Large-Scale Secondary Polarization of the Cosmic Microwave Background

Elinore Roebber

Master of Science

Department of Physics

McGill University Montr´eal,Qu´ebec July 31, 2012

A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Science c 2012 Elinore Roebber ACKNOWLEDGEMENTS

Firstly, I would like to thank my supervisor Gil Holder for giving me the opportunity to do research in such an interesting field, as well as his guidance of my work, patience in answering my questions, and extensive help in review- ing this thesis. I also greatly appreciate the research opportunities Pat Scott has given me, as well as his guidance and willingness to answer questions. Sec- ondly, I would like to thank my officemates for the lively conversations, good humor, and helpful suggestions; in particular, I would like to thank Jon and Alex for their good spirits in sharing their cosmological know-how with a less- informed master’s student and Gabrielle for the many helpful (if occasionally frustrating) hours of studying together and for helping me translate the ab- stract of this thesis. Lastly, I would like to thank my friends, family, landlady Katy, and most especially Olivier for all the support and encouragement they have given me over the course of writing this thesis.

ii ABSTRACT

We present two examples of secondary effects on the large-scale polariza- tion of the cosmic microwave background (CMB). We begin with a review of the standard model of cosmology, as well as the predictions of bulk flows by the linear theory of structure growth and the mathematical formalism of polariza- tion. Following this review, we explore the impact of the epoch of reionization on the large-scale polarization of the CMB, paying particular attention to the effects of an alternate reionization history containing dark stars. Subse- quently, we derive the CMB polarization signature of large-scale bulk flows and examine its potential for detection by future experiments.

iii RESUM´ E´

Nous pr´esentons deux exemples d’effets secondaires sur la polarisation du fond diffus cosmologique (CMB) `agrande ´echelle. Nous d´ebutonsavec une revue du mod`elestandard de la cosmologie, une pr´edictiondes mouvements d’ensemble des amas de galaxies par la th´eoriede la croissance lin´eairedes structures ainsi qu’une pr´esentation du formalisme math´ematiquede la polar- isation. Suivant cette revue, nous ´etudionsl’impact de l’´epoque de r´eionisation sur la polarisation `agrande ´echelle du CMB, en se concentrant sur les effets d’un mod`elealternatif de r´eionisationincluant des ´etoilessombres. Ensuite, nous d´erivons la signature en polarisation dans le CMB engendr´eepar un mou- vement d’ensemble `agrande ´echelle et examinons la mesurabilit´ede ce signal par des exp´eriencesfutures.

iv TABLE OF CONTENTS

ACKNOWLEDGEMENTS...... ii ABSTRACT...... iii RESUM´ E´...... iv LIST OF TABLES...... vii LIST OF FIGURES...... viii 1 Introduction: Cosmology...... 1 1.1 Introduction...... 1 1.2 FLRW Cosmologies...... 2 1.3 The ΛCDM Concordance Model of Cosmology...... 8 1.4 The Cosmic Microwave Background...... 11 2 Linear Theory of Structure Formation...... 17 2.1 Linear Density Perturbations...... 17 2.2 Large-Scale Structure...... 20 2.3 Peculiar Velocities and Bulk Flows...... 24 2.4 Recent Measurements of the Local Bulk Flow...... 26 3 CMB Polarization...... 31 3.1 Stokes Parameters...... 31 3.2 Spin-Weighted Spherical Harmonics...... 35 3.3 Thomson Scattering...... 38 3.4 E & B Modes...... 45 4 Polarization from the Epoch of Reionization...... 50 4.1 The Epoch of Reionization...... 50 4.2 CMB Polarization Generated at Reionization...... 54 4.3 Alternate Reionization Histories with Dark Stars...... 57 4.4 Measuring the Effects of Dark Star Reionizations..... 61 5 Secondary Polarization from Large-Scale Bulk Flows...... 68 5.1 Quadrupole Anisotropy...... 68 5.2 Polarization Signatures of Bulk Flow Measurements.... 74

v 6 Conclusions...... 78 REFERENCES...... 81

vi LIST OF TABLES Table page 1–1 Cosmological parameters...... 10

3–1 Spherical harmonic quadrupoles...... 38

5–1 Polarization signature for large-scale bulk flows...... 75

vii LIST OF FIGURES Figure page 1–1 CMB temperature power spectrum...... 15 2–1 Matter power spectrum...... 22 2–2 ΛCDM rms bulk velocity...... 27 3–1 Stokes Q and U for linearly polarized light...... 34 3–2 Production of polarized light by Thomson scattering...... 39 3–3 Thomson scattering of a single photon...... 40 3–4 Polarization due to a quadrupole temperature anisotropy... 44 3–5 E and B Fourier modes...... 46 3–6 E and B hot and cold spots...... 47 4–1 Fiducial ionization history of the universe...... 51 4–2 Effect of reionization on the CMB EE power spectrum.... 56 4–3 Reionization with dark stars...... 59 4–4 Optical depths for dark star reionizations...... 61 4–5 Dependence of CMB polarization on reionization histories... 62 4–6 Dark star reionization with varying astrophysical parameters. 64 5–1 Frequency dependence of the kinematic quadrupole...... 70 5–2 Thomson scattering by moving electrons...... 72 5–3 A χ2 fit for large-scale bulk flows...... 76

viii CHAPTER 1 Introduction: Cosmology 1.1 Introduction

In the last few decades, cosmology has undergone a revolution: with the appearance of experiments sensitive enough to strongly test the predictions of theoretical cosmology, a great deal of progress has been made in understand- ing the fundamental nature of the universe. In particular, measurements of the cosmic microwave background (CMB) have been key in determining the composition and history of growth of the universe. As instrumentation con- tinues to improve, similarly precise measurements of the polarization of the CMB are being made. It is therefore worthwhile to make predictions for new signals that we can expect to measure in the polarization of the CMB in the coming years. This thesis treats the large-scale, late-time polarization of the cosmic mi- crowave background. In order to develop the necessary background, the first three chapters are devoted to a review of the mathematical and physical foun- dations. The first chapter reviews the basics of modern cosmology, including the generation of the CMB and some experimental results supporting the stan- dard cosmological model. The second chapter examines a particular aspect of the standard cosmological model: the physics of structure formation from lin- ear perturbation theory. The third chapter describes how the CMB becomes polarized, reviewing in detail the mathematical formalisms commonly used in studying polarized light. The following two chapters present specific examples of effects which cause large-scale late-time secondary polarization of the CMB. The fourth chapter

1 examines the consequences of an epoch of reionization on the polarization of the CMB and uses examples of reionization affected by the presence of ‘dark stars’ to demonstrate how the large-scale polarization of the CMB may be used to constrain the reionization history of the universe. The fifth chapter investigates in detail a result of structure growth: the unique polarization signature resulting from large-scale coherent bulk flows. The prospects for future detection of this signal are examined. 1.2 FLRW Cosmologies

This chapter provides the background for the rest of the thesis, sum- marizing basic cosmology as it stands today. We begin with a summary of the basic general-relativistic mathematical models for an expanding universe (the FLRW cosmologies), and follow with a summary of the standard ΛCDM model for cosmology and some of the important experimental results, with a particular emphasis on recent and future polarization experiments. The final section provides an overview of one of the most important phenomenological effects for studying cosmology: the CMB. The basic cosmological model of the universe assumes that the universe is spatially homogeneous, isotropic, and expanding. These assumptions are incorporated into the Friedmann-Lemaˆıtre-Robertson-Walker metric [1]:

 dr2  ds2 = −c2dt2 + a2(t) + r2dΩ2 . (1.1) 1 − κr2/R2

The FLRW metric describes a spatially symmetric universe which expands with time and can be described by three parameters: a(t), R, and κ. The scale factor a(t) contains all information about the expansion. The following two parameters, R, the radius of curvature, and its sign κ together characterize

2 the geometry of the universe, where   −1 if the universe is open.   κ = 0 if the universe is flat. (1.2)    +1 if the universe is closed.

The standard Friedmann-Lemaˆıtre-Robertson-Walker cosmological mod- els are based on general relativity and are governed by the Einstein field equa- tions. If we consider the Einstein equations with the FLRW metric and an energy-momentum source term, we can determine the behavior of the scale factor in terms of the matter and energy contained within the universe. The Einstein equation can be written as [1]

8πG  1  R + Λg = T − T g (1.3) µν µν c2 µν 2 µν where Rµν is the Ricci tensor made up of the FLRW metric gµν and its first and second derivatives, Λ is an arbitrary constant (the ‘cosmological constant’), G is Newton’s gravitational constant, and T is the trace of the matter-energy source term Tµν. We will model the matter and energy as a perfect fluid with pressure p and density ρ such that   −ρ 0 0 0      0 p 0 0  µ   T ν =   (1.4)  0 0 p 0      0 0 0 p in the rest frame of the fluid, and therefore

T = −ρ + 3p. (1.5)

3 Using the explicit representation of the Ricci tensor [1]

a¨ R = −3 (1.6) 00 a aa¨ + 2˙a2 + 2κ/R2 R = , (1.7) 11 1 − κr2/R2 the Einstein equation (1.3) is equivalent to the following two equations:

a¨ 4πG Λ = − (ρ + 3p) + (1.8) a 3c2 3 from the time-time component and

a˙ 2 8πG κc2 Λ = ρ − + (1.9) a 3c2 a2R2 3 the first space-space component, simplified using (1.8). This second equation could have been derived from any other spatial component of the Einstein field equations due to the symmetries of the FLRW metric. Together, (1.8) and (1.9) are the Friedmann equations, although this name is often reserved for (1.9). The Friedmann equations allow us to determine the evolution of the scale factor in terms of the geometry and matter-energy composition of the universe.

To write this explicitly, first consider that the cosmological constant term Λgµν in (1.3) is mathematically equivalent to an additional perfect fluid term (a

2 ‘vacuum energy’) with ρΛ = −Λc /8πG and pΛ = −ρΛ. Moreover, the perfect

fluid described by Tµν describes all other matter and energy in the universe, which we can now separate according to type. In particular, we expect Tµν to include two distinct components: nonrelativistic matter with density ρm (normal ‘baryonic’ matter and nonbaryonic dark matter) as well as relativistic

‘radiation’ of density ρr (photons and light, energetic neutrinos). We can then

4 write (1.9) as 2 a˙  X 8πG κc2 = ρ − (1.10) a 3c2 i a2R2 i=r,m,Λ To further simplify this, consider the equation of state for a perfect fluid:

p = wρ. (1.11)

Nonrelativistic matter has a small thermal velocity such that kT  mc2. From

1 the ideal gas law, this implies that p  ρ, so wm ≈ 0 [2]. For photons wr = 3 ; this will be approximately true for relativistic matter as well. Finally, if we consider the cosmological constant as a perfect fluid, by necessity w = −1.

