from the 21-cm power spectrum atscales near the BAO scale. We direct the reader to Refs. [141, 142] for anin-depth analysis of the e\u00E2\u0086\u00B5ect of foreground subtraction on the recovered21-cm signal.In a sidereal day, we can create a map of a large fraction of the sky. Withadequate foreground subtraction, by relating angular and frequency scalesto comoving position, the power spectrum of the underlying density field\u0000(x) may be measured. If the response of our telescope to the overdensityfield is given by W (x)\u0000(x), where W (x) is a window function appropriatefor our experiment, then the \u00E2\u0080\u0098raw\u00E2\u0080\u0099 measured power spectrum \u00002(k) can bewritten as [143, 144]\u00002(k) = |W (k)|2P (k) + Pshot + PN , (7.18)where P (k) is the \u00E2\u0080\u0098true\u00E2\u0080\u0099 underlying power spectrum appearing as the samplevariance, Pshot is the shot noise, and PN is the additional noise introducedby the instrument.52 The power spectrum under investigation is the 21-cm brightness temperature power spectrum P21cm(k, z) = T\u00C2\u00AF 2b (z)b2Pm(k, z),where Pm is the matter power spectrum and b is the bias. The shot noise canbe expressed as Pshot = 1/n\u00C2\u00AF with n\u00C2\u00AF the expected average number density ofemitters detectable by the experiment.We can use P\u00CB\u0086 (k) = \u00002(k) \u0000 Pshot \u0000 PN as the estimator of P (k). As-suming Gaussian statistics, the covariance of P\u00CB\u0086 (k) is simply hP\u00CB\u0086 (k)P\u00CB\u0086 (k\u00CB\u009C)i =(P (k)|W (k)|2 +Pshot +PN )2\u0000k,k\u00CB\u009C. If in surveying a real space volume Vs wemeasure the average power in k within a k-space volume Vk, then we mayobserve Nk = VkVs/2(2\u00E2\u0087\u00A1)3 independent modes within the volume, and thusmay reduce the variance of our power spectrum measurement by a factor of1/Nk.53 The covariance matrix for the 21-cm power spectrum measurementsfor modes ki is thenCP(ki,kj) = 2(2\u00E2\u0087\u00A1)3VkiVs\u00EF\u00A3\u00BFT\u00C2\u00AF 2b (z)b2(P (ki)|W (ki)|2 + Pshot) + PN\u00002\u0000ij . (7.19)52In this chapter, we refer to the dimensionful power spectrum as simply the powerspectrum, unless otherwise stated, so that the power spectrum for field \u0000 is P\u0000(k) = V |\u0000k|2with V the volume.53The extra factor of 12 accounts for the fact that since the field is real valued, the modesk and \u0000k are not independent of one another.1127.5. Measuring the 21-cm Power SpectrumWe will now specify the parameter values necessary to compute the co-variance in Eq. (7.19) for our forecast. Since during the redshifts underconsideration we expect TS \u0000 T\u0000 , the mean brightness temperature simpli-fies to [145] \u0000T\u00C2\u00AFb \u00E2\u0087\u00A1 0.1 \u00E2\u008C\u00A6HI10\u00003 (1 + z)2H/H0 mK. (7.20)For the HI density parameter we set \u00E2\u008C\u00A6HIb = 6.2 \u00E2\u0087\u00A5 10\u00004 [146] and take thebias as b = 1. The matter power spectrum can be expressed as Pm(k, \u00C2\u00B5) =R(\u00C2\u00B5)Pm(k), where \u00C2\u00B5 is the cosine of the angle between k and the line of sight,R(\u00C2\u00B5) = (1 + \u0000\u00C2\u00B52)2 is the linear redshift-space distortion factor, \u0000 = f/b,and f = (a/G)dG/da is the linear growth rate. We use the transfer functionof Ref. [28] when computing the matter power spectrum.For our forecast, we will divide our coverage in frequency into bins as-suming constant cosmological values (i.e. H and DA) in each. For a redshiftbin of size \u0000z, the comoving real-space volume measured within the bin isgiven by Vs = D2A(1 + z)2H \u0000z\u00E2\u008C\u00A6s, (7.21)where \u00E2\u008C\u00A6s is the solid angle covered by the survey.The coverage in declination is determined by both the array configurationas well as the primary beams of the feeds. We can determine the locationsof outermost synthetic beams by used of Eq. (7.17), which for large Nf is 54sin \u00E2\u009C\u0093nmax = \u00002df . (7.22)Synthetic beams close to the horizon may be dampened by the primarybeam pattern A(p). To account for the decreasing sensitivity close to thehorizon, we approximate the survey solid angle by\u00E2\u008C\u00A6s \u00E2\u0087\u00A1 2\u00E2\u0087\u00A1 Z \u00E2\u009C\u0093max\u00E2\u009C\u0093min |A(\u00E2\u009C\u0093)|2 cos(\u00E2\u009C\u0093)d\u00E2\u009C\u0093, (7.23)where \u00E2\u009C\u0093min = min(\u00E2\u009C\u0093lat \u0000 \u00E2\u009C\u0093nmax/2,\u0000\u00E2\u0087\u00A1/2), \u00E2\u009C\u0093max = max(\u00E2\u009C\u0093lat + \u00E2\u009C\u0093nmax/2,\u00E2\u0087\u00A1/2),and \u00E2\u009C\u0093lat is the latitude of the telescope. We take the primary beam patternin the meridian as the used in Ref. [141], which is proportional to the flux54Note that Eq. (7.17) is more properly written with the addition of an arbitrary integersince this expression originates from inside the exponential in Eq. (7.16), which ensuresthat | sin \u00E2\u009C\u0093nmax | \u00EF\u00A3\u00BF 1. We set \u00E2\u009C\u0093nmax = \u00E2\u0087\u00A1/2 in cases where \u0000/2df > 1.1137.5. Measuring the 21-cm Power Spectrumpassing through the ground-plane, so that|A(\u00E2\u009C\u0093)|2 = (cos \u00E2\u009C\u0093 \u0000\u00E2\u0087\u00A1/2 \u00EF\u00A3\u00BF \u00E2\u009C\u0093 \u00EF\u00A3\u00BF \u00E2\u0087\u00A1/20 else, (7.24)The parameters for our telescope used for the forecast are listed in Ta-ble 7.1. From these values we see that for wavelengths \u0000 < 2df (\u00E2\u008C\u00AB & 484MHzin our case) the synthetic beams will be aliased with directions pointed closerto the horizon. However, since these aliases appear closer to the horizonwhere the sensitivity of the primary beam is reduced, the impact of thealiasing is diminished, although not completely removed.Parameter Value\u00E2\u009C\u0093lat 49.5\u0000Ncyl 5Nf 256df 31 cmTsys 50Kttot 2 yrs\u00E2\u008C\u00AB 400\u0000800MHzTable 7.1: Telescope parameters for CHIME used for BAO forecasting.We now specify the window function in Eq. (7.19), dealing with modesperpendicular and parallel to the line of sight separately. Assuming an ad-equately small angular resolution \u0000\u00E2\u009C\u0093, our resolution in comoving distancesperpendicular to the line of sight is \u0000x \u00E2\u0087\u00A1 (\u0000/L)DA(1 + z). We may thensample at the Nyquist rate 2\u00E2\u0087\u00A1(1/(2\u0000x)) = \u00E2\u0087\u00A1(L/\u0000)/DA(1+z), which acts asour low-scale cuto\u00E2\u0086\u00B5 mode k?max in the perpendicular direction. We assumea sharp cuto\u00E2\u0086\u00B5 at k?max and take W (k) to be a top hat function in this direc-tion. As the frequency resolution of our telescope of \u00E2\u0087\u00A0 1MHz corresponds toscales smaller than those relevant for measuring the BAO, we forego addingsuch a cuto\u00E2\u0086\u00B5 in the kk direction.Assuming uncorrelated noise between feeds, the variance on a real-spacepixel is given by the expression in Eq. (5.17), where we assume an identi-cal system temperature for all feeds. We can conservatively estimate theintegration time for a real-space pixel by assuming that a pixel is seen byeach cylinder for a fraction \u0000\u00E2\u009C\u0093/2\u00E2\u0087\u00A1 of each sidereal day so that the totalintegration time for a real-space pixel istint \u00E2\u0087\u00A1 \u0000\u00E2\u009C\u00932\u00E2\u0087\u00A1 Ncylttot, (7.25)1147.5. Measuring the 21-cm Power Spectrumwhere Ncyl is the number of cylinders and ttot is the total observation timeof the experiment.55 In addition, CHIME will be equipped with dual-polarization feeds, which in e\u00E2\u0086\u00B5ect doubles the integration time.The above approximations yield a position independent estimate of theinstrument noise on a real-space pixel. The final step is to Fourier trans-form to k-space. Although the real-space pixels are arranged on concentricspheres, we are only interested in smaller scales near the BAO scale and willtherefore use the flat-sky approximation here. As we approximate the real-space noise as being position independent, the Fourier transform to k-spaceis trivial and simply introduces a factor of 1/Npix, where Npix is the numberof real-space pixels measured. The instrument noise power spectrum thenbecomes PN = Vpix T 2systint\u0000\u00E2\u008C\u00AB , (7.26)where \u0000\u00E2\u008C\u00AB is frequencies resolution of our telescope and Vpix is a averagevolume of a real-space pixel.56We take the shot noise as constant over the times of interest and, as inRef. [139], set its value to n\u00C2\u00AF = 0.01h3Mpc\u00003 as inferred from the catalogueof 4315 extragalactic HI sources with z < 0.042 measured by the HIPASSsurvey [147].The di\u00E2\u0086\u00B5erent noise contributions to the measurement of the 21-cm powerspectrum are plotted in Fig. 7.1, where each curve corresponds to a termin the square brackets in Eq. (7.19). At low redshifts, the sample variancedominates at all relevant scales, while the instrumental noise dominates atsmaller scales for higher redshifts. As seen in Fig. 7.2, the contribution tothe total survey volume is higher at larger redshifts, allowing more modesto be measured at lower frequencies.55The integration time per real-space pixel in actuality varieties with declination. Theintegration time in Eq. (7.25) is an estimate of the lower bound and thus leads to aconservative estimate of the noise.56A volume factor was added to Eq. (7.26) to conform to our definition of the powerspectrum.1157.5. Measuring the 21-cm Power Spectrum0.0 0.1 0.2 0.3k (Mpc\u00001h)101102103noise(mK2Mpc3 h\u00003 )0.0 0.1 0.2 0.3k (Mpc\u00001h)101102103Figure 7.1: Contributions to the 21-cm power spectrum noise per mode asdescribed in Eq. (7.19) at z = 0.8 (left) and z = 2.5 (right). The curvescorrespond to the di\u00E2\u0086\u00B5erent terms in the square brackets in Eq. (7.19), wherethe blue curve is the sample variance term at \u00C2\u00B5 = 0 and the green and redcurves are the shot and instrument noise terms, respectively.1.0 1.5 2.0 2.5z30405060708090100110dV s/dz(Gpc3 h\u00003 )Figure 7.2: Survey volume per redshift over the CHIME band.1167.5. Measuring the 21-cm Power SpectrumWe are now in a position to calculate our forecasted power spectrumuncertainties. For our analysis, we divide our band into 16 redshift binsof equal size \u0000z \u00E2\u0087\u00A1 0.11. The uncertainties in the power spectrum \u0000P forspherically averaged k-bins with width \u0000k = 0.02Mpc\u00001h can be seen inFig. 