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The cosmic neutrino background and effects of Rayleigh scattering on the CMB and cosmic structure Alipour Khayer, Elham 2015

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The Cosmic Neutrino Background and Effects of RayleighScattering on the CMB and Cosmic StructurebyElham Alipour KhayerB.Sc. Sharif University of Technology, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)October 2015c© Elham Alipour Khayer, 2015AbstractThe Cosmic Microwave Background (CMB) radiation, photons free-streamingfrom their last scattering surface at z' 1090, is currently our main source of infor-mation about the origin and history of the Universe. The vast recent advancementin technology has led to new possibilities for gathering data especially detectingthe CMB with high accuracy. The goal of the two projects studied in this thesis isto improve the cosmological perturbation theory to better test cosmology with theupcoming data.In chapter 4 we explore the effect of Rayleigh scattering on the CMB and cos-mic structure. During and after recombination, in addition to Thomson scatteringwith free electrons, photons also coupled to neutral hydrogen and helium atomsthrough Rayleigh scattering. The frequency-dependence of the Rayleigh cross sec-tion breaks the thermal nature of CMB temperature and polarization anisotropiesand effectively doubles the number of variables needed to describe CMB intensityand polarization statistics, while the additional atomic coupling changes the matterdistribution and the lensing of the CMB. We introduce a new method to capturethe effects of Rayleigh scattering on cosmological power spectra. We show theRayleigh signal, especially the cross-spectra between the thermal (Rayleigh) E-polarization and Rayleigh (thermal) intensity signal, may be detectable with futureCMB missions even in the presence of foregrounds, and how this new informationmight help to better constrain the cosmological parameters.In chapter 5 we study the Cosmic Neutrino Background (CNB). In addition tothe CMB, the standard cosmological model also predicts that neutrinos were de-iicoupled from the rest of the cosmic plasma when the age of the Universe was lessthan one second, far earlier than the photons. We study the anisotropy of the CNBand for the first time present the full CNB anisotropy power spectrum at large andsmall scales both for a massless and massive neutrinos. We also show that howpresence of nonstandard neutrino self-interactions compatible with current cosmo-logical data alters the CNB power spectrum.iiiPrefaceThis thesis is partly based on one published paper and one manuscript that isclose to submission.A version of chapter 4 has been published. Aside from calculating the Rayleighscattering cross section in the necessary limit (section 4.2) which was done by KrisSigurdson and Christopher M. Hirata, I performed all the rest of the calculations,made the plots and drafted the manuscript. Professor Sigurdson provided guidanceand comments on the manuscript. The Boltzmann code (CAMB) used in this chap-ter was provided by Dr. Antony Lewis, however I heavily modified this code forthis project. Most of this chapter is contained in the following paper: E. Alipour,K. Sigurdson, Ch. M. Hirata, Physical Review D, 91, 083520 (2015)I led the study of the cosmic neutrino background presented in chapter 5. Allthe analytical and numerical calculations in this chapter were conducted by me butheavily influenced by significant consultation with my supervisor.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The standard cosmological model . . . . . . . . . . . . . . . . . 21.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Cosmic Microwave Background . . . . . . . . . . . . . . . . . . 51.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 92 The linear cosmological perturbation theory . . . . . . . . . . . . . 112.1 Background equations . . . . . . . . . . . . . . . . . . . . . . . 122.2 Metric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Linearized Einstein equations . . . . . . . . . . . . . . . . . . . . 152.4 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 17v3 The Cosmic Microwave Background anisotropy . . . . . . . . . . . 193.1 Temperature anisotropies . . . . . . . . . . . . . . . . . . . . . . 193.1.1 Temperature power spectrum . . . . . . . . . . . . . . . . 213.2 Polarization anisotropies . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Polarization power spectra . . . . . . . . . . . . . . . . . 233.3 Foregrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Effects of Rayleigh Scattering on the CMB and Cosmic Structure . 264.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Rayleigh scattering cross section . . . . . . . . . . . . . . . . . . 284.3 Cosmological equations and proposed method . . . . . . . . . . . 314.4 Matter power spectrum . . . . . . . . . . . . . . . . . . . . . . . 374.5 Photon power spectra . . . . . . . . . . . . . . . . . . . . . . . . 394.5.1 Convergence of the numerical code . . . . . . . . . . . . 504.6 Rayleigh Distorted Statistics . . . . . . . . . . . . . . . . . . . . 504.7 Detectibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.7.1 Signal to noise ratio of Rayleigh signal . . . . . . . . . . 584.7.2 Constraints on Cosmological Parameters . . . . . . . . . 614.7.3 Improvements to signal to noise ratio . . . . . . . . . . . 674.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 Cosmic Neutrino Background Anisotropy Spectrum . . . . . . . . . 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Evolution equations for neutrino anisotropies . . . . . . . . . . . 805.3 Neutrino power spectrum . . . . . . . . . . . . . . . . . . . . . . 875.4 Anisotropy spectrum for massive neutrinos . . . . . . . . . . . . 945.5 Neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 975.6 Averaging over momenta . . . . . . . . . . . . . . . . . . . . . . 1005.7 Extra neutrino interactions via an alternative Fermi constant . . . 1045.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111viA phase-space integration for driving neutrino evolution equations . . 117viiList of TablesTable 4.1 The cross-section coefficients b2k+4Ry2k+4 for H and He in theRydberg-based units that we adopt for this work. . . . . . . . . 30Table 4.2 The percentage constraints on cosmological parameters (100σpi/pi)for a hypothetical cosmic-variance limited case with and with-out accounting for the Rayleigh signal. Note that although theRayleigh signal is detectable with the PRISM-like experiment,this signal doesn’t add much constraining power for cosmolog-ical parameters as its accumulative signal-to-noise ratio is mod-est. The constraints on parameters with the PRISM-like ex-periment are nearly identical to the third (Primary CV limited)column in this table. . . . . . . . . . . . . . . . . . . . . . . . 66Table 5.1 The weak interactions involving electron neutrinos and theircorresponding squared amplitudes. . . . . . . . . . . . . . . . 82Table 5.2 The weak interactions involving µ and τ neutrinos and theircorresponding squared amplitudes. . . . . . . . . . . . . . . . 82Table A.1 Value of coefficients gi j for different values of l. . . . . . . . . 123viiiList of FiguresFigure 1.1 The exact solution and Saha approximation for the free elec-tron fraction as a function of redshift [1]. . . . . . . . . . . . 6Figure 1.2 The Cosmic microwave background temperature perturbationmap as seen by Planck [2]. . . . . . . . . . . . . . . . . . . . 6Figure 1.3 The CMB temperature anisotropy power spectrum as a func-tion of angular scale l observed by Planck [2]. The red curveon the upper panel shows the best fit ΛCDM theoretical spec-trum and the residuals with respect to this model are shown inthe lower panel. . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 3.1 The theoretical CMB power spectra. The blue (solid), red(dashed), green (dotted) and purple (dot dashed) are for thetemperature, E-polarization, B-polarization and Temperature-E polarization cross-correlation spectra respectively. . . . . . 24Figure 4.1 The Comoving opacity as a function of comoving time. Theblack (solid) line is for Thomson scattering while the blue(large dashed), red (small dashed), green (dot dashed) and brown(dotted) lines are for Rayleigh scattering at frequencies 857,545, 353, and 217 GHz respectively. . . . . . . . . . . . . . . 33Figure 4.2 The matter two-point correlation function, r2ξ (~r), as a func-tion of the distance between two over-densities for our fiducialcosmological parameters. . . . . . . . . . . . . . . . . . . . . 38ixFigure 4.3 The percentage change in the matter correlation function dueto Rayleigh scattering for our fiducial cosmological parameters. 38Figure 4.4 The redshift evaluation of a narrow Gaussian-shaped adiabaticdensity fluctuation in real space. The blue (solid) and red(dashed) lines are respectively the baryon and photon densitywaves. Panel(a) shows a snapshot at very early times whenbaryons and photons are tightly coupled and their density wavestravel together. In panels (b), redshift z = 1050, photons beginto decouple from baryons and the baryon density wave slowsdown compare to photon density wave due to the drop in thesound speed. . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.5 Same as 4.4 but for later redshifts. Panel (c) shows the densitywaves at redshift z= 500 where photons and baryons are com-pletely decoupled. The late time picture is presented in panel(d). The photons free stream to us and baryons cluster aroundthe initial over-density and in a shell at about 150 Mpc radius. 41Figure 4.6 The percentage change in physical baryon density fluctuationsin real space due to Rayleigh scattering at different redshifts.The blue (solid), red (dashed), green (dot dashed) and brown(dotted) lines correspond to redshifts 0, 100, 500 and 1050 re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 4.7 The total visibility function as a function of conformal timefor several frequencies. The black (solid), red (dotted), blue(dot dashed) and green (dashed) lines are the total visibilityfunction for frequencies 0, 545, 700 and 857 GHz respectively.The total photon visibility function shifts toward later timeswith increasing frequency. . . . . . . . . . . . . . . . . . . . 43Figure 4.8 The cross correlation temperature power spectrum CT T (2r,2r′)lof the Θ(2r)Il and Θ(2r′)Il intensity coefficients for the ν0, ν4 andν6 spectral distortions. . . . . . . . . . . . . . . . . . . . . . 46Figure 4.9 The cross correlation temperature power spectrum CEE(2r,2r′)lof the Θ(2r)El and Θ(2r′)El E-polarization coefficients for the ν0,ν4 and ν6 spectral distortions. . . . . . . . . . . . . . . . . . 47xFigure 4.10 Shown are a fractional measure, δCXYl /√CXXl CYYl , of the changeδCXYl in (lensed) scalar CMB anisotropy spectra due to Rayleighscattering. The blue (solid), red (dashed), green (dotted) andpurple (dot dashed) are for the temperature, E-polarization,B-polarization from lensing and Temperature-E polarizationcross-correlation spectra respectively. The upper and lowerpanels are for 217 and 353 GHz frequency channels respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.11 Same as 4.11 but for different frequency channels. The upperand lower panels are for 545 and 857 GHz frequency channelsrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 4.12 Shown are the difference in the fractional change, δCT Tl /CT Tlin (lensed) scalar temperature CMB anisotropy spectra due toRayleigh scattering for 545 GHZ frequency channel when weinclude up to νn in Eqs. 4.22 and 4.23 compared to when weinclude up to νn−2 terms. The blue (solid), red (dashed), green(dotted) and purple (dot dashed) are for when n = 4, n = 6,n= 8 and n= 10 respectively and they justify the convergenceof the expansions. . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.13 The non-zero power spectra in the Rayleigh distorted CMB co-variance matrix as a function of l. The upper and lower panelsshow the first eigenvalues of intensity and polarization spec-tra respectively which are almost proportional to the primarythermal signal. . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.14 Same as 4.13 but for the second eigenvalues of intensity (upperpanel) and polarization (lower panel). Note that the secondeigenvalues are purely Rayleigh signals which are uncorrelatedto the first eigenvalues. . . . . . . . . . . . . . . . . . . . . . 55Figure 4.15 Same as 4.13 but for temperature-polarization cross-spectra.The upper and lower panels present 〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉as a function of l respectively. . . . . . . . . . . . . . . . . . 56xiFigure 4.16 Same as 4.13 but for temperature-polarization cross-spectra.The upper and lower panels show 〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉as a function of l respectively. . . . . . . . . . . . . . . . . . 57Figure 4.17 The first eigenvalues of intensity 〈α I1lmα I1lm′〉 and polarization〈αE1lmαE1lm′〉 spectra (blue, solid) and their signal-to-noise ra-tio at each l (red, dashed) as well as the accumulative signal-to-noise ratio for the PRISM-like experiment. Note that thesignal-to-noise ratio for the temperature-polarization cross powerspectrum can be negative at some l values due to anti-correlationof the temperature and polarization. However the accumulativesignal-to-noise, added in quadrature, is always positive. . . . . 62Figure 4.18 Same as 4.17 but for the second eigenvalues of intensity 〈α I2lmα I2lm′〉and polarization 〈αE2lmαE2lm′〉 spectra. . . . . . . . . . . . . . . 63Figure 4.19 Same as 4.17 but for the temperature-polarization cross-spectra〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉. . . . . . . . . . . . . . . . . . . 64Figure 4.20 Same as 4.17 but for the temperature-polarization cross-spectra〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉. . . . . . . . . . . . . . . . . . . 65Figure 4.21 Accumulative signal-to-noise ratios for the first eigenvaluesof intensity 〈α I1lmα I1lm′〉 and polarization 〈αE1lmαE1lm′〉 spectra.The blue (solid), red (dashed), green (dotted) and black (dotdashed) lines are the signal-to-noise ratios respectively for aPRISM-like experiment, for Case I: improved foregrounds re-moval method, for Case II : improved detector noise, and forCase III which combines Case I and II. . . . . . . . . . . . . 69Figure 4.22 Same as 4.21 but for the second eigenvalues of intensity 〈α I2lmα I2lm′〉and polarization 〈αE2lmαE2lm′〉 spectra. . . . . . . . . . . . . . 70Figure 4.23 Same as 4.21 but for the temperature-polarization cross-spectra〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉. . . . . . . . . . . . . . . . . . . 71Figure 4.24 Same as 4.21 but for the temperature-polarization cross-spectra〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉. . . . . . . . . . . . . . . . . . . 72xiiFigure 4.25 The biases and constraints on cosmological parameters thatcould potentially occur if one ignores the Rayleigh signal. Theblue contours are the one-sigma and two-sigma constraints onparameters using only the primary signal centred at the fidu-cial value of the parameters. The red, green, orange and blackdots represent the bias introduced by ignoring the Rayleighsignal respectively in PRISM-like experiment, Case I (improv-ing foreground removal), Case II (reducing detector noise) andCase III (combination of both). . . . . . . . . . . . . . . . . 73Figure 4.26 The two-sigma constraints on cosmological parameters by con-sidering both the primary and Rayleigh signal. The small-est and darkest contour represents the cosmic-variance lim-ited case. The lighter contours show the Case III, Case II,Case I and the PRISM-like experiment respectively as we gofrom smallest-darkest to largest-lightest contours. Note thatthe largest contours essentially delineate the conventional (pri-mary only) cosmic variance limit, and smaller contours rep-resent an improvement in parameter constraints beyond thislimit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 5.1 Ratio of neutrino temperature to photon temperature as a func-tion of conformal time. The final asymptotic value of this ratiois 0.7161 which is a bit higher that the value in instantaneousneutrino decoupling scenario (( 411)1/3) and shows that neutri-nos receive some energy from the pair annihilation. . . . . . 88Figure 5.2 The visibility function as a function of conformal time. Thepeak of the visibility function occurs at τ = 5.75× 10−5Mpcor T ' 1.48 MeV . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 5.3 The anisotropy power spectrum for massless electron (blue)and µ or τ (red) neutrinos. Similar to the CMB power spec-trum, the acoustic oscillations and Silk damping are visible inthis plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94xiiiFigure 5.4 The comoving distance travelled by a massive neutrino fromthe last scattering surface to us as a function of mν for onevalue of the neutrino momentum (3Tν ). . . . . . . . . . . . . 96Figure 5.5 The anisotropy power spectrum at low l values for three dif-ferent neutrino masses with momentum p = 0.0005 eV: m1 =0 eV (blue, solid), m2 = 0.00894 eV (red, dashed) and m3 =0.05894 eV (green, dotted). Since the ISW effect is larger formassive neutrinos, there is a boost at low l angular power spec-trum for these neutrinos. . . . . . . . . . . . . . . . . . . . . 98Figure 5.6 The anisotropy power spectrum for massive neutrinos with mo-mentum p = 0.0005 eV before they are detected. The blue(solid), red (dashed) and green (dotted) curves are the powerspectra for m1 = 0 eV, m2 = 0.00894 eV and m3 = 0.05894 eVmass eigenstates respectively. . . . . . . . . . . . . . . . . . 99Figure 5.7 The anisotropy power spectrum for an electron neutrino (blue,solid) and a muon neutrino (red, dashed) at the detector. Theneutrino momentum is p = 0.0005 eV . . . . . . . . . . . . . 101Figure 5.8 The total anisotropy power spectra for three mass eigenstatesafter averaging over all the momenta. The blue (solid), red(dashed) and green (dotted) curves are the power spectra form1 = 0 eV, m2 = 0.00894 eV and m3 = 0.05894 eV masseigenstates respectively. . . . . . . . . . . . . . . . . . . . . 102Figure 5.9 The averaged anisotropy power spectrum over all momentaat low l values for three mass eigenstates: m1 = 0 eV (blue,solid), m2 = 0.00894 eV (red, dashed) and m3 = 0.05894 eV(green, dotted). Averaging over all momenta leads to smalleramplitude for the power spectra at smaller scales. . . . . . . 103Figure 5.10 The small-scale part of the anisotropy power spectrum for anelectron neutrino (blue, solid) and a muon neutrino (red, dashed)at the detector. Note that since the amplitude of the powerspectrum for massless neutrinos is significantly larger than forthe massive ones, they contribute the most. . . . . . . . . . . 105xivFigure 5.11 The large-scale part of the anisotropy power spectrum for anelectron neutrino (blue, solid) and a muon neutrino (red, dashed)at the detector. Note that since the amplitude of the powerspectrum for massive neutrinos is significantly larger than forthe massless ones, they contribute the most. . . . . . . . . . . 