The stress-energy tensor Tµν obeys local conservation of energy [1]

µ ∇µT 0 = 0. (1.12)

Using (1.12) together with (1.4) allows us to relate a to ρ and p:

a˙ 0 = −ρ˙ − 3 (ρ + p), (1.13) a with the equation of state this becomes

ρ˙ a˙ = −3(1 + w) . (1.14) ρ a

Since w is a constant for all of our simple cosmological fluids, this equation can be integrated to give the evolution of the density of each component of the fluid with the scale factor:

ρ ∝ a−3(1+w) (1.15) and therefore,

−4 −3 0 ρr ∝ a , ρm ∝ a , ρΛ ∝ a . (1.16)

5 Including this dependence in (1.10) will allow us to determine the evolu- tion of the scale factor, and thus the FLRW metric, in terms of the measured densities of matter and energy in the universe. Before we proceed to this, it is convenient to re-express the parameters of the Friedmann equation in a slightly different fashion. The first derivative of the scale factor is traditionally written in terms of the Hubble parameter a˙ H = . (1.17) a This is normalized so that at the present time, a = 1. The Hubble parameter then becomes the Hubble constant H0, which is often expressed in a unitless form as the reduced Hubble constant h:

−1 −1 H0 = 100h km s Mpc . (1.18)

The densities of the different fluid components are rescaled as

8πG ρ Ω ≡ 2 2 ρ = , (1.19) 3c H ρcrit where the critical density 3c2H2 ρ = (1.20) crit 8πG is the total density necessary to make the universe flat. Each of the Ωi’s are then defined in terms of the densities at the present time so that all time dependence is written explicitly in terms of a. The total present density is

Ω0 = Ωr + Ωm + ΩΛ, (1.21)

so that for a flat universe, Ω0 = 1. In terms of the present-day density parameters, (1.9) becomes

2 2 H Ωr Ωm κc 2 = 4 + 3 + ΩΛ − 2 2 . (1.22) H0 a a a R

6 For notational convenience, it is traditional to define a similar ‘curvature density’ for the last term in (1.22):

κc2 Ω = − , (1.23) k R2

Since Ω0 = 1 defines a flat universe,

Ω0 = Ωk − 1. (1.24)

The final form of the Friedmann equation is therefore

2 2 H 1 a˙ Ωr Ωm Ωk 2 = 2 2 = 4 + 3 + 2 + ΩΛ. (1.25) H0 H0 a a a a

With (1.25) we have a differential equation for the scale factor based on the measurable present-day densities of the radiation, matter and cosmological constant components of the universe as well as its curvature, and allowing us to determine its time evolution explicitly. It is also possible to introduce other components; this is often done for different dark energy models such as quintessence [3]. In this work we will confine ourselves to the basic components presented here. Since each component in (1.25)scales as a different power of a, for a universe that grows or contracts significantly, there will be different regimes in which the different components dominate. The behavior of the scale factor will consequently be different at different times, and complicated expansion and contraction behavior is possible. Although the scale factor provides all the necessary information about the expansion history of the universe, it is not directly experimentally measurable. It is useful to define the cosmological redshift, z, by [1,2]

λ 1 1 + z ≡ obs = . (1.26) λemit a

7 This is the physical redshift observed in photons due to the expansion of the universe: a photon emitted with wavelength λemit when the scale factor is a

−4 will lose energy as the universe expands since ρr ∝ a . When it is observed at the present time (with scale factor 1) it will have wavelength λobs. Since the redshift is an observable quantity and contains all of the same information as scale factor, it is commonly used to parameterize the expansion of the universe. We will use the redshift in this fashion throughout the rest of this work. Since a ≡ 1 at the present time, z ≈ 0 for very nearby objects and increases for objects that are farther away. In summary, the FLRW models for the universe are cosmologies that obey the FLRW metric (1.1). The unknown parameter a governing the scale of the universe can be determined at all times by the matter-energy and curvature of the universe through the Einstein field equations, which become the Friedmann equation (1.9) in an FLRW model. Knowing the equation of state for each component of the stress-energy tensor as well as its overall conservation of energy allow us to explicitly include all dependence on the scale factor in the Friedmann equation (1.25). As a result, we can describe the evolution of the universe completely if we know the present densities of the matter, radiation, and vacuum energy as well as the present curvature of the universe. 1.3 The ΛCDM Concordance Model of Cosmology

An important feature of multi-component FLRW universes is that they evolve differently according to epochs when different components dominate: since each density component evolves according to (1.16), depending on the relative scale factor, a different component will more or less dominate (1.25), affecting the rate of evolution of the scale factor. At the present, experimental evidence suggests that we are in a cosmological-constant-dominated universe, with an important matter component. Since the present radiation density is

8 negligible and the curvature is consistent with zero, Ωm + ΩΛ ≈ 1 for much of the recent history of the universe. During this era, Friedmann equation reduces to 2 H Ωm 2 = 3 + ΩΛ. (1.27) H0 a The standard model of modern cosmology, ΛCDM, is based on this FLRW model; it is named for its important cosmological constant and cold dark mat- ter components. The ΛCDM concordance model has a slightly different set of parameters from its FLRW equivalent, with six parameters in its simplest form. Spatial curvature and the radiation density are assumed to be negligible and are not included, and several parameters are added to represent cosmolog- ically significant factors not represented in the basic FLRW model. The first set of new parameters is simply a splitting of the matter component between normal, ‘baryonic’ matter (Ωb) and cold dark matter (Ωc). The two matter components are then multiplied by a factor of h2 to represent the physical

−2 matter densities; as a result of scaling by the critical density, Ωm ∝ ρmh . Furthermore, although the Hubble constant is fundamental to the Friedmann equation, since the ΛCDM parameterization incorporates a factor of h2 into the two matter components, it can be derived from the other parameters. The final three new parameters are the optical depth of the universe (τ), which will be discussed in chapter4, and two parameters governing the initial spectrum of perturbations away from homogeneity, as will be explained in chapter2. For completeness, we note here that these parameters describe the amplitude

2 (∆R) and slope (ns) of this spectrum of perturbations. Recent experimentally

9 Table 1–1: Cosmological parameters ΛCDM Other relevant parameters 2 +0.056 2 −5 Ωbh 0.02249−0.057 Ωrh 2.469 × 10 [5] 2 +0.0054 Ωch 0.1120 ± 0.0056 Ωk −0.0023−0.0056[5] +0.030 −1 −1 ΩΛ 0.727−0.029 H0 71.0 ± 2.5 km s Mpc ns 0.967 ± 0.014 YP 0.240 ± 0.006 [6] τ 0.088 ± 0.015 TCMB 2.725 ± 0.002 K [7] 2 −9 ∆R (2.43 ± 0.11) × 10 determined values of the parameters in ΛCDM are presented in the first col- umn of Table 1–1. Unless specified otherwise, all values in Table 1–1 are taken from the recommended parameter choices1 of the WMAP7 data release [4,5]. The six-parameter model is a simplification of the physical universe, ne- glecting many aspects which may be relevant for other calculations. Some pa- rameters representing these other aspects are presented in the second column of Table 1–1. The first three entries are the FLRW parameters not included in ΛCDM. The present-day radiation component is much smaller than the mat- ter and cosmological constant components, and has little effect on the growth of the universe except at very early times. The radiation density, which is related to TCMB below, is used as a fixed value for high-precision calculations. The curvature of the universe is consistent with zero. We also give the Hubble constant, which may be derived from the other parameters of ΛCDM. The next two entries of the second column of Table 1–1 represent quanti- ties which do not affect the expansion of the universe, but will be necessary for later calculations. For ΛCDM, the only distinction between different types of matter is the separation into ‘baryonic’ and ‘dark matter’ components, with

1 available on LAMBDA (the Legacy Archive for Microwave Background Data Analysis) at http://lambda.gsfc.nasa.gov/product/map/dr4/best_ params.cfm.

10 the general assumption that the non-dark matter is made up of ’baryons’, which are primarily hydrogen. Since the early universe is sufficiently hot and dense to allow nuclear fusion, there is also a fairly significant amount of he- lium [2,8]. YP gives the primordial helium fraction. The final parameter of Table 1–1 is the temperature of the cosmic microwave background, TCMB, and characterizes the radiation in the universe. We will discuss the cosmic microwave background in the next section. The rest of this work will employ the ΛCDM cosmological model, with this choice of parameters or similar values. The results derived in later chapters do not depend strongly on the values of the parameters. 1.4 The Cosmic Microwave Background

This section contains an explanation of the cosmic microwave background, a consequence of any physical model of the universe containing a hot big bang. We briefly review its temperature anisotropies and their importance to experimental cosmology. The presentation is based primarily on those by Peebles [9] and Dodelson [10]. Shortly after the big bang, the universe was very hot and dense and the baryonic matter was entirely ionized into a plasma of mainly protons and electrons, tightly coupled to the photons through Thomson scattering. As a result of the tight coupling between matter and radiation, the baryons and photons were in thermal equilibrium and the universe was optically thick to photons. As the universe expanded, it cooled sufficiently to allow protons and electrons to form neutral hydrogen according to

− p + e H + γ. (1.28)

About a quarter of the baryons take the form of helium, which recombines in a similar fashion. However, for simplicity, we will neglect the recombination

11 of helium during the following discussion. This creation of neutral atoms from the primordial plasma, is the defining process of the momentous epoch of recombination, which took place early in the history of the universe, at z ≈ 1100, when the universe was 350,000 years old, and not long after the period of matter-radiation equality. As the universe expanded, the radiation component was redshifted. Even- tually, there were not enough photons left at high enough energy to ionize the newly-formed hydrogen, and the forward direction of (1.28) dominated, caus- ing the rapid ‘recombination’ of nearly all the baryons into hydrogen and helium. See Fig. 4–1 for a graphical representation of the ionization history of the universe. There are several early ‘steps’ of baryon recombination, but most of the process takes place very suddenly after z ≈ 1100. As a major consequence of this period of recombination, few free elec- trons were left to interact with the photons, and the universe rapidly became optically thin to photons, which, decoupled from the baryons, began to free stream. These photons are the origin of the cosmic microwave background radi- ation (CMB), and carry information from their last interaction with electrons during a narrow range of redshifts, the so-called ‘surface of last scattering’. In particular, the photons in the CMB have a blackbody spectrum,

3 −2 3 0 2hν c Cx Iν = Bν = = x (1.29) exp(hν/kBTCMB) − 1 e − 1

3 for x = hν/kBTCMB, C = 2(kBTCMB) /hc, reflecting the spectrum of the hot baryon-photon plasma. Disregarding some secondary effects in the later uni- verse, all that happens to the CMB photons between last scattering and the present is that their thermal spectrum redshifts with the expansion of the uni- verse. In particular, it remains thermal, with the new associated temperature

12 TCMB defined in terms of the temperature at an early time by

TCMB = T0(1 + z). (1.30)

As a result, we observe it today as a cosmic microwave background; it was once much hotter, but has since redshifted considerably. The CMB was first discovered in 1965 by Penzias and Wilson [11], and provided the first important evidence for a hot Big Bang. This was further strongly supported when the results of COBE FIRAS [12,7] showed the CMB to be a blackbody to exquisite precision, with TCMB = 2.725 ± 0.002 K. Since background blackbody radiation is a key theoretical prediction of the Big Bang model, these results essentially ruled out the possibility of a steady- state universe. Conditions of the universe immediately prior to recombination are indelibly imprinted on the CMB, making it a powerful probe of the early universe and cosmology. This application of the CMB is outside the scope of this work, as are the conditions of the early universe; the remainder of this work will concentrate on later effects on the CMB. Because the CMB is, to extremely good accuracy, blackbody radiation, it is traditional to measure it in terms of ‘temperature units’, defined by the temperature associated with the given blackbody spectrum. Information in the CMB is encoded into fluctuations away from the isotropic radiation field. These anisotropies take the form of first-order departures from a pure black- body, thermal fluctuations ∆T , where

∂Bν Iν(TCMB) ≈ Bν(TCMB) + ∆T (1.31) ∂T TCMB  ∆T xex  = Bν(TCMB) 1 + x . (1.32) TCMB e − 1

The monopole has the blackbody frequency dependence of x3/(ex − 1) but that temperature anisotropies have an additional frequency dependence of

13 xex/(ex − 1). In the following, most discussion of the CMB will focus on anisotropies, and temperature units will be used. To describe the CMB anisotropies mathematically in terms of the their position on the sky, the temperature anisotropy field is expanded into spherical harmonics, so that X ∆T = a`mY`m. (1.33) `,m Anisotropies on different scales can then be analyzed in terms of the individual a`m’s measured. There is no way to predict the set of a`m’s that we will see, since initial conditions predict that they should be randomly distributed. Accordingly, we must use statistical measures to extract information from the anisotropy field. The most popular such statistical measure for the CMB

TT temperature fluctuations is the angular power spectrum, C` , defined by

T T ∗ TT ha`ma`0m0 i = C` δ``0 δmm0 , (1.34)

TT with the averaging occurring over m. A plot of C` as calculated by the numerical code CAMB2 is given in Fig. 1–1. Peaks in the power spectrum correspond to modes that are brighter on average, and are directly related to the physics of the baryon-photon plasma during the epoch of recombination. The era of measuring the temperature anisotropies of the CMB began with COBE DMR [13], which made the first measurements of the anisotropies across the full sky. Various balloon-borne and ground-based observations added to this throughout the 1990s, culminating in the first high-accuracy measure- ments of the temperature anisotropies by the WMAP satellite in 2003, which