7.3. The top panel shows the power spectrum uncertainties for z-binscentred at z = 0.83, 1.61, 2.50. At all redshifts, the trend of decreasinguncertainty with k at larger scales is due to the increased k-space volumeavailable for spherical shells at higher values of k. At smaller scales, theuncertainties decrease with redshift as larger volumes can be surveyed forz-bins of equal size at higher redshifts. The resolution of the array decreaseswith redshift and thus the power spectrum uncertainties worsen at smallerscales for higher redshifts as compared to lower redshifts. The bottom panelof Fig. 7.3 shows the power spectrum uncertainties at z = 1.61 for thefunction P/Psm, where Psm is the \u00E2\u0080\u0098smooth only\u00E2\u0080\u0099 power spectrum which hasthe baryonic oscillations removed (see Section 7.6 for more details).1177.6. The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 Method0123456s P(%)0.00 0.05 0.10 0.15 0.20 0.25k (Mpc\u00001h)0.951.001.051.10P/P smFigure 7.3: Forecasted power spectrum uncertainties for spherically averagedk-bins with width \u0000k = 0.02Mpc\u00001h for redshift bins of size \u0000z \u00E2\u0087\u00A1 0.11.Top: Power spectrum uncertainties at z = 0.83 (blue), 1.61 (red), and 2.50(green). Bottom: Total power spectrum over \u00E2\u0080\u0098smooth only\u00E2\u0080\u0099 power spectrumwith uncertainties at z = 1.61.7.6 The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 MethodOnce we have the uncertainties in the matter power spectrum, we wouldlike to propagate these to uncertainties in our measurements of the BAOscale. In other words, we would like to find the Jacobian Js to transformthe Fisher matrix FP \u00E2\u0087\u00A1 C\u00001P for the measurements of the power spectrum1187.6. The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 Methodinto the Fisher matrix Fs for the sound horizon. To accomplish this task,we use the \u00E2\u0080\u0098wiggles only\u00E2\u0080\u0099 method [140], which assumes we can remove thepower from scales much larger then the BAO scale and models the remaining\u00E2\u0080\u0098wiggly\u00E2\u0080\u0099 BAO power spectrum as Pb = P \u0000 Psm, where P is the full powerspectrum and Psm is the power spectrum smoothed on a scale larger thanthe BAO scale.7.6.1 Modelling the BAO Power SpectrumOur task is to build a model of the BAO-only power spectrum that willcontain the acoustic oscillations as well as e\u00E2\u0086\u00B5ects that degrade this signal,which we now outline.Acoustic OscillationsThe BAO manifests itself as preferred separation in the two-point corre-lation function at a particular angular and redshift separation, which canbe translated into the comoving lengths s? and sk, respectively. As such,the un-deteriorated BAO signal is approximated as a ellipsoidal Dirac deltafunction with semi-axes sk and s? parallel and perpendicular to the lineof sight, respectively, the Fourier transform of which is Pb / sinc(x) withx = q(k?s?)2 + (kksk)2.Silk DampingAlthough photons are tightly coupled to baryons at early times, near thetime of recombination the mean free path of the photons grows to a nonnegligible size, allowing them to steam out of overdense regions and intounderdense ones. This e\u00E2\u0086\u00B5ect, known as Silk damping, smooths out pertur-bations on small scales. We can make a rough estimate of the Silk scale \u0000silkby use of the mean free path \u0000mfp = (\u0000Tne)\u00001, where \u0000T is the Thomsoncross-section. As the relevant timescale is the Hubble time H\u00001, we expectthe number of collisions to be of the order H\u00001/\u0000mfp = \u0000TneH\u00001. For aGaussian random walk, the rms distance travelled is given by the product ofthe mean free path and the root of the number of steps, so we can estimatethe Silk scale by \u0000silk \u00E2\u0087\u00A0 (\u0000TneH)\u00001/2. Evaluating our estimate of \u0000silk justprior to recombination and using \u0000Tne \u00E2\u0087\u00A1 2.307\u00E2\u0087\u00A5 10\u00005 Mpc\u00001!ba\u00003 resultsin \u0000silk \u00E2\u0087\u00A0 57Mpc (ksilk \u00E2\u0087\u00A1 0.11Mpc\u00001). A more detailed examination can bepreformed using the Boltzmann equations, from which the following fitting1197.6. The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 Methodformula was derived [28]ksilk = 1.6!0.52b !0.73m [1 + (10.4!m)\u00000.95]Mpc\u00001, (7.27)where it was found that the baryonic power spectrum is dampened as Pb /Dsilk = exp(\u0000(k/ksilk)1.4).BAO Power Spectrum AmplitudeWe can now write our model for the linear BAO power spectrum amplitudeas Pb,lin(k) = p8\u00E2\u0087\u00A1A0P0.2sinc(x(k))Dsilk(k), (7.28)where is A0 is a normalization constant and P0.2 is the linear power spectrumevaluated at k = 0.2hMpc\u00001. To find the normalization constant A0, wefirst apply a low-pass filter with a sharp cuto\u00E2\u0086\u00B5 just below the BAO scaleto our fiducial linear power spectrum to estimate Psm and then subtract itfrom the total power spectrum to yield Pb. This estimate is subsequentlyfit to Eq. (7.28), which for the parameters used yields a best fit value ofA0 = 0.42.Nonlinear DampingNonlinear behaviour distorts the BAO signal by displacing matter at the\u00E2\u0087\u00A0 10Mpc scale, which smears out the BAO peak in the correlation func-tion, thereby damping the oscillations in the power spectrum. Ref. [151]found that the nonlinear displacement distribution is well approximated byan elliptical normal distribution with rms values \u00E2\u008C\u0083k and \u00E2\u008C\u0083? parallel andperpendicular to the line of sight, respectively. As the distorted correlationfunction is found by convolution with this elliptical Gaussian, the resultinge\u00E2\u0086\u00B5ect on the power spectrum Pb is a multiplication by the Fourier transformof the distorting elliptical normal distribution DnlDnl = exp \u0000(k?\u00E2\u008C\u0083?)22 \u0000 (kk\u00E2\u008C\u0083k)22 ! . (7.29)The rms displacements parallel and perpendicular to the line of sight werefound to be [151]\u00E2\u008C\u0083k = (1 + f)\u00E2\u008C\u0083?, (7.30)\u00E2\u008C\u0083? = 8.35h\u00001MpcG(z)G(0) \u000080.8 . (7.31)1207.6. The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 Method7.6.2 Distance UncertaintiesWith an analytical expression for the power spectrum as a function of skand s?, we can find the Jacobian Js to transform the Fisher matrix FPfor the power spectrum into the Fisher matrix Fs for the parameters \u00E2\u009C\u0093s =(ln s\u00001? , ln sk).Including the e\u00E2\u0086\u00B5ects of Silk and nonlinear damping, our model for Pbbecomes Pb(k) = p8\u00E2\u0087\u00A1A0P0.2sinc(x(k))Dsilk(k)Dnl(k). (7.32)The Jacobian Js is then found to be(Js)ij = @P (ki)@(\u00E2\u009C\u0093s)j=@Pb(ki)@ lnxi @ lnxi@(\u00E2\u009C\u0093s)j=p8\u00E2\u0087\u00A1A0P0.2 [cos(xi)\u0000 sinc(xi)]Dsilk(ki)Dnl(ki)@ lnxi@(\u00E2\u009C\u0093s)j , (7.33)where xi = x(ki) and @ lnx@ ln s\u00001? = \u00C2\u00B52 \u0000 1, (7.34a)@ lnx@ ln sk = \u00C2\u00B52. (7.34b)As sk and s? are equivalent to an angular and redshift separation, re-spectively, from Eqs. (7.8) we see thats? / rd/DA, sk / rdH, (7.35)so the Fisher matrix Fdist for the variables \u00E2\u009C\u0093dist = (lnDA, lnH,\u00E2\u008C\u00A6\u00E2\u0087\u00A4,\u00E2\u008C\u00A6k,!m,!b)can be found from Fdist = JTdistFsJdist, where the Jacobian Jdist is given by(Jdist)0j = @ ln s\u00001?@(\u00E2\u009C\u0093dist)j = @ lnDA@(\u00E2\u009C\u0093dist)j \u0000 @ ln rd@(\u00E2\u009C\u0093dist)j ,(Jdist)1j = @ ln sk@(\u00E2\u009C\u0093dist)j = @ lnH@(\u00E2\u009C\u0093dist)j + @ ln rd@(\u00E2\u009C\u0093dist)j , (7.36)where the derivatives of rd can be found using Eq. (7.5).The uncertainties on DA and H for our forecast as a function of redshiftcan be seen in Fig. 7.4. The fractional uncertainties on H show a decreasingtrend towards higher redshifts as greater volumes are contained within each1217.6. The \u00E2\u0080\u0098Wiggles Only\u00E2\u0080\u0099 Methodspherical shell with equal redshift spacing. The same trend would be seen inDA as well if not for the cuto\u00E2\u0086\u00B5 imposed on k?, which decreases at increasingredshift. The correlation between DA and H remains roughly at \u00E2\u0087\u00A2 \u00E2\u0087\u00A1 0.4,which is close to the value one would expect from a spherically symmetricmodel.1.01.52.02.53.0s(%)1.0 1.5 2.0 2.5z0.00.5rFigure 7.4: Top: Forecast uncertainties for DA (blue) and H (red) as afunction of redshift. Bottom: Correlation coe\u0000cient between DA and H.Most current BAO measurements only produce a high signal to noisemeasurement when averaged over angle, and therefore do not measure DAand H separately. The \u00E2\u0080\u0098dilation scale\u00E2\u0080\u0099 DV is often constrained instead,whose cube root consists of two factors of the angular diameter distanceand one factor of the proper radial distanceDV (z) = \u00EF\u00A3\u00BF(1 + z)2DA(z)2 czH(z)\u00001/3 . (7.37)The forecasted uncertainties on DV for CHIME can be seen in Fig. 7.5,along with uncertainties from current experiments. The CHIME band com-pliments these experiments by covering a higher redshift range, a band whichis particularly sensitive to the dark energy equation of state (see Fig. 7.6).1227.7. Dark Energy Constraints0.0 0.5 1.0 1.5 2.0 2.5z123456s D V/r d(%) 6dFGS BOSSSDSS WiggleZ CHIMEFigure 7.5: Measurement uncertainties on DV from CHIME forecasts as wellas from current detections from 6dFGS [13], SDSS [152], WiggleZ [153], andBOSS [15].7.7 Dark Energy ConstraintsWith the uncertainties on DA and H, we can finally forecast the uncertain-ties on the dark energy equation of state parameters w0 and wa. We form theFisher matrix FDE = JTDEFdistJDE for the variables \u00E2\u009C\u0093DE = (w0, wa,\u00E2\u008C\u00A6\u00E2\u0087\u00A4,\u00E2\u008C\u00A6k,!m,!b) with the Jacobian JDE given byJDE = @\u00E2\u009C\u0093dist@\u00E2\u009C\u0093DE , (7.38)where Eq. (7.2) can be used to compute derivatives of lnH and lnDA. Asseen in Fig. 7.6, the redshift range covered by CHIME encompasses regionswhere the derivatives of lnH and lnDA with respect to w0 and wa are large,allowing for tight constraints on w0 and wa.1237.7. Dark Energy Constraints0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0z\u00000.2\u00000.10.00.10.20.3 dlnH/dw0dlnH/dwadlnDA/dw0dlnDA/dwaFigure 7.6: Derivatives of lnH and lnDA with respect to w0 and wa, whichappear in JDE, as a function of redshift.The final steps in forecasting our constraint contours in the w0 \u0000 waplane is to add any relevant priors to FDE and then marginalize over all othervariables. The 95% CL contours in the w0\u0000wa plane can be seen