106Figure 5.12 The anisotropy power spectrum for massive neutrinos with mo-mentum p = 0.0005 eV with the self-interaction strength ofGeff ' 10−5 MeV−2. The blue (solid), red (dashed) and green(dotted) curves are the power spectra for m1 = 0 eV, m2 =0.00894 eV and m3 = 0.05894 eV mass eigenstates respectively.107xvGlossaryACT Atacama Cosmology TelescopeBAO Baryon Acoustic OscillationCIB Cosmic Infrared BackgroundCMB Cosmic Microwave BackgroundCNB Cosmic Neutrino BackgroundCOBE Cosmic Background ExplorerFRW Friedmann-Robertson-WalkerISW Integrated Sachs-WolfPIXIE The Primordial Inflation ExplorerPRISM Polarized Radiation Imaging and Spectroscopy MissionSPT South Pole TelescopeWMAP Wilkinson Microwave Anisotropy ProbexviAcknowledgmentsFirstly, I would like to thank my research supervisor Prof. Kris Sigurdsonfor the continuous support, guidance and motivation throughout my Phd, and forhelping me in all the time of research and writing of this thesis.I would also like to thank my supervisory committee, Prof. David Morrissey,Prof. Douglas Scott, Prof. Colin Gay and Prof. Mark Van Raamsdonk for theirencouragement, constructive feedback, and hard questions.I acknowledge the financial support from the University of British Columbia(UBC) through University Four Year Fellowship, PhD Tuition Fee Award and In-ternational Partial Tuition Scholarship.I will always be grateful to my beloved parents and my sisters, who have alwayssupported, encouraged and believed in me. To my friends who always inspired andmotivated me throughout my research.Most importantly, I would like to thank Hamed, my loving, supportive, en-couraging, and patient husband without whom this accomplishment would havenot been possible. Thank you.xviiChapter 1IntroductionWe are now entering a golden era of cosmology. Recent advances in tech-nology are opening up new possibilities for gathering data. Complementing that,there has been a dramatic theoretical development in the last couple of decades. Inparticular, comprehensive models, such as inflation, have emerged which describehow the Universe evolved in its earliest stages and how galaxies and other struc-tures began to form. Inflation predicts that the Universe exponentially expandedjust a short time after the Big Bang, and a scale-invariant spectrum of density fluc-tuations was generated. These primordial perturbations evolved and grew throughtime to form the distribution of large-scale structure that we see today. This theoryhas numerous characteristic signatures that allow it to be tested by observationaldata. One phenomenon which allows us to probe these signatures is the spectrumof temperature and polarization anisotropies in the Cosmic Microwave Background(CMB) radiation. The CMB is a form of polarized electromagnetic radiation fillingthe entire Universe at a temperature 2.725K that is almost, but not quite, uniform.The goal of my work is to improve cosmological perturbation theory in orderto be able to better test cosmology with the upcoming data. In the past decadethe Wilkinson Microwave Anisotropy Probe (WMAP) has provided us with precisemeasurements of CMB anisotropies [3] and, complemented by next-generationground based experiments such as South Pole Telescope (SPT) [4] and AtacamaCosmology Telescope (ACT) [5], the Planck satellite has now characterized themicrowave background anisotropies even to a higher precision [2]. Future mea-1surements may even probe CMB anisotropies with more frequencies and higherprecision (e.g., Polarized Radiation Imaging and Spectroscopy Mission (PRISM)[6] or The Primordial Inflation Explorer (PIXIE) [7]). This dramatic improvementin the observations challenges theorists to improve the level of their calculations.The focus of the projects in this thesis is to pursue this direction.We begin the thesis by briefly reviewing some elements of the standard cos-mology in section 1.1. In sections 1.2 and 1.3 we present an introduction to therecombination and the cosmic microwave background respectively. Finally wegive a brief summary of each thesis chapter. Note that throughout this thesis weuse units in whichh¯ = c = kB = 1. (1.1)1.1 The standard cosmological modelThe current well accepted model of cosmology is called the standard model ofBig Bang cosmology also known as Lambda Cold Dark Matter model (ΛCDM).This model describes an expanding Universe that on large enough scales, based oncosmological principle, is both homogeneous and isotropic. A homogeneous spaceis one that is translation invariant, or the same at every point. An isotropic spaceis one which is rotationally invariant, or the same in every direction. According tothis model, supported by observations ([8], [9], [10] and [11]), only a small fractionof the Universe consists of radiation and baryonic matter that we are familiar withfrom our everyday experience. The dominant components of the Universe todayare cold dark matter and dark energy.Cold dark matter is a form of matter that while gravitationally acts the sameway as normal visible matter, its particles interact very weakly with electromag-netic radiation (they are dark) and are non-relativistic (they are cold). Dark energyis the most accepted hypothesis to explain the observed acceleration of the expan-sion of the Universe. The letter Λ in ΛCDM model stands for cosmological con-stant, the constant energy density filling space homogeneously, which is a sampleproposed and often accepted form of dark energy.This standard observationally consistent cosmological model still however hassome unsolved problems. The Universe appears to be statistically homogenous and2isotropic at large scales. The observed CMB has the same temperature across theregions on the sky separated by large enough distances that according to this model,have never been in casual contact [12]. This homogeneity problem is often calledthe horizon problem. Another problem is the flatness problem. The current energydensity of the Universe is observed to be very close to the critical value required tomake the Universe flat [13]. Since the total density departs rapidly from the criticalvalue with time, the energy density would have to be fine-tuned to be extremelyclose to the critical density at the Big Bang.Inflation was initially introduced to solve the problems mentioned above. Ac-cording to the theory of inflation, the early Universe expanded exponentially fastfor a short time period after the Big Bang. Before this period of inflation, the entireUniverse could have been in causal contact and equilibrated to a common tem-perature, giving the same initial condition everywhere. Widely separated regionstoday were actually very close together in the early Universe, solving the horizonproblem. Additionally, an exponential expansion drives the Universe to flatnessresolving the flatness problem [14]. More importantly, inflation provides a mech-anism for generating the observed density fluctuations observed today. Since it isvery difficult to test a theory based on energy scales far beyond the reach of theaccelerators, we can not be certain that inflation is the mechanism explaining theinitial condition of the Universe. However, it is by far the most plausible explana-tion.The evolution of the Universe before 10−10 seconds (1 TeV) is not well un-derstood due to very high energy scales and therefore the physics of this era isspeculative. However, we know that we need a mechanism such as inflation in thevery early universe (∼ 10−34 seconds) to solve the horizon and flatness problemsand also to generate the seed for primordial fluctuations. We also need baryogene-sis, a process that produced an asymmetry between baryons and antibaryons in thevery early universe, and also a mechanism to create dark matter.The history of the Universe from 10−10 seconds to today is better understoodsince it is based on tested physical theories like the standard model of particlephysics, general relativity and fluid dynamics. Here, we summarize the cosmichistory of the Universe after this time on the basis of Ref. [14].Below 100 GeV, electroweak symmetry breaking occurs, the Z and W± bosons3acquire mass and the weak interaction weakens as the temperature of the Universedrops. Around 100 MeV, the temperature of the Universe is cool enough for quarksto form hadrons. Around 1 MeV neutrinos decouple from the rest of the plasma.As the temperature drops below the electron mass, 0.511 MeV, electron-positronannihilation happens. Around 0.1 MeV, during Big Bang nucleosynthesis, protonsand neutrons begin to combine into atomic nuclei in the process of nuclear fusion.The energy density of matter dominates over that of radiation at T ' 1 eV. Asthe temperature falls below ∼ 0.1 eV, the neutral hydrogen atoms begin to formand photons decouple. Therefore the relic photons travelling freely from this pointto us can tell us about the condition of the Universe when it was roughly 380000years old.1.2 RecombinationHydrogen and helium atoms begin to form about 380000 years after the BigBang. Before this time they are ionized. As the Universe expands and cools down,electrons get captured by the ions, forming electrically neutral atoms. This processis known as recombination.The cosmic ionization history is generally described in terms of the free elec-tron fraction xe as a function of time:xe =nene+nH, (1.2)where ne is the number density of free electrons and nH that of atomic hydrogen.While the recombination process (p+H−↔H+γ) is fast compared to the expan-sion of the Universe, the ionization fraction obeys an equilibrium distribution. Theequilibrium situation is described by the Saha equation [1]:x2e1− xe =1ne+nH(meT2pi)3/2e−B/T , (1.3)where me is the mass of electron, T is the temperature of plasma and B = 13.6 eVis the binding energy.Due to high photon-baryon ratio (109), at T = B = 13.6 eV, any produced hy-drogen atoms will be instantaneously ionized. When the temperature drops below40.3 eV at redshift z = 1000, far below the binding energy, the ionization rate be-comes too slow to keep the equilibrium since there are not enough photons in theWien tail to keep ionizing the hydrogen atoms.The Saha equation is good for describing the initial departure from full ioniza-tion, but the equilibrium breaks down shortly after recombination starts. Thereforethe full Boltzmann equation must be solved to find the evolution of the free elec-tron fraction. The direct recombination to the ground state is not relevant sincephotons emitted from direct recombination can easily re-ionize the nearby neutralatoms. The only way for recombination to happen is via capture to one of the ex-cited states of hydrogen. Photons emitted by electrons going from first excited stateto the ground state can escape re-absorption by redshifting out of the line, but theprobability for this is very low. The other way for electrons to move from the firstexcited state to the ground state is through the 2s-ls two-photon transition. Photonsemitted by this transition can escape re-absorption. The exact solution and Sahaapproximation for the free electron fraction are plotted in Figure 1.1.[1]1.3 Cosmic Microwave BackgroundThe Cosmic Microwave Background, photons free-streaming from their lastscattering surface at redshift z ' 1100, is currently our main source of informa-tion about the origin and history of the Universe. The CMB has a thermal blackbody spectrum at temperature of 2.725 K. Historically, the CMB was accidentallyobserved first at 1964 by Penzias and Wilson who noticed excess noise in theirreceiver while working on an antenna to detect radio waves bounced off metallicballoons in the atmosphere [15]. In 1992, the Cosmic Background Explorer (COBE)[16] observed small fluctuations of order one in 105 in temperature of the CMB.Later, more advanced experiments such as WMAP [3] and Planck [2] have char-acterized the anisotropies with higher accuracy. In Figure 1.2 we show the CMBtemperature perturbation map seen by Planck.According to the inflationary cosmological model, the quantum fluctuations inthe inflation field provided seed for the density perturbations. Below a temperatureof ∼ 0.1 MeV, the early universe was made of a hot interacting plasma of photons,electron and protons. The photons and electrons were tightly coupled by Thomson5Figure 1.1: The exact solution and Saha approximation for the free electronfraction as a function of redshift [1].Figure 1.2: The Cosmic microwave background temperature perturbationmap as seen by Planck [2].6scattering and since the mean free path of photons (the mean distance each photoncould travel before encountering an electron) was short, the plasma was opaqueto the electromagnetic radiation. As the Universe expanded and the temperaturedropped, the electrons and protons combined to form hydrogen atoms. At thispoint, which is called recombination or the epoch of last scattering, the photonsstopped interacting with the electrically neutral hydrogen atoms and they travelledthrough the Universe without interacting with matter. Hence, the Universe be-came transparent. After photon decoupling, the Universe continued expanding andcooling and therefore the wavelengths of photons have redshifted to roughly onemillimetre and their temperature has cooled down to∼ 2.725 K. These photons arethe CMB photons that we observe today.Radiation pressure prevents photon anisotropies from growing due to gravi-tational instability (in contrast to matter inhomogeneities), therefore the photonperturbations remain small and their evolution falls almost entirely in the domainof linear perturbation theory. They are only influenced a little by the non-linearprocesses of galaxy formation and therefore we can compute them very precisely.This makes the CMB an excellent source of information about the early Universesince it basically gives a snapshot of the Universe when it was only 380000 yearsold.The CMB temperature map is close to a Gaussian random field and thereforeto the extent that it is Gaussian, the statistical properties of the temperature mapcan be described by its power spectrum. The CMB temperature power spectrumobserved by Planck is shown in Figure 1.3 [2].The temperature power spectrum has a series of peaks and troughs. This char-acteristic peak structure arises due to the physics of the photon-baryon plasma inthe early Universe. The pressure of photons tends to erase anisotropies whereasthe gravitational attraction of baryons makes them tend to collapse. The compe-tition between these two effects leads to the acoustic oscillations. The peaks andtroughs in the spectrum are a signature of this oscillations in the plasma. The peakscome from waves at an extrema of their oscillation, either maximally compressedor rarefied.The peaks contain interesting physical information. The angular scale of thefirst peak tells us about the curvature of the Universe [17], [18]. The first peak cor-70100020003000400050006000DTT`[µK2]30 500 1000 1500 2000 2500`-60-3003060DTT`2 10-600-3000300600Figure 1.3: The CMB temperature anisotropy power spectrum as a functionof angular scale l observed by Planck [2]. The red curve on the upperpanel shows the best fit ΛCDM theoretical spectrum and the residualswith respect to this model are shown in the lower panel.responds to the fundamental acoustic mode, the first mode that reached maximumat the time of recombination. The physical scale of this mode is well understood sothe angular scale that we observe on the sky depends on the angular diameter dis-tance to the last scattering surface which depends on the curvature of the Universe.The ratio of the even peaks to the odd peaks can be used to determine thebaryon density of the Universe [17], [18]. Baryons load down the photon-baryonoscillations which means the compressions in potential wells got enhanced overrarefactions. This means that the amplitudes of the odd peaks are enhanced overeven peaks.Another feature of the CMB temperature power spectrum is damping of theoscillations at smaller scales, which is called Silk damping or diffusion damping.Silk damping is caused by the finite thickness of the last scattering surface; sincephotons move a mean distance during the decoupling, any perturbation on scales8smaller than this distance will be washed out.In addition to the intensity of the CMB, its polarization also gives to furtherinformation. There are two types of polarization patterns: the E-mode (curl-freecomponent of the polarizations); and B-mode (curl component). The E-mode, sim-ilar to the temperature fluctuations, reflects the recombination history. The detec-tion of the B-mode polarization, which is not sourced by standard scalar type per-turbations, can tell us about inflation and physics beyond the standard model ofparticle physics.Studying the CMB temperature and polarization anisotropy power spectra pro-vides us with much information about many important cosmological parameterssuch as the curvature of the Universe, the dark matter and baryon densities, theamplitude and slope of the primordial power spectrum, and the number of rela-tivistic species present at photon decoupling.1.4 Outline of the thesisThis thesis is organized as follows:In chapter 2 we briefly review the linear cosmological perturbation theorywhich is a very powerful tool to describe how primordially-generated perturbationsin matter and radiation grow due to gravitational instability to form the structuresthat we see in the Universe today. First we present the background equations gov-erning the evolution of a homogenous and isotopic universe filled with radiation,matter and dark energy. Then we derive the linearized governing equations fora perfect fluid in a perturbed universe in synchronous and conformal Newtoniangauges.Chapter 3 presents an overview of the CMB anisotropy and the derivation ofits temperature and polarization angular power spectrum. We also discuss howforeground contamination is an obstacle in detecting the CMB anisotropies. Fore-grounds are any emissions that confuse the primordial CMB signal after the timeof last scattering.In chapter 4 we study the effects of Rayleigh scattering on the CMB and cosmicstructure. Rayleigh scattering from neutral hydrogen during and after recombina-tion causes the CMB anisotropies to be frequency dependent and alters the distri-9bution of matter in the Universe. We introduce a new method to capture the effectsof Rayleigh scattering on cosmological power spectra. Also a discussion about thedetectibility of Rayleigh signal with future CMB missions even in the presence offoregrounds has been given in this chapter.In addition to the CMB, the standard model also provides us with cosmic neu-trino background (CNB), the relic neutrinos which travelled to us freely fromwhen the Universe was only less than one second old. In chapter 5 we studythe anisotropy of the Cosmic Neutrino Background (CNB) and derive the fullCNB anisotropy power spectrum at large and small scales both for a massless andmassive neutrinos. We also discuss how presence of nonstandard neutrino self-interactions which are compatible with the current cosmological data modifies theCNB power spectrum.10Chapter 2The linear cosmologicalperturbation theoryLinear cosmological perturbation theory is an extremely useful tool to describehow primordially-generated fluctuations in matter and radiation grow to form thestructures that exist in the Universe today through gravitational instability. Linearperturbation theory can be formulated since these fluctuations remain small for alarge part of cosmic history.In this theory, to compute the growth of small perturbations in the context ofgeneral relativity, one need to solve the Einstein equations linearized about an ex-panding background. The use of general relativity introduces some complicationsrelated to gauge freedom, i.e., the choice of coordinate system; Physical variablesmust be independent of this choice. Lifshitz [19] pioneered the early work on per-turbations. He adopted the synchronous gauge for his coordinate system which isstill used in many recent works due to the fact that equations in this particular gaugeare more numerically stable than in other gauges. However there are some com-plications associated with using this gauge such as the appearance of coordinateartifacts or gauge modes. One way to deal with this gauge problem is to carefullykeep track of physical and gauge modes. A different approach to this problemwas taken by Bardeen [20] by introducing gauge-invariant variables and thereforeremoving the gauge artifacts. A thorough review of gauge-invariant perturbationtheory is given in Ref. [21].11According to the decomposition theorem, linear perturbations to the homoge-nous and isotropic metric can be divided into three types: scalar, vector and tensor.This classification refers to the way each type transforms under spatial transfor-mations. In this thesis we only follow the evolution of scalar perturbations andsince each of these types of perturbations evolve independently we do not need toworry about the possible vector or tensor perturbations. For scalar perturbations,the most convenient and common gauge to use is the conformal Newtonian gaugein which the two scalar fields describing the metric perturbations are identical tothe gauge-invariant variables constructed by Bardeen.In this section we present the Einstein, Boltzmann and fluid equations for themetric and density perturbations in the two most common gauges: synchronousgauge and conformal Newtonian gauge. We largely follow the notation and equa-tions of Ref. [22].2.1 Background equationsThe metric describing the geometry of an isotropic and homogenous spacetimeis the Friedmann-Robertson-Walker (FRW) metric gµν [1]. The line element isgiven byds2 = gµνdxµdxν =−dt2+a2(t)γi jdxidx j, (2.1)where t is the coordinate time, a(t) is a function that only depends on time and iscalled the scale factor and γi j is the metric tensor on spatial hypersurfaces. Sincecurrent observations are consistent with a flat or very nearly flat universe we adoptthis choice henceforth and write the line element asds2 =−dt2+a2(t)δi jdxidx j, (2.2)where δi j is the Kronecker delta. An important quantity characterizing the FRWspacetime is the Hubble rate H which measures how rapidly the scale factor changes:H =da/dta. (2.3)The Hubble rate has unit of inverse time. The Hubble time (t ∼H−1) sets the scalefor the age of the Universe and the Hubble length (d ∼ cH−1) sets the size of the12observable universe. The dynamic evolution of the scale factor a(t) is determinedby the Einstein equation:Gµν = 8piGTµν , (2.4)where the left-hand side Gµν is the Einstein tensor which denotes the curvatureof the Universe and is very straight-forward to obtain for a homogenous FRWspacetime [1]. G is the the Newton’s gravitational constant and Tµν is the energymomentum tensor.For a perfect fluid the energy momentum tensor has the following form:Tµν = Pgµν +(ρ+P)UµUν , (2.5)where ρ is the energy density of the fluid, P is the pressure and Uµ = dxµ/√−ds2is the four-velocity of the fluid.The Einstein equations give the following evolution equations for the scalefactor which are called the Friedmann equations:H2 = (da/dta)2 =8piG3ρ (2.6)d2a/dt2a=−8piG6(ρ+3P) (2.7)The second Friedmann equation implies that in an expanding Universe with ordi-nary matter (ρ + 3P > 0), d2a/dt2 < 0 which means there is a singularity in thefinite past (a(t = 0) = 0). This could be an indication of the breakdown of GeneralRelativity at high energy scales.The two Friedmann equations can be combined to obtain the continuity equa-tiondρdt+3H(ρ+P) = 0. (2.8)Defining the equation of state parameter w = P/ρ , the continuity equation may beintegrated to giveρ ∝ a−3(1+w), (2.9)13and the scale factor evolves asa ∝{t2/3(1+w) w 6=−1eHt w =−1 (2.10)Now we highlight the evolution of scale factor and the energy density in differ-ent eras of the Universe. In a radiation dominated universe which is filled withrelativistic particles, the equation of state parameter is w = 1/3 which givesρ ∝ a−4 and a ∝ t1/2. (2.11)The next epoch is the matter dominated era where the Universe is filled with non-relativistic matter (w = 0). In this epoch:ρ ∝ a−3 and a ∝ t2/3. (2.12)The last epoch is dominated by cosmological constant (w =−1) which results in:ρ ∝ a0 and a ∝ eHt . (2.13)2.2 Metric perturbationsNow we allow for small perturbations in the flat FRW metric that break homo-geneity and isotropy. It’s often convenient to use, instead of t, the conformal timecoordinate τ defined as dτ = dt/a. In synchronous gauge the line element is givenbyds2 = a2(τ){−dτ2+(δi j +hi j)dxidx j}. (2.14)In this gauge the g00 and g0i components of the metric are left unperturbed andonly the spatial part of it has perturbations hi j. Therefore τ defines proper timefor all comoving observers. In this section we will be working in Fourier spacewith variable k. According to the decomposition theorem, the metric perturbationhi j can be decomposed to scalar, vector and tensors modes. By introducing twoscalar fields h(~k,τ) and η(~k,τ) in k-space, we can write the scalar mode of hi j as a14Fourier integral [22]:hi j(~x,τ) =∫d3kei~k.