2 http://camb.info

14 104 ) 2 K µ ( TT ` 103 π C 2 / + 1) ` ( `

102 10 100 1000 `

Figure 1–1: The theoretical power spectrum of the CMB temperature anisotropies. Note that small ` corresponds to large modes on the sky and vice-versa. confirmed the ΛCDM model. Since then, accurate measurements of the tem- perature power spectrum have been made at all scales; in the next few years the Planck satellite [14]is expected to improve on WMAP large-scale measure- ments. Overall, studying cosmology with the CMB temperature anisotropies has been a resounding success. Thus far we have only discussed the CMB in terms of its intensity and frequency, but photons also carry information in their polarization. Since the CMB is only weakly polarized, research in this area is much less advanced than the temperature fluctuations; however, one can do the same sorts of analyses for polarization. This is discussed in more detail in chapter3, and will be a central theme of this thesis. In fact, although research in the field of CMB polarization is still in its early stages, it is becoming increasingly popular, with many important results

15 expected soon. Notable past and current experiments investigating large scales include DASI [15], the first to make a clear detection of the CMB polariza- tion,and WMAP [16,4], which measured polarization across the sky, and was able to make the first measurement of τ from large-scale polarization. Future experiments include Planck [14], which is expected to measure the polarization power spectrum in detail. There are also many proposals for future large-scale sensitive experiments, e.g. PIXIE [17], as well as later-generation extremely sensitive ‘CMBPol’ experiments [18, 19]. The basic FLRW models of the universe presented in section 1.2 are the foundation upon which the rest of cosmology is built. From the FLRW mod- els, we get the standard working model of modern cosmology: the ΛCDM concordance model. ΛCDM has at its base an FLRW model dominated by a cosmological constant and matter. It also considers additional effects: the existence of inhomogeneities in the fluids, the different behaviors of baryonic and dark matter, and the optical depth of the universe to photons. This model has been enormously successful, having emerged as the standard choice about a decade ago, and being repeatedly confirmed since then by a great number of experiments. In particular, the properties of the anisotropies in the cos- mic microwave background have proven to be a particularly powerful probe of cosmology. Similarly, the polarization of the CMB is expected to be a valu- able resource in the near future. We will illustrate this in chapters4 and5 in which we develop two examples where the large-scale polarization of the CMB is instrumental in providing constraints.

16 CHAPTER 2 Linear Theory of Structure Formation This chapter gives an overview of the physics of structure formation from first-order perturbation theory. The departure from the perfect homogeneity of the FLRW models allows us to explain the large-scale structure we observe in the physical universe. The first two sections constitute a review of the basic mathematics of linear perturbation theory, showing how the growth of large- scale structure follows from the basic fluid equations governing perturbations in the matter field. In the third section we derive the ΛCDM predictions for pe- culiar velocity fields on very large scales as a consequence of structure growth, and in the final section we summarize some recent experimental measurements of large-scale peculiar velocities. 2.1 Linear Density Perturbations

Gravity causes matter to clump together. Since it is a purely attractive force this leads to gravitational instability: matter overdensities will always continue to grow. In an expanding universe it is more complicated: if the universe is expanding more quickly than the matter can clump together, the growth of structure can be slowed. (Pressure also prevents growth, but since the dark matter is pressureless, and the baryonic matter is only a small frac- tion of the total mass, we will neglect the effects of the baryons. The effects of pressure are unimportant on large scales). Therefore, to understand the growth of structure in our universe, it is important to understand the evo- lution of density perturbations in an expanding background. For simplicity, we will work with Newtonian gravity in an expanding background with scale

17 factor given by the Friedmann equation (1.27). This is an accurate approxi- mation in the linear regime (very weak gravitational fields) and on sub-horizon scales compared to the full general-relativistic calculation [20, 21]. Introducing linear density perturbations into our expanding background is the first step in breaking the perfect homogeneity assumed in the FLRW model in favor of a more realistic cosmological model. If we consider the case of small perturbations in a field of pressureless dark matter, the basic fluid equations in proper (physical) coordinates are given by [8, 22, 23]: the Euler equation of motion,

(∂t + u · ∇)u = −∇Φ (2.1) the mass/energy conservation equation,

(∂t + u · ∇)ρ = −ρ∇ · u (2.2) and the Poisson equation, 4πG ∇2Φ = ρ, (2.3) c2 where Φ is the gravitational potential, and u = x˙ is the velocity field.

Consider first order perturbations in the fluid so that ρ ≈ ρ0 + δρ, u ≈ u0 + δu,Φ ≈ Φ0 + δΦ. The background matter density ρ0(t) is the homogeneous and isotropic matter density introduced in section 1.2; in proper

3 coordinates it evolves as ρ0(t) ∝ a (t). Similarly, u and Φ represent the ho- mogeneous and isotropic background velocity field and gravitational potential. Substituting into the above equations and considering only first-order terms (dropping terms such as δu∇(δρ) and subtracting the zeroth-order equations)

18 gives the linearized fluid equations

(∂t + u0 · ∇)δu + (δu · ∇)u0 = −∇(δΦ) (2.4)

(∂t + u0 · ∇)δρ + δu · (∇ρ0) = −(δρ)∇ · u − ρ0∇ · (δu) (2.5) 4πG ∇2(δΦ) = δρ. (2.6) c2

These equations can be simplified by considering the homogeneity of the background, and recalling that the Hubble flow is given by u0 = Hx0. This allows us to write ∂i(u0)j = Hδij [8]. Moreover, since we are particularly concerned with matter fluctuations, we will simplify the notation by writing

δρ δ(x, t) = (2.7) ρ0

For underdensities (ρ < ρ0), δ is negative, with a minimum value of −1 at

ρ = 0. For overdensities (ρ > ρ0), δ is positive, with no upper limit, although perturbation theory only holds when |δ|  1. The linearized equations are then

(∂t + u0 · ∇)(δu) = −H(δu) − ∇(δΦ) (2.8)

(∂t + u0 · ∇)δ = −∇ · (δu) (2.9) 4πG ∇2(δΦ) = ρ δ. (2.10) c2 0

Now, we will convert to comoving coordinates, absorbing the scale fac- tor into our coordinates by x(t) = a(t)r(t), so that the new coordinates r do not expand with the universe. As a result, all our factors of a will be explic- itly written, and the background density will no longer evolve with time. In comoving coordinates, the new velocity field v defined by δu(t) = a(t)v(t) represents peculiar velocities relative to the Hubble flow. Furthermore, the time derivative for a comoving observer is given by d/dt = ∂t + u0 · ∇. The

19 fluid equations become

a˙ ∇(δΦ) v˙ + 2 v = − (2.11) a a2 δ˙ = −∇ · v (2.12) 4πG ∇2(δΦ) = ρ a2δ. (2.13) c2 0

These equations may be combined to eliminate the factors of v and δΦ, giving an expression solely in terms of δ:

a˙ 4πG δ¨ + 2 δ˙ = ρ δ. (2.14) a c2 0

This equation may be interpreted physically as allowing density perturbations to grow with the source term on the right, but are impeded by the Hubble friction term 2(˙a/a)δ˙. Recalling that δ = δ(r, t), it can be shown [9, 23,8] that the solution to (2.14) is of the form

δ(r, t) = A(r)D1(t) + B(r)D2(t), (2.15)

where D1 is a growing mode and D2 is a decaying mode. We will ignore

D2, since it will quickly decay to 0, giving δ(t) ∝ D1(t). As a result, the growth factor D1(t) gives us the time evolution of our initial density field A(x). This relation will allow us to track the growth of the matter perturbations independent of their spatial distribution. 2.2 Large-Scale Structure

The growth of perturbations under gravitational instability inevitably leads to the creation of structure–the clustering of matter into galaxies and galaxies into galaxy clusters. In order to maintain the linear approximation, this discussion will focus on larger scales than galaxy clusters, corresponding to scales larger than ∼ 10 Mpc in the density field.As the initial conditions

20 which lead to the physical realization of large scale structure are believed to be set by a random process, any probe of structure must necessarily be statistical in nature. A popular statistical measure is the autocorrelation function, which measures how density perturbations are related spatially to one another:

ξ(r) = hδ(r0)δ(r0 + r)i, (2.16) where the average is taken over r0. A second popular choice is the matter power spectrum, which is given by the Fourier transform of the autocorrelation function. Using the Fourier transform convention

1 Z δ(r) = dk δ˜(k)e−ik·r, (2.17) (2π)3 the matter power spectrum is defined by

P (k) = h|δ˜(k)|2i, (2.18) averaging over all possible values of k. The matter power spectrum is plotted in Fig. 2–1. As in the case of the CMB power spectra, the advantage of doing structure calculations in Fourier space is that it allows us to keep track of specific scales, which is relevant to the physics involved since different scales evolve independently in the linear regime. The power spectrum is an important measure in cosmology as it gives the degree to which matter is clustered on different scales. The shape given in Fig. 2–1 is the result of the evolution of different scales in the universe. A thorough explanation of the shape of the matter power spectrum is given by Dodelson [10]; we give a short of summary of the relevant features here, although a detailed description of the physics involved in producing the observed shape of the matter power spectrum is outside the scope of this work.

21 105 z = 0 z = 1 z = 5 104 3 103 Mpc) 1 − 102 ) (h k ( 1

P 10

100

10−4 10−3 10−2 10−1 100 k (h Mpc−1)

Figure 2–1: The ΛCDM matter power spectrum as calculated by CAMB at redshifts z = 0, 1, 5, showing the effects of the growth factor.

It is expected that the initial conditions of the universe would lead to

ns primordial perturbations of all scales with a spectrum given by Pin(k) ∝ k .

The exponent ns is one of the six basic parameters of the standard ΛCDM cosmological model, and is close to one. For the present-day power spectrum, this relation continues to hold at large scales. However, at k ∼ 10−2, there is a turnover in the power spectrum, at the scale corresponding to the horizon size at the time of matter-radiation equality. Modes smaller than this are suppressed, because became causal at earlier times, when the universe was radiation dominated. In the radiation-dominated universe, matter perturba- tions could not grow, but any modes that were causal could be suppressed [9,8]. Note that our calculations for the growth of perturbations are for uni- verses dominated by matter and Λ, and so do not show this behavior. The final visible feature in the matter power spectrum is the bumps seen on scales

22 just smaller than the turnover. These are due to oscillations in the baryon- photon plasma before recombination (the ‘baryon acoustic oscillations’), and are closely related to the peaks in the temperature and polarization CMB power spectra.