in Fig. 7.7,which shows the forecasts for CHIME combined with Planck, Stage II, andBOSS data. Stage II priors, which represent the anticipated constraints thatwill be available after currently running experiments are complete [137],are comprise of cluster, supernovae, and weak lensing surveys and wereestimated using the DETFast 57 software. The BOSS survey aims to measureBAO over the redshift range 0.15 < z < 0.7 and anticipates a precision of1.0% and 1.8% on the measured values ofDA andH, respectively, at z = 0.35and of 1.0% and 1.7% at z = 0.6, with a correlation coe\u0000cient of 0.4 betweenDA and H at both redshifts [155]. As seen in Fig. 7.7, adding the anticipatedconstraints from CHIME to Planck+Stage II drastically improves the darkenergy equation of state constraints, which may be further improved uponby adding the lower redshift BAO measurements from BOSS.57http://www.physics.ucdavis.edu/DETFast/1247.7. Dark Energy ConstraintsFigure 7.7: Forecasted constraints in the w0\u0000wa plane for CHIME+Planck(green, FOM = 42.2), CHIME+Planck+Stage II (red, FOM =217.0), CHIME+Planck+Stage II+BOSS (orange, FOM = 270.9), andPlanck+Stage II (blue, FOM = 53.3). Each contour represents the 95% CLcontours of each forecast.The figure of merit (FOM) for the w0 \u0000 wa contour is defined as beingproportional to the inverse of the area of the contour, with a larger FOMindicating a more constraining measurement. We use the normalizationof the FOM given in Ref. [137], which can be easily computed from themarginalized Fisher matrix FMDE byFOM =qdet(FMDE). (7.39)For CHIME+Planck+Stage II the FOM is approximately 217.0 and in-creases to 270.9 when BOSS is added as a prior. The constraints fromCHIME are competitive with previous forecasts for Planck+Stage II+Stage III,58 which estimate a FOM of 279.5 [145]. Fig. 7.8 shows the rela-tive improvement of the figure of merit over the fiducial value FOM0 = 85.858As defined in Ref. [137], Stage III refers to near-future, medium cost experiments.1257.7. Dark Energy Constraintsfrom the forecast for Planck+Stage II+BOSS. Each curve represent a pos-sible survey with a di\u00E2\u0086\u00B5erent lowest redshift zmin probed as a function of thesurvey\u00E2\u0080\u0099s maximum redshift zmax. The chosen CHIME band has been indi-cated. Fig. 7.8 shows that there would be little constraining power gainedby extending the CHIME band to cover lower redshifts, as these redshiftshave already been or will soon be probed by existing surveys and that thereis limited benefit extending the band to higher redshifts where the changesin DA and H with w0, wa are smaller than at lower redshifts, as indicatedin Fig. 7.6.0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5zmax1.01.52.02.53.03.5FOM/FOM 0 CHIMEzmin0.00.81.31.72.02.22.52.73.03.23.5Figure 7.8: Relative improvement of figure of merit FOM with CHIME overfiducial value FOM0 as a function of redshift coverage. Each curve representa survey with a di\u00E2\u0086\u00B5erent lowest redshift measured zmin as a function ofmaximum redshift zmax. The fiducial figure of merit FOM0 = 85.8 is takenas the forecast for Planck+Stage II+BOSS. The black point denotes theCHIME band.We can now ask at which redshift our experiment best constrains wDE(z).Longer term projects such as the SKA are included in Stage IV.1267.7. Dark Energy ConstraintsTo this end, instead of expanding wDE in a Taylor series about a = 1 as donein Eq. (7.1), we expand around the point a = ap, so thatwDE(a) = wp + wa(ap \u0000 a), (7.40)where wp = w0 + wa(1 \u0000 ap). ap is defined to be the scale factor whenthe uncertainty in wDE is minimized, referred to as the pivot point. In thiscase, the pivot point coincides with the point when the uncertainty in wp isminimized. By use of the JacobianJpivot = \u00E2\u009C\u00931 ap \u0000 10 1 \u00E2\u0097\u0086 , (7.41)to change from the parameters (w0, wa) to (wp, wa), the uncertainty in wpis found to be\u00002wp = \u00002w0 + (1\u0000 ap)2\u00002wa + 2(1\u0000 ap)\u00E2\u0087\u00A2w0,wa\u0000w0\u0000wa , (7.42)where \u00E2\u0087\u00A2w0,wa is the correlation coe\u0000cient between w0 and wa, so that theminimum in \u00002wp occurs at ap = 1 + \u00E2\u0087\u00A2w0,wa\u0000w0\u0000wa , (7.43)and \u00002wp then simplifies to\u00002wp = \u00002w0(1\u0000 \u00E2\u0087\u00A22w0,wa). (7.44)It is easily verified that wp and wa are uncorrelated. In addition, sincedet(JTpivotFMDEJpivot) = det(JTpivot)det(FMDE)det(Jpivot) = det(FMDE), the FOMfor the wp \u0000wa contour will be the same as the w0 \u0000wp contour. The con-straints on wDE for CHIME+Planck+Stage II as a function of redshift canbe seen in Fig. 7.9, where the pivot point occurs at redshift zp = 0.49, nearwhere changes in wDE produce the largest changes in H and DA.1277.8. Conclusions0.0 0.5 1.0 1.5 2.0 2.5 3.0z\u00001.2\u00001.1\u00001.0\u00000.9\u00000.8w DEFigure 7.9: Constraints on wDE from the forecast forCHIME+Planck+Stage II. Filled regions denote the 1 \u0000 \u0000 and 95% CLbounds. The constraints are most stringent at the pivot point occurring atredshift zp = 0.49.7.8 ConclusionsThe forecasting methods and models presented in this chapter provide arapid and easily implementable method for generating dark energy forecastsfor 21-cm intensity mapping experiments such as CHIME. These forecastsallow one to optimize the design of 21-cm intensity mapping experimentsfor the detection of the BAO at di\u00E2\u0086\u00B5erent redshifts as well as how to bestconstrain the dark energy equation of state in combination with existingand future experiments. These methods may be extended to allow for moredetailed forecasts which include e\u00E2\u0086\u00B5ects due to foreground subtraction [141,142]. We have shown that CHIME will add competitive and complimentarybounds on the dark energy equation of state compared to Stage II and IIIexperiments.128Chapter 8Redundant BaselineCalibration8.1 IntroductionRadio interferometric telescopes with a large number of elements will forman important category of radio telescopes in the near future, with manyexperiments currently being built or planned for in the next decade [156,157, 158, 159, 160, 161]. These experiments will typically have to removebright foregrounds. For example, 21-cm intensity mapping experiments willbe required to remove foregrounds that are four orders of magnitude largerthan the desired 21-cm signal. These requirements will introduce manyoperational challenges for these projects, such as the precise calibration ofsystem gains, characterization of the beam shapes, and removal of radiofrequency interference, amongst others.There exist many calibration techniques already employed by radio inter-ferometers, for example the use of noise injection from a known noise sourceor the use of calibrator sources in the sky. While these methods have provensuccessful in the past, their application to 21-cm mapping experiments willintroduce many new uncertainties, especially when using calibrator sources,as many of these experiments will operate at frequencies where few precisemaps of the sky exist. Due to the need for great precision in the calibrationof these telescopes, we expect that a variety of new and existing calibra-tion techniques will be used. In this vein, in this chapter we examine newtechniques for the calibration of the direction-independent gains of interfer-ometric arrays that contain a large number of redundant baselines.Many array designs for new telescopes contain many redundant or nearlyredundant baselines. Redundant baseline calibration exploits this redun-dancy to extract both the antenna gains and calibrated sky visibilities withminimal a priori information about the sky. As such, redundant baselinecalibration may be a valuable calibration technique for such experiments.Redundant baseline calibration has been tested on the Westerbork Syn-1298.2. Calibration Requirements for CHIMEthesis Radio Telescope [162] as well as LOFAR [163] and several di\u00E2\u0086\u00B5erentalgorithms have been developed [164, 165, 166, 167, 168]. The commonfactor among algorithms is the assumption that all primary beams in thearray are identical, so that (up to noise levels and excluding the e\u00E2\u0086\u00B5ect ofthe di\u00E2\u0086\u00B5erent gains of the system) visibilities formed from perfectly redun-dant baselines will be identical, excluding other unintended signals such ascrosstalk. In this model, the di\u00E2\u0086\u00B5erences between measured visibilities withthe same baseline will be due to variation of the gains used to form thesevisibilities. Redundant baseline algorithms essentially compare the mea-sured visibilities that share a baseline to one another to achieve a nearly skyindependent measurement of the gains.An essential assumption of the basic implementation of the redundantbaseline algorithm is that all primary beams are identical. In actuality,the beams will di\u00E2\u0086\u00B5er from one another to some degree. If these di\u00E2\u0086\u00B5erencesare significant, calibrations done with the redundant baseline algorithm willbe poor. In this chapter, we examine the e\u00E2\u0086\u00B5ects of introducing variationsbetween primary beams in the array on redundant baseline calibrations. Wedevelop a model of redundant baseline calibration for the gain amplitudesthat accounts for the variation between beams which can result in improvedcalibrations.This chapter is organized as follows: The model of the gains is introducedin Section 8.3. We will use di\u00E2\u0086\u00B5erent algorithms for the gain amplitude andphase calibrations, which are examined in Sections 8.4 and 8.5, respectively.The di\u00E2\u0086\u00B5erent redundant baselines calibration algorithms discussed in thischapter will be tested on simulated data, the generation of which is describedin Section 8.4.4.8.2 Calibration Requirements for CHIMEAlthough our redundant baseline calibration algorithm can be applied toany multi-element interferometric telescope that contains enough redundantbaselines, we will be particularly interested in comparing the performance ofthe calibration algorithms that will be examined to the calibration require-ments for CHIME. Here we will give a brief overview of these requirementsbefore discussing calibration algorithms.In Ref. [142], simulated calibration errors were propagated through theCHIME pipeline, the results of which show that if the gain amplitude iscalibrated to an accuracy of a few percent and the phase calibrated to anaccuracy