~x{kˆikˆ jh(~k,τ)+(kˆikˆ j− 13δi j)6η(~k,τ)}. (2.15)Note that h is the trace of hi j. The drawback of this gauge is that it is not uniquelydefined by the synchronous gauge condition. There is still the freedom to make fur-ther transformations and remain within the gauge since the spacelike hypersurfaceat t = 0 can be chosen arbitrarily. This can introduce, in addition to the physicalmodes, coordinate artifacts.The line element in the conformal Newtonian gauge is given asds2 = a2(τ){−(1+2ψ)dτ2+(1−2φ)dxidxi}, (2.16)where φ and ψ are two scalar potentials. The Newtonian coordinates and metricperturbations in this gauge are defined uniquely, which means there are no gaugemodes in this gauge and all solutions are physical. This can also be realized bynoticing that the two scalar potentials in this gauge are equal to the gauge-invariantvariables ΦA and ΦH introduced by Bardeen in Ref. [20].ψ =ΦA, φ =−ΦH . (2.17)Another advantage of working in this gauge is that ψ plays the role of gravitationalpotential in the Newtonian limit. Also since the perturbed metric in this gauge isdiagonal, calculations in this gauge are simpler.2.3 Linearized Einstein equationsThe background evolution equations for an homogenous and isotropic universeare presented in section 2.1. In this section we derive the linearized governingequations for a perfect fluid in a perturbed FRW universe in the synchronous gaugeand conformal Newtonian gauge.To solve the linearized Einstein equations, we first perturb the energy-momentum15tensor of a perfect fluid given in 2.5 asδT 00 = −δρ, (2.18)δT 0i = (ρ+P)vi,δT ij = δPδij +Σij,where δρ and δP are the energy density and pressure perturbations respectively,vi is the velocity perturbation and Σij is the anisotropic shear perturbation which isthe traceless component of T ij .We solve the Einstein equations in Fourier space k. The linearized Einsteinequations in both synchronous and conformal Newtonian gauges are given as [22]:In synchronous gauge :k2η− 12a˙ah˙ = −4piGa2δρ(Syn), (2.19)k2η˙ = 4piGa2(ρ+P)θ(Syn),h¨+2a˙ah˙−2k2η = −24piGa2δP(Syn,)h¨+6η¨+2a˙a(h˙+6η˙)−2k2η = −24piGa2(ρ+P)σ(Syn).In con f ormal Newtonian gauge :k2φ +3a˙a(φ˙ +a˙aψ) = −4piGa2δρ(Con), (2.20)k2(φ˙ +a˙aψ) = 4piGa2(ρ+P)θ(Con),φ¨ +a˙a(ψ˙+2φ˙)+(2a¨a− a˙2a2)ψ+k23(φ −ψ) = 4piGa2δP(Con),k2(φ −ψ) = 12piGa2(ρ+P)σ(Con).The label “Syn” and “Con” is used to distinguish the components of energy-momentumtensor in the two gauges from each other. Here, overdots denote derivatives withrespect to the conformal time τ . The variable θ is the divergence of the fluid ve-locity θ = ik jv j and σ is defined as (ρ+P)σ =−(kˆikˆ j− 13δi j)Σij.162.4 Boltzmann equationThe Boltzmann equation gives us the evolution of the perturbed phase spacedistribution functions. The phase space distribution of the particles gives the num-ber of particles per unit volume in single-particle phase space. The equation relateshow the distribution function of each component evolves with time taking into ac-count the interactions with other species. The Boltzmann equation states:d fdτ=∂ f∂τ+∂ f∂xi∂xi∂τ+∂ f∂ pµ∂ pµ∂τ=C[ f ]. (2.21)The right-hand side, C[ f ], contains all possible collision terms, xi is the position ofparticles and pµ is the proper momentum, the momentum measured by an observerat fixed spatial coordinates. For convenience, we introduce the comoving momen-tum qi = api and the comoving energy ε = a√p2+m2 where m is the mass of theparticle. Moreover, we write the comoving momentum in terms of its magnitudeand its direction ~q = qnˆ. Expanding the distribution function up to the first orderleads to:f (~x,q, nˆ,τ) = f0(q)[1+Ψ(~x,q, nˆ,τ)], (2.22)where f0(q) is the zeroth-order distribution function and Ψ(~x,q, nˆ,τ) is the pertur-bation to this distribution function. The function f0(q) for fermions is the Fermi-Dirac distribution (positive sign in Eq. 2.23) and for bosons is the Bose-Einsteindistribution (negative sign in Eq. 2.23):f0(q) =1eε/kBT0±1 , (2.23)where kB is the Boltzmann constant and T0 = aT is the temperature of the particlestoday. Using the geodesic equations, one can estimate ∂ pµ/∂τ in Eq. 2.21 inthe two gauges we are interested in. Then the Boltzmann equation leads to thefollowing evolution equations for the perturbation of distribution function in k-space:In synchronous gauge :∂Ψ∂τ+ iqε(~k.nˆ)Ψ+dln f0dlnq[η˙− h˙+6η˙2(~k.nˆ)2] =1f0C[ f ]. (2.24)17In con f ormal Newtonian gauge :∂Ψ∂τ+ iqε(~k.nˆ)Ψ+dln f0dlnq[φ˙ − iεq(~k.nˆ)ψ] =1f0C[ f ]. (2.25)Since in the previous section we wrote Einstein equations in terms of the per-turbations in energy-momentum tensor δT µν , we need to find the relation betweenδT µν and the perturbation in the distribution function Ψ. The general energy-momentum tensor written in terms of the distribution function is given by [22]:Tµν =∫dP1dP2dP3(−g)−1/2 PµPνP0f (~x,~P,τ), (2.26)where Pµ is the canonical momentum and g denotes the determinant of the metricgµν . Using the perturbed distribution function, the components of the energy-momentum tensor to linear order in the perturbations can be written asT 00 = −(ρ+δρ) =−a−4∫q2dqdΩ√q2+m2a2 f0(q)(1+Ψ), (2.27)T 0i = (ρ+P)vi = a−4∫q2dqdΩqni f0(q)Ψ,T ij = (P+δP)δij +Σij = a−4∫q2dqdΩq2nin j√q2+m2a2f0(q)(1+Ψ),where dΩ is the solid angle associated with direction nˆ.18Chapter 3The Cosmic MicrowaveBackground anisotropyIn this chapter we briefly review the physics and the statistical interpretation ofCMB temperature and polarization fluctuations.3.1 Temperature anisotropiesPhotons evolve differently before and after recombination. Before recombina-tion, the rate of the Thomson scattering, which couples photons and electrons, ismuch larger than the rate of the expansion of the Universe. As a result, photonsand baryons behave as a single tightly coupled fluid. The differential cross sec-tion dσ/dΩ, defined as the radiated intensity per unit solid angle divided by theincoming intensity per unit area, for Thomson scattering is given by [22]:dσdΩ=3σT16pi(1+ cos2α), (3.1)where σT = 0.6652× 10−24cm2 is the total Thomson cross section and α is thescattering angle.After recombination, the rate of the Thomson scattering becomes smaller thanthe rate of the expansion of the Universe, photons travel almost freely and theUniverse becomes transparent.19To quantify the evolution of the CMB temperature anisotropies, we expand thephoton distribution function about its zero-order Bose-Einstein value:f (~x, p, nˆ,τ) = [epT (τ)[1+θI (~x,nˆ,τ)] −1]−1, (3.2)where~x, p and nˆ are the position, the magnitude of the momentum and the directionof the momentum respectively. T (τ) is the temperature of photons and θI(~x, nˆ,τ)is the intensity or temperature perturbation. The temperature perturbation does notdepend on p since the magnitude of the photon momentum is virtually unchangedduring a Thomson scattering. The temperature perturbation θI is related to thedistribution perturbation Ψ in Eq. 2.22 byθI =−[dln f0dlnp ]−1Ψ. (3.3)We solve the evolution of the temperature perturbation θI in Fourier k space. Weexpand θI in Legendre series:θI(~k, nˆ,τ) =∞∑l=0(−i)l(2l+1)θIl(~k,τ)Pl(µ). (3.4)The cosine of the angle between the wavenumber ~k and the photon direction nˆis defined to be µ = kˆ · nˆ and Pl(µ) is the Legendre polynomial of order l. Themonopole of photon distribution θI0 correspond to l = 0, dipole θI1 to l = 1, etc.Now we can write the Boltzmann equations for photons in both synchronousand conformal Newtonian gauges which govern the evolution of photon tempera-ture perturbations [1]:In synchronous gauge :θ˙I + ikµθI− (η˙− h˙+6η˙2 µ2) = neσT a[θI0−θI +µvb− 12P2(µ)Π]. (3.5)In con f ormal Newtonian gauge :θ˙I + ikµθI− φ˙ + ikµψ = neσT a[θI0−θI +µvb− 12P2(µ)Π]. (3.6)20h and η are the two scalar metric perturbations in synchronous gauge and φ and ψare metric perturbations in conformal Newtonian gauge, ne and vb are the propermean density and velocity of electrons, and Π is defined asΠ= θI2+θE0+θE2, (3.7)where θE0 and θE2 are monopole and quadrupole of E-polarization perturbation.3.1.1 Temperature power spectrumTo extract information about the Universe from the observational data, the Leg-endre transformation of the two-point correlation function of the CMB fluctua-tions, the CMB power spectrum, is often used. To connect the observed anisotropypower spectrum to the θIl variable, we expand the temperature perturbation fieldθI(~k, nˆ,τ) in terms of spherical harmonics Ylm:θI(~k, nˆ,τ) =∞∑l=1l∑m=−lalmYlm(nˆ). (3.8)Since the spherical harmonics are a complete basis for functions on the surface ofa sphere, all the information in the temperature field is contained in the amplitudesalm. The mean value of alm’s is zero but they have non-zero variance (angularpower spectrum) which is defined as:Cl =12l+1∑m〈alma∗lm〉, or 〈alma∗l′m′〉= δll′δmm′Cl. (3.9)The angular power spectrum Cl is an important tool in the statistical analysis of theCMB. This power spectrum gives the cosmological information contained in themillions of pixels of a CMB map in terms of a much more compact data represen-tation. The relation between Cl and θl is given by [1]:Cl = 4pi∫ ∞0k2dkPψ(k)|θIl(k,τ = τ0)|2, (3.10)where Pψ(k) is the primordial potential fluctuation power spectrum which containsinformation about the initial condition of the Universe.213.2 Polarization anisotropiesIn addition to anisotropies in the CMB temperature, we expect the CMB tobecome polarized via Thomson scattering. Thomson scattering allows all the radi-ation transverse to the outgoing direction to pass through unimpeded, while stop-ping any radiation parallel to the outgoing direction. If in the rest frame of theelectron, the incident radiation has the same intensity in every direction, then theoutgoing radiation remains unpolarized because orthogonal polarization directionscancel out. Only if the incoming radiation field has a quadrupole component, theoutgoing incident will be polarized [1] and [23]. As before photon decouplingthe electrons and photons are tightly coupled, the photon quadrupole is relativelysmall and we expect the polarization perturbations to be smaller than the tempera-ture anisotropies.Polarized light is usually described in terms of the Stokes parameters. Considera monochromatic plane electromagnetic wave with frequencyω0 propagating in thez direction. Its electric field vector at any given point in space can be written asEx = Ax(t)cos[ω0t−φx(t)], Ey = Ay(t)cos[ω0t−φy(t)], (3.11)where Ax, Ay and φx, φy describe the amplitude and phases in the xˆ− yˆ plane re-spectively.The Stokes parameters are defined as [23]:I = 〈A2x〉+ 〈A2y〉, (3.12)Q = 〈A2x〉−〈A2y〉, (3.13)U = 〈2AxAy cos(φx−φy)〉, (3.14)V = 〈2AxAy sin(φx−φy)〉. (3.15)The parameter I measures the intensity of radiation and is always positive. Theother three parameters describe the polarization state of radiation and can be eitherpositive or negative. Q and U quantify the magnitude of the linear polarization,and V parametrizes the circular polarization. I is invariant under a rotation in xˆ− yˆplane and therefore can be expanded in terms of scalar (spin-0) spherical harmonics22Eq. 3.8. However Q and U transform under rotation by angle φ such that [14](Q± iU)−→ e∓2iφ (Q± iU). (3.16)This implies that the linear combinations (Q± iU) are spin-2 quantities and weneed to expand them on a sphere in terms of tensor (spin-2) spherical harmonics[14]:(Q± iU)(nˆ) =∑l,ma±2,lm ±2Ylm(nˆ). (3.17)Instead of a±2,lm, it’s convenient to introduce their linear combinationaE,lm =−12(a2,lm+a−2,lm), aB,lm =−12i(a2,lm−a−2,lm). (3.18)Now instead of spin-2 quantities we define two scalar fields such that:E(nˆ) =∑lmaE,lmYlm(nˆ), B(nˆ) =∑lmaB,lmYlm(nˆ). (3.19)In analogy with electric and magnetic fields, these E and B field represent the“curl-free” and “divergence-free” component of the polarization field and theycompletely specify the linear polarization field [14].3.2.1 Polarization power spectraAnalyzing both temperature and polarization anisotropies at the same timeleads to four types of non-vanishing correlations: the auto-correlation of tempera-ture and E- and B-modes polarization perturbations and also the cross-correlationof temperature and E-mode polarization fluctuations. The cross-correlation of B-mode and temperature anisotropies and also B-mode and E-mode vanish due toparity [14].The angular power spectra are defined as beforeCXYl =12l+1∑m〈a∗X ,lmaY,lm〉, (3.20)where X and Y are I for intensity, E for E-mode polarization or B for B-mode235 10 50 100 500 100010-40.011100104llHl+1LCl2Π@Μk2DFigure 3.1: The theoretical CMB power spectra. The blue (solid), red(dashed), green (dotted) and purple (dot dashed) are for the temperature,E-polarization, B-polarization and Temperature-E polarization cross-correlation spectra respectively.polarization. Figure 3.1 shows the theoretical prediction for these power spectracalculated by CAMB [24].The temperature and E-mode auto-correlation as well as T E cross-correlationare dominated by the scalar perturbations, while the B-mode is only generated bytensor perturbations. As discussed in section 1.3, analyzing the observed CMBpower spectra provides us with information about the early universe.3.3 ForegroundsOne of the biggest challenges in observing CMB anisotropies are foregrounds,which are any other sources of radiation in the path of CMB photons that alsoemit at microwave frequencies. Dust, synchrotron radiation (relativistic electronsin galactic magnetic field) and free-free or bremsstrahlung emission (electrons ac-24celerated in ionized gas) from our galaxy as well as extragalactic point sourcescould be potential problems in detecting the CMB anisotropies [25].The understanding and removal of CMB foregrounds has become an importanttopic in CMB data analysis. There are a couple of reasons why foregrounds can bemanaged. Firstly the spectral shapes of all the foregrounds are different from oneanother and from the black body shape of the CMB. For example dust emissionhas a spectrum which rises with frequency, while free-free and synchrotron emis-sion have falling frequency spectra. Therefore by observing the CMB at differentfrequency channels we can extract the CMB signal from the foregrounds. Also theforeground amplitudes are found to be smaller than the CMB amplitude in a fairlywide frequency window. There are a number of papers such as Ref.[25], [26],[27] and [28] which present a comprehensive treatment of foreground problem andsuggest an optimal way to remove foreground contamination from the CMB signal.25Chapter 4Effects of Rayleigh Scattering onthe CMB and Cosmic Structure4.1 IntroductionMost descriptions of the Cosmic Microwave Background (CMB) anisotropiesassume that before recombination at z ' 1090, photons are tightly coupled tobaryons through Thomson scattering with electrons and afterwards free streamfrom the surface of last scattering to us [21], [22],[23]. However, in fact pho-tons were coupled not only to free electrons through Thomson scattering, but alsoto neutral hydrogen and helium through Rayleigh scattering. The Rayleigh scatter-ing cross section depends approximately on photon frequency to the fourth powerand, since it modifies the opacity near decoupling at the few percent level [29], hasbeen neglected in most of the literature to simplify the analysis. In this chapter werevisit the impact of Rayleigh scattering on cosmological perturbations, quantifyits effects, and suggest potential ways that this effect may be detected in the future.In the past decade the Wilkinson Microwave Anisotropy Probe (WMAP) hasprovided us with precise measurements of CMB anisotropies [3] and, comple-mented by next-generation ground based experiments (SPT [4], ACT [5]), thePlanck satellite has now characterized the microwave background anisotropies toan even higher precision [30]. Future measurements may even probe CMB anisotropieswith more frequencies and higher precision (e.g., PRISM [6] or PIXIE [7]). With26this dramatic improvement in experimental capability in mind it is timely to in-clude the physics of Rayleigh scattering in cosmological perturbation theory bothto find accurate solutions and to forecast whether these effects might be measuredwith proposed instruments.A conceptually straightforward method to calculate the effect of Rayleigh scat-tering on photon perturbations, as its cross section is frequency dependent, is toconsider separate Boltzmann hierarchies with different scattering sources and visi-bility functions at each frequency of interest. While this captures the effects of theextra opacity that photons experience, it does not account for either the momentumtransferred to the atoms nor the effect of spectral distortion on gravitational pertur-bations. In order to model these effects, the photon perturbation at each frequencymust be integrated over to determine the photon density and momentum densitywhich influence gravitation perturbations and the photon-baryon coupling. Exist-ing work has modelled the effect of Rayleigh scattering on CMB anisotropies buthas avoided determining the baryonic back reaction in detail [31, 32]. We intro-duce here a new approach to solve this problem and accurately treat baryons andfrequency-dependent photon perturbations simultaneously, allowing us to quantifythe impact of Rayleigh scattering on matter perturbations for the first time. We alsovalidate the results of existing CMB anisotropy calculations. The key innovationin our approach is to track perturbations in photon spectral-distortions rather thanphoton perturbations at a particular frequency.Rayleigh scattering changes the rate at which photons and baryons decouplefrom each other, and extra photon drag modifies how baryon perturbations are in-fluenced by photon perturbations. As we quantify below, this alters the shape of thematter correlation function and makes a small shift to the Baryon Acoustic Oscil-lation (BAO) scale. Like prior work on this subject we find that Rayleigh scatteringresults in percent level frequency-dependent distortions to CMB power spectra.These distortions break the thermal nature of CMB temperature and polarizationanisotropies so that primary CMB intensity and polarization patterns at differentfrequencies are not perfectly correlated with each other. We show below that to avery good approximation this effectively doubles the number of random variablesneeded to completely describe