Since the density perturbations evolve as δ(r, z) = A(r)D1(z), the matter power spectrum (2.18) will vary with redshift proportionally to the growth factor D1. This allows us to consider the power spectrum at some redshift z:

D (z)2 P (k, z) = 1 P (k, 0), (2.19) D1(0) so that the matter power spectrum grows as the square of the density per- turbations. In order to track the growth of structure, then we will need an expression for the growth factor, which in a model where the scale factor evolves as (1.27) is given by [22,9, 24]

Z a Z ∞ 3 0 a˙ −3 5 H(z) 0 H0 (1 + z ) D1(z) = a˙ da = Ωm(z) dz 3 0 . (2.20) a 0 2 H0 z H (z )

The evolution of the matter density with redshift is given in a ΛCDM model

(assuming Ωk = 0) by

3 Ωm(1 + z) Ωm(z) = 3 . (2.21) Ωm(1 + z) + ΩΛ

In general, this is not an analytic expression. An accurate approximation for the growth factor for the matter power spectrum was derived by Carroll et al. [24], following Lahav et al. [25] as

Ω (z)   1   1  D (z) ∝ m Ω4/7(z) − Ω (z) + 1 + Ω (z) 1 + Ω (z) , (2.22) 1 1 + z m Λ 2 m 70 Λ with Ωm(z) as in (2.21) and

ΩΛ ΩΛ(z) = 1 − Ωm(z) = 3 . (2.23) Ωm(1 + z) + ΩΛ

23 Although exact expressions for the growth factor have been derived [26], the error in the approximation is less than 2%, and so this approximation will be used here for simplicity. 2.3 Peculiar Velocities and Bulk Flows

As structures form, the matter falling into the potential wells picks up a velocity. This leads to the matter developing peculiar velocities, which can be large and coherent even over relatively large volumes. These large-scale peculiar velocities away from the Hubble flow produced are often referred to as bulk flows, and are an important probe of structure growth. When the growing mode dominates, we can rewrite the mass conservation equation

(2.12) in terms of D1 and the peculiar velocities v as

D˙ ∇ · v = −aδ 1 = −aHfδ. (2.24) D1 where the dimensionless linear velocity factor f is given by

a dD f = 1 . (2.25) D1 da

This gives us a differential expression for the velocity field in terms of the growth of the density field and the scale factor. As before, an accurate ap- proximation for the velocity factor is given by [25, 24]

1  1  f ≈ Ω4/7(z) + 1 − Ω (z)[1 + Ω (z)] . (2.26) m 70 2 m Λ

Using (2.24) we can derive the ΛCDM predictions for peculiar velocities on linear scales. As in electrostatics (see for example chapter 2 of Griffiths [27]), this equation can be inverted to solve for v, giving, in proper coordinates:

afH Z x0 − x v(x) = d3x0 δ(x0). (2.27) 4π |x0 − x|3

24 To calculate the predictions for bulk flows on a certain scale, we will consider the rms velocity within a window function of the desired scale. This is given by a convolution of the velocity field with a window WR of radius R: Z 3 0 vs = d x WR(x − x )v(x). (2.28)

Rewriting using Fourier transforms, we find

Z 1 3 −ik·r vs = d k WfR(k)v˜(k)e . (2.29) (2π)3

Now, using (2.27), we have that the Fourier transform of the velocity field is

0 afH Z x − x 0 v˜(k) = d3x0 d3k δ(x0)eik·x (2.30) 4π |x0 − x|3 k = −ifHa δ˜(k). (2.31) k2

The squared mean of the smoothed velocity field is then

2 *Z 2+ 2 (−afH) 3 k ˜ hv i = d k WfR iδ(k) (2.32) (2π)6 k2 2 Z 0  (afH) 3 3 0 0 k · k ˜ ˜ 0 = d k d k WfR(k)WR(k ) δ(k)δ(k ) (2.33) (2π)6 k2k02 and using the definition of the power spectrum (2.18) gives

(aHf)2 Z 1 hv2i = d3k P (k) Wf2 (k). (2.34) s (2π)3 k2 R

Therefore, the final expression for the rms velocity smoothed with a window W is given by  2 Z 1/2 (aHf) 2 vrms = dk P (k) Wf (k) . (2.35) 2π2 R

Note that vrms is a function of both the redshift (from a, H, and f) and size of the window. A simple choice of window is given by a spherical top-hat of

25 radius R; its Fourier transform is

3j1(kR) WfR(k) = , (2.36) kR

2 where j1(x) = (sin x − x cos x)/x is the first spherical Bessel function. A graphical representation of (2.35) is given in Fig. 2–2. With the results of (2.35), we have the ΛCDM predictions for the magni- tudes of bulk flows on large scales. As seen in the first plot of Fig. 2–2, as the scales increase, we slowly return to homogeneity, and the rms bulk flow goes to zero. Peculiar velocity fields of considerable magnitude are nonetheless pre- dicted on large scales. These large-scale predictions for bulk flows constitute an important test of the ΛCDM paradigm. 2.4 Recent Measurements of the Local Bulk Flow

This section presents a summary of some recent measurements of the bulk flow, compared to the ΛCDM predictions for bulk flows derived in section 2.3. Since bulk flows measurements complement direct probes of large scale struc- ture, peculiar velocity measurements have been of great interest for a long time. The most popular techniques for measuring large-scale peculiar veloci- ties involve comparing the redshift expected for a galaxy of a given luminosity distance with its actual redshift. The peculiar velocity fields are then deter- mined by accounting for the Doppler-shift necessary to match the measured redshift. Several ‘standard candles’ are frequently used [22]: the Tully-Fisher distance measurement for spiral galaxies [28], the Faber-Jackson [29] and Fun- damental Plane [30, 31] distance measurements for elliptical galaxies, and light curves from Type Ia supernovae [32]. The latter are becoming more popular, due to their decreased error and recent increases in data collection. Other techniques often work by bypassing the need for standard candles while probing the Doppler shifts due to the bulk flows. These techniques

26 200

z = 0 150

100

RMS velocity (km/s) 50

0 0 100 200 300 400 500 600 700 800 900 1000 Radius of window (h−1Mpc) 200

R = 100 h−1Mpc 150

100

RMS velocity (km/s) 50

0 0 1 2 3 4 5 z

Figure 2–2: The ΛCDM linear theory prediction for bulk flows smoothed over a spherical square window with radius R at redshift z. Note that the magnitude of the bulk flow only changes slightly with the evolution of the scale factor, and is much smaller than the variation due to choice of scale.

27 include the direct measurement of the CMB dipole [33], thought to result from the local peculiar velocity, and measurements of the kinetic Sunyaev-Zel’dovich (kSZ) effect [34]. In chapter5 we present a new method for measuring large- scale bulk flows by indirectly measuring Doppler shifts. We will now briefly summarize some recent results, arranged by method. This list of results is not exhaustive; it has been chosen to represent the state of the field over the last several years. Additionally, due to our interest in large local bulk flows, we restrict this list to measurements on scales of at least 50 h−1Mpc, well into the linear regime. We note that although these measurements present considerable disagreement about the size of the local bulk flow, the directions measured are quite similar. • Lavaux et al. [35] reconstruct the velocity field of galaxies in the Two- Micron all-sky Redshift Survey (2MRS) using the 3k catalogue of galaxy distance measurements. They measure a velocity of ∼ 500 km/s on scales of ∼ 100 h−1Mpc towards (l, b) ≈ (230◦, 30◦), and report that their result is in tension with ΛCDM. • Nusser & Davis [36] use the SFI++ Tully-Fisher survey of spiral galaxies. They measure a bulk velocity of 257 ± 44 km/s toward (l, b) = (279◦ ± 6◦, 10◦ ± 6◦) within 100 h−1Mpc and report consistency with ΛCDM. • Feldman, Watkins, and Hudson [37] use a ‘minimum variance’ weighting scheme to compile data from ‘all major peculiar velocity surveys’, with the exception of one outlier: the results of Lauer & Postman [38] are in- consistent with the other results. These surveys use a variety of different measurement techniques, primarily Tully-Fisher, Fundamental Plane, and Type Ia SNe. The authors measure a bulk flow of 416 ± 78 km/s towards (l, b) = (282◦ ± 11◦, 6◦ ± 6◦) using a 100 h−1Mpc Gaussian win- dow. This builds on an earlier result [39] which measures a velocity of

28 407 ± 80 km/s toward (l, b) = (287◦ ± 9◦, 8◦ ± 6◦) using a 50 h−1Mpc Gaussian window. In both cases the bulk flow measured is considered to be in tension with ΛCDM predictions. • Turnbull et al. [40] use the ‘First Amendment’ compilation of Type Ia SNe within 200 h−1Mpc. They claim consistency with ΛCDM, measuring a bulk velocity of 249 ± 76 km/s towards (l, b) = (319◦ ± 18◦, 7◦ ± 14◦). • Colin et al. [41] use the Union 2 SNe dataset, which overlaps with the data used by Turnbull et al. [40], Feldman et al. [37], and Watkins et al. [39]. Within 180 h−1Mpc, they measure a bulk flow of 260 km/s towards

◦ +62◦ ◦ +34◦ (l, b) = (287 −48◦ , 21 −52◦ ), claiming tension with ΛCDM. • Dai, Kinney, and Stojkovic [42] also perform a reanalysis of the Union 2

+119 SNe dataset. They measure a velocity of 188−103 km/s towards (l, b) =

◦ +39◦ ◦ ◦ −1 (290 −31◦ , 20 ± 32 ) within 150 h Mpc, consistent with the predictions of ΛCDM. On larger scales they report that there is no indication of a bulk flow. • Weyant et al. [43] reanalyze several sets of SNe Ia data (the data overlap with the other supernova analyses). They measure 538±36 km/s towards (l, b) = (258◦ ± 10◦, 36◦ ± 11◦) using a ‘weighted least squares’ analysis, and 446 ± 101 km/s towards (l, b) = (273◦ ± 11◦, 46◦ ± 8◦) for their ‘coefficient unbiased’ method. • A series of papers from Kashlinsky et al. [44, 45, 46] measure the bulk flow from the kSZ effect produced by hot gas in galaxy clusters. They measure a bulk flow of 1000 km/s on scales of ∼ 550 h−1Mpc. This is inconsistent with ΛCDM predictions, although the significance of this measurement has been contested [47, 48, 49]. Additionally, the authors

29 propose that they may have measured a ‘dark flow’, that is, a pecu- liar velocity field extending across the horizon. Such a flow would be consistent with the theoretical predictions for a ‘tilted’ universe [50]. • The CMB dipole as measured by COBE DMR is ∆T = 3.365±0.027 mK [33]. It is generally assumed that this dipole is entirely the result of the Doppler-shifting of the CMB monopole due to our peculiar velocity with respect to the CMB. If true, this would imply that the peculiar velocity of the local group with respect to the CMB is 627 ± 22 km/s toward l = 276◦ ± 3◦, b = 30◦ ± 3◦ [33]. It is common for bulk flow surveys to attempt to reconstruct this velocity, but none have yet succeeded conclusively. In summary, due to gravitational instability, small perturbations in the matter density field inevitably grow, causing the formation of large-scale struc- ture. A consequence of this is that the clumps of matter forming structure develop peculiar velocities. The ΛCDM model predicts the existence of bulk flows even on very large scales, although the magnitude is expected to be small. Seeking to test these predictions, many research groups have attempted to measure the local bulk flow; as their results vary significantly, the possibility remains that unexpectedly large bulk flows could be a feature of the universe. Such large scale bulk flows, either as a consequence of structure growth, or due to more exotic physics, will prove to be important to us in chapter5.

30 CHAPTER 3 CMB Polarization This chapter gives a mathematical description of polarized light, with emphasis on the polarization of the CMB. The standard mathematical po- larization formalism will be discussed, including Stokes parameters, the spin- weighted spherical harmonics, and the E & B mode linear combinations of the spin-2 spherical harmonics. The generation of CMB polarization from the Thomson scattering of a quadrupolar temperature anisotropy in the CMB will also be demonstrated. 3.1 Stokes Parameters

In this section, we develop the basic mathematical formalism for describ- ing polarized light. The material presented here is primarily based on the treatments by Kosowsky [51] and Rybicki and Lightman [52]. The solutions to Maxwell’s equations in a vacuum are the electromagnetic plane waves

˜ i(k·r−ωt) E = E0e (3.1)

˜ i(k·r−ωt) B = B0e . (3.2)

The magnetic field component can be completely determined by its corre- sponding electric field. Therefore, for mathematical simplicity, we will only consider the electric part. The polarization of a transverse wave such as light describes the orienta- tion of the wave (in this case given by the electric field vector) as a function of time. Consider the the wave at a fixed position r = 0 in a basis where

31 kˆ = ˆz. A general monochromatic electromagnetic wave as in (3.1) can then be written in terms of its (complex) x and y components

iφx iφy −iωt E˜(t) = (Exe ˆx + Eye ˆy)e (3.3) such that the real part of E˜(t) is the superposition of two sinusoids:

E(t) = Ex cos(ωt − φx)ˆx + Ey cos(ωt − φy)ˆy (3.4)

Since (3.4) traces out an ellipse in the x-y plane, the light is considered to be elliptically polarized. Special cases of (3.4) include linearly polarized light:

E(t) = (Exˆx ± Ey ˆy) cos(ωt − φ), (3.5) and circularly polarized light:

E(t) = E [cos(ωt − φ)ˆx ∓ sin(ωt − φ)ˆy] . (3.6)