of a few degrees, then the 21-cm power spectrum can be con-1308.3. Gain Modelstructed to an accuracy of \u00E2\u0087\u00A0 10% or better. At higher levels of calibrationerror, the extraction of the 21-cm power spectrum is significantly degraded,with \u00E2\u0087\u00A0 10% gain calibration errors resulting in systematic errors dominatingover statistical uncertainties. As such, we desire a gain amplitude calibra-tion accuracy of at least a few percent and a phase gain calibration accuracyof a few degrees. Furthermore, by achieving a phase calibration accurate toa few degrees, we ensure that such phase errors are subdominant to thoseinduced by a warping of the reflector on the scale of \u00E2\u0087\u00A0 1 cm, the expectedscale to which CHIME\u00E2\u0080\u0099s reflector could be considered accurate to.8.3 Gain ModelWe model the response Si of antenna i within an array of receivers as Si =giFi + ni, where gi is the (complex) gain of feed i and ni is an additivenoise term. Taking the time-averaged correlation between feeds producesthe measured visibilitiesV measij = hS\u00E2\u0087\u00A4i Sji = g\u00E2\u0087\u00A4i gjVij + nij , (8.1)where nij is the noise on the measured visibility and Vij are the \u00E2\u0080\u0098true\u00E2\u0080\u0099 visibil-ities. Each feed is sensitive to a particular polarization. As we will be con-sidering only visibilities formed by correlating feeds that measure the samepolarization, we do not add any references in our notation to the particularpolarization measured. For arrays that measure two polarization states, thecalibration algorithms to be described may be done on each polarizationseparately. The true visibility can be represented asVij = 1p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j Z d2n\u00CB\u0086A\u00E2\u0087\u00A4i (n\u00CB\u0086)Aj(n\u00CB\u0086)e2\u00E2\u0087\u00A1in\u00CB\u0086\u00C2\u00B7uijT (n\u00CB\u0086), (8.2)where Ai(n\u00CB\u0086) is the primary beam shape of feed i in the direction n\u00CB\u0086 and T (n\u00CB\u0086)is the sky intensity for the polarization of interest. uij is the feed separationbetween feeds i and j and \u00E2\u008C\u00A6i = R d2n\u00CB\u0086|Ai(n\u00CB\u0086)|2 is the beam solid angle.If all primary beams are identical, Vij is dependent only on the separationuij and thus Vij will be identical for pairs of feeds separated by the samedistance. In this case, we use the notation of Ref. [164] to write Vij asVi\u0000j to emphasize this point, although the subscript i\u0000j should not be takenliterally.In the standard redundant baseline algorithm where all primary beamsare identical, the measured visibilities V measij are used to give a simultaneousestimate of both the gains gi and true visibilities Vi\u0000j . For an array with N1318.4. Amplitude Calibrationfeeds, we have N(N \u0000 1)/2 correlations (excluding autocorrelations) and Ngains. If all baselines are unique, then we will have N(N \u0000 1)/2 true visibil-ities and solving for both gains and true visibilities is an underdeterminedproblem. On the other hand, if enough of the baselines in the array are thesame, the problem will be overdetermined. In particular, a regular array isoverdetermined for sizes larger than a few elements.Removing the assumption that all beams are identical, we can write thebeams in terms of a set of basis functions {A\u00C2\u00B5(n\u00CB\u0086)}, where \u00C2\u00B5 labels each basisfunction in the set. By expanding beam i with coe\u0000cients a\u00C2\u00B5i asAi(n\u00CB\u0086) = X\u00C2\u00B5 a\u00C2\u00B5i A\u00C2\u00B5(n\u00CB\u0086), (8.3)the true visibilities Vij are then given byVij = 1p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j X\u00C2\u00B5\u00E2\u008C\u00AB a\u00C2\u00B5\u00E2\u0087\u00A4i a\u00E2\u008C\u00ABjV \u00C2\u00B5\u00E2\u008C\u00ABi\u0000j , (8.4)where V \u00C2\u00B5\u00E2\u008C\u00ABi\u0000j = Z d2n\u00CB\u0086A\u00C2\u00B5\u00E2\u0087\u00A4(n\u00CB\u0086)A\u00E2\u008C\u00AB(n\u00CB\u0086)e2\u00E2\u0087\u00A1in\u00CB\u0086\u00C2\u00B7uijT (n\u00CB\u0086). (8.5)In this case, although the visibilities Vij are no longer identical for pairsof feeds with the same baseline, the quantities V \u00C2\u00B5\u00E2\u008C\u00ABi\u0000j are the same for thesepairs.If the beam can be well approximated by a small number of basis func-tions with known coe\u0000cients a\u00C2\u00B5i , we may employ a calibration algorithmvery similar to the standard redundant baseline algorithm, except we mustnow solve for the parameters V \u00C2\u00B5\u00E2\u008C\u00ABi\u0000j instead of Vi\u0000j in addition to the gains.8.4 Amplitude CalibrationIn this section we will examine redundant baseline algorithms useful for gainamplitude calibrations. We first review the so-called \u00E2\u0080\u0098logarithm method\u00E2\u0080\u0099that assumes that all beams are identical and subsequently describe a novelextension to this method that can account for variation between beams.8.4.1 The Logarithm MethodThe logarithm method provides a straightforward way to solve for the modelparameters, as well as for determining any degeneracies that are presentbetween parameters.1328.4. Amplitude CalibrationWe can linearize the calibration problem by taking the logarithm ofEq. (8.1), which will allow us to use well known linear least-squares solu-tions to estimate the model parameters. After taking the logarithm, we canseparate the real and imaginary parts asln |V measij | = ln |gi|+ ln |gj |+ ln |Vij |+ Re(\u00E2\u008C\u0098ij), (8.6a)arg(Vmeasij ) = \u0000arg(gi) + arg(gj) + arg(Vij) + Im(\u00E2\u008C\u0098ij) + 2\u00E2\u0087\u00A1c, (8.6b)where \u00E2\u008C\u0098ij = ln(1 + nij/g\u00E2\u0087\u00A4i gjVij) and c is an arbitrary integer. By takingthe logarithm, we have turned the product in Eq. (8.1) into a sum and haveseparated the amplitudes and phases into two separate equations, althoughthe amplitude and phase equations are weakly coupled through the noiseterm \u00E2\u008C\u0098ij . We can then solve for the gain amplitudes and phases separately.As noted in Ref. [164], while the logarithm method can accurately re-covers the gain amplitudes, large errors may appear in its estimate of thephases. This problem arises from the freedom of c in Eq. (8.6b) to assumethe value of any integer, which if a least-squares estimate is to be formedcreates an ambiguity in the phases of the gains. As a result, while the loga-rithm method may be useful in calibrating the gain amplitudes, in its currentformulation it is ill-suited for producing estimates of the phases and we willonly examine its performance for amplitude calibrations for this method.8.4.2 Identical BeamsIn the case where all primary beams are identical, we have Vij = Vi\u0000j soEq. (8.6a) becomesln |V measij | = ln |gi|+ ln |gj |+ ln |Vi\u0000j |+ Re(\u00E2\u008C\u0098ij). (8.7)We can write Eq. (8.7) as the matrix equationd = Mx+ \u00E2\u008C\u0098, (8.8)where the vector d holds the logarithms of the measured visibilities, x con-tains the logarithms of both the gains and true visibilities, and the noiseterms are put into \u00E2\u008C\u0098. The information regarding the array configuration isincorporated into the matrix M. For example, for a regular one-dimensionalarray with feeds numbered sequentially down the array, the amplitude equa-1338.4. Amplitude Calibrationtion could be written as0BBB@ln |V meas12 |ln |V meas23 |ln |V meas13 |...1CCCA=0BBB@1 1 0 1 00 1 1 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 1 0 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B71 0 1 0 1....... . .1CCCA0BBBBBBBBBB@ln |g1|ln |g2|ln |g3|...ln |V1|ln |V2|...1CCCCCCCCCCA+0BBB@Re(\u00E2\u008C\u009812)Re(\u00E2\u008C\u009823)Re(\u00E2\u008C\u009813)...1CCCA,(8.9)where the subscript b on Vb here labels the baseline lengths in units of thesmallest baseline bmin.Once the matrixM is formed for our array configuration, we can estimatethe gains and true visibility amplitudes contained in the vector x with theleast-squares estimatorx\u00CB\u0086 = (MTN\u00001M)\u00001MTN\u00001d, (8.10)where N = h\u00E2\u008C\u0098\u00E2\u008C\u0098T i is the (logarithmic) noise covariance matrix and thecovariance of the estimator x\u00CB\u0086 is given byCx\u00CB\u0086 = (MTN\u00001M)\u00001. (8.11)Although redundant baseline calibration is nearly independent of thesky, degeneracies in the model require a few additional constraints to beadded in order to fully solve for the system. In particular, the calibration isnot sensitive to the overall absolute gain of the array as the transformationsgi ! Cgi and Vij ! Vij/C2, where C is a real constant, leave the measuredvisibilities unchanged. To allow us to solve for x\u00CB\u0086 via Eq. (8.10), one maysupplement the series of linear equations by a \u00E2\u0080\u0098gauge-fixing\u00E2\u0080\u0099 equations, suchas Xi ln |gi| = 0, (8.12)or by adding a condition to fix the value of a particular gain or calibratedvisibility. In some situations only the relative gains are desired, so adjustingthe overall absolute gain of the calibration is unimportant. If an absolutecalibration is required, then the post-calibrated gains may be adjusted viathe above transformations to fit additional information.1348.4. Amplitude Calibration8.4.3 Nonidentical BeamsWe now adapt the logarithm method to accommodate beams that varyfrom one another, where the departure of the beams from each other willbe treated as a perturbation. As a shorthand, in later sections we will referto this method as the extended algorithm and the algorithm that modelseach beam (as identical as described in the previous section) as the basicalgorithm.We characterize the beams by two real-valued functions A0(n\u00CB\u0086) and A1(n\u00CB\u0086)and will set a0i = 1, as well as relabel the coe\u0000cients \u0000i \u00E2\u008C\u0098 a1i , which are takento be real. It will be assumed that the beam profiles (and thus {\u0000i} and {\u00E2\u008C\u00A6i})are known to some degree of accuracy. With this, Eq. (8.6a) becomesln |V measij p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j | = ln |gi|+ ln |gj |+ ln |V 00i\u0000j |+ ln |1 + \u00E2\u009C\u008Fij |+Re(\u00E2\u008C\u0098ij), (8.13)where \u00E2\u009C\u008Fij = (\u0000i + \u0000j)V 01i\u0000jV 00i\u0000j + \u0000i\u0000j V 11i\u0000jV 00i\u0000j , (8.14)and note that we have V 01i\u0000j = V 10i\u0000j . We will assume that the deviationof each primary beam from Ai(n\u00CB\u0086) = A0(n\u00CB\u0086) is small and that |\u00E2\u009C\u008Fij | \u00E2\u008C\u00A7 1.Approximating the second last