the CMB sky, and determine for the first time theset of intensity and E-polarization eigenspectra needed to capture this statistical in-27formation. Finally, we forecast how well future CMB missions might detect theseeigenspectra and show that a PRISM-like experiment may be able to detect theRayleigh signal.This chapter is organized as follows: In Section 4.2, the relevant Rayleigh scat-tering cross sections for hydrogen and helium are presented. Section 4.3 reviewsthe cosmological equations governing the evolution of perturbations in the pres-ence of Rayleigh scattering and presents our new method to calculate the effect ofthis additional frequency-dependent opacity. The effect of Rayleigh scattering onthe matter two-point correlation function and on the CMB power spectra is calcu-lated in Section 4.4 and 4.5 respectively. In Section 4.6, we present the two sets ofvariables needed to describe the CMB intensity and E-polarization statistics. Sec-tion 4.7 investigates the possibility of detecting the Rayleigh signal and Section 4.8concludes.4.2 Rayleigh scattering cross sectionAn electron of charge e and mass m which is part of an atomic system acts likea harmonic oscillator with frequency ν0, the characteristic transition frequency.When this oscillator is subjected to a plane wave radiation of frequency ν whichis much smaller than ν0, the total energy radiated per unit time in all directions is[33]:P =8pi3(e2mc2)2ν4(ν20 −ν2)2cU, (4.1)where c is the speed of light, U = E20/4pi is the energy density of the incident fieldand E0 is the amplitude of the electric field. The scattering cross section is definedasσ =PU=8pi3(e2mc2)2ν4(ν20 −ν2)2. (4.2)The corresponding quantum mechanical expression for the Rayleigh scatteringcross section isσR(ν) = σT |S|2, (4.3)28where σT is the Thomson cross section and the dimensionless scattering amplitude,S, is given by [33]S =∞∑j=2f1 jν2ν21 j−ν2. (4.4)Here ν1 j is the Lyman series frequencies, and f1 j is the Lyman series oscillatorstrength which expresses the probability of the absorption or emission of a photonin transition between the ground state and excited states of an atom. Note that thesummation includes an implied integration over unbound states j.At the time of recombination, when T ' 0.25 eV, typical photon frequenciesare much smaller than ν1 j and it is therefore appropriate to Taylor-expand the di-mensionless scattering amplitude asS =∞∑k=0a2k+2(hν)2k+2, (4.5)where the coefficients area2k+2 = ∑j≥2f1 j(hν1 j)−2k−2+∫ ∞EId fdEE−2k−2dE. (4.6)Here we have written the integral over continuum states explicitly. The integralstarts at the ionization energy EI of the relevant atom. The Rayleigh scatteringcross section is thenσR = σT∞∑k=0b2k+4(hν)2k+4, (4.7)whereb2k+4 =k∑p=0a2p+2a2(k−p)+2. (4.8)The coefficients can be evaluated provided that the oscillator strength distributionsare known. For H, these are known exactly: for the discrete spectrum (1s→ np),the oscillator strengths are [34]f1s,np =256n5(n−1)2n−43(n+1)2n+4, (4.9)with hν1s,np = (1− n−2)Ry. Above EI = 1Ry = 13.6 eV there is a continuous292k+4 H He4 1.265625 0.1207986 3.738281 0.0672438 8.813931 0.03158510 19.153795 0.01415312 39.923032 0.006226Table 4.1: The cross-section coefficients b2k+4Ry2k+4 for H and He in theRydberg-based units that we adopt for this work.spectrum of oscillator strengths,d fdE=128e−4varctan(v−1)3(E/Ry)4(1− e−2piv)Ry−1, (4.10)where v= (E/Ry−1)−1/2 is the principal quantum number of the continuum state.Helium is a multi-electron atom, for which to show the angular momentumquantum numbers we use the following term symbol:2S+1L, (4.11)where S is the total spin quantum number and L is the orbital quantum number inspectroscopic notation which means L = 0 correspond to letter S and L = 1 cor-respond to letter P. For He, the electric dipole selection rules allow the ground1s2 1S state to have nonzero oscillator strength only with the 1P discrete and con-tinuum states. We have taken the oscillator strengths and energies for the 1s2 1S→1snp 1P transitions from Refs. [35, 36] for n ≤ 9 and used the asymptotic for-mula of Ref. [37] for n > 9. For the continuum states we used the TOPbase crosssections [38], which are trivially converted into oscillator strengths. The resultingb2k+4 coefficients that we adopt for the rest of this work are shown in Table 4.1.The radiative transfer equations also require the angular distribution and po-larization of Rayleigh-scattered radiation. For scattering with initial and finalstates of zero orbital angular momentum (S→ S), and neglecting spin-orbit cou-pling, the scattering is of a pure “scalar” nature (in the language of Ref. [39] §61)30and has the same angular and polarization dependence as Thomson scattering,dP/dΩ ∝ 1+ cos2 θ . Near a resonance such as Lyman-α , fine structure splittingmakes the electron spin important, and the scattering by hydrogen takes on a differ-ent form that is a combination of scalar, anti-symmetric, and symmetric scattering;the full equations for the angular scattering distribution as a function of frequencythrough the resonance can be found in e.g. Appendix B of Ref. [40]. The equationsin Appendix B of Ref. [40] show that the angular distribution approaches the scalarcase with corrections of order ∆ν2fs/(νLyα −ν)2 as one moves away from the reso-nance, where the fine structure splitting is ∆νfs ∼ 11GHz. For cases considered inthis paper (frequencies up to 857 GHz observer frame, or 0.52νLyα at z = 1500),we are thus safely below the lowest resonant frequency, and the scalar angular dis-tribution – already incorporated in the CMB Boltzmann hierarchy formalism – isapplicable.4.3 Cosmological equations and proposed methodTo include the effects of Rayleigh scattering on cosmological perturbations, wemust modify the evolution equations for photon temperature, photon polarizationand baryon velocity perturbations. We use synchronous gauge in this paper as itis convenient for most numerical computations. The full cosmological evolutionequations in this gauge are given in a number of papers [22, 41], and thereforewe only explicitly show the equations that need modification. In particular, us-ing the Boltzmann equation in this gauge we find the evolution equations for the31temperature perturbation, ΘI , and E-polarization, ΘE , hierarchies areΘ˙I0 = −kΘI1+ a˙aν∂ΘI0∂ν− h˙6, (4.12)Θ˙I1 =k3ΘI0− 2k3 ΘI2+a˙aν∂ΘI1∂ν−κ˙[−ΘI1+ 13vb], (4.13)Θ˙I2 =2k5ΘI1− 3k5 ΘI3+a˙aν∂ΘI2∂ν+h˙+6η˙15−κ˙[−ΘI2+ 110Π], (4.14)Θ˙Il =k2l+1[lΘI(l−1)− (l+1)ΘI(l+1)]+a˙aν∂ΘIl∂ν+ κ˙ΘIl l ≥ 3, (4.15)Θ˙E2 =2k5ΘE1− k3ΘE3+a˙aν∂ΘE2∂ν+κ˙(ΘE2− 25Π), (4.16)Θ˙El =k2l+1[lΘE(l−1)−(l+3)(l−1)l+1ΘE(l+1)]+a˙aν∂ΘEl∂ν+ κ˙ΘEl l ≥ 3, (4.17)where an overdot denotes derivatives with respect to conformal time τ , k is thewavenumber of the perturbations, h and η are the synchronous gauge metric pertur-bations, Π is the combination ΘI2+ 32ΘE2, a the scale factor and κ˙ is the comovingopacity defined as−κ˙ = −κ˙T − κ˙R= neσT a+nHσHR a+nHeσHeR a. (4.18)Here ne, nH and nHe are respectively the number densities of free electrons, neutralhydrogen and helium atoms. The comoving opacity for Rayleigh and Thomsonscattering as a function of conformal time is plotted in Figure 4.1 for a couple ofobserved frequencies.32200 300 400 500 60010-810-610-40.011Τ HMpc LΚ HMpcL-1Figure 4.1: The Comoving opacity as a function of comoving time. The black(solid) line is for Thomson scattering while the blue (large dashed),red (small dashed), green (dot dashed) and brown (dotted) lines are forRayleigh scattering at frequencies 857, 545, 353, and 217 GHz respec-tively.In standard case when opacity does not depend on frequency, the baryonsevolve according to equationsδ˙b =−kvb− 12 h˙, (4.19)v˙b+a˙avb− kc2sδb =1ρ¯b∫ d3 p(2pi)3(−pµ)C[ f (~p)]=4ρ¯γ3ρ¯bκ˙(−3ΘI1+ vb), (4.20)where δb and vb are baryon overdensity and velocity, cs is the intrinsic baryonsound speed, f (p) is photon distribution function, C[ f (~p)] = d fdt is the collisionterm in the Boltzmann equation for photon temperature perturbations, µ = pˆ · kˆ,and ρ¯γ and ρ¯b are the mean photon and baryon energy densities.33Including Rayleigh scattering will make the opacity frequency-dependent, there-fore the scattering term in the baryon velocity must be modified to∫ d3 p(2pi)3(−pµ)C[ f (~p)] =∫ d3 p(2pi)3p2∂ f∂ pµκ˙(p)(ΘI0(p)−ΘI(p)+µvb). (4.21)As discussed above, a straightforward method to solve the above system ofequations is to consider a separate Boltzmann hierarchy for each frequency of inter-est, each with different scattering sources and visibility function, and then integrateover each photon frequency bin to get the total baryon-photon coupling [31, 32].However there is another computationally efficient method that can be used. If atthe times that atoms are present the typical CMB photon energies are much smallerthan Rydberg energy hνRy, then we can writeΘIl andΘEl as Taylor series in thecomoving frequency aν where each term in the series describes spectral-distortionperturbations that scale with increasing powers of frequency. Specifically we write:ΘIl(ν) =∞∑r=0Θ(2r)Il(ahνa∗Ry)2r, (4.22)ΘEl(ν) =∞∑r=0Θ(2r)El(ahνa∗Ry)2r. (4.23)Note that only even powers of ν appear because the scattering cross section con-tains only even powers of ν . We expanded the perturbations in terms of ahν/a∗Rybecause this ratio does not evolve with time for a given photon and a∗ = 0.001 is areference epoch for normalizing the coefficients in the series expansion (its valuehas no physical consequences). Similarly we can write the opacity asκ˙(ν) =∞∑r=0κ˙2r(ahνa∗Ry)2r, (4.24)where κ˙0 =−neσT a is the standard Thomson scattering rate, κ˙1 = 0 and−κ˙2r = (nHbH2r +nHebHe2r )σT a(a∗a)2r. (4.25)34Substituting these Taylor expansions into evolution equations for photon tem-perature and polarization perturbations leads to the following evolution equationsfor each Θ(2n)Il and Θ(2n)El terms:Θ˙(2n)I0 = −kΘ(2n)I1 −h˙6δn,0, (4.26)Θ˙(2n)I1 =k3Θ(2n)I0 −2k3Θ(2n)I2−n∑r=0κ˙2r[−Θ2(n−r)I1 +vb3δn−r,0], (4.27)Θ˙(2n)I2 =2k5Θ(2n)I1 −3k5Θ(2n)I3 +h˙+6η˙15δn,0−n∑r=0κ˙2r[−Θ2(n−r)I2 +Π2(n−r)10], (4.28)Θ˙(2n)Il =k2l+1[lΘ(2n)I(l−1)− (l+1)Θ(2n)I(l+1)]+n∑r=0κ˙2rΘ2(n−r)Il l ≥ 3, (4.29)Θ˙(2n)E2 =2k5Θ(2n)E1 −k3Θ(2n)E3+n∑r=0κ˙2r[Θ2(n−r)E2 −25Π2(n−r)], (4.30)Θ˙(2n)El =k2l+1[lΘ(2n)E(l−1)−(l+3)(l−1)l+1Θ(2n)E(l+1)]+n∑r=0κ˙2rΘ2(n−r)El l ≥ 3. (4.31)To find the evolution equation for baryon velocity we first must calculate the fol-lowing integralIn = − 14ρ¯γT n∫ ∞0dν2pi2νn+4∂ f∂ν=154pi4(n+4)!ζ [n+4], (4.32)35where ζ is the Riemann ζ -function. Therefore the baryon velocity in the presenceof Rayleigh scattering evolves according tov˙b = − a˙avb+ kc2sδb+4ρ¯γ3ρ¯b∞∑r=0κ˙2r[−3∞∑n=0Θ(2n)I1 I2(n+r)(aTa∗Ry)2(n+r)+vbI2r(aTa∗Ry)2r]. (4.33)As shown in Equation 4.4, the Rayleigh cross section blows up near the reso-nant frequencies. Therefore photons with these frequencies remain tightly coupledto baryons. Photons do not self interact so these resonant photons are unlikely tochange the CMB power spectrum. However they do enhance the pressure or soundspeed of baryons. There is typically of order 1 photon per baryon near the Lyman-α line and since the photon energy is 10.2 eV, and the baryon mass is 1 GeV, thebaryon sound speed increases by roughly 10−8. This only alters perturbations atvery small scales below those of interest in this work.Since metric perturbation evolution depends on the total photon overdensityand velocity, the final modification is to calculate the change in the photon stress-energy tensor in the presence of frequency dependent scattering. The fractionalphoton energy density perturbation isδγ = − 1ρ¯γ∫ν4dν∂ f∂νΘI0(ν)= 4∞∑r=0Θ(2r)I0 I2r(aTa∗Ry)2r, (4.34)and the photon momentum density isΘγ = − 3k4ρ¯γ∫ν4dν∂ f∂νΘI1(ν)= 3k∞∑r=0Θ(2r)I1 I2r(aTa∗Ry)2r. (4.35)This appears to replace the problem of summing over many perturbations at dif-36ferent frequencies with summing over many perturbations with different spectral-distortion shapes. However, we find in practice that these sums rapidly convergeafter including only a few of the spectral-distortion terms which allows the entiresystem to be solved for efficiently and accurately.For numerical computations we modified CAMB [24], a public Boltzmanncode for anisotropies in CMB which calculates the theoretical matter and CMBpower spectra given the cosmological parameters. We added the additional opac-ity due to Rayleigh scattering to this code as well as the evolution equations forspectral distortion terms Θ(2n)Il and Θ(2n)El .4.4 Matter power spectrumOne of the physical effects of Rayleigh scattering is a change in matter two-point correlation function. The matter correlation function is the excess probability,compared with what expected from a random distribution, of finding a matter over-density at a distance~r apart and its Fourier transform is the matter power spectrum,ξ (~r) = 〈δ (~x)δ (~x+~r)〉=∫ d3k(2pi)3P(k)ei~k.~r. (4.36)Rayleigh scattering increases the total baryon-photon coupling which delaysthe time of recombination. As shown in Figure 4.2, the correlation function hasa peak near a radius of ∼ 150 Mpc, the BAO scale, which represents the soundhorizon at the time of recombination. This changes due to the delay in the timeof photon-baryon decoupling. The percentage change in the two-point correlationfunction due to Rayleigh scattering is plotted in Figure 4.3. Adding Rayleigh scat-tering to the opacity changes the correlation function by up to ∼ 0.3%. Unlessotherwise stated we show all results in a fiducial model where we adopt the best-fitparameters from PLANCK [30].Another way of visualizing how much the matter power spectrum is changed inthe presence of Rayleigh scattering is by looking at the evolution of a concentratedmatter over-density in real space. In Figures 4.4 and 4.5, the redshift evaluation ofa narrow Gaussian-shaped adiabatic density fluctuation in real space is displayedfor baryons (blue) and photons (red).370 50 100 1500.000.010.020.030.040.05rH MpcLr2ΞHrLFigure 4.2: The matter two-point correlation function, r2ξ (~r), as a functionof the distance between two over-densities for our fiducial cosmologicalparameters.0 50 100 150- 0.3- 0.2- 0.10.00.1rH MpcL100∆ΐΞFigure 4.3: The percentage change in the matter correlation function due toRayleigh scattering for our fiducial cosmological parameters.38At very early stages, when the photons and baryons were tightly coupled, panel(a), the baryon-photon plasma density wave travels outward from the initial over-density. Panel (b) shows a snapshot of the density waves at redshift z = 1050. Atthis time the temperature is low enough that neutral atoms can form, therefore thephotons begin to decouple from baryons and the sound speed starts to drop. Thusthe baryon density wave slows down compared to the photon density wave. Inpanel (c), the waves are shown at z= 500 when photons and baryons are completelydecoupled. The photon perturbation smooths itself out at the speed of light. Butbecause the sound speed is much smaller than speed of light the baryon densitywave stalls. Panel (d) present the late time picture. The photons free-stream untilnow when we can observe them as the cosmic microwave background and thebaryon perturbation clusters around the initial over-density and in a shell of about∼ 150 Mpc radius.In Figure 4.6, the percentage change in physical baryon density fluctuationsin real space due to Rayleigh scattering is plotted at different redshifts. Note thatwhile ∆δ/δ is up to 0.6% at some points the percentage change in the location ofthe peak in baryon density wave or the BAO scale due to Rayleigh scattering is lessthan 0.01% in this example, and so the detailed effect of Rayleigh scattering is notwell modelled as a simple shift in the BAO scale.4.5 Photon power spectraTo calculate the power spectra for both photon temperature and E-polarizationperturbations, we use the line of sight integration approach of Ref. [42]. In thisapproach, the solutions of Eqs. 4.26 to 4.31 can be written as an integral over theproduct of a source term and a geometrical term which is just the spherical Besselfunction,ΘIl(τ0) =∫ τ00dτSI(k,τ) jl[k(τ0− τ)], (4.37)ΘEl(τ0) =∫ τ00dτSE(k,τ) jl[k(τ0− τ)]. (4.38)3910 1005020 3015 150700.0000.0020.0040.0060.008r @ h- 1Mpc D∆HrLH aL1005020 30 150700.0000.0020.0040.0060.008r @ h- 1Mpc D∆HrLH bLFigure 4.4: The redshift evaluation of a narrow Gaussian-shaped adiabaticdensity fluctuation in real space. The blue (solid) and red (dashed) linesare respectively the baryon and photon density waves. Panel(a) showsa snapshot at very early times when baryons and photons are tightlycoupled and their density waves travel together. In panels (b), redshiftz = 1050, photons begin to decouple from baryons and the baryon den-sity wave slows down compare to photon density wave due to the dropin the sound speed. 40100 2001500.0000.0020.0040.0060.0080.010r @ h- 1Mpc D∆HrLH cL100 2001500.0000.0050.0100.0150.0200.0250.030r @ h- 1Mpc D∆HrLH d LFigure 4.5: Same as 4.4 but for later redshifts. Panel (c) shows the densitywaves at redshift z = 500 where photons and baryons are completelydecoupled. The late time picture is presented in panel (d). The photonsfree stream to us and baryons cluster around the initial over-density andin a shell at about 150 Mpc radius.4110 1005020 2003015 15070-0.20.00.20.40.6r @h -1MpcD100D∆∆Figure 4.6: The percentage change in physical baryon density fluctuations inreal space due to Rayleigh scattering at different redshifts. The blue(solid), red (dashed), green (dot dashed) and brown (dotted) lines corre-spond to redshifts 0, 100, 500 and 1050 respectively.The source functions for temperature and E-polarization perturbations are given inmany previous studies [41, 42].SI(k,τ) = e−κ [− h˙6 +k3σ +σ¨k] (4.39)+g(τ)[2σ˙k+ΘI0+v˙bk+Π4+34k2Π¨]+g˙(τ)[σk+vbk+34k22Π˙]+ g¨(τ)34k2Π(0),SE(k,τ) = g(τ)34Π1[k(τ0− τ)]2 , (4.40)where σ = (h˙+ 6η˙)/2k and g(τ) = −κ˙e−κ is the visibility function. In the pres-ence of Rayleigh scattering the visibility function is frequency dependent and canbe written as a Taylor series in ahν/a∗Ry. The total visibility function for severalfrequencies is plotted in Figure 4.7. Note that the total photon visibility functionshifts toward later time with increasing frequencies.Substituting the Taylor expansions of visibility function and temperature and42240 260 280 300 320 340 3600.0000.0050.0100.0150.020ΤH MpcLgHΤLFigure 4.7: The total visibility function as a function of conformal time forseveral frequencies. The black (solid), red (dotted), blue (dot dashed)and green (dashed) lines are the total visibility function for frequencies0, 545, 700 and 857 GHz respectively. The total photon visibility func-tion shifts toward later times with increasing frequency.E-polarization perturbations into the above equations gives the source functions foreach of the Θ(2n)Il and Θ(2n)El terms,Θ(2n)Il (τ0) =∫ τ00dτS(2n)I (k,τ) jl[k(τ0− τ)], (4.41)Θ(2n)El (τ0) =∫ τ00dτS(2n)E (k,τ) jl[k(τ0− τ)], (4.42)43whereS(0)I = e−κ0 [− h˙6+k3σ +σ¨k]+g0[2σ˙k+Θ(0)I0 +v˙bk+Π(0)4+34k2Π¨(0)]+g˙0[σk+vbk+34k22Π˙(0)]+ g¨034k2Π(0), (4.43)S(4)I = e−κ0 [− h˙6+k3σ +σ¨k](−κ4)+(g0(−κ4)+g4)×[2 σ˙k+Θ(0)I0 +v˙bk+Π(0)4+34k2Π¨(0)]+g0[Θ(4)I0 +Π(4)4+34k2Π¨(4)]+(g˙0(−κ4)+g0(−κ˙4)+ g˙4)[σk +vbk+34k22Π˙(0)]+ g˙034k22Π˙(4)+g¨034k2Π(4)+[g¨0(−κ4)+2g˙0(−κ˙4)+g0(−κ¨4)+ g¨4] 34k2 2Π(0),(4.44)S(6)I = e−κ0 [− h˙6+k3σ +σ¨k](−κ6)+(g0(−κ6)+g6)×[2 σ˙k+Θ(0)I0 +v˙bk+Π(0)4+34k2Π¨(0)]+g0[Θ(6)I0 +Π(6)4+34k2Π¨(6)]+(g˙0(−κ6)+g0(−κ˙6)+ g˙6)[σk +vbk+34k22Π˙(0)]+ g˙034k22Π˙(6)+g¨034k2Π(6)+[g¨0(−κ6)+2g˙0(−κ˙6)+g0(−κ¨6)+ g¨6] 34k2 2Π(0),(4.45)S(0)E =34[k(τ0− τ)]2 g0Π(0), (4.46)S(4)E =34[k(τ0− τ)]2 (g0[Π(4)+Π(0)(−κ4)]+g4pi(0)), (4.47)S(6)E =34[k(τ0− τ)]2 (g0[Π(6)+Π(0)(−κ6)]+g6pi(0)). (4.48)Here g2r =−κ˙2re−κ0 . The anisotropy spectrum can be obtained by integratingover the initial power spectrum of the metric perturbation, Pψ(k):CXYl (ν ,ν′) =∫ ∞0k2dkPψ(k)(ΘXl(ν ,k)ΘY l(ν ′,k))(4.49)=∞∑r,r′=0CXY (2r,2r′)l(ahνa∗Ry)2r( ahν ′a∗Ry)2r′,44whereCXY (2r,2r′)l =∫ ∞0k2dkPψ(k)(Θ(2r)Xl (k)Θ(2r′)Y l (k)). (4.50)We used the modified version of CAMB [24] to numerically calculate CT T (2r,2r′)land CEE(2r,2r′)l power spectra. These results are shown in Figure 4.8 and Figure 4.9.Note that while Eq. 4.50 describes unlensed power spectra from the surface of lastscattering, here and elsewhere, these power spectra include the effect of gravita-tional lensing from structure along the line of sight implemented in CAMB.Using Eq. 4.50, the relative difference in the (lensed) scalar CMB power spec-tra due to Rayleigh scattering is calculated for four different frequencies and pre-sented in Figures 4.10 and 4.11. As expected, the relative difference in CMB powerspectrum is bigger for higher frequencies. In the limit of very low frequenciesthe only modification in these power spectra arises from the increase in the totalbaryon-photon coupling due to Rayleigh scattering which is of order 0.05%.On small scales, Rayleigh scattering