Elliptically polarized light, as in (3.4), may also be considered in a ge- ometric fashion, by choosing three parameters to describe the ellipse. A ge- ometrically convenient parameterization would include an overall magnitude, an angle between an axis of the coordinate system and an axis of the ellipse, and a second angle giving the helicity (whether the ellipse is traced in a clock- wise or counterclockwise fashion) as well as the ratio between the major and minor axes [52]. Although this set of ellipse parameters has a clear geometric interpreta- tion, a better choice of parameterization comes from the Stokes parameters I,

32 Q, U, and V , which have useful physical interpretations. The Stokes parame- ters are defined in terms of (3.4) by

2 2 I = Ex + Ey (3.7)

2 2 Q = Ex − Ey (3.8)

U = 2ExEy cos(φx − φy) (3.9)

V = 2ExEy sin(φx − φy) (3.10)

so that I gives the total intensity, Q the degree to which Ex and Ey differ (the orientation of the ellipse), U the degree to which they are in phase (also orientation), and V the degree to which they are 90◦ out of phase (circular- ity). Since Q and U are measurements of the orientation of the ellipse, the magnitudes of each for any particular wave depend on the coordinate system chosen. Important special cases of polarization, as before, are circularly po- larized light, defined by Q = U = 0, as in (3.6), linearly polarized light, when V = 0, as in (3.5), and unpolarized light, where Q = U = V = 0. The physical interpretation of the Stokes parameters is then: I gives the total intensity, Q and U give the linearly polarized part, and V gives the circularly polarized part. An important property of the Stokes parameters is that they are additive for the superposition of incoherent sources of light: unlike the basic geometric parameters [52, 51], they continue to be useful when we consider multiple sources of light. The definitions of the Stokes parameters have been written for monochromatic waves, but they may be adapted [52, 51] for waves that are nearly monochromatic by averaging over several periods. Varying amplitudes and phases are therefore permitted as long as the variation time is long compared to the period of the waves. In practice, this simply means that we should consider light in narrow bands.

33 (a) Positive and negative Stokes Q (b) Positive and negative Stokes U

Figure 3–1: Linearly polarized light decomposed into pure Q and U. The solid black lines represent the path the electric field vector traces in the x-y plane separately for each sign of the Stokes parameters. Note that Q and U may be transformed into each other under a 45◦ rotation of the x-y coordinate system.

Since only three parameters are necessary to completely describe an el- lipse, for purely elliptically polarized waves (i.e. no superposed unpolarized light) the Stokes parameters satisfy

I2 = Q2 + U 2 + V 2. (3.11)

If instead the light measured includes an unpolarized component, the left-hand side of the equation will increase without affecting the right and the equality will be broken. In general, then,

I2 ≥ Q2 + U 2 + V 2. (3.12)

For the rest of this work, we will primarily consider linearly polarized light, assuming that V = 0. This will allow us to work entirely in terms of Q and U. Since (3.11) relates the intensity of the polarized component to the linear polarization parameters, we can extend Q and U to temperature units following the standard conversion for intensity outlined in section 1.4.

34 3.2 Spin-Weighted Spherical Harmonics

To describe polarization on the surface of a sphere, we will need a more complicated mathematical formalism analogous to the spherical harmonics that describe scalar fields on a sphere. This section is primarily based on the treatments by Lin and Wandelt [53] and Hu and White [54]. Since the Stokes Q and U parameters both represent linearly polarized light, there is no unique decomposition of a polarized field into Q and U. Each decomposition requires a choice of coordinate system, as shown in Fig. 3–1. In particular, a rotation of the coordinate system will convert Q into U and vice-versa. Specifically, under a rotation of the x-y axes by α,

Q0 = Q cos 2α + U sin 2α (3.13)

U 0 = −Q sin 2α + U cos 2α. (3.14)

This can be more compactly expressed as

(Q ± iU)0 = e∓2iα(Q ± iU). (3.15)

Note that this relation is spin-2: because of the coefficient of α, it only requires a rotation by π for Q ± iU to return to its initial value, rather than the full 2π rotation needed for a vector field. This is a fundamental property of polarization, and can be seen from Fig. 3–1, or by considering the rotation symmetries of an ellipse. As a result, polarization calculations done in Stokes Q and U depend on the choice of coordinate system. This may desirable for some calculations or experimental measurements where there is often a natural choice of coordinate system, and so the Stokes parameters are a good choice in these circumstances. They are ill-suited for calculations such as cosmological power spectra, which are intended to be coordinate-independent.

35 In 1997, two independent research teams [55, 56] discovered a better method for full-sky calculations of CMB polarization anisotropies, namely, by expressing the polarization in terms of spin-weighted spherical harmonics [55] and tensor harmonics [56]. Although the mathematical formalisms are different, the results are equivalent. In the last fifteen years, spin-weighted spherical harmonics [55] have become the more popular choice; we will use them in the following. The spherical harmonics form a complete, orthonormal basis for scalar functions on the surface of the sphere. The spin-weighted spherical harmonics, first described by Newman and Penrose [57] and Goldberg et al. [58] in the mid-1960s, are directly analogous to the usual ‘spin-0’ spherical harmonics, forming a complete basis for spin-weighted fields on a sphere. In particular, since polarization is a spin-2 quantity, it may be represented in terms of the spin-2 spherical harmonics. In order to define the spin-weighted spherical harmonics, we first define a set of raising and lowering operators, ð and ð, which will allow us to raise and lower the spin weights of functions. We follow the treatments by Lin and Wandelt [53], Hu and White [54], and Goldberg et al. [58]; note that the sign conventions in different references vary by a factor of (−1)m. Our sign choices are chosen to be consistent with CMB polarization conventions as in Lin and Wandelt [53] and Hu and White [54].

Analogously to (3.15), for a generic function sF of spin weight s, rotating the axes by α results in a new expression for the function:

0 isα sF = e sF. (3.16)

36 After applying ð or ð, a similar rotation of the coordinate system yields

0 i(s+1)α ð(sF ) = e ð(sF ) (3.17)

0 i(s−1)α ð(sF ) = e ð(sF ), (3.18) so that the new functions ð(sF ), ð(sF ) act as functions of higher or lower spin. Specifically, acting with a raising/lowering operator on a spherical harmonic of spin weight s gives [53],

 i  ( Y ) = − sins θ ∂ + ∂ (sin−s θ Y ) ð s `m θ sin θ φ s `m p = (` − s)(` + s + 1) s+1Y`m (3.19)  i  ( Y ) = − sin−s θ ∂ − ∂ (sins θ Y ) ð s `m θ sin θ φ s `m p = − (` + s)(` − s + 1) s−1Y`m (3.20) so that we have a set of ladder relations between the spherical harmonics of different spin. As a result, we can define the desired spin-2 spherical harmonics in terms of the well-known spin-0 (scalar) ones by applying the raising or lowering operator twice. Explicitly [54], s (` − 2)!  2i 1  Y = ∂2 − cot θ∂ ± (∂ − cot θ)∂ − ∂2 Y . ±2 `m (` + 2)! θ θ sin θ θ φ sin2 θ φ `m (3.21) For reference, the spin-0 and spin-2 spherical harmonic quadrupoles have been written in Table 3–1. As in the case of the scalar spherical harmonics, the spin-weighted spheri- cal harmonics have a variety of useful properties [54, 58]. A few such properties that we will make use of later are: their complex-conjugation rule,

∗ sY`m = −sY`,−m, (3.22)

37 Table 3–1: Spherical harmonic quadrupoles Spin-0 Spin-2 1 q 15 2 2iφ 1 q 5 2 2iφ Y2,2 = 2 8π sin θ e ±2Y2,2 = 8 π (1 ∓ cos θ) e q 15 iφ 1 q 5 iφ Y2,1 = 8π sin θ cos θ e ±2Y2,1 = ± 4 π sin θ(1 ∓ cos θ)e 1 q 5 2 3 q 5 2 Y2,0 = 2 4π (3 cos θ − 1) ±2Y2,0 = 4 6π sin θ q 15 −iφ 1 q 5 −iφ Y2,−1 = − 8π sin θ cos θ e ±2Y2,−1 = ± 4 π sin θ(1 ± cos θ)e 1 q 15 2 −2iφ 1 q 5 2 −2iφ Y2,−2 = 2 8π sin θ e ±2Y2,−2 = 8 π (1 ± cos θ) e their orthonormality condition,

Z ∗ dΩ sY`m sY`0m0 = δ``0 δmm0 , (3.23) and their relationship to the Wigner rotations,

r 4π D` (φ, θ, ψ) = Y (θ, φ)e−isψ, (3.24) −s,m 2` + 1 s `m

` where the Wigner rotations Dsm(φ, θ, ψ) represent a rotation by the Euler angles φ (measured from the x-axis), θ (measured from the z-axis), and ψ (measured from the y-axis). 3.3 Thomson Scattering

Thomson scattering of photons off free electrons is an important astro- physical mechanism for producing polarized light and the primary mechanism by which the CMB becomes polarized [51, 54]. Since free electrons are required for the interaction, cosmological polarization is generated during epochs in which there are a large number of free electrons. In particular, these are the surface of last scattering (see section 1.4), when the universe first becomes op- tically thin to photons but recombination has not yet finished, and the epoch of reionization (described in greater detail in section 4.1), which occurs at much later times, after the first stars begin producing enough high-energy photons to ionize the intergalactic medium.

38 Figure 3–2: A single Thomson scattering of unpolarized light into polarized light. The initial polarization of the light can be separated into components parallel and perpendicular to the outgoing wavevector. The perpendicular component is scattered and the parallel component is not, so the resulting light is polarized.In this case, the resulting light is completely polarized, since the scattering occurs at a 90◦ angle. In general it would be only partially polarized.

The process of Thomson scattering is the simplest case of a photon scat- tering off a free electron; unlike Compton scattering, the photon energy does not change. The cross section for this interaction is polarization-dependent, so it can create polarized light. Due to the geometry of the interaction, Thomson scattering is incapable of generating circular polarization, so the resulting light will be linearly polarized. To see why the cross section is dependent on the linear polarization of the photon, consider an interaction between an electromagnetic wave with an electron. As the incoming wave interacts with the electron, it shakes it along the direction of its electric field vector, which corresponds to the direction in which the light is polarized. Moving the electron up and down creates an oscillating dipole moment, leading to radiation. Since the dipole is aligned with the initial polarization of the electromagnetic wave, the outgoing radia- tion is affected by the initial polarization. Consequently, the cross section for

39 Figure 3–3: Thomson scattering of a single photon into the ˆz direction, with 0 ˆ initial polarized component ˆ1. The outgoing photon has k = ˆz and ˆ1 = ˆy. The polarized scattering illustrated can represent either the only polarized component, as in (3.25), or, for initially unpolarized light, the component chosen in the plane of scattering.