term in Eq. (8.13) by a Taylor series to firstorder in \u00E2\u009C\u008Fij yields the linear equationln |V measij p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j | \u00E2\u0087\u00A1 ln |gi|+ ln |gj |+ ln |V 00i\u0000j |+ (\u0000i + \u0000j)\u0000(0)i\u0000j + \u0000i\u0000j\u0000(1)i\u0000j + Re(\u00E2\u008C\u0098ij), (8.15)where\u0000(0)i\u0000j = Re V01i\u0000jV00i\u0000j! , (8.16a)\u0000(1)i\u0000j = Re V11i\u0000jV00i\u0000j! (8.16b)are unknown sky parameters. Note that we can write \u00E2\u008C\u00A6i as\u00E2\u008C\u00A6i = \u00E2\u008C\u00A600 + 2\u0000i\u00E2\u008C\u00A601 + \u00002i\u00E2\u008C\u00A611, (8.17)where\u00E2\u008C\u00A6\u00C2\u00B5\u00E2\u008C\u00AB = Z d2n\u00CB\u0086A\u00C2\u00B5(n\u00CB\u0086)A\u00E2\u008C\u00AB(n\u00CB\u0086). (8.18)1358.4. Amplitude CalibrationWith a preexisting estimate of the \u0000s, we can write Eq. (8.15) as a matrixequation analogous to Eq. (8.8), which we write asd\u00CB\u009C = M\u00CB\u009Cx\u00CB\u009C+ \u00E2\u008C\u0098. (8.19)As in the basic algorithm, the vector x\u00CB\u009C holds all model parameters, butnow also includes the parameters \u0000(0)i\u0000j and \u0000(1)i\u0000j and may be packed as x\u00CB\u009C =({ln |gi|}, {ln |V 00i\u0000j |}, {\u0000(0)i\u0000j}, {\u0000(1)i\u0000j}). The vector d\u00CB\u009C(ij) = ln |V measij p\u00E2\u008C\u00A6i\u00E2\u008C\u00A6j |still holds the logarithms of the measured visibilities, but now has factors ofp\u00E2\u008C\u00A6i added. M\u00CB\u009C is constructed in a similar manner to M, but now includeselements dependent on the \u0000s.As we have reduced our model to a set of linear equations, we can employthe same least-squares solution in Eq. (8.10) that was used to solve for themodel parameters in the basic algorithm. However, as will be discussedin the following section, there exists an additional degeneracy between themodel parameters that must be fixed in some way before the least-squaressolution may be applied.Lastly, we note that when using the basic algorithm in a situation withnonidentical primary beams, the vector d used to solve Eq. (8.10) can bereplaced by the vector d\u00CB\u009C. This accounts for the variability of the beam solidangle between feeds while not changing the basic algorithm as described inSection 8.4.2, which will be taken into account when showing calibrationresults in Section 8.4.4.Fixing the DegeneraciesA number of additional degeneracies are introduced by having expandedthe number of model parameters, which may be found by calculating thenull space of M\u00CB\u009C. However, not all of these degeneracies a\u00E2\u0086\u00B5ect the gainsand as such will not be of concern here. For example, for a regular lineararray, there are three additional distinct degeneracies that leave the gainsuna\u00E2\u0086\u00B5ected. On the other hand, the degeneracies that do a\u00E2\u0086\u00B5ect the gains areln |V 00i\u0000j |! ln |V 00i\u0000j |+ 1 and ln |gi|! ln |gi|\u0000 12 , (8.20a)\u0000(0)i\u0000j ! \u0000(0)i\u0000j + 1 and ln |gi|! ln |gi|\u0000 \u0000i. (8.20b)The degeneracy in Eq. (8.20a) is the perturbed version of the overall ampli-tude degeneracy described in Section 8.4.2, which may be fixed in the samemanner as done in the identical beams model. On the other hand, as the1368.4. Amplitude Calibrationdegeneracy of Eq. (8.20b) is not present in the identical beams model, wewill require an additional \u00E2\u0080\u0098gauge-fixing\u00E2\u0080\u0099 condition.We can fix all degeneracies (both those that do and do not a\u00E2\u0086\u00B5ect thegains) by using the matrix M\u00CB\u009Cf in Eq. (8.10) instead of M\u00CB\u009C, which is con-structed by appending the null space M\u00CB\u009Cnull = Null(M\u00CB\u009C) to M\u00CB\u009CM\u00CB\u009Cf = \u00E2\u009C\u0093 M\u00CB\u009CM\u00CB\u009Cnull\u00E2\u0097\u0086 , (8.21)The vector d\u00CB\u009C must also be appended by a vector of length equal to thetotal number of degeneracies. For this part of the calculation, we chooseto set these additional values to zero. For the degeneracies that do nota\u00E2\u0086\u00B5ect the gains, setting those values in d\u00CB\u009C to zero is inconsequential. In boththe identical and nonidentical beams algorithms we will enforce the overallgain amplitude condition given in Eq. (8.12) and will only be interestedin the relative gain calibration. The only remain degeneracy is that ofEq. (8.20b), which we will use to adjust our initial estimate of x\u00CB\u009C, found byuse of Eq. (8.21), to fit to prior information (the details of which will bediscussed in the following section).8.4.4 SimulationTo test of our calibration algorithms, we simulate the response of a 12 feedregular linear array aligned in the North-South direction with beams pointedat zenith. The primary beam of each feed is modelled as being nearly Gaus-sian, with a narrow width in the East-West direction and wide field of viewin the NS direction, to mimic the response of feeds placed along the focal lineof a cylindrical reflector oriented with its axis along the NS direction. Thezeroth order beam A0 is taken as a two dimensional Gaussian with widths\u0000u and \u0000v in the EW and NS directions, respectively. Our beam basis func-tions are chosen from the Hermite functions, with zeroth order function ofwhich is a Gaussian. We choose the perturbing first order beam A1 to bethe second Hermite function.59 Specifically, we have A0(x2) / exp(\u0000x2/2)and A1(x2) = (2x2 \u0000 1)A0(x2)/p2, where x2 = (n\u00CB\u0086 \u00C2\u00B7 u\u00CB\u0086)2/\u00002u + (n\u00CB\u0086 \u00C2\u00B7 v\u00CB\u0086)2/\u00002v ,with u\u00CB\u0086 and v\u00CB\u0086 pointing in the East and North directions, respectively. With59We have chosen A1 as the second Hermite function as there has been a significantvariability in the beam widths of the CHIME pathfinder observed. Choosing the firstHermite function instead would correspond to the feeds having di\u00E2\u0086\u00B5erent pointings, whichhas been observed as well in the CHIME pathfinder. As the conclusions are the same ineither case, we only examine the single case where the second Herminte function is used.1378.4. Amplitude Calibration\u00004 \u00003 \u00002 \u00001 0 1 2 3 4x\u00000.6\u00000.4\u00000.20.00.20.40.60.8A\u00C2\u00B5 A0A1A0+0.1A1Figure 8.1: Beam basis functions chosen as the zeroth and second orderHermite functions. A linear combination of the two beam basis function isdisplayed, illustrating that for small values of \u0000i, the chosen basis functionsperturb the width of a Gaussian beam.this basis, having a nonzero value of \u0000i perturbs the beam width, as seen inFig. 8.1.The visibilities used for the test calibrations in this section where gener-ated using the 408 MHz Haslam map [169].60 These visibilities were gener-ated assuming a feed separation of 30 cm, where the latitude of the telescopewas chosen to be at 45\u0000. The time step used to generate our test set of vis-ibilities was taken with a transiting right ascension of 0\u0000.We set the EW beam width as \u0000u = \u0000/W , where \u0000 is the wavelengthcorresponding to the frequency of 408 MHz and W is a length scale that inthe present context can be thought of as approximating the width of ourcylindrical reflector, chosen to be W = 20m. The NS beam width is set as\u0000v = 10\u0000u so that each feed has a large field of view in that direction.The noise on the visibility V measij is constructed with a variance of (\u00002n)ij =|gi||gj |T 2sys/tint\u0000\u00E2\u008C\u00AB and is uncorrelated between di\u00E2\u0086\u00B5erent visibilities. Thenoise covariance matrix N for the logarithmic noise variables \u00E2\u008C\u0098ij can be60The map used is available at http://lambda.gsfc.nasa.gov/product/foreground/haslam408.cfm1388.4. Amplitude Calibrationapproximated by N(ij),(kl) = \u00002n|V measij |2 \u0000(ij),(kl) (8.22)where (ij) specifies the correlation index number between feeds i and j and\u0000(ij),(kl) is the Kronecker delta function. Although in our model \u0000n dependson the gain amplitudes, as we do not assume detailed a priori informationabout the gain amplitudes, we do not vary our estimate of \u00002n for eachvisibility when forming N for use in Eq. (8.10). Instead, an expected averagegain amplitude level between all feeds may be incorporated into \u00002n so thatit remains constant for all visibilities, although including variations of \u00002nbetween di\u00E2\u0086\u00B5erent visibilities may easily be incorporated if such informationis available. For the generation of the noise we use a system temperatureof Tsys = 200K, a bandwidth of \u0000\u00E2\u008C\u00AB = 1MHz, and integration time oftint = 10 s.61We select the true values of the gain amplitudes randomly from a uniformdistribution between \u00000.5 and 1.5 and assign a random phase, and pick the\u0000s from a uniform random distribution from \u0000\u0000max to \u0000max, the particularchosen values of which can be seen in Fig. 8.2. For \u0000max we use a fiducialvalue of \u0000max,fid = 0.1.8.4.5 Amplitude Calibration ResultsAs described in Section 8.4.3, we require at least one additional piece ofinformation to properly fix the extra relevant degeneracy in Eq. (8.20b)arising in the nonidentical beam model. To do this, we will provide a prioron \u0000(0)bmin for the smallest baseline bmin. This choice of priors may be usefulwhen accurate prior sky information is available for only large scales probedby only the smallest baseline of an array, while small scale information thatcontributes significantly to the signal measured by larger baselines is poorlyconstrained by prior information.To begin with, we assume that both the \u0000 parameters and the prior on\u0000(0)bmin are known to arbitrary accuracy, both of which will later be relaxed.The calibration results for both basic and extended redundant baseline al-gorithms can be seen in Fig. 8.3, which shows the relative bias and standarddeviation for each feed in the array over a set of visibilities taken with 100di\u00E2\u0086\u00B5erent realizations of the noise.62 The set of visibilities used for this cal-61The value of Tsys = 200K was chosen to stress test the algorithm and as such is a bithigher than the nominal value of Tsys = 50K for CHIME.62The same set of 100 noise realizations are used for all example calibrations throughoutthe chapter.1398.4. Amplitude Calibration0.81.01.21.4|g i|\u0000180\u00000\u0000180\u0000arg(g i)0 2 4 6 8 10 12feed number i\u00000.10.00.1d iFigure 8.2: Fiducial simulated values of the gains gi and the beam pertur-bation parameters \u0000i with \u0000max,fid = 0.1.1408.4. Amplitude Calibration\u00005\u00004\u00003\u00002\u0000101234bias(%)0 2 4 6 8 10feed number0.000.050.100.150.20std(%)Figure 8.3: Relative calibrated gain amplitude |gi| bias and standard devi-ation over a set of 100 noise realizations for the basic (blue, Section 8.4.2)and extended (red, Section 8.4.3) redundant baseline calibration methods.Simulated data was generated with the fiducial values for the gains and \u0000s asseen in Fig. 8.2. The extended calibration algorithm was performed with the\u0000s known to arbitrary accuracy and by fixing \u0000(0)bmin for the smallest baselineto a prior that is known without error.ibration was generated using the fiducial values of the \u0000s, as depicted inFig. 