leads to damping of both temperature andpolarization anisotropies. Rayleigh scattering increases the rate of photon-baryoninteraction and hence it reduces the photon-diffusion length. Since the amplitudeof Silk damping depends on the integrated photon-diffusion length, it is also re-duced by Rayleigh scattering. But there is another reason why the small-scaleanisotropies are more damped in the presence of Rayleigh scattering. The damp-ing factor at a given wave number is weighted by the photon visibility function.As we have seen above, adding Rayleigh scattering shifts the visibility functiontoward lower redshifts where Silk damping is more important and as a result, theanisotropy spectra at small scale decreases.We also find Rayleigh scattering leads to a boost in large-scale E-polarization.The reason for this is that the low-multipole polarization signal is sourced by theCMB quadrupole. Since the visibility function is shifted toward later time, wherethe quadrupole is larger, by Rayleigh scattering the low-multipole E-polarizationsignal is increased. In contrast, the effect of Rayleigh scattering on the lensing Bmodes is significantly smaller at low-multipole because these modes are producedby the gravitational lensing of E modes from a wide range of scales, so the Rayleighcontribution for them partly averages out.4510 20 50 100 200 500 1000 2000010002000300040005000llHl+1LclTT002Π@Μk2D10 20 50 100 200 500 1000 2000- 35000- 30000- 25000- 20000- 15000- 10000- 50000llHl+1LclTT042Π@Μk2D10 20 50 100 200 500 1000 20000100000200000300000400000500000llHl+1LclTT442Π@Μk2D10 20 50 100 200 500 1000 2000- 60000- 50000- 40000- 30000- 20000- 100000llHl+1LclTT062Π@Μk2D10 20 50 100 200 500 1000 200002000004000006000008000001.0 ´ 1061.2 ´ 106llHl+1LclTT462Π@Μk2D10 20 50 100 200 500 1000 200005000001.0 ´ 1061.5 ´ 1062.0 ´ 1062.5 ´ 106llHl+1LclTT662Π@Μk2DFigure 4.8: The cross correlation temperature power spectrum CT T (2r,2r′)l ofthe Θ(2r)Il and Θ(2r′)Il intensity coefficients for the ν0, ν4 and ν6 spectraldistortions.4610 20 50 100 200 500 1000 2000010203040llHl+1LclEE002Π@Μk2D10 20 50 100 200 500 1000 2000- 500- 400- 300- 200- 1000100200llHl+1LclEE042Π@Μk2D10 20 50 100 200 500 1000 2000020004000600080001000012000llHl+1LclEE442Π@Μk2D10 20 50 100 200 500 1000 2000- 1000- 800- 600- 400- 2000200400llHl+1LclEE062Π@Μk2D10 20 50 100 200 500 1000 20000500010000150002000025000llHl+1LclEE462Π@Μk2D10 20 50 100 200 500 1000 20000100002000030000400005000060000llHl+1LclEE662Π@Μk2DFigure 4.9: The cross correlation temperature power spectrum CEE(2r,2r′)l ofthe Θ(2r)El and Θ(2r′)El E-polarization coefficients for the ν0, ν4 and ν6spectral distortions.475 10 50 100 500 1000- 0.0010- 0.00050.00000.00050.00100.0015l∆clcl217 GHz5 10 50 100 500 1000- 0.0050.0000.0050.010l∆clcl353 GHzFigure 4.10: Shown are a fractional measure, δCXYl /√CXXl CYYl , of thechange δCXYl in (lensed) scalar CMB anisotropy spectra due toRayleigh scattering. The blue (solid), red (dashed), green (dot-ted) and purple (dot dashed) are for the temperature, E-polarization,B-polarization from lensing and Temperature-E polarization cross-correlation spectra respectively. The upper and lower panels are for217 and 353 GHz frequency channels respectively.485 10 50 100 500 1000- 0.050.000.05l∆clcl545 GHz5 10 50 100 500 1000- 0.3- 0.2- 0.10.00.10.20.30.4l∆clcl857 GHzFigure 4.11: Same as 4.11 but for different frequency channels. The upperand lower panels are for 545 and 857 GHz frequency channels respec-tively.49Another effect worth noting is that, the oscillations of δCl/Cl show that thepeaks in anisotropy spectra are shifted in the presence of Rayleigh scattering. Sincethe photon cross section is frequency dependent, the location of the surface of lastscattering τR+T∗ will depend on frequency too and the higher the frequency, thebigger τR+T∗ (k,ν). Therefore the sound horizon at the last scattering,rR+Ts =∫ τR+T∗0csdτ, (4.51)will be larger than the sound horizon at last scattering when we only include theThomson scattering rTs and it will increase with increasing frequencies. The shiftin the location of the peaks will beδ l/l = δk/k = 1− rR+Ts (τR+T∗ )/rTs (τT∗ ) (4.52)in the direction of decreasing l.4.5.1 Convergence of the numerical codeWe verify the convergence of the modified version of CAMB [24] by runningseveral computations with increasing accuracy i.e. by increasing the accuracy-boost parameter in the code which decreases the time steps, affects the sampling,etc. In addition we justify the number of spectral distortion terms that we needto keep in Eqs. 4.22 and 4.23 by showing that keeping only the first five leadingterms in these expansions results in an error smaller than 0.1%. To show this con-vergence, in 4.12 we plot the difference in the fractional change in the temperaturepower spectra when we include up to νn in Eqs. 4.22 and 4.23 compared to whenwe include up to νn−2 terms and as shown, by increasing the terms kept in theexpansion this difference becomes smaller and smaller.4.6 Rayleigh Distorted StatisticsSince the terms in the expansion of temperature and E-polarization perturba-tions, Eqs. 4.22 and 4.23, fall off quickly like (ahν/a∗Ry)2 only the two leadingterms play an important role at frequencies smaller than 800 GHZ. We therefore505 10 50 100 500 1000- 0.06- 0.05- 0.04- 0.03- 0.02- 0.010.00lH∆clHnL -∆clHn-2L Lc l(0)545 GHzFigure 4.12: Shown are the difference in the fractional change, δCT Tl /CT Tl in(lensed) scalar temperature CMB anisotropy spectra due to Rayleighscattering for 545 GHZ frequency channel when we include up to νnin Eqs. 4.22 and 4.23 compared to when we include up to νn−2 terms.The blue (solid), red (dashed), green (dotted) and purple (dot dashed)are for when n = 4, n = 6, n = 8 and n = 10 respectively and theyjustify the convergence of the expansions.effectively need two sets of random variables to describe the statistics of tempera-ture and E-polarization. In this section we find a compressed representation of thepower spectra for independent random variables. First we introduce the antennatemperature which is defined asTant(ν) = 2piν f (ν), (4.53)where f (ν) is the photon phase space distribution function and ν is the frequency.For the CMB, the antenna temperature has the formTant(ν)T=hν/kBTehν/kBT −1 +Θ(hν/kBT )2ehν/kBT(ehν/kBT −1)2 . (4.54)51The first term is the monopole which does not interest us here and we ignore it. Thesecond term gives the spectral shape of CMB anisotropies. Keeping only the firsttwo non-zero terms in Eqs. 4.22 and 4.23, the antenna temperature for the CMB isT Xant(ν)T=Θ(0)X F(0)(ν)+Θ(4)X F(4)(ν), (4.55)where F(0)(ν) = (hν/kBT )2ehν/kBT(ehν/kBT−1)2 is the black body shape function and F(4)(ν) =(hνRy )4F(0)(ν) is the shape function for the Rayleigh signal and X is either I forintensity perturbations or E for E-polarization perturbations. The angular powerspectrum covariance matrix for the antenna temperature isCXXl (ν ,ν′) =CXX(00)l F(0)(ν)F(0)(ν ′)+ CXX(04)l (F(0)(ν)F(4)(ν ′)+F(4)(ν)F(0)(ν ′))+ CXX(44)l F(4)(ν)F(4)(ν ′). (4.56)This structure indicates that Tant(ν) and Tant(ν ′) are correlated to each other butare not perfectly correlated like in the standard thermal case. We diagonalize theanisotropy spectrum in frequency space for a given X ∈ {I,E} to obtain the twouncorrelated eigenvalues:λXX1,2 (l) =12[CXX(00)l G00+2CXX(04)l G04+CXX(44)l G44 (4.57)±√(CXX(00)l G00+2CXX(04)l G04+CXX(44)l G44)2−4((CXX(04)l )2−CXX(00)l CXX(44)l )((G04)2−G00G44)],where Gi j =∫F(i)(ν)F( j)(ν)dν . The two orthogonal eigenvectors arevX1,2l(ν) = NX1,2[(CXX(04)l λXX1,2 +CXX(00)l CXX(44)l G04 (4.58)− (CXX(04)l )2G04)F(0)(ν)+(CXX(44)l λXX1,2− CXX(00)l CXX(44)l G00+(CXX(04)l )2G00)F(4)(ν)],52where NX1,2 is the normalization factor. If we expand the antenna temperature interms of spherical harmonics,T Xant(ν)/T =∞∑l=1l∑m=−laXlmYlm, (4.59)then we can write the coefficients aXlm in the new basis spanned by the eigenvectors{vT1l(ν),vT2l(ν),vE1l(ν),vE2l(ν)},aXlm = αX1lmvX1l(ν)+αX2lmvX2l(ν). (4.60)The covariance matrix in this new basis takes the compact formClδm,m′ =(CIl CIElCIEl CEl)δm,m′ = (4.61)〈α I1lmα I1lm′〉 0 〈α I1lmαE1lm′〉 〈α I1lmαE2lm′〉0 〈α I2lmα I2lm′〉 〈α I2lmαE1lm′〉 〈α I2lmαE2lm′〉〈αE1lmα I1lm′〉 〈αE1lmα I2lm′〉 〈αE1lmαE1lm′〉 0〈αE2lmα I1lm′〉 〈αE2lmα I2lm′〉 0 〈αE2lmαE2lm′〉 .Using this diagonalization, we reduced the number of power spectra needed todescribe the theoretical CMB covariance matrix from 10 to 8. These 8 non-zeroelements in the covariance matrix are shown in Figures 4.13 to 4.16. 〈α I1lmα I1lm′〉and 〈αE1lmαE1lm′〉 are almost proportional to the primary thermal signal (no Rayleighscattering included) and we call them the primary temperature and polarization sig-nal. The second eigenvalues of intensity and polarization spectra 〈α I2lmα I2lm′〉 and〈αE2lmαE2lm′〉, which are due purely to Rayleigh scattering and uncorrelated to thefirst eigenvalues, we call the Rayleigh intensity and E-polarization signal. Note thatsince intensity and E-polarization perturbations must be separately diagonalizedtheir eigenvectors are not orthogonal to each other. Thus all possible temperature-polarization cross-spectra are non-zero and present in Figures 4.15 and 4.16.53Figure 4.13: The non-zero power spectra in the Rayleigh distorted CMB co-variance matrix as a function of l. The upper and lower panels showthe first eigenvalues of intensity and polarization spectra respectivelywhich are almost proportional to the primary thermal signal.54Figure 4.14: Same as 4.13 but for the second eigenvalues of intensity (upperpanel) and polarization (lower panel). Note that the second eigenvaluesare purely Rayleigh signals which are uncorrelated to the first eigen-values.55Figure 4.15: Same as 4.13 but for temperature-polarization cross-spectra.The upper and lower panels present 〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉 as afunction of l respectively.56Figure 4.16: Same as 4.13 but for temperature-polarization cross-spectra.The upper and lower panels show 〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉 as afunction of l respectively.574.7 DetectibilityMeasurement of the Rayleigh signal is very challenging since at high frequen-cies that Rayleigh scattering becomes important, there are very few photons andvery high levels of foreground contamination including Galactic dust, and the Cos-mic Infrared Background (CIB). Yet if many high frequency channels are measuredin future CMB missions, in principle, foregrounds can be removed. The reason forthis is that the spectral shape of foregrounds are different from one another andfrom the spectral shape of the Rayleigh signal. In addition, the Rayleigh powerspectrum looks very different from all the foregrounds since it’s oscillatory and itspans the full range of scales whereas most of the foregrounds are important eitherat lower or higher l values. For example, the CIB and thermal Sunyaev-Zel’dovichhave small amplitudes at large scales but larger amplitudes at smaller scales, whileGalactic dust is important at lower l and is less so at higher l. A future CMB mis-sion that could be a candidate for detecting the Rayleigh signal is one similar to theproposed PRISM experiment [6], which has many high frequency bands with morethan 7000 detectors. In this section we take a PRISM-like experiment as an exam-ple of what capabilities a next generation CMB satellite might have and explorethe detectibility of the Rayleigh signal with this experiment.4.7.1 Signal to noise ratio of Rayleigh signalOur goal is to find the signal-to-noise ratio for the 8 non-zero elements of theCMB covariance matrix. As an example, we use the foreground removal methoddescribed in Ref. [25] and closely follow its notation. In this method, the fore-grounds are treated as an additional source of noise which is correlated betweenfrequency channels. If the frequency dependence, the scale dependence and alsothe variation in frequency dependence across sky are known for each physical com-ponent of foregrounds, this leads to a natural way of removing them.Let’s say that our experiment has F frequency channels. The F-dimensionalvectors aIlm and aElm, which are the measured multipoles at F different frequencies,are assumed to be composed of signal plus noise:ylm = Alxlm+nlm, (4.62)58ylm =(aIlmaElm), xlm =α I1lmα I2lmαE1lmαE2lm , (4.63)Al =(vI1l(ν) vI2l(ν) 0 00 0 vE1l(ν) vE2l(ν)). (4.64)Al is the 2F × 4 scan strategy matrix for a given (l,m). nlm is the sum of detectornoise and K different foreground components such as Galactic dust, synchrotonemission or CIB. The covariance matrix for the noise is obtained byNl =(NIl NIElNIEl NEl), (4.65)where NXl =∑K+1k=1 CXl (k) is a F×F matrix. CXl (k = 1) is the covariance matrix fordetector noise, and CXl (k) is the angular power spectrum for different foregroundcomponents.To see how accurately we can remove the foregrounds and measure the CMBpower spectra xlm, we need to invert the noisy linear problem of Eq. 4.62. It’sshown in Ref. [43] that the minimum-variance estimate of the xlm is x˜lm =WtlylmwhereWl = N−1l Al[AtlN−1l Al]−1=(wI1l wI2l wE1l wE2lwI′1l wI′2l wE ′1l wE ′2l). (4.66)wXil are the F-dimensional weight vectors whereα˜ Iilm = wItil aIlm+wI′til aElm,α˜Eilm = wEtil aElm+wE ′til aIlm. (4.67)The weight vectors are different for each l-value, so that at each angular scale, thefrequency channels with smaller foregrounds contribution have more weight.59The estimated solution x˜lm is unbiased such that 〈x˜lm〉= xlm and the covariancematrix of the pixel noise ε lm = x˜lm−xlm is Σlδm,m′ = 〈ε lmε tlm′〉 whereΣl = [AtlN−1l Al]−1 =(N˜Il N˜IElN˜IEl N˜El). (4.68)Here N˜Il , N˜El and N˜IEl are 2×2 cleaned power spectrum matrices of the non-cosmicsignals. The covariance matrix of our estimate x˜lm isC˜lδm,m′ = 〈x˜∗lmx˜tlm〉=(C˜Il C˜IElC˜IEl C˜El)δm,m′ , (4.69)where C˜Xl = CXl + N˜Xl is the total power spectrum in the cleaned maps. To findhow accurately we can measure any of the eight non-zero element of cosmic powerspectrum, we must compute the 8×8 Fisher matrix:Flαβ =12Tr[C˜−1l∂ C˜l∂αC˜−1l∂ C˜l∂β], (4.70)where α and β could be any of the 8 non-zero elements. Up this point, we haveused only one multipole xlm to calculate the Fisher matrix, but for each l-valuewe have (2l+1) fsky independent modes where fsky is the fraction of sky covered.Therefore the full Fisher matrix is (2l+1) fsky times what we calculated in Eq 4.70.Inverting this matrix gives the constraints on the 8 non-zero elements of the cosmiccovariance matrix.We compute this Fisher matrix for a PRISM-like experiment with the samefrequency channels between 30GHz and 800GHz as PRISM. For the noise, wechoose the resolution to be 1 arc min and the sensitivity to be 1nK for channelswith frequencies less than 500GHz and 10nK for channels with frequencies higherthan 500GHz. For the dominant foreground components, the temperature and E-polarization power spectra of Galactic dust and the temperature power spectra ofCIB, we used the power spectra given in a series of Planck papers [44–47]. Forother foreground components which are subdominant for detecting the Rayleighsignal, we used the power spectra given in Table 2 of Ref. [25]. The eight non-zero60elements and their signal-to-noise ratio for each l value as well as accumulativesignal-to-noise ratio are plotted in Figures 4.17 to 4.20.Since the power spectra 〈α I1lmα I1lm′〉, 〈αE1lmαE1lm′〉 and 〈α I1lmαE1lm′〉 are almostthe same as the primary thermal signal, their signal-to-noise ratio is huge. Forthe auto correlation of the primary temperature and E-polarization, the signal-to-noise ratio is almost equal to the cosmic variance limit up to l = 2000. Among theremaining elements, 〈α I1lmαE2lm′〉 and 〈α I2lmαE1lm′〉 have larger accumulative signal-to-noise ratios and these two are detectable for this PRISM-like experiment. Theaccumulative signal-to-noise for 〈α I1lmαE2lm′〉 is almost 5.4 and for 〈α I2lmαE1lm′〉 isaround 5.2. If we do not include the Rayleigh scattering, these two signals arezero. Detecting them to be non-zero would be an interesting and non-trivial crosscheck of the CMB physics and the assumed cosmological model.4.7.2 Constraints on Cosmological ParametersThere is independent information contained in the Rayleigh signal which mighthelp to better constrain the cosmological parameters. To show how much potentialinformation we can get from the Rayleigh signal, we consider an ideal experimentwith no foregrounds and negligible detector noise so that the signal-to-noise ratiosfor both the primary and Rayleigh signals are cosmic-variance limited. To find theconstraints on seven cosmological parameters, Ωb,Ωc,τ,ns,As,H,Yp, we calculatethe Fisher matrix using the standard equation:Fi j =lmax∑l(2l+1) fsky12Tr[C˜−1l∂ C˜l∂ piC˜−1l∂ C˜l∂ p j], (4.71)where pi and p j could be any of the seven cosmological parameters considered.The constraints on cosmological parameters for the cosmic-variance limited exper-iment are presented in Table 4.2. Note that in this calculation we only includedmoments up to lmax = 2000. In principle the extra information contained in theRayleigh sky is quite powerful. For instance, adding the Rayleigh signal poten-tially could help to improve the constraint on the helium fraction Yp by a factorof four. Furthermore, the fundamental limit on ns from the CMB only is less than10−3 which could be of interest for inflation studies.61Figure 4.17: The first eigenvalues of intensity 〈α I1lmα I1lm′〉 and polarization〈αE1lmαE1lm′〉 spectra (blue, solid) and their signal-to-noise ratio at eachl (red, dashed) as well as the accumulative signal-to-noise ratio forthe PRISM-like experiment. Note that the signal-to-noise ratio forthe temperature-polarization cross power spectrum can be negative atsome l values due to anti-correlation of the temperature and polariza-tion. However the accumulative signal-to-noise, added in quadrature,is always positive.62Figure 4.18: Same as 4.17 but for the second eigenvalues of intensity〈α I2lmα I2lm′〉 and polarization 〈αE2lmαE2lm′〉 spectra.63Figure 4.19: Same as 4.17 but for the temperature-polarization cross-spectra〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉.64Figure 4.20: Same as 4.17 but for the temperature-polarization cross-spectra〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉.65parameter values Primary Primary+RayleighPlanck+WP CV Limited CV LimitedΩbh2 0.02205 0.25657 0.10136Ωch2 0.1199 0.3570 0.1149τ 0.089 2.4033 1.0887ns 0.9603 0.2623 0.0950As 2.1955×10−9 0.4009 0.1829H 67.3 0.2667 0.0870Yp 0.24770 1.4288 0.3375Table 4.2: The percentage constraints on cosmological parameters(100σpi/pi) for a hypothetical cosmic-variance limited case withand without accounting for the Rayleigh signal. Note that althoughthe Rayleigh signal is detectable with the PRISM-like experiment, thissignal doesn’t add much constraining power for cosmological parametersas its accumulative signal-to-noise ratio is modest. The constraints onparameters with the PRISM-like experiment are nearly identical to thethird (Primary CV limited) column in this table.We also calculate how much of a constraint one can except for the PRISM-likeexperiment. In this case, although the Rayleigh signal is detectable, the Rayleighsignal adds very little constraining power for cosmological parameters as its accu-mulative signal-to-noise ratio is small.It’s also reasonable to ask how biased each cosmological parameter will be byignoring the Rayleigh scattering. These biases will move the central measured val-ues of each parameter relative to their actual values. The observed power spectrumis a sum of the primary power spectrum, Rayleigh power spectrum and generalizednoise (including foregrounds)C˜l = CPrimaryl +CRayleighl + N˜l. (4.72)To calculate the bias, we need to find the difference between expectation valueof the parameter estimator, 〈pˆi〉, and the true value pi, usingbi = 〈pˆi〉− pi = F(00)−1i j B j, (4.73)66where F(00)−1i j and B j are Fisher matrix and bias vector respectively for the powerspectrum CPl = CPrimaryl + N˜lF(00)−1i j =lmax∑l(2l+1)12Tr[CP−1l∂CPl∂ piCP−1l∂CPl∂ p j], (4.74)B j =lmax∑l(2l+1)12Tr[CP−1l∂CPl∂ p jCP−1l CRayleighl ]. (4.75)The biases (relative to standard deviation) introduced by ignoring the Rayleighscattering for the PRISM-like experiment arebi/σi = {−0.13,0.08,−0.06,−0.20,−0.02,−0.18,−0.28} (4.76)for the set of parameters {Ωb,Ωc,τ,ns,As,H,Yp}. While these potential biasesare worrisome and Rayleigh scattering should be incorporated into future analysis,they are still smaller than the forecasted constraints on each parameter.The potential constraints that could be achieved using a cosmic-variance lim-ited experiment, motivate us to consider how larger signal-to-noise measurementsmight be made.4.7.3 Improvements to signal to noise ratioThere are a few ways to improve the signal-to-noise ratio of the Rayleigh sig-nal and bring it closer to the idealized cosmic-variance limit. One is to