Thomson scattering is polarization-dependent. For initially polarized light, the expression for the differential cross section is given by dipole radiation [52]:  dσ  3σ 3σ = T sin2 Θ = T cos2 θ (3.25) dΩ pol 8π 8π 2 2 −24 2 where σT = 8πe /3mc ≈ 0.665 × 10 cm is the Thomson cross section, Θ is the angle between the initial polarization direction ˆ0 and the outgoing direction kˆ, and θ = π/2−Θ is the angle between the directions of the incoming and outgoing photons (kˆ0 and kˆ). See Fig. 3–3. Since we are interested in the polarization signal imprinted on the CMB, we must investigate how Thomson scattering produces polarized light from unpolarized light. Incoming unpolarized light can be considered as the super-

0 0 position of two polarized waves, with arbitrary polarization directions ˆ1, ˆ2,

40 0 0 0 0 as long as ˆ1 ⊥ ˆ2. If we choose ˆ1 to lie in the scattering plane, ˆ2 will be 0 ˆ perpendicular to the scattering plane. Therefore, ˆ2 ⊥ k, and Θ2 = π/2. The

0 scattering of the first polarized component, ˆ1, can then be treated as before, with Θ1 = Θ = π/2 − θ. Having this, we can derive the Stokes parameters for the outcome of Thomson scattering of initially unpolarized light. Consider a scattering in the y-z plane, with outgoing direction ˆz. We will write the outgoing light in the basis of its polarization vectors ˆ1 = ˆy and ˆ2 = ˆx. Similarly, the incoming light

0 will be expressed in the basis of its polarization vectors ˆ1 = ˆy cos θ + ˆz sin θ

0 and ˆ2 = ˆx. If the incoming light of intensity I0 is unpolarized, we may write r I0 E0 = (ˆ0 cos ωt + ˆ0 sin ωt) 2 1 2 r I0 = (ˆx sin ωt + ˆy cos θ cos ωt + ˆz sin θ cos ωt). (3.26) 2

From (3.25), the scattered light has components

 1/2 X 3σT E = |ˆ0 · ˆ |2 E0 ˆ (3.27) i 8π j i j i j so that

r 3σ E = I0 T (ˆ cos θ cos ωt + ˆ sin ωt) 16π 1 2 r 3σ = I0 T (ˆx sin ωt + ˆy cos θ cos ωt). (3.28) 16π

Using the definition of the Stokes parameters (3.7)–(3.10), the light resulting from a single scattering of initially unpolarized light with intensity I0 can be

41 written as

3σ I = T I0(1 + cos2 θ) (3.29) 16π 3σ Q = T I0 sin2 θ (3.30) 16π U = 0 (3.31)

V = 0 (3.32)

The resulting light is partially polarized, with the degree of polarization de- pendent on the direction of the incoming light. Since V = 0, the polarization is linear. From our choice of coordinates, there is no U component, and the polarization is entirely Stokes Q. To measure the polarization induced in the CMB, we will need to extend the calculations for a single scattering to the scattering of an initial radiation field across the entire sky. Furthermore, for the whole-sky calculation, we will need to choose a consistent coordinate system for Q and U and represent the polarization in terms of the spin-2 spherical harmonics. From (3.30) and (3.31), we can express the total polarization generated by a single incoming wave in the y-z plane by

3σ Q ± iU = T I0 sin2 θ. (3.33) 16π

We use (3.15) to change to a general coordinate system where the incident photon is at some angle φ from the z axis, giving

3σ Q ± iU = T I0 sin2 θ e∓2iφ. (3.34) 16π

42 Integrating over all incoming radiation gives the outgoing Stokes param- eters for light scattered from the whole sky into the ˆz direction:

3σ Z I(ˆz) = T dΩ (1 + cos2 θ)I0(θ, φ) (3.35) 16π 3σ Z Q(ˆz) ± iU(ˆz) = T dΩ sin2 θe∓2iφI0(θ, φ) (3.36) 16π

Now, expanding the incident radiation field in spherical harmonics so that

0 P I (θ, φ) = a`mY`m, the previous set of equations becomes Z   σT 1 X I(ˆz) = √ dΩ Y + √ Y a Y (3.37) 2 π 00 20 `m `m 4 5 `,m r 3 Z X Q(ˆz) ± iU(ˆz) = σ dΩ Y a Y . (3.38) T 40π 2,∓2 `m `m `,m

Using the orthonormality of spherical harmonics, we can write the resulting Stokes parameters in terms of the moments of the initial radiation field. In particular, note that (3.38) becomes

r 3 Q(ˆz) ± iU(ˆz) = σ a , (3.39) T 40π 2,∓2 so that the polarized component of the outgoing radiation is entirely generated by a quadrupole moment in the initial field. An illustration of how linear polarization results from a temperature quadrupole is presented in Fig. 3–4. The result in (3.39) only holds for outgoing light aligned with the z-axis, but polarization will be generated by a quadrupole in any coordinate system. To characterize the polarization seen by an observer across the sky, it will be helpful to calculate the polarization produced by an arbitrary direction for some initial radiation field. Following Kosowsky [51], to find the polariza- tion generated in the direction (θ, φ), we will rotate (3.39) so that ˆz points along (θ, φ) and find the new quadrupole (˜a22) that will generate polarized

43 Figure 3–4: Polarization produced by an initial quadrupole temperature anisotropy scattering off a single electron. The resulting light has a net ver- tical polarization since light coming from the sides has a greater amplitude than the top or bottom. This the multiple-scattering equivalent to Fig. 3–2. Figure adapted from ref. [59]. light in this direction. As an example, consider an incoming radiation field independent of φ. The more general case will be given in chapter5. For an azimuthally-symmetric initial radiation field, a generic outgoing direction can be chosen by rotating the initial coordinate system through an Euler angle θ measured from the z-axis. Using the relationship between Eu- ler angles, Wigner rotations, and spin-weighted spherical harmonics given by (3.24), we find that the moments of the incoming radiation field in the rotated system are related to their unrotated counterparts by

m Z X ` ∗ ∗ 0 a˜`m = Dm0m(θ, 0, 0) dΩ Y`m0 I (θ, φ). (3.40) m0=−m

44 As before, we will expand the initial radiation field in spherical harmonics, using the orthonormality of the spherical harmonics to simplify. In particular, the quadrupoles that generate polarization, as in (3.39), become √ r4π 6 a˜ = D∓2 ∗(θ) a = Y (θ) a = sin2 θ a (3.41) 2,∓2 2,0 20 5 ±2 20 20 4 20 so that the polarized light emitted at an angle θ to the z-axis is given by

3σT 2 Q(θ) ± iU(θ) = √ sin θ a20. (3.42) 8 5π

The initial (unrotated) intensity quadrupole moment a20 generates all the polarized light emitted. Note that since we have assumed an azimuthally symmetric initial radiation field, a20 represents the total initial quadrupole signal and the resultant polarization is entirely Stokes Q. 3.4 E & B Modes

Using the formalism developed in the previous sections, we can now apply the description of polarized light on a sphere to the CMB. This will allow us to develop the standard coordinate-independent polarization basis originally proposed by Zaldarriaga and Seljak [55]. Our presentation will follow their original work and the review by Lin and Wandelt [53]. The first step in choosing the new basis is to expand the polarization along some direction ˆn in spin-2 spherical harmonics, so that

X Q(ˆn) ± iU(ˆn) = ±2a`m ±2Y`m(ˆn). (3.43)

We will then use the raising and lowering operators defined in (3.19) and (3.20) on this expansion to produce spin-zero quantities, which can be mathemat- ically treated like the temperature field. We can construct two independent

45 (a) E mode polarization (b) B mode polarization

Figure 3–5: E and B polarization for a single plane wave Fourier mode. Note that the polarization in E is alternately parallel and perpendicular to the direction of the wavevector, whereas the polarization in B is at ±45◦ from kˆ. Figure adapted from ref. [60]. linear combinations of these quantities, which will form our new spin-0 polar- ization basis, the E and B modes:

1 h 2 i E(ˆn) ≡ − (Q + iU)(ˆn) + 2(Q − iU)(ˆn) (3.44) 2 ð ð 1 h 2 i B(ˆn) ≡ − (Q + iU)(ˆn) − 2(Q − iU)(ˆn) . (3.45) 2i ð ð

Since the E and B modes are spin-0 quantities, we can express them in terms of the spin-0 spherical harmonics:

X E E(ˆn) = a`mY`m(ˆn) (3.46) `,m

X B B(ˆn) = a`mY`m(ˆn) (3.47) `,m

46 Figure 3–6: ‘Hot’ and ‘cold’ spots in E and B mode polarization for the super- position of many Fourier modes as in Fig. 3–5, but with different directions for the wavevectors. Note the parity of the E and B modes, and that the points with nonzero E exhibit curl-free divergence, but that the points with nonzero B exhibit divergence-free curl. Figure adapted from ref. [18].

E B where the coefficients a`m and a`m are defined by s (` + 2)! a + a aE ≡ − 2 `m −2 `m (3.48) `m (` − 2)! 2 s (` + 2)! a − a aB ≡ − 2 `m −2 `m . (3.49) `m (` − 2)! 2i

Since this definition is written in terms of the ±2a`m of (3.43), it allows con- version between the Stokes parameter description of the polarization field and the E and B mode description. Physically, the E and B modes correspond to the curl-free and divergence- free components of the polarization field [55]. See Fig. 3–6. It is from these properties that they get their names: by analogy with electromagnetism, the curl-free ‘gradient’ mode is called the ‘E’ mode, and the divergence-free ‘curl’

47 mode is the ‘B’ mode. A related important property of E and B modes is that they have opposite parities, due to the different choices of sign in (3.48) and (3.49). Unlike the E modes, B modes have an associated handedness, also il- lustrated in Fig. 3–6. Since the distribution of electrons in the universe is expected to be the result of scalar density perturbations, which do not have a handedness, symmetry implies that Thomson scattering of CMB photons cannot produce a polarization field with B modes. This will continue to hold for linear evolution of scalar perturbations [51]. As a result, the majority of the cosmological polarization will be in E. B modes are generated by vector and tensor modes (rather than the scalar modes generating E modes) in the early universe, and are of interest for fundamental cosmology, but are not rel- evant to the topics pursued here. Correspondingly, the rest of this work will be primarily concerned with the E mode polarization of the CMB. The E and B modes form a convenient basis for full-sky polarization compared to Stokes Q and U because they are linearly independent, and, like the temperature fields, the E and B fields are spin-0 and no longer necessitate choosing a fixed set of frames for all calculations. The most useful consequence of writing the polarization in terms of the spin-0 E and B modes is to allow us to calculate polarization power spectra in the same way as the temperature power spectrum (see section 1.4). We will be most interested in the EE power

EE spectrum C` , defined by

E E ∗ EE ha`ma`0m0 i = δ``0 δmm0 C` , (3.50)

BB with averaging over the m’s. Note that we can also define C` as well as

TE EB TB EB TB the cross power spectra C` , C` , and C` . However, C` and C` are

BB expected to be zero for reasons of parity, and C` provides limited information

48 during our period of interest, so we will disregard them in the following. The

TE cross power spectrum C` gives the relationship between the polarization and the temperature anisotropies, and is defined by

E T ∗ TE ha`ma`0m0 i = δ``0 δmm0 C` . (3.51)

The EE, TE, and BB polarization power spectra are the canonical choices for analysis of CMB polarization;they play the same role for the study of CMB polarization as the TT polarization power spectra plays for the study of the CMB temperature anisotropies. For a plot of the theoretical EE polarization power spectrum, see Fig. 4–2. As in Fig. 1–1, the peaks at higher ` are the result of oscillations in the primordial plasma; they are directly related to the peaks in the temperature power spectrum and the bumps in the matter power spectrum. In summary, we know that the CMB becomes polarized as a result of Thomson scattering of its primordial quadrupole during eras with many free electrons. This polarization can be expressed either in terms of the spin-2 Stokes parameters, which are convenient for experimental measurements and calculations with fixed coordinate systems, or can be converted into spin-0 E and B modes, which are often more convenient for full-sky, coordinate- independent calculations such as the calculation of power spectra. Both pa- rameterizations of the polarization will be helpful in the coming chapters, as we investigate several effects that lead to the polarization of the cosmic mi- crowave background.