8.2. It is clear from Fig. 8.3 that in this situation, the calibrated gainamplitudes produced by the basic redundant baseline algorithm are highlybiased, while the extended algorithm results in a small bias. In general,the resulting biases in the basic algorithm will be correlated with the beamperturbation parameters \u0000i, which for this particular calibration has a corre-lation coe\u0000cient of \u00000.86. As the extended algorithm uses the same amountof information to fit for more parameters, we expect that its gain calibrationswill have a larger variance, as can be seen in the lower panel of Fig. 8.3.The improvement of the extended algorithm over the basic algorithm de-pends on the size of the beam perturbations. Fig. 8.4 shows the calibrationresults of the basic and extended algorithms as a function of the maximumbeam perturbation parameter \u0000max, where the beam perturbation variablesare given by \u0000i = (\u0000fid)i\u00E2\u0087\u00A5 \u0000max/\u0000max,fid. The absolute value of the bias aver-aged over all feeds (we denoted the average over all feeds by h. . .if) is plottedwith errors bars that represent the square root of h\u00002gif , which is the variance1418.4. Amplitude Calibration\u00002g of the calibrated gain amplitudes over the set of noise realizations aver-aged over all feeds (note that the error bars do not represent the varianceof the estimate of the bias. Error bars in all further plots in this chaptershould be interpreted in this way). Since in our example the beam pertur-bations act to perturb the width of the beams, the value of each \u0000i can betranslated into a perturbation of the full width at half maximum (FWHM)of the beam. The relative di\u00E2\u0086\u00B5erence between the largest and average beamwidth, FWHMmax and FWHM0, respectively, is displayed on the top hor-izontal axis of Fig. 8.4.63 If the perturbations are negligible, the extendedalgorithm will produce similar averages for the calibrated gain amplitudes,but with a higher variance. With large perturbations, the truncated termsin the Taylor series in Eq. (8.15) become more important and the bias inthe extended algorithm grows, although the calibration may still show sig-nificant improvement over the basic algorithm. Intermediate values of the\u0000s show a significant improvement in the calibrations of the extended overthe basic algorithm, with little bias appearing in the extended calibrations.Although using the logarithm method as described by Eq. (8.15) canyield accurate estimates of the gain amplitudes, as with the basic logarithmicredundant baseline algorithm, if the noise nij is normally distributed, thelogarithmic noise term \u00E2\u008C\u0098ij will not be, leading to a statistical bias in the gainamplitude calibration. However, with noise levels attainable with modernlow noise amplifiers and integration times on the order of a second or greater,the distributions of \u00E2\u008C\u0098ij will be very close to normal, resulting in only a smallstatistical bias.We now relax the assumption that the prior on \u0000(0)bmin is known exactly andrun the extended calibration algorithm with an error added to this prior, theresults of which can be seen in Fig. 8.5. Although obscuring our knowledgeof the prior degrades the calibration, only after adding a large error to theprior does the basic algorithm outperform the extended algorithm.As a final test of the extended algorithm, the assumption that the beamperturbation parameters are known exactly is relaxed by adding a randomerror to our knowledge of each \u0000i, which are used to construct the matrixM\u00CB\u009C. To generate this error, we draw values for the fractional error of eachbeam width from a normal distribution centred on zero with variance \u00002ln \u0000.The resulting calibrations can be seen in Fig. 8.6 for the extended algorithmwith no errors on the prior for \u0000(0)bmin and as well as for 50% and 100% errorson this prior. The top horizontal axis in Fig. 8.6 displays the RMS error63Based on preliminary analysis, the beam widths of the CHIME pathfinder can varyby \u00E2\u0087\u00A0 15%, which corresponds to a value of \u0000 slightly less than \u0000max,fid = 0.1.1428.4. Amplitude Calibration0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16dmax0.00.51.01.52.02.53.03.5h|bias|i f(%) 0 5 10 15 20 25(FWHMmax\u0000FWHM0)/FWHM0 (%)Figure 8.4: Gain amplitude calibration for the basic (blue) and extended(red) algorithms as a function of the maximum beam perturbation parameter\u0000max. At each value of \u0000max the beam perturbation parameters are scaled as\u0000i = (\u0000fid)i \u00E2\u0087\u00A5 \u0000max/\u0000max,fid. Each point is the absolute value of the relativebias averaged over all of the feeds. The error bars represent the square rootof the variance in the calibrated gain amplitude \u00002g averaged over all feedsqh\u00002gif .1438.4. Amplitude Calibration0 20 40 60 80 100 120 140 160prior error on \u0000(0)bmin (%)0.00.51.01.52.02.53.03.5h|bias|i f(%)Figure 8.5: Amplitude calibration as a function of error on the prior of \u0000(0)bminused to fix the extra degeneracy in the extended algorithm. The extendedalgorithm is shown in red and the basic algorithm, which is independentof the prior on \u0000(0)bmin , is shown as the blue band. Fiducial values of the\u0000s are used and are assumed to be known perfectly. Error bars are to beinterpreted as in Fig. 8.4.1448.5. Phase Calibrationon the beam FWHM over all the feeds in the array. With the beam widthsknown to a few percent and a reasonably accurate prior on \u0000(0)bmin , we can seethat the mean bias on the gain amplitude calibration performed with theextended algorithm remains near the percent level or better.8.5 Phase CalibrationIn the previous section, we examined the performance of the logarithmmethod (and its extension to accommodate for varying primary beams) forthe calibration of the gain amplitudes of our array. As mentioned in Sec-tion 8.4.1, the logarithm method has di\u0000cultly in estimating the phases ofthe gains and thus we look for another method better suited for the phasecalibration using redundant baselines. In this section we examine such acalibration algorithm, although no attempt is made to accommodate forvariations between primary beams, which we leave for future work.8.5.1 The Eigenvector MethodThe eigenvector method is an iterative calibration technique close in veinto the self-calibration algorithm. We begin by seeking an estimate of thegain matrix G = |gihg|. To form our estimate, we divide V measij with anestimate of the true visibilities (we will soon see that this estimate can bemade arbitrary), yielding the matrix G\u00CB\u0086ij = V measij /Vij . We would now liketo find the gain vector |gi that minimizes the chi-squared\u00002 = Xijkl(G\u00CB\u0086ij \u0000Gij)\u00E2\u0087\u00A4C\u00001(ij),(kl)(G\u00CB\u0086kl \u0000Gkl), (8.23)where C(ij),(kl) is the covariance matrix for G\u00CB\u0086. We assume that the co-variance matrix is uncorrelated between di\u00E2\u0086\u00B5erent baselines and weights allbaselines with the same variance \u00002. Under these assumptions, Eq. (8.23)simplifies to \u00002 = 1\u00002 Xij |G\u00CB\u0086ij \u0000Gij |2. (8.24)Before we attempt to minimize the above chi-squared, we note that sinceG|gi = hg|gi|gi, then |gi is an eigenvector of G with eigenvalue hg|gi.The matrix G\u00CB\u0086 is Hermitian by construction and therefore has an eigen-decomposition G =Pi \u0000i|\u0000iih\u0000i|, where \u0000i is the ith eigenvalue of G\u00CB\u0086 cor-responding to eigenvector |\u0000ii. As G is the outer product of a vector with1458.5. Phase Calibration0 10 20 30 40slnd (%)0.00.51.01.52.02.53.0h|bias|i f(%) 0.0 0.9 1.8 2.8 3.7RMS beam FWHM error (%)Figure 8.6: Amplitude calibration as a function of the level of uncertaintyon the beam perturbation parameters \u0000i. The accuracy to which the beamsare known is parameterized by \u0000ln \u0000. The extended algorithm is performedboth with a perfect prior on \u0000(0)bmin (red) as well as with 50% (orange) and100% (green) error on this prior. The basic algorithm is shown in blue. Thetop horizontal axis shows the corresponding RMS error on the FWHM ofthe beams.1468.5. Phase Calibrationitself, we can use an eigenvector |\u0000ii of G\u00CB\u0086 (with some overall normaliza-tion) as an estimator for the gain vector |gi. The question is now whicheigenvector of G\u00CB\u0086 is the best estimate of the gain vector?Since G\u00CB\u0086 is Hermitian and positive semi-definite, its eigen-decompositioncoincides with its singular value decomposition (SVD). This recognition isvery useful, as the SVD can be used to find a lower rank approximation fora matrix. As G is a rank-1 matrix, we would like to find the rank-1 matrixclosest to G\u00CB\u0086. We can see that the chi-square in Eq. (8.24) is proportionalto the square of the Frobenius norm of the matrix G\u00CB\u0086 \u0000G.64 The rank-mmatrix that minimizes the Frobenius norm between it and a rank-n matrixwith n > m can be found by decomposing the rank-n matrix with SVD andthen replacing its lowest n \u0000 m singular values by zeros. Reducing G\u00CB\u0086 toa rank-1 matrix via the SVD is equivalent to minimizing the chi-square inEq. (8.24). Since for G\u00CB\u0086 its eigen-decomposition and SVD are the same, itssingular values are its eigenvalues, so the eigenvector that will be our bestestimate of the gains will be the eigenvector with the largest correspondingeigenvalue.Once we have an estimate for the gains, we can refine our estimate ofthe true visibilities by minimizing Eq. (8.23) with respect to Vi\u0000j , which,from di\u00E2\u0086\u00B5erentiating with respect to the visibility Vb with baselines b, givesV \u00E2\u0087\u00A4b = Pi gig\u00E2\u0087\u00A4i+bV measi,i+bPi |gi|2|gi+b|2 . (8.25)The process of solving for the gains and subsequently the true visibilities asdescribed above can be done iteratively until the desired level of convergenceis reached.A shortcoming of the eigenvector method as formulated above is that wehad to assume that the covariance matrix C(ij),(kl) in Eq. (8.23) was pro-portional to the identity matrix and thereby have weighted all correlationsequally, including autocorrelations. In this case, we would need to solvefor the system temperature of each feed, in addition to the other parame-ters in the model. This method is then not ideal for a redundant baselinecalibration of the gain amplitudes, since we would have to specify the diag-onal elements of the matrix G\u00CB\u0086. However, if we are only concerned with thephases of the gains, we can make the substitution G\u00CB\u0086ij ! G\u00CB\u0086ij/|G\u00CB\u0086ij | beforepreforming the SVD, so that the matrix G\u00CB\u0086 contains only complex elementswith unity norms. The diagonal elements of G\u00CB\u0086, which estimate the auto-correlation, are then all scaled to a value of one in all cases. Therefore, we64The Frobenius norm of a matrix is the square root of the sum of the squares of eachelement in the matrix.1478.5. Phase Calibrationcan calibrate the phases of the gains using the eigenvector method withouthaving to solve for the system temperature of each feed.8.5.2 Phase DegeneraciesThe phase solution of the redundant baseline scheme is insensitive to thedegeneracies Vi\u0000j ! ei\u00E2\u0086\u00B5\u00C2\u00B7(ri\u0000rj)Vi\u0000j and gi ! ei\u00E2\u0086\u00B5\u00C2\u00B7rigi, (8.26)where ri is the position of feed i and \u00E2\u0086\u00B5 is an arbitrary vector. This trans-formation corresponds to a rotation of the sky in either direction or equiv-alently tilting the entire array by a certain angle. Note that unlike withthe logarithm method, an initial arbitrary \u00E2\u0080\u0098gauge-fixing\u00E2\u0080\u0099 condition, such asthat added in Eq. (8.21), does not need to be employed with our iterativemethod for determining the phases.Additionally, adding a constant phase to the gains leaves the measuredvisibilites unchanged. However, since we are only interested in the prod-uct g\u00E2\u0087\u00A4i gj , which is invariant under this transformation, this degeneracy isinconsequential and may be fixed arbitrarily.8.5.3 Phase Calibration ResultsTo illustrate the performance of the eigenvector redundant baseline phasecalibration, we employ the same array configuration and simulated signalsas were used for the gain amplitude calibration, which were described inSection 8.4.4, and perform the phase calibrations over the same set of 100noise realizations.Since we are considering a one-dimensional array, the only phase degen-eracy of consequence is a tilt of the array in a plane containing the baselinevectors. As with the amplitudes, we fix this degeneracy by fitting to a priorof the phase of the visibility with the smallest baseline Vbmin .To start the iterative process, we assume that the true sky visibilities arepurely real. Although in practice it may be sensible to begin with a morerealistic set of phases, for an array of this size, since each iteration can becomputed quickly and the solution converges with a small number of itera-tion, the starting point is of little consequence. This can be seen in Fig. 8.7,which shows the estimated calibrated visibilities after each iteration of thealgorithm. For this calibration, all primary beams were taken to be identicaland given by the two-dimensional Gaussian described in Section 8.4.4 (inother words \u0000i = 0 for all beams).1488.5. Phase Calibration0 1 2 3 4 5 6 7 8 9iteration\u000030\u0000\u000020\u0000\u000010\u00000\u000010\u000020\u000030\u000040\u000050\u0000arg(V b)baseline1234567891011Figure 8.7: Phases of calibrated visibilities for the example calibration aftereach iteration of the phase calibration algorithm. Each curve represents thesolution for a visibility with a particular baseline, labelled in multiples ofthe smallest baseline length bmin. The simulated measured visibilities wereproduced with all primary beams identical.1498.5. Phase Calibration0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16dmax0\u00002\u00004\u00006\u00008\u000010\u0000h|bias|i f 0 5 10 15 20 25(FWHMmax\u0000FWHM0)/FWHM0 (%)Figure 8.8: Phase calibration as a function of maximum beam perturbationparameter \u0000max. Each point is the absolute value of the bias in the phaseaveraged over all of the feeds. The error bars represent the square root ofthe variance in the calibrated gain phases averaged over all feeds.Although the phase calibration does not account for variations amongthe beams, it is crucial to evaluate its performance when the beams dovary. The bias and variance of the phase calibration as a function of theamplitude of the beam perturbations can be seen in Fig. 8.8, where thephase degeneracy is fixed using a perfect prior on the phase of Vbmin . Aswith the gain amplitudes, the data points plotted are the absolute value ofthe bias averaged over all feeds and the error bar represent the square rootof the variance averaged over all feeds. With all beams identical, little biasis seen in the phase calibration, but when the di\u00E2\u0086\u00B5erences between beamsare increased the bias grows. From Fig. 8.8, we can see that to achieve aphase calibration accurate to a few degrees, the widths of the beams may notvary by more than a few percent (in the absence of any other complicatingfactors).The e\u00E2\u0086\u00B5ect of introducing an error on the prior of the phase of Vbmin (used1508.6. Conclusions0\u0000 0.5\u0000 1\u0000 1.5\u0000 2\u0000 2.5\u0000 3\u0000prior error on arg(Vbmin)0\u00002\u00004\u00006\u00008\u000010\u0000h|bias|i fFigure 8.9: Phase calibrations as a function of the error on the phase priorof the visibility Vbmin for the smallest baseline bmin. The black points arecalibrations for which the simulated visibilities have all primary beams iden-tical, while the blue and red points are for calibrations that have \u0000max = 0.04and 0.08, respectively, corresponding to a variability of the beam widths of(FWHMmax \u0000 FWHM0)/FWHM0 = 6.8% and 13.6%.to fix the phase degeneracy) can be seen in Fig. 8.9, which shows the e\u00E2\u0086\u00B5ectof increasing this error for the case of identical beams as well as when thewidths di\u00E2\u0086\u00B5er by as much as 6.8% and 13.6% from the unperturbed beam.With all beams identical, there is room for a prior error of around a degreeto achieve a phase calibration biased by only a few degrees or less, whilethere is little leeway for prior errors to achieve this accuracy when the beamwidths di\u00E2\u0086\u00B5er from one another by more than a few percent.8.6 ConclusionsWith many new interferometric radio telescopes being built currently orplanned for the near future, containing a large number of feeds and many1518.6. Conclusionsredundant baselines, redundant baseline calibration may prove to be a valu-able tool for calibration. The strongest appeal of redundant baseline cali-bration is that the calibration is done nearly independently of our knowledgeof the sky, a feature which may be important when mapping regions withlittle prior knowledge of the sky at the frequencies under examination.A crucial assumption in the basic redundant baseline calibration algo-rithm is that each primary beam in the array is identical. However, theactual beams will be at least slightly di\u00E2\u0086\u00B5erent from one another. We haveshown that by decomposing the beams into a basis of functions that canrepresent the beams accurately with only a small number of functions, wecan adapt the amplitude redundant baseline calibration algorithm to ac-commodate beams that vary from feed to feed. As this introduces moreparameters into our model, a larger array with more visibilities is requiredfor this extension compared to the basic algorithm.The extended algorithm for the amplitude calibration can yield betterresults than the basic algorithm when a small perturbation is added to thebeams that is well represented by a single additional function. However,the extended algorithm requires as input a reasonably accurate model ofthe beams. In addition, the extended algorithm requires an additional pieceof prior information (about either the sky or the gains) to fix an extradegeneracy that appears when the beam model is expanded to account fortwo di\u00E2\u0086\u00B5erent beam basis functions.The logarithm method provides a simple approach for solving for thecalibration model parameters. As with the basic algorithm, a drawbackof this method is that with the split into amplitude and phase equations,the cyclic nature of the phase equations leads to phase errors in a naiveimplementation of this calibration algorithm. However, as the amplitude andphase equations may be solved independently of one another, this methodmay still be used successfully for amplitude calibration.The eigenvector redundant baseline calibration algorithm uses an itera-tive scheme to solve for both the gains and true sky visibilities that treatsboth the gain amplitudes and phases simultaneously. However, due to a dif-ficulty in assigning di\u00E2\u0086\u00B5erent weights to measured visibilities, for our calibra-tion tasks this algorithm is not well suited for estimating the gain amplitudesas this would require solving for the system temperature of each feed in addi-tion to the other parameters in the model. On the other hand, by modifyingthis algorithm to use only complex unit vectors, accurate phase calibrationsmay be produced when all beams are nearly identical. By simulating vis-ibilities from an array with varying beam widths, we have estimated thedegree of variance between the widths that is allowed if a phase calibration1528.6. Conclusionsaccuracy of a few degrees is to be achieved.153Chapter 9ConclusionsIn this thesis, we have developed models for the behaviour and detectionof inflation, dark matter, and dark energy. This work only represents afraction of that needed to fully understand the nature of these mysterioussubstances/events, which will likely be a major focus of both theoretical andexperimental work in cosmology for many years to come.Precision measurements of the CMB will