have amore effective foreground removal method. The scheme we discussed assumes anisotropic power spectrum for each foreground component and aims to detect thesignal in the presence of foregrounds using only this knowledge. Since Rayleighscattering is more important at frequencies higher than 300GHz and at high fre-quencies the dominant foregrounds are Galactic dust and CIB, one might do abetter job at foreground removal by measuring Galactic dust and CIB maps atvery high frequency, (for example higher than 600GHz), and then extrapolatingtheir spectrum and removing them at the map level from lower frequencies suchas 300GHz or 400 GHz. While we will still be left with some residual foregroundpower spectra they should have a smaller amplitude than the original foreground67power spectra. Furthermore, as long as the Rayleigh signal in not limited by cos-mic variance, instead of probing the whole sky one could concentrate observingtime on regions of the sky where foreground contamination is less.Another way to enhance the signal-to-noise ratio is to improve the experiment.To do so, we can either reduce the detector noise by having more detectors (bettersensitivity) or by including more frequency channels so that we can model fore-grounds with higher fidelity and remove them more effectively.To examine how sensitive the signal-to-noise ratio of the Rayleigh signal isto each of these improvements, we study three cases: Case I. In the first case,we keep the specification of the experiment the same as our PRISM-like experi-ment but imagine a more effective foreground removal method. More specifically,in this case, by measuring the foregrounds at very high frequencies or optimiz-ing observation to low foreground region, we assume we can remove most of theforeground contamination from lower frequencies and are left with only 5% of theoriginal foreground spectra as residuals. Case II. In the second case we use thesame normal foreground levels but improve the specification of the experiment.For illustrating purposes we consider an extremely ambitious experiment with 50frequency channels between 30 GHz and 800 GHz and a noise in each frequencychannel of 0.01 nK. Case III. The third case is the combination of I and II.In Figures 4.21 to 4.24, we show the accumulative signal-to-noise ratios for theall eight non-zero elements of CMB covariance matrix for these improved cases.The blue, red, green and black lines are the signal-to-noise ratios respectively fora PRISM-like experiment, Case I, Case II and Case III. For example, the accumu-lative signal-to-noise ratio for the cross spectra between the primary temperaturesignal and Rayleigh E-polarization signal which was around 5 for the PRISM-likeexperiment, is amplified to 26 by improving the foregrounds removal method (CaseI), to 71 by decreasing the detector noise (Case II) and to 218 by combining CaseI and II (Case III). As can be seen from this graph, in Case III the accumulativesignal-to-noise ratio of all the power spectra are greater than 100 and could provideus with valuable information about cosmological parameters.The effects of improving the signal-to-noise ratio on cosmological parametersare illustrated in Figures 4.25 and 4.26. In Figure 4.25 we plotted the 1σ and2σ constraints on cosmological parameters using only the primary signal. Since68Figure 4.21: Accumulative signal-to-noise ratios for the first eigenvalues ofintensity 〈α I1lmα I1lm′〉 and polarization 〈αE1lmαE1lm′〉 spectra. The blue(solid), red (dashed), green (dotted) and black (dot dashed) lines are thesignal-to-noise ratios respectively for a PRISM-like experiment, forCase I: improved foregrounds removal method, for Case II : improveddetector noise, and for Case III which combines Case I and II.69Figure 4.22: Same as 4.21 but for the second eigenvalues of intensity〈α I2lmα I2lm′〉 and polarization 〈αE2lmαE2lm′〉 spectra.the signal-to-noise ratio for the primary signal is cosmic-variance limited in allthe cases considered here, the constraints on the parameters remain the same forall cases. We also show the bias introduced by ignoring the Rayleigh signal inthis Figure. In almost all the cases (save for one) the bias for each parameter isless than one sigma and only when the foreground contamination is large and thedetector noise is small, Case II, we are left with biases larger than two sigma forsome parameters.70Figure 4.23: Same as 4.21 but for the temperature-polarization cross-spectra〈α I1lmαE1lm′〉 and 〈α I1lmαE2lm′〉.71Figure 4.24: Same as 4.21 but for the temperature-polarization cross-spectra〈α I2lmαE1lm′〉 and 〈α I2lmαE2lm′〉.In Figure 4.26, we plotted the two-sigma constraints on cosmological param-eters using both the primary and Rayleigh signal and show that by improving thePRISM-like experiment, as we go through Case I, II and III, the constraints onparameters become smaller since the signal-to-noise ratio of the Rayleigh signalbecomes larger. For instance, the percentage error on Yp in case III is half theconstraint of the PRISM-like experiment.72Ypæàìòô0.02195 0.02205 0.022150.2420.2440.2460.2480.2500.2520.254æàìòô0.1190 0.1195 0.1200 0.12050.2420.2440.2460.2480.2500.2520.254Wb h2 Wc h2Ypæàìòô0.086 0.088 0.090 0.0920.2420.2440.2460.2480.2500.2520.254æàìòô0.956 0.958 0.960 0.962 0.9640.2420.2440.2460.2480.2500.2520.254Τ nsYpæàìòô3.085 3.090 3.0950.2420.2440.2460.2480.2500.2520.254æàìòô63.4 63.5 63.6 63.7 63.8 63.9 64.00.2420.2440.2460.2480.2500.2520.254lnH1010 As L HFigure 4.25: The biases and constraints on cosmological parameters thatcould potentially occur if one ignores the Rayleigh signal. The bluecontours are the one-sigma and two-sigma constraints on parametersusing only the primary signal centred at the fiducial value of the pa-rameters. The red, green, orange and black dots represent the bias in-troduced by ignoring the Rayleigh signal respectively in PRISM-likeexperiment, Case I (improving foreground removal), Case II (reducingdetector noise) and Case III (combination of both).73Figure 4.26: The two-sigma constraints on cosmological parameters by con-sidering both the primary and Rayleigh signal. The smallest and dark-est contour represents the cosmic-variance limited case. The lightercontours show the Case III, Case II, Case I and the PRISM-like exper-iment respectively as we go from smallest-darkest to largest-lightestcontours. Note that the largest contours essentially delineate the con-ventional (primary only) cosmic variance limit, and smaller contoursrepresent an improvement in parameter constraints beyond this limit.744.8 ConclusionsIn this chapter, we have calculated the effect of Rayleigh scattering on CMBtemperature and polarization anisotropies as well as the impact on cosmic struc-ture. We also have investigated the possibility of detecting the Rayleigh signal inthe CMB. A new method was introduced to account for the frequency-dependenceof the Rayleigh cross section by solving for a hierarchy of spectral distortion per-turbations, which allows for an accurate treatment of Rayleigh scattering includingits back-reaction on baryon perturbations with only a few spectral-distortion hier-archies. We have found that Rayleigh scattering modifies the distribution of matterin the Universe at the 0.3% level.Since the Rayleigh cross section is frequency-dependent, the CMB tempera-ture and polarization anisotropies depend on frequency too. For each frequency ofinterest, Rayleigh scattering reduces the Cl power spectrum at high l multipoles be-cause the visibility function shifts to lower redshifts when the Silk damping is moreimportant. For reference, at 857 GHz, the highest frequency of the Planck experi-ment, both temperature and E-polarization anisotropies decrease as much as 20%near l ∼ 1000 and at 353 GHz they decrease as much as 0.6%. Low-multipole E-polarization anisotropies increase because the visibility function shifts toward latertime when the CMB quadrupole is larger. The increase in E-polarization signal atl ∼ 50 is 35% at 857 GHz and 0.8% at 353 GHz.We showed that due to these distortions, the primary intensity and E-polarizationpower spectra at different frequencies are not perfectly correlated with each otherlike in standard treatments of the CMB. Furthermore we have found, to a very goodapproximation, we need two sets of random variables to completely describe thestatistics of primordial intensity and E-polarization patterns on the sky we observe.There is a second Rayleigh-distorted CMB sky beyond the primary CMB sky thatcontains additional information. We have determined a compressed representationof the joint power spectra of these two temperature/intensity and E-polarizationskies.Detecting the Rayleigh signal is very challenging because at high frequenciesthe number of CMB photons is low and the signal is contaminated by foregrounds.However since both the spectral shape and power spectra of the Rayleigh sky are75different from all the foregrounds, the Rayleigh signal might be detectable if manyhigh frequency channels are included in future CMB missions. We have shown thatwith a PRISM-like experiment that has many frequency bands, and using a simplepower spectrum based foregrounds removal method, the cross spectrum betweenthe primary E-polarization and Rayleigh temperature signal and the cross spec-trum between the primary temperature and Rayleigh E-polarization signal shouldbe detectable with accumulative signal-to-noise ratio of 5.2 and 5.4 respectively.Measuring the Rayleigh signal could provide powerful constraints on cosmo-logical parameters including the helium fraction and scalar spectral index. A moreambitious experiment either observing in low foreground contaminated regions orusing a more sophisticated foreground removal method might detect the RayleighCMB sky at high signal-to-noise. This would tighten CMB constraints on cosmo-logical parameters beyond what was, even in principle, previously thought possi-ble.Furthermore, as the Rayleigh opacity is more prominent at high frequencies,one may be able to use this signal to constrain other new physics such as darkmatter annihilation into Standard Model particles that inject energy into primordialgas and/or distort the high energy CMB spectrum (e.g., Ref [48]). If these effectsalter the Rayleigh signal then detecting it may provide corroborating evidence forsuch new physics. The detail of such effects and how the Rayleigh signal mightconstrain them is an interesting topic for future studies.76Chapter 5Cosmic Neutrino BackgroundAnisotropy Spectrum5.1 IntroductionStudying the Cosmic Microwave Background anisotropies provides us withdetailed information about the physical content of the Universe [2, 3]. These relicphotons have travelled to us from the last scattering surface at redshift z ' 1090and thus they can inform us about the condition of the Universe when it was only∼ 380,000 years old.The standard model also predicts a cosmic neutrino background (CNB); relicneutrinos which have free-streamed to us freely from when the Universe was only∼ 1s old. Although the direct detection of the CNB is extraordinary challenging,serious possibilities have been discussed in Ref. [49–52]. One of the recently pro-posed methods is the process of neutrino capture on a tritium target [52]. In thisprocess, a relic neutrino is captured by a tritium nucleus, and an electron and a 3Henucleus are emitted in the final state. This process can be distinguished from tri-tium β -decay background by observation of electron kinetic energies emitted froma tritium target that are above the β -decay endpoint by twice the mass of neutrinosand so is sensitive to the CNB provided the energy resolution is comparable to mν .Moreover in Ref. [53] a new method is proposed to use the annual modulation ofthe CNB by the gravitational focusing of the sun to detect its dipole moment. If77techniques like this prove successful in the future it might be possible to detectCNB anisotropies. With this renewed interest in the detection of relic neutrinos, itis interesting to consider the theory of the CNB, both in the standard model and intheories with non-standard neutrino interactions.The CNB started to form when neutrinos decoupled from the rest of the cosmicplasma. In the standard model, neutrino decoupling is assumed to happen almostinstantaneously at the temperature 1-2 MeV. In this scenario, before decouplingthe neutrinos were in contact with the rest of the plasma through weak interactions(Γ ∼ G2FT 5) and they had a Fermi-Dirac equilibrium distribution function. Afterdecoupling, when the rate of the interactions became much smaller than the Hubblerate (Γ H), the neutrino distribution function maintained its equilibrium shapewith the temperature decreasing as the inverse of scale factor Tν ∼ 1a(t) . The tem-perature of electromagnetic plasma scaled in the same way until it reached the elec-tron mass, Tγ = me ≈ 0.5 MeV. When pair annihilation occurred, the photons andbaryons were heated up relative to the neutrinos as the latter didn’t receive any sig-nificant energy from the electron-positron pairs since they were already decoupledand the ratio TγTν increased to the well known asymptotic valueTγTν' (114 )1/3 = 1.401[1, 54].However to calculate the CNB anisotropies properly, a more detailed study ofnon-instantaneous neutrino decoupling is needed. It is well known that the neu-trino plasma receives a small energy contribution from pair annihilation and itsfinal energy density is a bit higher than the instantaneous case. Since neutrinoswith higher momentum are heated more, then neutrino spectra have a momentumdependent distortion. There have been a number of papers considering the effectof non-instantaneous decoupling on the background energy density of neutrinosand photons [55–57]. For example, in the most recent paper [56], the final ratio ofphoton temperature to neutrino temperature is found to be TγTν = 1.399.The goal of this chapter is to study the impact of the non-instantaneous de-coupling on the evolution of cosmological perturbations and specifically to cal-culate the CNB anisotropy power spectrum. Some existing works have alreadyderived the massless [58] and massive [59] CNB power spectrum at only largescales assuming neutrino decoupling takes place instantaneously. The main sourceof anisotropies for the large scale modes is the Integrated Sachs-Wolf (ISW) effect78which is caused by the gravitational redshift occurring between the last scatteringsurface and the Earth. For massive neutrinos, the ISW effect is momentum de-pendent and larger than for massless neutrinos as they don’t move at the speed oflight and are eventually nonrelativistic. Also the distance from the last scatteringsurface to us is smaller for massive neutrinos [60]. Using a modified version ofCAMB [24], we rederive the low l part of the spectrum and show that for mass-less neutrinos our result is similar to Ref. [58] but we find a different result thanRef. [59] for massive neutrinos. We believe there are two main reasons for thisdifference. First, for massive neutrinos the power spectrum is momentum depen-dent and the method used in this work to average over all momenta to present thetotal power spectrum, is different than the method used in Ref.[59]. Second, in thiswork we include the effect of the variation of the distance from the last scatteringsurface for massive neutrinos, which it appears was not included in Ref.[59]. Wediscuss this further in section 5.6.In addition to the low l part of the power spectrum, we write a numerical code tosolve the Boltzmann equations and calculate the small scale part for both masslessand massive neutrinos with allowing neutrinos to oscillate. At small scales, similarto the CMB power spectrum, acoustic oscillations and silk damping are visiblein the CNB power spectrum. The smaller momentum-dependent distance fromthe last scattering surface for the massive neutrinos leads to a shift in the powerspectrum toward lower l values and makes the spectrum depend on the momentumof neutrinos.A recent study [61] showed that the CMB data is compatible with a neutrinoself-interaction strength that is orders of magnitude larger than the standard Fermiconstant which can delay the neutrino free-streaming to a much later time. Weshow how the presence of such strong self-interactions modifies the CNB powerspectrum.This chapter is organized as follows: In Section 5.2 the set of the equationsevolving the neutrino anisotropies are presented. In Section 5.3, we derive the CNBanisotropy power spectrum for massless neutrinos. The CNB anisotropy powerspectrum for massive neutrinos and a discussion on neutrino oscillation are given inSections 5.4 and 5.5 respectively. Section 5.6 presents a method for averaging overmomenta to get the total CNB power spectrum, Section 5.7 investigates the effect79of nonstandard strong self-interactions on the CNB power spectrum and Section5.8 concludes.5.2 Evolution equations for neutrino anisotropiesAs discussed above, since neutrino decoupling is non-instantaneous, neutri-nos with higher momentum will receive more energy during pair annihilation andtherefore the neutrino distribution function has a non-thermal distortion. Trackingthis non-thermal distribution function makes the calculation rather complicated.1Therefore in the first approximation, we assume that the distribution function ofneutrinos are still the thermal Fermi-Dirac distribution function:f (~x, p, pˆ, t) = [epT (t)[1+θ(~x,pˆ,t)] +1]−1, (5.1)where ~x is the position, ~p is the momentum, T is the temperature and θ is thetemperature perturbation. We assume a Universe filled with neutrinos, photons,electrons, positrons and dark matter. At the time of neutrino decoupling the pho-tons, electrons and positrons were tightly coupled. Thus we can treat them as onesingle perfect fluid with the energy density of ργe = pi215 T4γ (1+74χρ) and pressureof Pγe = pi245 T4γ (1+74χp) [62]. χρ and χp are functions of temperature and electronmass, they are equal to one when the temperature is much higher than the electronmass and electrons and positrons are relativistic and they vanish as the tempera-ture drops down and pair annihilation happens. Using the Einstein and Boltzmannequations given in chapter 2, one can solve for the evolution of perturbations.In this chapter the Newtonian Conformal gauge is used since variables in thisgauge are directly related to the physical quantities. The line element in this par-ticular gauge is:ds2 = a2(τ)[−(1+2ψ)dτ2+(1−2φ)dxidxi], (5.2)where φ and ψ are the gravitational potentials, τ is the conformal time anda(τ) is the scale factor. The phase-space neutrino distribution evolves according to1For an example of how to proceed in such an energy dependent case see Chapter 4.80the Boltzmann equation.d fdt=C[ f (~p)]. (5.3)The left-hand side of the Boltzmann equation is the full time derivative of thedistribution function and the right-hand side contains all possible collision terms.The first order part of d f/dt in Newtonian Conformal gauge can be written asd fdt= −p∂ f0∂ p[∂θ∂ t+ikµaθ − ∂φ∂ t+ikµaψ] (5.4)−p(θ −ψ)∂ (C0/p)∂ p,where k is the wavenumber of the perturbations, µ = pˆ · kˆ and C0 is the zero ordercollision term.The right hand side or the collision term can be written asC[ f (~p)] =12E(p)∫dΠqdΠq′dΠp′∑spin|M|2(2pi)4 (5.5)×δ 4(p+q− p′−q′){ f (~q′) f (~p′)[1− f (~q)][1− f (~p)]− f (~q) f (~p)[1− f (~q′)][1− f (~p′)]}.In this equation, dΠq = d3q(2pi)32E(q) is the Lorentz-invariant phase-space volume ele-ment, δ 4(p+q− p′−q′) enforces the energy-momentum conservation and M is theprocess amplitude. Calculating these integrals is extremely complicated especiallywhen we include the Pauli blocking terms. For simplicity, we assume that all theparticles involved in these interactions are massless and use Maxwell-Boltzmannstatistics for all particles during neutrino decoupling. With these assumptions wecan ignore the Pauli blocking terms while maintaining detailed balance. Note thatwe only use Maxwell-Boltzmann statistics to obtain the collision terms which arefunctions of only temperature. For the rest of the calculations we use the Fermi-Dirac distribution function for the neutrino plasma.Around the time of neutrino decoupling, the weak reactions that keep neutri-nos in contact with the rest of plasma are scattering and annihilation processes withelectrons and positrons which heat up neutrinos, and also scattering and annihila-tion processes involving only neutrinos which thermalize the neutrino distributions.81processes ∑ |M|2νe+ ν¯e→ e−+ e+ 8G2F(bwt2+awu2)νe+ ν¯e→ νi+ ν¯i 8G2Fu2νe+ e−→ νe+ e− 8G2F(aws2+bwu2)νe+ e+→ νe+ e+ 8G2F(bws2+awu2)νe+νe→ νe+νe 8G2Fs2νe+ ν¯e→ νe+ ν¯e 8G2F(4u2)νe+νi→ νe+νi 8G2Fs2νe+ ν¯i→ νe+ ν¯i 8G2Fu2Table 5.1: The weak interactions involving electron neutrinos and their cor-responding squared amplitudes.processes ∑ |M|2νi+ ν¯i→ e−+ e+ 8G2F(bwt2+ cwu2)νi+ ν¯i→ νe+ ν¯e 8G2Fu2νi+ ν¯i→ ν j + ν¯ j 8G2Fu2νi+ e−→ νi+ e− 8G2F(cws2+bwu2)νi+ e+→ νi+ e+ 8G2F(bws2+ cwu2)νi+νe→ νi+νe 8G2Fs2νi+ ν¯e→ νi+ ν¯e 8G2Fu2νi+νi→ νi+νi 8G2Fs2νi+ν j→ νi+ν j 8G2Fs2νi+ ν¯i→ νi+ ν¯i 8G2F(4u2)νi+ ν¯ j→ νi+ ν¯ j 8G2Fu2Table 5.2: The weak interactions involving µ and τ neutrinos and their cor-responding squared amplitudes.All the relevant weak interactions and corresponding squared amplitudes involvingelectron neutrinos and muon or tau neutrinos are displayed in Table 5.1 and 5.2respectively [55].The interactions involving µ and τ-neutrino are identical but different frominteractions involving electron neutrinos since electron neutrinos have both chargedand neutral current interactions. In our notation, GF is the Fermi constant, p isthe four-momentum of the incoming neutrino, q the four-momentum of the other82incoming particle, p′ the four-momentum of the outgoing neutrino or lepton and q′is the four-momentum of the other outgoing particle. s = (p+q)2 is the square ofthe center of mass energy, u=(p−q′)2 and t =(p− p′)2 are the squares of the four-momentum transfer. Also aw = (2sin2 θw +1)2 ' 2.13, bw = (2sin2 θw)2 ' 0.212and cw = (2sin2 θw−1)2 ' 0.292 where θw is the Weinberg angle.We can rewrite the first order Boltzmann equation governing the neutrinosanisotropies aspT[−p∂ f0∂ p(∂θ∂τ+ ikµθ − ∂φ∂τ+ ikµψ)− p(θ −ψ)∂ (C0)p]= 4G2FT5ν a[p∂ f 0∂ pθ(pˆ)A(p)+B(p)+C(p)+D(p)]. (5.6)Similar to Ref. [55], we categorize the terms arising from the expansion of[ f (~q′) f (~p′)− f (~q) f (~p)] into four different types of terms in equation 5.6. The “Aterms” represent the disappearing of a neutrino of momentum p and arise from allthe processes, e.g., ν(p)+e−→ ν+e−. “B terms” come from all the processes thatjust change the momentum of neutrinos from p to p′, e.g., ν(p)+e−→ ν(p′)+e−and involve an integration over θν(p′). “C terms” are similar to B terms except theyarise from interactions of electron neutrinos with µ or τ neutrinos and vice versa,e.g., νe(p)+ ν¯e → νµ(p′)+ ν¯µ . And finally “D terms” arise from interactions ofneutrinos with electrons or positrons which heat up the neutrinos and they involvean integration over θe. These terms for electron neutrinos are:83Ae(p) =1T 6ν∫dΛ[ fe0(q)(aw+bw)(u2+ s2) (5.7)+ fν0(q)(bwt2+(8+aw)u2+3s2)],Be(p) =1T 6ν∫dΛ[ fν0(p)q∂ fν0∂qθνe(qˆ)[s2+bwt2+(aw+6)u2] (5.8)−p′ ∂ fν0∂ p′θνe(pˆ′)[(aw+bw)(s2+u2) fe0(p′)+(6u2+3s2) fν0(p′)]− fν0(p′)q′ ∂ fν0∂q′ θνe(qˆ′)[s2+4u2]],Ce(p) =1T 6ν∫dΛ[ fν0(p)q∂ fν0∂qθνµ (qˆ)[2s2+2u2] (5.9)− fν0(q′)p′ ∂ fν0∂ p′ θνµ (pˆ′)[2u2]− fν0(p′)q′ ∂ fν0∂q′ θνµ (qˆ′)[2s2+4u2]],De(p) =1T 6ν∫dΛ[ fν0(p)q∂ fe0∂qθe(qˆ)[(aw+bw)s2+(aw+bw)u2] (5.10)− fe0(q′)p′ ∂ fe0∂ p′ θe(pˆ′)[bwt2+awu2]−q′ ∂ fe0∂q′θe(qˆ′)[(aw+bw)(u2+ s2) fν0(p′)+(bwt2+awu2) fe0(p′)]],where dΛ = dΠqdΠq′dΠp′(2pi)4δ 4(p+ q− p′− q′) is a nine-dimensional phase-space volume element.The µ− and τ−neutrino phase-space distribution functions are identical andfor them these four terms are:84Aµ(p) =1T 6ν∫dΛ[ fe0(q)(cw+bw)(u2+ s2) (5.11)+ fν0(q)(bwt2+(8+ cw)u2+3s2)],Bµ(p) =1T 6ν∫dΛ[ fν0(p)q∂ fν0∂qθνµ (qˆ)[2s2+bwt2+(cw+7)u2] (5.12)−p′ ∂ fν0∂ p′θνµ (pˆ′)[(cw+bw)(s2+u2) fe0(p′)+(7u2+3s2) fν0(p′)]− fν0(p′)q′ ∂ fν0∂q′ θνµ (qˆ′)[2s2+6u2]],Cµ(p) =1T 6ν∫dΛ[ fν0(p)q∂ fν0∂qθνe(qˆ)[s2+u2] (5.13)− fν0(q′)p′ ∂ fν0∂ p′ θνe(pˆ′)[u2]− fν0(p′)q′ ∂ fν0∂q′ θνe(qˆ′)[s2+2u2]],Dµ(p) =1T 6ν∫dΛ[ fν0(p)q∂ fe0∂qθe(qˆ)[(cw+bw)s2+(cw+bw)u2] (5.14)− fe0(q′)p′ ∂ fe0∂ p′ θe(pˆ′)[bwt2+ cwu2]−q′ ∂ fe0∂q′θe(qˆ′)[(cw+bw)(u2+ s2) fν0(p′)+(bwt2+ cwu2) fe0(p′)]].Evaluation of these terms Ai, Bi, Ci and Di are rather tedious and the details ofhow to calculate them are presented in the Appendix A.By Integrating the Boltzmann equation over p2d p and dividing it by∫p3d p f0(p),we can write the evolution equation for temperature perturbations:∂θν∂τ+ ikµθν − ∂φ∂τ + ikµψ+(θν −ψ/4)Qνρν= 4G2FT5ν a×[−Aˆθν + 124∫ ∞0(B(z,µ)+C(z,µ)+D(z,µ))z2dz] (5.15)where Aˆ = 5 A(p/Tν )2is just a number and Qν ≡∫d3 ppC0(p) is the zero order energytransfer. Expanding the angular dependence of perturbations in a series of Legen-dre polynomials θν = ∑(−i)l(2l+1)θν lPl(µ), one can find an evolution equationfor each θν l by multiplying the Boltzmann equation by 1(−i)l∫ 1−1dµ2 Pl(µ) and inte-85grating:θ˙νel + kl+12l+1θνel+1−kl2l+1θνel−1− φ˙δl,0−k3ψδl,1+θνelQνeρνe−δl,0ψ4Qνeρνe= 4G2FT5ν a[−Aˆeθνel−1pi3[δl,043((aw+bw+9)θνe0+8θνµ0+4(aw+bw)r−4θe0)−δl,1 23((aw+bw+9)θνe1+8θνµ1+4(aw+bw)r−4θe1)+δl,2215((aw+bw+9)θνe2+8θνµ2+4(aw+bw)r−4θe2)]+1384pi3[θνelcνe(l,r)+θνµ lcνµ (l,r)+θelce(l,r)]], (5.16)θ˙νµ l + kl+12l+1θνµ l+1−kl2l+1θνµ l−1− φ˙δl,0−k3ψδl,1+θνµ lQνµρνµ−δl,0ψ4Qνµρνµ= 4G2FT5ν a[−Aˆµθνµ l−1pi3[δl,043((cw+bw+13)θνµ0+4θνe0+4(cw+bw)r−4θe0)−δl,1 23((cw+bw+13)θνµ1+4θνe1+4(cw+bw)r−4θe1)+δl,2215((cw+bw+13)θνµ2+4θνe2+4(cw+bw)r−4θe2)]+1384pi3[θνµ ldνµ (l,r)+θνeldνe(l,r)+θelde(l,r)]], (5.17)where r= TνTγ is the ratio of neutrino temperature to photon temperature and cνe , cνµ ,ce, dνe , dνµ ,de are functions of both l and r and they are defined in the Appendix.The neutrino energy over-density δν , velocity θν and anisotropic stress σν arerelated to the first three moments of temperature perturbation: θν0 = δν4 , θν1 =13kθν and θν2 =σν2 [22]. Since the photon-electron-positron plasma is a perfectfluid, we only need two variables to describe it: θe0 =ργeTγdργe/dTγ δγe and θe1 =13kθγe. The evolution equations for δγe and θγe is found using energy conservation.These equations are similar to the ones for neutrinos except that the momentum-energy conservation implies that the collision term for the l = 0 moment must bemultiplied by ρνργe and for l = 1 must be multiplied byρν+Pνργe+Pγe where ρα and Pα arethe energy density and pressure density of α-fluid respectively .Also note that the background energy-momentum conservation implies that86Qν = Qνe +2Qνµ =−Qγe [63]. To close this set of equations and solve them, oneneed to include the zero order energy conservation equations for each fluid:ρ˙α =−3H(ρα +Pα)+Qα . (5.18)We solve this set of equations by writing a numerical code and utilizing theLSODA solver from ODEPACK [64] library which implements a variety of solversfor ordinary differential equations. The LSODA solver is based on the backwarddifferentiation formula method and automatically switches between nonstiff andstiff solvers depending on the behaviour of the problem.It’s interesting to first calculate the final value of ratio of neutrino temperatureto photon temperature TνTγ to see how much it differs from the well known value( 411)1/3 = 0.7138, i.e. how much neutrino energy density has increased due toheating caused by the pair annihilation. We plot the ratio of neutrino temperatureto photon temperature as a function of conformal time in Figure 5.1. At early timesthe neutrino temperature and photon temperature are the same, and as electron-positron annihilation happens, the photons receive most of the energy and the ratioof neutrino temperature to photon temperature decreases to the final asymptoticvalue of TνTγ = 0.7161 which is a bit higher than (411)1/3 since neutrinos share someof the heat as well.One can also express this increase in terms of an excess in the effective numbersof neutrinos, Neff, which is defined asρν =78(411)4/3Neffργ . (5.19)Therefore the effective number of neutrinos when we include the increase in neu-trino energy due to pair annihilation is Neff = 3.0399 which is consistent with theresult of Ref. [56].5.3 Neutrino power spectrumTo calculate the neutrino anisotropy power spectrum, we use the line of sightintegration approach [42]. In this approach, the temperature anisotropy can bewritten as an integral over the product of a source term and a geometrical term.87 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1e-05  0.0001  0.001  0.01T ν/Tγτ(Mpc)Figure 5.1: Ratio of neutrino temperature to photon temperature as a functionof conformal time. The final asymptotic value of this ratio is 0.7161which is a bit higher that the value in instantaneous neutrino decouplingscenario (( 411)1/3) and shows that neutrinos receive some energy fromthe pair annihilation.Defining the opacity as κ˙ =−4G2FT 5ν a, we write the solution of Equation 5.15 asθνe(k,µ,τ0) =∫ τ00dτ S˜e(k,µ,τ)eikµ(τ−τ0)−κ , (5.20)where the source term isS˜e = φ˙ − ikµψ− (Qνeρνe+ κ˙(1− Aˆe)) (5.21)×∞∑l=0(−i)l(2l+1)pl(µ)θνel +ψ4Qνeρνe−κ˙ 307pi4∫ ∞0(Be(z,µ)+Ce(z,µ)+De(z,µ))z2dz].88We can eliminate the angle µ in the integrand by integrating by parts, and thendrop the boundary terms since at τ → 0 it vanishes, and at τ = τ0 it doesn’t haveany angular dependence. Therefore we can replace µ with 1ikddτ everywhere in theintegrand. Then by multiplying the equation by 1(−i)l∫ 1−1dµ2 Pl(µ) and taking theintegral, we get the following evolution equation for each moment of the neutrinotemperature perturbation.θνel(k,τ0) =∫ τ00dτSe(k,τ) jl[k(τ0− τ)]. (5.22)The line of sight integration method helps us to have fewer coupled differentialequations since the geometrical term is just the Bessel function and the sourceterm, which depends on cosmological perturbations, can be calculated using asmall number of differential equations.We can even simplify the source term further and move the time derivatives tothe Bessel functions by integrating by parts and then use the recurrence relationsof the Bessel functions. After simplification, the source function for the electron89neutrino isSe(k,τ, l) = e−κ(φ˙ + ψ˙) (5.23)+ g(τ)[ψ− ψ4Qνeρνe κ˙− 1pi3[(9+aw+bw)(43θνe0+12θνe2)+ 8(43θνµ0+12θνµ2)+4(aw+bw)r−4(43θe0+12θe2)]+ 21pi3[(9+aw+bw)θνe1+8θνµ1+4(aw+bw)r−4θe1](− l+1x +1)− 321pi3[(9+aw+bw)θνe2+8θνµ2]((l+1)(l+2)x2− 2x−1)+ (Qνeρνe κ˙+1− Aˆe)[θνe0+3θνe1(−l+1x+1)− 5θνe2(−32(l+1)(l+2)x2+3x+1)+ 7θνe3(−52(l+1)(l+2)(l+3)x3+52l2+ l+6x2+l+6x−1)+ ...]+1384pi3[ f (0,r)+3 f (1,r)(− l+1x+1)− 5 f (2,r)(−32(l+1)(l+2)x2+3x+1)+ 7 f (3,r)(−52(l+1)(l+2)(l+3)x3+52l2+ l+6x2+l+6x−1)+ ...]].In this equation, g(τ)=−κ˙e−κ is the visibility function, the probability densityfunction for the time at which neutrinos decoupled from the rest of the plasma andf (l,r) = θνelCνe(l)+θνµ lCνµ (l)+θelCe(l). (5.24)There are two terms in the source function, one is proportional to the derivativesof gravitational potentials and the other is proportional to the visibility function.The visibility function is a very sharp function and it is only nonzero during a verybrief period around decoupling (∆τ/τ0 ' 10−7). The behaviour of this visibilityfunction is shown in Figure 5.2. After this short period, the only nonzero term in thesource function is the term proportional to φ˙ and ψ˙ which is called the IntegratedSachs-Wolfe term.Note that since the Universe is radiation-dominated at the time of neutrino902 ´10-5 5 ´10-5 1 ´10-4 2 ´10-4 5 ´10-4024681012ΤH MpcLgHΤLFigure 5.2: The visibility function as a function of conformal time. Thepeak of the visibility function occurs at τ = 5.75× 10−5Mpc or T '1.48 MeV.decoupling, the gravitational potentials for the modes that enter the horizon at thistime evolve more dramatically compared to the modes that enter the horizon at thetime of photon decoupling. Therefore we expect the ISW term to play a larger rolefor the neutrino power spectrum than for the CMB power spectrum.As discussed earlier, the visibility function is sharply peaked during decouplingand since during this time τ/τ0 is smaller than 10−7, the spherical Bessel functionjl[k(τ0−τ)] is a very smooth function compare to the visibility function. Thereforeθνel(k,τ0) = jl[k(τ0)]∫ τ00dτSe(k,τ, l) (5.25)= jl[k(τ0)]Sˆe(k, l),where the Sˆe(k, l) is just the time integral over the source function.The anisotropy spectrum can be obtained by integrating over the initial power91spectrum of the metric perturbation:Cl = 4pi∫ ∞0k2dkPψ(k)|θl(k,τ = τ0)|2 (5.26)= 4piA∫ ∞0dkkSˆ2e(k, l) j2l [kτ0]. (5.27)In this equation Pψ(k) = Akns−4 is the primordial potential fluctuation power spec-trum. For the power law index, we assumed a flat Harrison-Zeldovich spectrumns = 1. We anticipate the peaks of the power spectrum to appear at very smallscales (l > 108). In Ref. [65] it’s shown that for high Bessel index l, the integral inEq. 5.27 is equivalent toCl =piA(l+1/2)2∫ ∞0Sˆ2e [l+1/2τ0√1+ y, l]√y(1+ y)3dy (5.28)Similarly the same calculation can be done for µ and τ neutrinos:θνµ l(k,τ0) =∫ τ00dτSµ(k,τ, l) jl[k(τ0− τ)], (5.29)92whereSµ(k,τ, l) = e−κ(φ˙ + ψ˙) (5.30)+ g(τ)[ψ− ψ4Qνeρνe κ˙− 1pi3[(13+ cw+bw)(43θνµ0+12θνµ2)+ 4(43θνe0+12θνe2)+4(cw+bw)r−4(43θe0+12θe2)]+ 21pi3[(13+ cw+bw)θνµ1+4θνe1+4(cw+bw)r−4θe1](− l+1x +1)− 321pi3[(13+ cw+bw)θνµ2+4θνe2]((l+1)(l+2)x2− 2x−1)+ (Qνµρνµ κ˙+1− Aˆµ)[θνµ0+3θνµ1(−l+1x+1)− 5θνµ2(−32(l+1)(l+2)x2+3x+1)+ 7θνµ3(−52(l+1)(l+2)(l+3)x3+52l2+ l+6x2+l+6x−1)+ ...]+1384pi3[ fµ(0,r)+3 fµ(1,r)(− l+1x +1)− 5 fµ(2,r)(−32(l+1)(l+2)x2+3x+1)+ 7 fµ(3,r)(−52(l+1)(l+2)(l+3)x3+52l2+ l+6x2+l+6x−1)+ ...]],We numerically calculate the anisotropy spectrum for electron and µ or τ neutrinos.The result is shown in Figure 5.3.Similar to the CMB power spectrum, the acoustic oscillations and Silk dampingare visible in neutrino anisotropy spectrum. The acoustic oscillations arise from thecompetition between gravitational collapse and neutrino pressure which tends toerase the anisotropies. This competition leads to this pattern of peaks and troughsin the power spectrum.The Silk damping is caused by the finite thickness of the last scattering, sinceneutrinos move a mean distance during the decoupling, any perturbation on scalessmaller than this distance will be washed out, which results in damping of higherk modes, ie higher l values.In contrast to the CMB power spectrum, in the CNB spectrum the height of the931 ´ 108 2 ´ 108 5 ´ 108 1 ´ 109 2 ´ 109 5 ´ 1091101001000104llHl+1LCl2Π@Μk2DFigure 5.3: The anisotropy power spectrum for massless electron (blue) andµ or τ (red) neutrinos. Similar to the CMB power spectrum, the acousticoscillations and Silk damping are visible in this plot.peaks do not alternate because there is no non-relativistic particle during neutrinodecoupling to resemble baryons during photon last scattering.The peaks show the characteristic structure of coherent oscillations and appearat roughly lp = npi τ0rs where rs =∫ τ0 dτcs is the sound horizon at the time of decou-pling.Another feature worth noticing is that 3000 . l . 3× 108 values correspondto the modes that were outside the horizon at the time of decoupling but enteredthe horizon in the radiation-dominated era. For these modes the ISW term is thedominant term which leads to an almost flat power spectrum.5.4 Anisotropy spectrum for massive neutrinosIn the last section, we assumed that neutrinos were massless. Yet, based oncosmological constraints and measured neutrino mass differences from neutrinooscillation experiments, only one of the three species of neutrinos is potentially94relativistic and the other two must have masses greater than 10−2 eV [66]. Thusit’s important to also find the anisotropy spectrum for massive neutrinos.We start by writing the evolution equation for temperature perturbations for amassive neutrino:θ˙ν +(ikqεµ− κ˙)θν = S˜(k,τ)+∆S˜(k,τ,m). (5.31)We follow the notation of Ref. [22] where ε =√q2+m2a2 and q is the comovingmomentum. In the above equation S˜ is the source term for massless neutrinosand ∆S˜ is the difference in source between massive and massless neutrinos. Sinceduring the neutrino decoupling the temperature of the universe is much greaterthan any mass allowed for neutrinos, all the three species of CNB neutrinos arerelativistic and the only difference between massive and massless neutrinos is inthe ISW term:∆S˜ = (φ˙ − ikεqµψ)− (φ˙ − ikµψ), (5.32)which is only nonzero for large-scale modes that entered the horizon at the timewhen the neutrinos are un-relativistic.Again we use the line of sight integration approach to find θν :θν =∫dτ S˜(k,τ)eikµ(∫ τ0qε dτ′−∫ τ00 qε dτ ′) (5.33)+∫dτ[ikµψ− ikεqµψ]eikµ(∫ τ0qε dτ′−∫ τ00 qε dτ ′).Note that∫ τ0qε dτ′ = χ(τ) is the comoving distance travelled by a massive neutrinofrom big bang to τ and χ0 = χ(τ0). The distance from the last scattering surface tous (χ0− χ(τdec)) depends on both the mass of neutrinos and their momentum. InFigure 5.4. this distance is plotted as a function of mν for when the momentum ofneutrinos at present is 3Tν . Note that for masses above 10−4 eV, the last scatteringsurface is much closer to us.Following the same steps that we did for the massless neutrinos in Eq. 5.20 to9510- 5 10- 4 0.001 0.01 0.1 1100010 00050002000300015007000mΝ H eVLΧHMpchLFigure 5.4: The comoving distance travelled by a massive neutrino from thelast scattering surface to us as a function of mν for one value of theneutrino momentum (3Tν ).5.25 leads toθν l(k,τ0,q) =∫dτ(S+∆S) jl[k(χ0−χ(τ))] (5.34)= jl[k(χ0]∫dτ(S(k,τ)+∆S(k,τ)),where∆S(k,τ) =ddτ(ψε2q2)− ψ˙ (5.35)= ψ˙(ε2q2−1)+ψ 2m2aa˙q2.First note that now θl is momentum-dependent, but we can still calculate theneutrino angular power spectrum as a function of momentum.For large values of l which correspond to the modes entering the horizon in the96radiation-dominated era, all the neutrinos can be treated as massless and therefore∆S for these modes is negligible. Thus, apart from neutrino oscillation, the onlydifference for massive neutrinos would be that the distance from the last scatteringsurface for massive neutrinos are smaller than for massless neutrinos which leads toa shift in the angular power spectrum toward lower l values for massive neutrinos.For l . 107 which correspond to the modes that entered the horizon long afterneutrino decoupling, the only relevant source is the ISW term. Using a modifiedversion of CAMB [24], the angular power spectrum at low l values is plotted forthree different neutrino masses in Figure 5.5 . In this chapter we assume the normalhierarchy and the squared mass difference to be ∆m21,2 = 8×10−5 eV2 and ∆m22,3 =2.5× 10−3 eV2. Assuming that the lightest neutrino is massless then m1 = 0 eV,m2 = 0.00894 eV and m3 = 0.05894 eV. For low l values, in addition to a shifttoward lower l which is due to the fact that the distance from the last scattering formassive neutrinos is smaller, we observe a boost because of the larger ISW effectfor massive neutrinos.5.5 Neutrino oscillationNeutrinos are produced and detected in flavor eigenstates but they propagatein mass eigenstates. Since the unitary matrix that transforms from the mass basisto the flavor basis is not orthogonal, in calculating the neutrinos anisotropy powerspectrum, we must take into account that neutrinos can oscillate.We are primarily interested in the anisotropy power spectrum for an electronneutrino at the detector. For a neutrino that is travelling in the m1 mass eigenstate,we can define an effective visibility function or equivalently an effective sourcefunction:Seffm1(k,τ) = |Ue1|2Se(k,τ)+2|Uµ1|2Sµ(k,τ) (5.36)where |Ue1|2 = |〈νe|ν1〉|2 is the probability that the m1 mass eigenstate has electronflavour and |Uµ1|2 = |〈νµ |ν1〉|2 is the probability that it has muon flavour. Thenθ eff,m1,pl (k,τ0) =∫dτSeffm1(k,τ) jl[k(χm1,p0 −χm1,p(τ))] = Sˆeffm1(k) jl[kχm1,p0 ] (5.37)Note that χm1,p(τ) depends both on the mass of neutrino m1 and momentum p971 5 10 50 100 500100500100050001 ´ 1045 ´ 1041 ´ 105llHl+1LCl2Π@Μk2DFigure 5.5: The anisotropy power spectrum at low l values for three differentneutrino masses with momentum p = 0.0005 eV: m1 = 0 eV (blue,solid), m2 = 0.00894 eV (red, dashed) and m3 = 0.05894 eV (green,dotted). Since the ISW effect is larger for massive neutrinos, there is aboost at low l angular power spectrum for these neutrinos.of it at present. Using the a similar procedure as for the massless neutrinos wecan derive the equation to calculate the anisotropy power spectrum for a neutrinotravelling in one of the mass eigenstates.Ceff,mi,pl =piA(l+1/2)2∫ ∞0Sˆeffmi [l+1/2χmi,p0√1+ y]dy√y(1+ y)3(5.38)The anisotropy power spectrum for the three mass eigenstates are plotted in Figure5.6. As discussed previously, the distance from the last scattering surface is smallerfor massive neutrinos and the angular power spectrum for massive neutrinos areshifted toward lower l values.Neutrinos propagate in these mass eigenstates until they reach a detector wherethey are detected in flavour eigenstates. The effective θl(k,τ = τ0) for an electron981 ´ 108 2 ´ 108 5 ´ 108 1 ´ 109 2 ´ 109 5 ´ 1091101001000104llHl+1LCl2Π@Μk2DFigure 5.6: The anisotropy power spectrum for massive neutrinos with mo-mentum p = 0.0005 eV before they are detected. The blue (solid), red(dashed) and green (dotted) curves are the power spectra for m1 = 0 eV,m2 = 0.00894 eV and m3 = 0.05894 eV mass eigenstates respectively.neutrino at the detector isθ eff,e,pl (k,τ0) = |Ue1|2θ eff,m1,p0l + |Ue2|2θ eff,m2,pl (5.39)+|Ue2|2θ eff,m2,p0l .Then the effective anisotropy power spectrum for an electron neutrino at the detec-tor isCeff,e,pql = 4pi∫dkk2P(k)(θ eff,e,pl θeff,e,ql ) (5.40)=4pi∑3i=1 |Uei|4∫dkk2P(k)|θ eff,mi,pl |2, if p = q,4pi∑i6= j |Uei|2|Ue j|2∫dkk2P(k)(θ eff,mi,pl (k)θeff,m j,ql (k)), p 6= q..To