49 CHAPTER 4 Polarization from the Epoch of Reionization Having developed the background, in this chapter we present the first example of secondary effects affecting the polarization of the CMB. Large- scale polarization of the CMB is generated during the epoch of reionization; we will show how different models for reionization can affect this. 4.1 The Epoch of Reionization

At early times, when the universe was very dense and hot, the matter existed in the form of a hot plasma. As the universe expanded, this plasma cooled; at z ≈ 1100 it became possible for the protons and electrons to form neutral hydrogen, leading to a rapid decrease in the ionization fraction of the universe. This event also resulted in the the formation of the CMB, as explained in section 1.4. After this rapid period of recombination, the universe was almost entirely neutral. Much later, likely between z = 15 and z = 5, the hydrogen and helium present in the intergalactic medium became ionized again, during a period known as the epoch of reionization (EoR). For an illustration of the ionization history of the universe, see Fig. 4–1. Since hydrogen is the most abundant element in the universe, much of the physics can be understood by considering the conversion of neutral hydrogen (HI) into ionized hydrogen (HII); we will also examine the effects from helium since the helium fraction is fairly large (∼ 24%). The helium atom has two electrons; it is ionized in two stages according to the tightness of the bonds. Since the first helium ionization (HeI becoming HeII) has a similar ionization potential and recombination rate as hydrogen, it is generally expected to follow the hydrogen reionization closely [61, 62]. The second helium electron has a

50 1.2

1 e x 0.8

0.6

0.4 Ionization fraction

0.2

0 100 101 102 103 104 z

Figure 4–1: Fiducial ionization history of the universe, as used by CAMB. Sev- eral steps of recombination occur between z ≈ 1500 and z ≈ 1000. Hydrogen and neutral helium reionize at z ≈ 10, with singly-ionized helium reionizing at z ≈ 3. Note that here the ionization fraction has been normalized by the hydrogen number density so that if all the hydrogen is ionized xe = 1, and values above 1 are due to the contribution of electrons from ionized helium. stronger bond than the first, so it requires higher-energy photons to ionize. As a result, the second helium ionization (HeII becoming HeIII) is expected to occur after the period of hydrogen and neutral helium reionization has finished [62]. The reionization of the universe is believed to be a result of the emission of high-energy (UV) photons by the first stars. As the ionizing photons escape the galaxies hosting the stars, they create bubbles of ionized hydrogen and helium around the galaxies, which grow until essentially all the intergalactic medium is ionized [61]. This model of reionization, in which the massive first (Pop III) stars emit UV photons, leading to reionization, is somewhat naive. It is also thought that quasars and AGN or more exotic processes such as

51 decaying particles may aid reionization, although it is still expected that most of the effects are due to star-forming galaxies [61, 63]. Additionally, although most ionizing photons are expected to be in the ultraviolet range, a small amount of x-ray radiation is possible, potentially having a significant impact on the bubble formation [61, 63, 64]. In general there is a great deal of uncertainty in theoretical models for the EoR. Even if we assume that reionization is driven by stars emitting UV photons, several important parameters are not well known. In particular, there exists a wide range of models for stellar formation; different star formation rates lead to quite different amounts of ionizing radiation emitted at different times. Moreover, the fraction of photons that escape the galaxies and are available to reionize the IGM is poorly constrained [62]. These unknowns affect the time at which reionization occurs as well as the rate at which it proceeds. In light of these uncertainties, it is unclear how the ionization history ought to be modelled. For generic reionization, a simple step-function reionization is often used to limit the number of parameters. In particular, the reionization model in Fig. 4–1, the model used by the popular cosmological Boltzmann code CAMB [65], is a smoothed step function, with the center chosen to match the WMAP constraints [4]. It is also assumed that HeI reionization occurs with the HI reionization, and that HeII reionization follows later, with a similar step function at z ∼ 3.5. A relevant cosmological parameter for studying the epoch of reionization is the optical depth τ measured between the end of recombination and the present era. The optical depth is a measure of the degree to which a beam of light is absorbed as it travels along its path; for CMB radiation undergoing Thomson scattering off of free electrons, it is equivalent to a measurement of

52 the integrated ionization fraction between last scattering and now:

Z τ = dr a(r)ne(r)xe(r)σT (4.1)

Since no contribution is made to τ when the universe is neutral, the optical depth is therefore solely due to the reionized universe. In ΛCDM, the total optical depth to an instantaneous period of reionization occurring at zreion is given by [66]

  2 Ωb p 3 τ(zreion) = σT (1 − YP ) hxe Ωm(1 + zreion) + ΩΛ − 1 , (4.2) 3 Ωm where YP is the primordial helium abundance, xe the current ionization frac- tion, and σT is the Thomson cross section. Note that (4.2) is dominated by contributions at higher redshifts, making τ sensitive to the redshift of reion- ization. Since the optical depth is an integrated measure, the duration of reionization ∆z has a much weaker impact on τ. Current constraints on reionization primarily come from two sources. The WMAP constraint on τ from the large scale polarization of the CMB suggests a large amount of ionization out to z ∼ 10.5 [4]. This constraint is derived as the redshift of an instantaneous reionization, as a result of the sensitivity of τ to the beginning of reionization, but not to its duration. The second constraint comes from the measurement of a Gunn-Peterson trough in quasar absorption spectra [67, 61], which suggests an end to hydrogen reionization at z ∼ 6. The Gunn-Peterson trough is a measurement of Ly-α lines in the quasar spectra, and is sensitive to neutral hydrogen, and therefor to the end of reionization. This difference between the redshift suggested by the quasar data and the CMB data implies the possibility of an extended period of reionization: the

53 universe must have begun ionizing by z ∼ 10.5 and finished after z ∼ 6. Con- straints on the duration of reionization can be derived from measurements of the kinetic Sunyaev-Zel’dovich effect produced by the ionized regions. Results from the South Pole Telescope [68] show a conservative upper limit on the duration of reionization of ∆z ≤ 7.9, possibly much lower. This approach is sensitive to the properties of the ionized bubbles, which could be affected by significant x-ray radiation [64]. Better constraints on the EoR are also expected in upcoming years from 21 cm experiments [63]. Like the measurements of the Ly-α line in the current quasar data, the 21 cm line is a transition in the neutral hydrogen atom, and so provides a complementary measure to the data derived from the CMB, which is primarily sensitive to electrons, and therefore to the ionized hydrogen. As a result, large-scale CMB polarization will continue to provide useful constraints on the epoch of reionization in the decades to come. CMB constraints on optical depth are usually calculated assuming homo- geneous models of reionization. This is a naive viewpoint, since the expected model is patchy, with HII bubbles growing around star-forming galaxies [63]. However, as the scales of interest are large compared to the expected bubble sizes [61], the spatial dependence of reionization will not have a strong effect on the large-scale CMB polarization. For simplicity, we will ignore it in the following. 4.2 CMB Polarization Generated at Reionization

This section gives a theoretical overview on how the large-scale CMB po- larization signal is produced during the epoch of reionization. Polarization cannot be produced unless there are free electrons, and the epoch of reioniza- tion is the first time since recombination that free electrons exist in sufficient

54 quantities to produce measurable polarization. CMB polarization thus con- tains information about the very different epochs of recombination and reion- ization. The effects of the two eras on the polarization power spectrum can be separated to some extent, making the CMB polarization a valuable probe for both. The new polarization created during the EoR can be measured in the EE power spectrum (see section 3.4). By contrast, the temperature power spectrum is mostly sensitive to the epoch of recombination. Since the epoch of reionization has only a small effect on the large-scale temperature power spectrum, the TE cross power spectrum is less powerful for constraining reion- ization. The new polarization signal is produced by Thomson scattering of the primordial quadrupole as in section 3.3. This is the same mechanism which created the polarization power spectrum at z ∼ 1100, but due to the growth of the universe in the intervening time, the effect is very different. An impor- tant difference between the two epochs is that at last scattering, there was tight coupling between the electrons and photons, due in part to the high density of the electrons. Between recombination and reionization the universe grew significantly, making the density of electrons during the later epoch much smaller. As a result, the post-reionization intergalactic medium is optically thin, and only a small percentage of the photons experience Thomson scat- tering. However, since after recombination the photons are uncoupled to the electrons, they propagate freely, causing the temperature quadrupole seen by the electrons at reionization to be larger than the quadrupole present at re- combination [69]. The main results of reionization are that a small signal is added to the power spectrum at large scales, while the primordial signal is degraded on small scales. These effects are illustrated in Fig. 4–2, which gives the EE power spectrum for τ = 0, 0.088, corresponding to the power spectrum

55 102

101

) 0 2 10 K µ ( 10−1 EE ` 10−2 π C 2 / 10−3 + 1)

` −4

( 10 `

10−5

10−6 10 100 1000 `

Figure 4–2: The effect of reionization on the CMB E mode polarization power spectrum, as calculated using the numerical code CAMB. The solid line corre- sponds to the EE power spectrum with instantaneous reionization at z = 10.5; the dashed line has the same normalization, but no reionization. measured in a universe where reionization never occurred, and one with the fiducial reionization. The low ` peak of the EE power spectrum is frequently referred to as the ‘reionization bump’, since essentially all polarization at these scales is created during reionization. This is an effect of the relative sizes of the horizons during recombination and reionization. Areas in causal contact create a signal in the power spectrum on scales corresponding to the angle subtended by the size of the causal region. During recombination, this corresponds to scales of ` ∼ 100. The primordial power spectrum has its first peak near ` ∼ 100 and rapidly declines for lower `, since the acausal regions represented by these scales cannot be correlated. The epoch of reionization occurred much later,

56 and so the horizon has had time to grow. Since larger areas are in causal contact, polarization is created on larger scales. The second effect of reionization on the EE power spectrum is the damping of its amplitude by a factor of e−2τ on smaller scales. As before, this depen- dence on τ is actually a dependence on the amount of Thomson scattering that has occurred since the beginning of reionization. As photons are scat- tered, they are redirected from their original paths, causing the signals from different regions to mix. This produces an averaging effect, reducing the over- all signal. This effect can be clearly seen on scales smaller than the reionization bump in Fig. 4–2, which compares power spectra with the same amplitude, but optical depths of τ = 0, τ = 0.088. This effect is not polarization-specific, unlike the reionization bump. The temperature power spectrum is similarly damped at small scales. Since the only effect of reionization at scales smaller than ` & 20 is to damp the amplitude of the power spectra, this creates a par- tial degeneracy between the optical depth τ, and the original normalization of

2 the power spectra, ∆R. This partial degeneracy is broken by the existence of the reionization bump in the polarization power spectrum, which provides a

2 method for measuring τ separately from ∆R. 4.3 Alternate Reionization Histories with Dark Stars

Since the details of reionization are largely unknown, a wide range of models are possible. At the present time, a strong constraint on alternate reionization histories is the large-scale polarization of the CMB. Although the CMB polarization is mostly sensitive to the total number of electrons from the amplitude of the reionization bump, the details of how reionization occurs can affect the shape of the bump as well [69, 70, 71]. In particular, the amplitude of the bump is proportional to the optical depth to reionization, since a larger optical depth requires more Thomson scattering, and therefore results in a

57 stronger imprint of the new signal. Additionally, the position of the bump and any smaller ‘bumpy’ features are related to the redshift of reionization. In this section, we will present an example of how more exotic reionization scenarios affect the large-scale polarization of the CMB. Material in this section and the following section is primarily original research done for ref. [72]. Our model producing alternate reionization histories [72] includes an early population of ‘dark stars’ which contain a substantial component of self- annihilating dark matter. As the first stars begin to form, the self-annihilation of the dark matter at their cores creates enough energy to slow down their col- lapse and prevent the onset of nuclear fusion expected for normal stars. The end product is a semi-collapsed state supported by dark matter annihilation in the core. The surface is cooler than that of a regular star, and so produces fewer ionizing photons, potentially slowing down the process of reionization. Once the dark matter is mostly gone, and can no longer support the dark star through self-annihilation, it collapses into a near-main-sequence star. At this point, it begins to emit a normal amount of ionizing radiation, and the process of reionization begins in earnest. For this work, we used dark star reionization histories generated by col- laborator A. Venkatesan [73]. The models permit a great deal of flexibility in how long the ‘dark star proper’ phase lasts, and what proportion of Pop III stars are dark stars. These two factors produce similar effects on the epoch of reionization: higher proportions of dark stars and longer lasting dark stars will both work to delay reionization. For illustration, we will first examine a set of models displaying greater and lesser effects on the reionization history by choosing a relatively long dark star proper phase, tDSP = 500 Myr, and a widely varying dark star fraction, fDS = (0, 0.4, 0.6, 0.8, 0.9, 1). Accordingly, the dark star fraction ranges from having no dark stars at all (a fiducial model