likely continue to be an essentialtool for constraining models of inflation well into the foreseeable future. Alarge part of this work will be refining polarization measurements that aresensitive to primordial gravitational waves.The examination of macroscopic models of inflation may provide a usefulframework in which to view inflation. Of particular interest is the realizationthat there exist inflationary elastic models that have an equation of statefar from w = \u00001. This work highlights the di\u00E2\u0086\u00B5erent superhorizon behaviourthat may occur in some models of inflation and demonstrates that the \u00E2\u0080\u0098sep-arate universe approach\u00E2\u0080\u0099 is a useful tool in understanding this behaviour.The potential benefit of developing new 21-cm radiation experiments tocosmology has been highlighted throughout this thesis. It may prove to beone of the most important tools for exploring the physics of the high-redshiftUniverse. Although predicting the pre-reionization 21-cm signal is complex,since it is influenced by both astrophysical sources as well as cosmologi-cal phenomena, measuring both its mean value and power spectrum mayprovide a rich set of data from which we may learn about early structureformation, the dark ages, and the properties of dark matter.In the later stages of the Universe\u00E2\u0080\u0099s evolution, 21-cm intensity mappingcan potentially measure the BAO scale during a wide range of redshifts inwhich the Universe was transitioning into a dark energy dominated state.Such experiments will complement BAO detections made at lower redshifts,which when combined with data from a variety of other experimental meth-ods, are expected to provide new strong constraints on the dark energyequation of state.Although the potential benefit of these 21-cm experiments to cosmologymay be immense, a number of challenges must be overcome before rele-154Chapter 9. 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II-Theatlas of contour maps,\u00E2\u0080\u009D A. & A.S., vol. 47, p. 1, 1982.169Appendix ASupplemental Details forElastic Solid Model ofInflationA.1 Equations of Motion for Scalar and TensorPerturbationsIn Section 4.4, we derived the equations of motion for the mode functions\u0000 and X for the scalar and tensor linear perturbations, respectively, fromthe action, which can then be used to find the equations of motion for Rand hTij . Alternatively, we can derive these equations of motion directlyfrom the Einstein equations and the properties of the elastic solid givenin Eq. (4.20). Although we can derive the equations of motion for u andUp in this manner, it is only through the action in which we can properlyidentify u and Up as the canonical variables for the scalar and tensor linearperturbations, respectively.As before, to derive the equations of motion for the scalar perturba-tions, it is convenient to work in the comoving gauge and will also move toFourier space. We can combine the Fourier space versions of Eqs. (4.31a)and (4.31c) to eliminate \u0000 and then combine the derivative of this equationwith Eqs. (4.9a), (4.9c), and (4.9d) in the comoving gauge to eliminate \u0000,E0 and E00, then use Eq. (4.31b) and its derivative to eliminate \u0000 and \u00000,which gives 00k + \u00EF\u00A3\u00BF\u00E2\u009C\u00932 + 3 \u00EF\u00A3\u00BFw \u0000 dPd\u00E2\u0087\u00A2 \u0000\u00E2\u0097\u0086H\u0000 \u00E2\u009C\u0093ln \u00EF\u00A3\u00BFdPd\u00E2\u0087\u00A2 \u0000\u00E2\u0097\u00860\u0000 0k + dPd\u00E2\u0087\u00A2 k2 k+H3(1 + w)dPd\u00E2\u0087\u00A2 \u00EF\u00A3\u00BF3HdPd\u00E2\u0087\u00A2 \u00E2\u009C\u00933w + w2 \u0000 2dPd\u00E2\u0087\u00A2 \u00E2\u0097\u0086\u0000 2w\u00E2\u009C\u0093dPd\u00E2\u0087\u00A2 \u00E2\u0097\u00860\u0000\u00E2\u0087\u00A7k+23w1 + wH\u00E2\u0087\u00A70k = 0. (A.1)170A.2. Multicomponent System with Energy-Momentum TransferThis result does not assume any properties of the substance occupying ourspacetime, other than being able to write its stress-energy tensor in the formin Eq. (4.8).We now specify the elastic properties of our substance by employingEq. (4.22), which can be written in terms of k by use of the Fourier versionof Eq. (4.35). Substituting this into Eq. (A.1) yieldsR00k + 2z0z R0k + \u00EF\u00A3\u00BFc2sk2 +m2e\u00E2\u0086\u00B5,S + z00z \u0000Rk = 0, (A.2)where z and m2e\u00E2\u0086\u00B5,S were defined in Eqs. (4.37) and (4.40), respectively, andhave used the fact that R = in the comoving gauge. By substituting u forR using Eq. (4.38), we can recover the equation of motion for u in Eq. (4.41).Obtaining the equation of motion for the tensor perturbations is verystraightforward, since there is only one Einstein equation for the tensorperturbations, given in Eq. (4.12). With the tensor part of the anisotropicstress for the elastic solid in Eq. (4.24), the equation of motion for the tensorperturbations is(hTk )i00j + 2H(hTk )i0j + (k2 + 4c2v\u0000)(hTk )ij = 0, (A.3)where we have switched to Fourier space.A.2 Multicomponent System withEnergy-Momentum TransferIn this appendix, we review the equations governing the energy-momentumtransfer between multiple \u00E2\u0080\u0098fluid-like\u00E2\u0080\u0099 substances.65 This discussion will fol-low the work of Ref. [65]. We will begin by stating the general equationsfor energy-momentum transfer between any number of \u00E2\u0080\u0098fluid-like\u00E2\u0080\u0099 substancesand then specialize to the case of an elastic solid decaying into a perfect fluid.As in Section 4.8, we will use the coordinate time instead of conformal timein this section.The energy-momentum transfer 4-vector Q\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) that appears in Eq. (4.114)must vanish when summed over all substances \u00E2\u0086\u00B5,X\u00E2\u0086\u00B5 Q\u00E2\u008C\u00AB(\u00E2\u0086\u00B5) = 0, (A.4)65By \u00E2\u0080\u0098fluid-like\u00E2\u0080\u0099, we are not referring to a perfect fluid, but instead any substance whosestress-energy tensor can be parameterized by Eq. (4.8), which includes an elastic solid.171A.2. Multicomponent System with Energy-Momentum Transferfor the total stress-energy tensor to be covariantly conserved. The conser-vation equation for the background energy density \u00E2\u0087\u00A2\u00E2\u0086\u00B5 of substance \u00E2\u0086\u00B5 is\u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5 = \u00003H(\u00E2\u0087\u00A2\u00E2\u0086\u00B5 + P\u00E2\u0086\u00B5) +Q\u00E2\u0086\u00B5, (A.5)where Q\u00E2\u0086\u00B5 = Q0(\u00E2\u0086\u00B5). Importantly, from Eq. (A.4), the conservation equationfor the total energy density given in Eq. (4.4) still holds.If all substances have vanishing intrinsic nonadiabatic pressure (\u0000P\u00E2\u0086\u00B5 =(dP\u00E2\u0086\u00B5/d\u00E2\u0087\u00A2\u00E2\u0086\u00B5)\u0000\u00E2\u0087\u00A2\u00E2\u0086\u00B5) then the equations of motion for \u00E2\u0087\u00A3\u00E2\u0086\u00B5 and R\u00E2\u0086\u00B5, defined inEqs. (4.115) and (4.117), are given by\u00E2\u0087\u00A3\u00CB\u0099\u00E2\u0086\u00B5 = \u0000 H\u00CB\u0099\u00E2\u0087\u00A2\u00E2\u0086\u00B5 (\u0000Qintr,\u00E2\u0086\u00B5 + \u0000Qrel,\u00E2\u0086\u00B5)\u0000 H\u00CB\u0099H (R\u0000 \u00E2\u0087\u00A3) + k23a2H \u00E2\u009C\u00931\u0000 Q\u00E2\u0086\u00B5\u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5 \u00E2\u0097\u0086R\u00E2\u0086\u00B5 (A.6)andR\u00CB\u0099\u00E2\u0086\u00B5 = H\u00CB\u0099H (R\u00E2\u0086\u00B5 \u0000R)\u0000 \u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5\u00E2\u0087\u00A2\u00E2\u0086\u00B5 + P\u00E2\u0086\u00B5 dP\u00E2\u0086\u00B5d\u00E2\u0087\u00A2\u00E2\u0086\u00B5 (R\u00E2\u0086\u00B5 \u0000 \u00E2\u0087\u00A3\u00E2\u0086\u00B5)\u0000H\u00E2\u0087\u00A2\u00E2\u0086\u00B5 + P\u00E2\u0086\u00B5 \u00E2\u009C\u009323P\u00E2\u0086\u00B5\u00E2\u0087\u00A7\u00E2\u0086\u00B5 + frel,\u00E2\u0086\u00B5\u00E2\u0097\u0086 , (A.7)where we have written the equations in Fourier space, but neglected towrite the subscript k to minimize the number of subscripts written on eachvariable. In the above equations, \u0000Qintr,\u00E2\u0086\u00B5 and \u0000Qrel,\u00E2\u0086\u00B5 are the intrinsic andrelative nonadiabatic energy transfer perturbations, respectively, and frel,\u00E2\u0086\u00B5is the relative momentum transfer, which are defined as\u0000Qintr,\u00E2\u0086\u00B5 \u00E2\u008C\u0098 \u0000Q\u00E2\u0086\u00B5 \u0000 Q\u00CB\u0099\u00E2\u0086\u00B5\u00E2\u0087\u00A2\u00CB\u0099\u00E2\u0086\u00B5 \u0000\u00E2\u0087\u00A2\u00E2\u0086\u00B5, (A.8a)\u0000Qrel,\u00E2\u0086\u00B5 \u00E2\u008C\u0098 \u0000 Q\u00E2\u0086\u00B56H\u00E2\u0087\u00A2 X\u0000 \u00E2\u0087\u00A2\u00CB\u0099\u0000S\u00E2\u0086\u00B5\u0000 , (A.8b)frel,\u00E2\u0086\u00B5 \u00E2\u008C\u0098 aQ\u00E2\u0086\u00B5X\u0000 \u00E2\u0087\u00A2\u0000 + P\u0000\u00E2\u0087\u00A2 + P v\u00E2\u0086\u00B5\u0000 , (A.8c)where \u0000Q\u00E2\u0086\u00B5 is the perturbation to Q\u00E2\u0086\u00B5.With Q\u00E2\u008C\u00ABe for the decay of the elastic solid specified in Eq. (4.121), wehave Qe = \u0000Qf = \u0000\u0000\u00E2\u0087\u00A2e(1 + we). Since Qe is a function of \u00E2\u0087\u00A2e, \u0000Qintr,e = 0.Using this fact and \u0000Qf = \u0000\u0000Qe, we find that \u0000Qintr,f = Q\u00CB\u0099eSef/3H.172A.3. Scalar AmplitudeA.3 Scalar AmplitudeIn this appendix we give detailed expressions for the scalar amplitude evalu-ated at the pivot scale kp, as given in Eq. (4.126). We first examine the casewhere the sound speeds and equation of state are varying slowly in time and\u00E2\u009C\u008F0 = cs0 = 0. From Eqs. (4.103) and (4.125), \u0000 is given by\u0000 = 0.11\u00E2\u0087\u00A5 105.23(ns\u00000.96)\u00002(\u0000S)(3\u0000 18\u00E2\u009C\u008F1 \u0000 8c2s1\u00E2\u009C\u008F1 \u0000 2\u00E2\u008C\u00A7s + 2\u00E2\u008C\u00A7\u00E2\u009C\u008F)\u00E2\u0087\u00A5 g 1\u0000ns3s0 (T0/kp)1\u0000ns , (A.9)where we have assumed that the reheating process is very rapid. Choosingthe pivot scale as kp = 0.002Mpc\u00001, and using the CMB temperature T0 =2.725K and gs0 = 43/11, \u0000 becomes\u0000 = 1.53\u00E2\u0087\u00A5 10\u000028.5(ns\u00000.96)\u00002(\u0000S)(3\u0000 18\u00E2\u009C\u008F1 \u0000 8c2s1\u00E2\u009C\u008F1 \u0000 2\u00E2\u008C\u00A7s + 2\u00E2\u008C\u00A7\u00E2\u009C\u008F). (A.10)In the case of constant sound speeds and equation of state, \u0000 is foundto be\u0000 = 5.82\u00E2\u0087\u00A5 10\u00005\u000022.9(ns\u00001)\u00002(\u00E2\u008C\u00AB)(21 + ns(ns \u0000 10))2|1 + 3w|7\u0000ns . (A.11)173"@en .
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