get the last line in the above equation, we use the completeness relation of99the spherical Bessel functions [67]:∫dkk2F(k) jl[kr] jl[kr′]' pi2rδ (r− r′)F(lr), (5.41)where the assumption is that F(k) is a slowly varying function. Therefore :∫dkk2P(k)(θ eff,mi,pl (k)θeff,m j,ql (k)) (5.42)= A∫ dkkSˆeffmi (k)Sˆeffm j(k) jl[kχmi,p0 ] jl[kχm j,q0 ]∝ δ [χmi,p0 −χm j,q0 ].This condition tells us that the cross spectra between the mass eigenstates are onlynon-zero when p and q are such that χmi,p0 = χm j,q0 , in other words they are onlynon-zero when the neutrinos with momenta p and q are coming from the samedistance.In Figure 5.7 the anisotropy power spectra for an electron neutrino and a muonneutrino at the detector, Ceff,e,pl and Ceff,τ,pl are plotted for p = 0.0005 eV. Sincethe most massive mass eigenstate only contributes to µ or τ neutrinos, the effec-tive distance from the last scattering surface for these neutrinos is smaller than forelectron neutrinos and thus the anisotropy power spectrum for these neutrinos isshifted toward lower l values.5.6 Averaging over momentaWe should emphasize here that the neutrino temperature perturbation is mo-mentum dependent since the the comoving distance travelled by massive neutrinosfrom decoupling surface to us and the late time ISW effect depend on momen-tum. The CNB anisotropy spectrum plotted in Fig. 5.6 and 5.5 are the small-scaleand large-scale part of the mass eigenstates spectra for a neutrino with momentump= 0.0005 eV. If the momentum of the neutrino is lower then since the distance tothe last scattering surface becomes smaller for massive neutrinos, the spectra shifttoward even lower l values. Also lower momentum leads to larger late time ISWeffect which means the large-scale parts of the spectra enhance even more.The total temperature perturbation for a neutrino travelling in a mass eigenstate1001 ´ 108 2 ´ 108 5 ´ 108 1 ´ 109 2 ´ 109105010050010005000llHl+1LCl2Π@Μk2DFigure 5.7: The anisotropy power spectrum for an electron neutrino (blue,solid) and a muon neutrino (red, dashed) at the detector. The neutrinomomentum is p = 0.0005 eVθ eff,mil (k) then is found by averaging over momenta at the present time:θ eff,mil (k) =∫d pp2 f0(p)θ eff,mi,pl (k)∫d pp2 f0(p), (5.43)where f0(p) is the relativistic Fermi-Dirac distribution function. At the time ofdecoupling, the neutrinos follow the relativistic Fermi-Dirac distribution and sinceparticle number is conserved after decoupling, this distribution hold even whenneutrinos become non-relativistic. Then the total angular power spectrum for aneutrino travelling in a mass eigenstate is1011 ´ 108 2 ´ 108 5 ´ 108 1 ´ 109 2 ´ 109 5 ´ 10910- 60.00111000llHl+1LCl2Π@Μk2DFigure 5.8: The total anisotropy power spectra for three mass eigenstates af-ter averaging over all the momenta. The blue (solid), red (dashed)and green (dotted) curves are the power spectra for m1 = 0 eV, m2 =0.00894 eV and m3 = 0.05894 eV mass eigenstates respectively.Ceff,mil = 4piA∫ dkk|θ eff,mil (k)|2 (5.44)=4piA(∫d pp2 f0(p))2∫d pp2 f0(p)∫dqq2 f0(q)×∫ dkk[Sˆeffmi (k)]2 jl[kχmi,p0 ] jl[kχmi,q0 ]=2pi2Al3(∫d pp2 f0(p))2∫d pp4 f 20 (p)χmi,p0χ′mi,p0[Sˆeffmi (lχmi,p0)]2.To get the last line, we again used the completeness relation of the bessel functionsto carry out the integral over wave number k. χ′mi,p0 in here is the derivative of thecomoving distance with respect to the momentum.The total effective anisotropy power spectra for the three mass eigenstates for1022 5 10 20 500.1101000105llHl+1LCl2Π@Μk2DFigure 5.9: The averaged anisotropy power spectrum over all momenta atlow l values for three mass eigenstates: m1 = 0 eV (blue, solid),m2 = 0.00894 eV (red, dashed) and m3 = 0.05894 eV (green, dotted).Averaging over all momenta leads to smaller amplitude for the powerspectra at smaller scales.the small scales are plotted in Figure 5.8. The amplitude of the power spectrumfor the massive neutrinos are significantly smaller than for the massless one. Thereason for this is that the massive neutrinos with different momenta come fromdifferent distances and if the range of the distances is much larger than the mattercorrelation length (' 7Mpc) then the perturbations for different momenta are un-correlated and their contribution partly averages out giving a significantly smalleranisotropy power spectrum.For the large-scale part of the power spectrum we used a modified version ofCAMB [24] to calculate the temperature perturbations for different momenta andused Eq. 5.43 to average over all momenta to get the total power spectrum. Theresult is shown in Figure 5.9. Similar to the small-scale part, averaging over allmomenta leads to a smaller amplitude for the power spectrum at smaller scales.103The large-scale part of the anisotropy power spectra for massive neutrinos haspreviously been calculated in Ref. [59] but our result is different than theirs. Thefirst reason for this difference is that using Eq. 5.43 we averaged over momenta atthe perturbation level, which we believe is the correct way to deal with this prob-lem, but in Ref. [59], first the power spectra at different momenta are calculatedand then they are averaged over momenta. The second reason is that we have in-cluded the effect of the variation of the distance from the last scattering surface formassive neutrinos, which apparently was not included in Ref. [59], and as we dis-cussed above, neutrinos with different momenta coming from different distances isthe main reason for having smaller power spectra at smaller scales.We use Eq. 5.43 to evaluate the total power spectrum for an electron or tauneutrino at the detector. For the small-scale part of the power spectrum, since theamplitude anisotropy power spectrum for the massless neutrinos is much largerthan the massive ones, they contribute the most to the power spectrum for flavoureigenstates. In contrast, for the large-scale part the amplitude of the massive neu-trinos is significantly larger than the amplitude for the massless ones, thereforethe massive neutrinos contribute the most for flavour eigenstates. The results areshown in Figures 5.10 and 5.11.5.7 Extra neutrino interactions via an alternative FermiconstantIn previous sections we assumed the only relevant interactions for neutrinos areweak interactions in the standard model which results in neutrino free-streamingfrom the last scattering surface at the weak decoupling epoch. But a nonstan-dard neutrino self-interaction may delay the neutrino free-streaming until a muchlater time. A recent study [61] showed that the CMB data allows for a neutrinoself-interaction strength Geff that is orders of magnitude larger than the standardFermi constant. In fact Geff . 10−5 MeV−2 has almost no impact on the CMBand can delay the neutrino free-streaming until their temperature is as low as∼ 200 eV. Even an alternative cosmology with strongly self-interacting neutri-nos (Geff . 10−2 MeV−2) in which start free-streaming very close to the matter-radiation equality is compatible with CMB.1041 ´ 108 2 ´ 108 5 ´ 108 1 ´ 109 2 ´ 109 5 ´ 1090.11101001000104llHl+1LCl2Π@Μk2DFigure 5.10: The small-scale part of the anisotropy power spectrum for anelectron neutrino (blue, solid) and a muon neutrino (red, dashed) atthe detector. Note that since the amplitude of the power spectrum formassless neutrinos is significantly larger than for the massive ones,they contribute the most.To show how the neutrino anisotropy power spectrum can be affected by theneutrino self-interaction strength, we evaluated it for a larger value of Geff whichis still compatible with the CMB data. The anisotropy power spectra for masseigenstates for when the self-interaction strength is Geff ' 10−5 MeV−2 are plottedin Figure 5.12.The stronger self-interactions delay the neutrino decoupling until T ' 240 eVfor Geff ' 10−5 MeV−2 which lead to a shift for the small-scale part of the powerspectrum toward much lower l values.5.8 ConclusionIn this chapter, we have studied the anisotropies of the cosmic neutrino back-ground for both massless and massive neutrinos. Assuming the normal hierarchy1052 5 10 20 50101001000104105llHl+1LCl2Π@Μk2DFigure 5.11: The large-scale part of the anisotropy power spectrum for anelectron neutrino (blue, solid) and a muon neutrino (red, dashed) atthe detector. Note that since the amplitude of the power spectrum formassive neutrinos is significantly larger than for the massless ones,they contribute the most.for the neutrino masses, the CNB power spectrum is calculated with allowing neu-trinos to oscillate and it’s shown that there is some resemblance between neutrinopower spectrum and the usual CMB spectrum as both have acoustic oscillationsand silk damping. The acoustic oscillations are due to the competition between thegravitational collapse and pressure of relativistic neutrinos and the Silk dampingis due to the finite thickness of the last scattering surface. In contrast to the CMBpower spectrum, in the CNB spectrum the height of the peaks do not alternate be-cause there is no non-relativistic particles during neutrino decoupling to resemblebaryons during photons last scattering.The anisotropy power spectra of massless and massive neutrinos differ fromeach other in a few ways: first the ISW effect for massive neutrinos is larger thanfor the massless ones which leads to a boost in the large scales for massive neu-10610 100 1000 104 1051100104106108llHl+1LCl2Π@Μk2DFigure 5.12: The anisotropy power spectrum for massive neutrinos with mo-mentum p = 0.0005 eV with the self-interaction strength of Geff '10−5 MeV−2. The blue (solid), red (dashed) and green (dotted)curves are the power spectra for m1 = 0 eV, m2 = 0.00894 eV andm3 = 0.05894 eV mass eigenstates respectively.trinos. Also the distance from the last scattering surface for the massive neutrinosis smaller which means that the power spectrum shifts toward lower l values forthem. Moreover since this distance depends on the momentum, the anisotropypower spectrum for massive neutrinos are momentum-dependent. We also aver-aged over all the momenta to get the total anisotropy power spectra for mass eigen-states and showed that the amplitude of the power spectra for massive neutrinosare much smaller than massless ones since the massive neutrinos with differentmomenta come from different distances and because the range of the distances thatthey come from is larger than the matter correlation length, they are uncorrelatedand their contribution partly averages out and gives a much smaller amplitude.In addition we investigated the possibility of having strong nonstandard self-interactions in neutrino sector which is compatible with the current cosmological107data and showed how these strong self-interactions delay the time of neutrino de-coupling and therefore shift the small-scale part of the spectrum toward lower lvalues. While detecting these anisotropies is far beyond any technology we canimagine, in principle, detecting small-scale anisotropies in the CNB would putvery tight limits on new physics in the neutrino sector.108Chapter 6ConclusionIn this thesis we studied two projects in the pursuit of improving cosmologicalperturbation theory and therefore our knowledge about the Universe around us. Wesummarize the main results obtained in this two projects.In chapter 4 we calculated the effect of Rayleigh scattering on CMB tempera-ture and polarization anisotropies as well as the impact on cosmic structure. Thefrequency dependence of the Rayleigh cross section breaks the thermal natureof CMB temperature and polarization anisotropies and makes them frequency-dependent. We introduced a new method to capture the effect of the frequency-dependence of the Rayleigh cross section by tracking the spectral distortion pertur-bations rather than photon perturbations at a particular frequency, which allows foran accurate treatment of Rayleigh scattering including its back-reaction on baryonperturbations with only a few spectral-distortion hierarchies. We have found thatRayleigh scattering modifies the distribution of matter in the Universe at the 0.3%level.We displayed the effect of Rayleigh scattering on the CMB power spectra forfour different frequencies and showed that for each frequency of interest, Rayleighscattering reduces the Cl power spectrum at high l multipoles because the visibilityfunction shifts to lower redshifts when the Silk damping is more important. Inaddition, the shift of the visibility function toward later times leads to a boost inLow-multipole E-polarization anisotropies because the CMB quadrupole is largerat later times. For reference, at 857 GHz, the highest frequency of the Planck109experiment, both temperature and E-polarization anisotropies decrease as much as20% near l ∼ 1000 and the increase in E-polarization signal at l ∼ 50 is 35%.We also investigated the possibility of detecting the Rayleigh signal in the CMBand showed that with a future CMB mission with many high frequency channelslike a PRISM-like experiment the Rayleigh signal might be detectable. Measuringthe Rayleigh signal could provide powerful constraints on cosmological parame-ters including the helium fraction and scalar spectral index. We also investigatehow more ambitious experiments either observing in low foreground contaminatedregions or using a more sophisticated foreground removal method might detectthe Rayleigh CMB sky at high signal-to-noise which would tighten CMB con-straints on cosmological parameters beyond what was, even in principle, previouslythought possible.In chapter 5, we studied the anisotropies of the Cosmic Neutrino Backgroundradiation. We derived the angular power spectrum for both massless and massiveneutrinos with allowing them to oscillate. Similar to the CMB power spectrum,the CNB spectrum has the same peak structure which is due to the fact that radia-tion pressure from the neutrinos resists the gravitational compression into potentialwells and sets up acoustic oscillations. Also due to the finite thickness of the neu-trino last scattering surface, small scales in the power spectrum are damped.We also studied the difference between the anisotropy power spectra for mass-less and massive neutrinos and showed that since the last scattering surface for themassive neutrinos is closer to us, the locations of the peaks in the power spectrumshift to lower l values. Also the larger ISW term for massive neutrinos enhances thelarge-scale part of the power spectrum. Moreover the anisotropy power spectrumfor massive neutrinos is momentum-dependent and averaging over all momentaleads to a smaller amplitude for smaller scales since the range of distances thatmassive neutrinos are coming from is larger than the matter correlation length, andthus they are uncorrelated.In addition, we discussed the effect of extra nonstandard neutrino self-interactionsand showed that the delay in the time of neutrino decoupling caused by these ex-tra self-interactions which are allowed by the current cosmological data makes thelocations of the peaks in the power spectrum shift toward much lower l values.110Bibliography[1] S. Dodelson. Modern Cosmology. Academic Press. Academic Press, 2003.ISBN: 9780122191411. → pages ix, 4, 5, 6, 12, 13, 20, 21, 22, 78[2] Planck Collaboration, R. Adam, P. A. R. Ade, N. Aghanim, Y. Akrami,M. I. R. Alves, M. Arnaud, F. Arroja, J. Aumont, C. Baccigalupi, and et al.Planck 2015 results. I. Overview of products and scientific results. 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Imprint of Reionization on the Cosmic MicrowaveBackground Bispectrum. apj, 534:533–550, May 2000. → pages 100116Appendix Aphase-space integration fordriving neutrino evolutionequationsIn this Appendix, we calculate the terms Ai, Bi, Ci and Di in the Boltzmannequations that govern the evolution of neutrino anisotropies, equation 5.6 in chapter5.One can evaluate some of the integrals in these terms very easily since portionsof the integrand in them are Lorentz-invariant and thus some part of the integrationcan be carried out in the center of mass frame. These integrals are∫dΛs2 f0(q) = 3∫dΛt2 f0(q) = 3∫dΛu2 f0(q) =p2T 4pi3,∫dΛ f0(p)q∂ f0∂qθν(qˆ)s2 = 3∫dΛ f0(p)q∂ f0∂qθν(qˆ)t2= 3∫dΛ f0(p)q∂ f0∂qθν(qˆ)u2=−3p2T 4pi3f0(p)(43θν0+2iµθν1− 3µ2−13θν2). (A.1)For the rest of the integrals, one must carry out all of the integrations in FRWframe. For these integrals, the following method can be used: First we use the117momentum part of the energy momentum delta function to carry out the dΠq inte-gration. Then dΛ can be expressed as:dΛ=1256pi5d3q′q′δ (µ ′−µ0)d p′dµ ′dφ ′|~p−~q′| , (A.2)where µ ′ = cosθ ′, θ ′ is the angle between ~p′ and (~p−~q′), φ ′ is the angle betweenthe plane defined by p and q′ and that defined by p′ and q, and µ0 is given byµ0 =−~p.~q′+ p′(p−q′)+ pq′p′|~p−~q′| . (A.3)The limits of d p′ integration are p′− < p′ < ∞ wherep′− =|~p−~q′|+(p−q′)2. (A.4)Using this method, if we write Pl(η) = ∑ln=0 hn,lηn, the following integrals can beevaluated:∫dΛ fy0(p′)q′∂ fx0∂q′θx(qˆ′)u2 = (A.5)− p2T 5y /Tx16pi3Pl(µ)∞∑l=0(−i)l(2l+1)θell∑n=0hn,lJ1(l,n,pTy,r),∫dΛ fy0(p′)q′∂ fx0∂q′θx(qˆ′)t2 = (A.6)− p2T 5y /Tx32pi3Pl(µ)∞∑l=0(−i)l(2l+1)θell∑n=0hn,lJ2(l,n,pTy,r),∫dΛ fy0(p′)q′∂ fx0∂q′θx(qˆ′)ut = (A.7)− p2T 5y /Tx16pi3Pl(µ)∞∑l=0(−i)l(2l+1)θell∑n=0hn,lJ3(l,n,pTy,r).In here we define µ = pˆ.kˆ, z = pTν , r =TνTγand118J1(l,n,z,r) =∫ 1−1dη(1−η)2ηn∫ ∞0dxx4e−xr√z2+ x2−2xzη (A.8)×Exp[−√z2+ x2−2xzη+ z− x2],J2(l,n,z,r) =∫ 1−1dηηn∫ ∞0dxx2e−xr√z2+ x2−2xzη (A.9)×([3− v2+β 2(3v2−1)−4βv][y2+2y+2]+[2αβ (3v2−1)−4αv][1+ y]+α2[3v2−1])e−y,J3(l,n,z,r) =∫ 1−1(1−η)dηηn∫ ∞0dxx3e−xr√z2+ x2−2xzη (A.10)×[(1−βv)(1+ y)−αv]e−y,whereα =zx(1−η)√z2+ x2−2xzη , (A.11)β =z− x√z2+ x2−2xzη , (A.12)y =12(√z2+ x2−2xzη+ z− x), (A.13)v =z− xη√z2+ x2−2xzη . (A.14)Finally we are able to evaluate all the terms Ai, Bi, Ci and Di:119Ae(pTν) =13pi3(pTν)2[4(aw+bw)(TγTν)4+aw+bw+17], (A.15)Be(pTν,µ) = −(aw+bw+9)pi3(pTν)2 fν0(p)[43θνe0+2iµθνe1−3µ2−13θνe2] (A.16)+( pTν )216pi3∞∑l=0(−i)l(2l+1)θνelPl(µ)l∑n=0hn,l[8J1(l,n,z,1)+5J2(l,n,z,1)+8J3(l,n,z,1)+ r−10(aw+bw)(J1(l,n,z,r−1)+ J2(l,n,z,r−1)+2J3(l,n,z,r−1))],Ce(pTν,µ) = − 8pi3(pTν)2 fν0(p)[43θνµ0+2iµθνµ1−3µ2−13θνµ2] (A.17)+( pTν )216pi3∞∑l=0(−i)l(2l+1)θνµ lPl(µ)l∑n=0hn,l[(6J1(l,n,z,1))+2J2(l,n,z,1)+4J3(l,n,z,1)],De(pTν,µ) = −4(aw+bw)pi3(pTν)2 fν0(p)[43θνe0+2iµθνe1−3µ2−13θνe2] (A.18)+( pTν )216pi3(aw+bw)r−4∞∑l=0(−i)l(2l+1)θelPl(µ)l∑n=0hn,l[r−4(J1(l,n,pTγ,1)+12J2(l,n,pTγ,1))+ r( j1(l,n,z,r)+12J2(l,n,z,r)+2J3(l,n,z,r))],120Aµ(pTν) =13pi3(pTν)2[4(cw+bw)(TγTν)4+ cw+bw+17], (A.19)Bµ(pTν,µ) = −(cw+bw+13)pi3(pTν)2 fν0(p)[43θνµ0+2iµθνµ1−3µ2−13θνµ2] (A.20)+( pTν )216pi3∞∑l=0(−i)l(2l+1)θνµ lPl(µ)l∑n=0hn,l[11J1(l,n,z,1)+6J2(l,n,z,1)+10J3(l,n,z,1)+ r−10(cw+bw)(J1(l,n,z,r−1)+ J2(l,n,z,r−1)+2J3(l,n,z,r−1))],Cµ(pTν,µ) = − 4pi3(pTν)2 fν0(p)[43θνe0+2iµθνe1−3µ2−13θνe2] (A.21)+( pTν )216pi3∞∑l=0(−i)l(2l+1)θνelPl(µ)l∑n=0hn,l[(3J1(l,n,z,1))+ J2(l,n, ,z,1)+2J3(l,n,z,1)],Dµ(pT,µ) = −4(cw+bw)pi3(pTν)2 fν0(p)[43θνe0+2iµθνe1−3µ2−13θνe2] (A.22)+( pTν )216pi3(cw+bw)r−4∞∑l=0(−i)l(2l+1)θelPl(µ)l∑n=0hn,l[r−4(J1(l,n,pTγ,1)+12J2(l,n,pTγ,1))+ r( j1(l,n,z,r)+12J2(l,n,z,r)+2J3(l,n,z,r))].To get the Boltzmann equations governing the neutrinos anisotropies, we also121need to define the following functions:cνe(l,r) =l∑n=0hn,l[8I1(l,n,1)+5I2(l,n,1)+8I3(l,n,1) (A.23)+r−10(aw+bw)(I1(l,n,r−1)+ I2(l,n,r−1)+2I3(l,n,r−1))],cνµ (l) =l∑n=0hn,l[6I1(l,n,1)+2I2(l,n,1)+4I3(l,n,1)],ce(l,r) =l∑n=0hn,l(aw+bw)[r−9(I1(l,n,1)+12I2(l,n,1))+r(I1(l,n,r)+12I2(l,n,r)+2I3(l,n,r))],dνµ (l,r) =l∑n=0hn,l[11I1(l,n,1)+6I2(l,n,1)+10I3(l,n,1)+r−10(cw+bw)(I1(l,n,r−1)+ I2(l,n,r−1)+2I3(l,n,r−1))],dνe(l) =l∑n=0hn,l[3I1(l,n,1)+ I2(l,n,1)+2I3(l,n,1)],de(l,r) =l∑n=0hn,l(cw+bw)[r−9(I1(l,n,1)+12I2(l,n,1))+r(I1(l,n,r)+12I2(l,n,r)+2I3(l,n,r))],whereI1(l,n,r) =∫ ∞0z2J1(l,n,z,r)dz =g10[l]+g11[l]r+g12[l]r2r6, (A.24)I2(l,n,r) =∫ ∞0z2J2(l,n,z,r)dz =g20[l]+g21[l]r+g22[l]r2r6, (A.25)I3(l,n,r) =∫ ∞0z2J3(l,n,z,r)dz =g30[l]+g31[l]r+g32[l]r2r6. (A.26)Some of these coefficients,gi j, can be calculated analytically and the rest numeri-cally. The result is shown in Table A.1.122l l=0 l=1 l=2 l=3 l=4 l=5 l=6 l=7 l=8 l=9 l=10g10 160 160 86.698 85.4802 55.1955 55.2245 38.4266 39.3688 28.3084 29.8438 21.6903g11 384 -192 24.8824 -112.902-7.34499-78.3898-11.5957-59.2652-10.7893-47.2298-9.04102g12 0 0 40.1714 0 33.8388 0 27.3112 0 22.3304 0 18.5982g20 960 960 914.817 872.476 830.967 792.751 757.999 725.744 696.718 669.492 644.982g21 256 -128 -277.308-359.741-412.057-438.336-458.874-465.633-472.85 -471.336-472.246g22 0 0 81.2602 86.6739 123.392 122.681 142.786 139.207 151.415 146.591 154.511g30 160 160 130.391 118.536 101.624 93.03 81.9376 75.8088 67.9416 63.4381 57.5606g31 128 -64 -60.4575-84.7234-75.4815-80.7843-72.2895-73.0258-65.9128-65.2953-59.4069g32 0 0 81.2602 86.6739 123.392 122.681 142.786 139.207 151.415 146.591 154.511Table A.1: Value of coefficients gi j for different values of l.123

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