58 1 fDS = 0 fDS = 0.4 fDS = 0.6 0.8 fDS = 0.8 fDS = 0.8 fDS = 1 0.6

0.4 HII fraction

0.2

0 0 2 4 6 8 10 12 14 16 18 20 z

Figure 4–3: Reionization with dark stars. The solid line represents reionization in the fiducial model, which contains only normal Pop. II and III stars. Lines to the left have increasing fractions of dark stars of lifetime tDSP = 500 Myr, causing delayed reionization. where reionization is driven entirely by normal Pop III and Pop II stars) to the case where the entire first generation of stars begins as dark stars. The reionization histories resulting from this choice of models are plotted in Fig. 4–3. We have chosen these models so that they form a representative sample of reionization histories, rather than dark star physics. They include a relatively wide range of reionization times; in particular, the fiducial model (fDS = 0) has a significantly earlier reionization than in the CAMB parameterization, and the model where all stars are dark (fDS = 1) reionizes considerably later. We calculate the effects of the different reionization histories on the CMB by modifying the Boltzmann code CAMB [65] (using the version from July 2011) to use the reionization histories plotted in Fig. 4–3 rather than the de- fault parameterization (shown in Fig. 4–1). Since the generated reionization

59 histories only treat the reionization of hydrogen, we included helium by assum- ing that reionization of HeI follows the reionization of HI and that the reion- ization of HeII follows a smoothed step function at z ∼ 3.5, as in the CAMB

−4 default. We also added a small (xe = 2.1 × 10 ) constant residual ionization fraction from recombination to ensure a smooth transition between the end of recombination and the epoch of reionization. This value corresponds to the residual ionization fraction calculated by the RECFAST module in CAMB at z = 20. We used a slightly different set of ΛCDM parameters1 than in the rest of this work, and we took YP = 0.25. The difference between this choice of parameters and those of Table 1–1 is small and should have a minimal effect on the results. We used CAMB’s highest accuracy settings to calculate the optical depths and power spectra in the different models; the results for the optical depths are given in Fig. 4–4 and the EE power spectra are shown in Fig. 4–5. TT and TE power spectra are not plotted, since the effects (other than the damping at high `) are smaller. The key signatures of the different reionization histories lie in the shape and amplitude of the reionization bump, with later reionizations having a much less pronounced bump than the earlier reionizations provide. There is also a small change in the location of the features. The second effect of the multiple reionization histories on the power spectra is the e−2τ damping at small scales. This damping is visible in the much-better measured temperature power spectrum, as well as the polariza- tion power spectrum. It is difficult to disentangle the effect of the damping

2 from a change in ∆R using only the temperature power spectrum. Since the

1 we used the WMAP7 parameter set from http://lambda.gsfc.nasa. gov/product/map/current/params/lcdm_sz_lens_wmap7.cfm.

60 0.13 tDSP = 500 Myr tDSP = 150 Myr 0.12 tDSP = 50 Myr WMAP7 τ 0.11 1-σ error on WMAP7 τ τ 0.1 0.09 0.08

Optical depth 0.07 0.06 0.05 0.04 0 0.2 0.4 0.6 0.8 1

Dark star fraction fDS Figure 4–4: Optical depths for dark star reionizations, with different lengths of dark star proper phase tDSP. Since dark stars with a shorter tDSP have a smaller effect, the other plots in this section refer to the case where tDSP = 500 Myr. Error bars indicate the variation due to choice of astrophysical parameters; this effect will be examined in the following section. original amplitude is not known independently, it falls to the amplitude of the reionization bump to differentiate these effects, as it is less affected by changes to the initial amplitude of the power spectrum. We can therefore account for the effects of the damping by incorporating the difference into the initial amplitude, as shown in the second plot of Fig. 4–5. 4.4 Measuring the Effects of Dark Star Reionizations

The optical depths calculated have a wide range; several reionization his- tories are already well outside the WMAP constraints of τ = 0.088 ± 0.015. This includes the fiducial theoretical model, as a result of its very early reion- ization. However, as will be explained below, the choice of astrophysical pa- rameters in the dark star model has a strong effect on the redshift of reion- ization. Therefore, the failure of the fiducial model to match the WMAP

61 100 ) 2 K µ ( 10−1 EE ` π C 2 /

−2 fDS = 0 + 1) 10

` fDS = 0.4 (

` fDS = 0.6 fDS = 0.8 fDS = 0.9 fDS = 1 cosmic variance 100 ) 2 K µ ( 10−1 EE ` π C 2 /

−2 + 1) 10 ` ( `

WMAP7 10 100 ` Figure 4–5: The effects on the reionization bump from the reionization his- tories in Fig. 4–3, with tDSP = 500 Myr. The thick solid line is the fiducial model, with colored dashed, dotted, and dot-dashed lines below corresponding to the delayed reionizations. The shading represents cosmic variance around the fiducial model. All power spectra in the first plot have the same initial 2 choice of ∆R, and so have different normalizations at high `; the power spectra in the second plot have been normalized to match WMAP7 measurements at 2 high `, and so have different initial ∆R. For reference, the second plot includes the WMAP7 measurements of the EE power spectrum.

62 constraints cannot be construed as evidence for dark stars. It is likely due to the current uncertainty in the values of the astrophysical parameters. There are two astrophysical parameters which have a strong impact on the dark star models: fesc, the fraction of photons that escape the galaxies to reionize the intergalactic medium, and f?, the star formation efficiency; they enter the model in the combination f?fesc. These are standard reionization parameters, with considerable theoretical uncertainty. They are thought to lie in the ranges 0.01 < f? < 0.1, 0.01 < fesc < 0.2 [72, 73]. The values used in the models presented are f? = 0.05, fesc = 0.1, such that f?fesc = 0.005; these values have not been strongly motivated by empirical evidence. The effect of varying these parameters for a given dark star model can be seen in the reionization histories and the resulting power spectra in Fig. 4–6, where the product of the astrophysical parameter ranges from f?fesc = 0.02(f? = 0.1,

−4 fesc = 0.2) to f?fesc = 10 (f? = fesc = 0.01). The effects are illustrated for the fiducial model with fDS = 0 and a model with fDS = 1 for dark stars with a fairly long dark star proper phase, tSP = 150 Myr. Changing the astrophysical parameters has considerable effect on the reionization histories produced: they can greatly increase or decrease zreion and therefore τ, although this effect is reduced in models with a significant fraction of dark stars. Since the actual values of the astrophysical parame- ters are so uncertain, we cannot use the CMB constraints on the reionization histories as an indication of the presence of dark stars. An important conse- quence of this is that the presence of dark stars could permit a wider range of astrophysical parameters than would be expected from the CMB constraints. The degeneracy between the astrophysical parameters and the effects of dark stars cannot be resolved without better measurements of the values of the

63 1

0.8

0.6

0.4 HII fraction

0.2 fDS = 0 f?fesc = 0.005 f?fesc = 0.02 f f = 10−4 0 ? esc 0 2 4 6 8 10 12 14 16 18 20 z

100 ) 2 K µ ( 10−1 EE ` π C 2 / 10−2 + 1) `

( tDSP = 150 Myr, fDS = 1 ` f?fesc = 0.005 f?fesc = 0.02 −4 f?fesc = 10 10 100 `

Figure 4–6: Reionization with varying astrophysical parameters. Thick lines correspond to no dark stars; thin lines are for a model with tDSP = 150 Myr, fDS = 1. Solid lines represent the canonical choice of astrophysical parameters f?fesc = 0.005, while dot-dashed lines represent the lower limit with f?fesc = −4 10 and dashed lines the upper limit with f?fesc = 0.02. Shading represents the cosmic variance about the two models with f?fesc = 0.005. Changing the astrophysical parameters greatly affects the redshift of reionization for the model without dark stars, but has a less dramatic effect when dark stars are important.

64 astrophysical parameters. Nevertheless, it would still be very interesting to place tight constraints on the reionization history. In order to place stronger constraints on the epoch of reionization, it will be necessary to carefully measure the large-scale EE polarization power spectrum. Although current WMAP data on large-scale polarization does not provide much information about the shape of the EE power spectrum at low ` beyond a measure of the optical depth (see the second plot of Fig. 4–5), we can expect better results from Planck, which should have the capacity to accurately measure the EE power spectrum, as well as next-generation polarization satellites such as PIXIE. A limit on the utility of the reionization bump for our future ability to constrain the properties of the epoch of reionization comes from cosmic vari- ance, a fundamental uncertainty in the information in the power spectrum.

The C`’s making up the power spectrum are defined as the variance in the distribution from which the a`m’s are drawn. On large scales, there are only a small number of a`m’s; they do not form a large enough sample to accurately estimate the underlying power spectrum. The cosmic variance scales as [10]

∆C r 2 ` = , (4.3) C` 2` + 1 greatly decreasing our ability to measure small differences in the power spec- trum in at low `. In particular, cosmic variance limits the measurable effects of alternate reionization histories, since for many models, the changes to the power spectrum are small compared to cosmic variance. Nevertheless, as can be seen from Fig. 4–5, cosmic variance does not eliminate the potential for the shape of the reionization bump to carry helpful information. In short, of ionization of the z ≤ 15 universe allows the CMB photons to once again interact with free electrons via Thomson scattering. This period

65 has a smaller impact than recombination, but leads to a noticeable effect on the CMB power spectrum. All of the primordial CMB power spectra are damped at small scales from the new interaction with electrons, and a new polarization signal is created at large scales, with an amplitude proportional to the optical depth. This new signal is sensitive to how reionization proceeds; different models for the emission of ionizing photons by sources lead to different ionization histories. The model we used to test this, which proposes an early population of cool dark stars, allows for considerable flexibility in the time of reionization, depending on the lifetime and number fraction of the dark stars, as well as the star formation efficiency and photon escape fraction. The effect of the dark stars is to delay reionization except where there is a considerable shift in the astrophysical parameters. In this case, the presence of dark stars allows for a wider range of parameters to compensate for current limits on zreion. The effect of a shift in reionization time can be seen in the low-` signal of the polarization power spectrum: earlier reionizations create a stronger signal in the polarization power spectrum, and cause more damping at small scales; later reionizations have a weaker signal and damp less. This effect can be reproduced by a different choice of astrophysical parameters. Since little is known about the true values of the astrophysical parameters, the degeneracy between them and the dark star fraction makes it impossible to separate the effects. However, as our understanding of the relevant astrophysics improves, we may be able to learn more. Two effects conspire to limit our ability to extract information from the CMB power spectra. First, a partial degeneracy between the initial amplitude and the optical depth means that the damping signal cannot itself constrain

66 reionization. Second, cosmic variance limits the degree to which the large- scale polarization can distinguish between models. Despite these drawbacks, useful information remains, and the large-scale polarization of the CMB will continue to be a valuable resource for constraining the epoch of reionization.

67 CHAPTER 5 Secondary Polarization from Large-Scale Bulk Flows In this chapter we present a second example of a mechanism which gen- erates large-scale secondary polarization. The effect is a second-order pro- duction of polarization by moving electrons after reionization, and may be significant in the case of large local bulk flows. This is a secondary effect due to the quadrupole term produced by Doppler shifting of the CMB tempera- ture, rather than the primordial quadrupole (cf. chapter4). The material in this chapter is original work. 5.1 Quadrupole Anisotropy

The spectrum of the CMB monopole is given by blackbody radiation:

2hν3c−2 Cx3 Iν = = x (5.1) exp(hν/kBTCMB) − 1 e − 1

3 where, as before, x = hν/kBTCMB and C = 2(kBTCMB) /hc. An observer moving with respect to this background with velocity β = v/c will see this spectrum Doppler-shifted to

" ! #−1 3 1 + βµ Iν = Cx exp x − 1 (5.2) p1 − β2 where µ is the cosine of the angle between the incoming photon direction and the direction of the velocity in the reference frame of the observer. Expanding this to second-order in velocity gives

Cx3  xex xex x x 1  I = 1 − µβ + coth µ2 + β2 + O(β3) . (5.3) ν ex − 1 ex − 1 ex − 1 2 2 2

68 From the dependence on µ we see that the zeroth order term is the CMB monopole and that the term first-order in β contains a thermal dipole with the same frequency dependence as the temperature perturbations in section 1.4. The term second-order in β contains a quadrupole and monopole with both thermal and nonthermal frequency dependences. As shown in section 3.3, intensity quadrupoles scattering off electrons will generate polarized light. Therefore, for electrons moving with velocity β with respect to the CMB, the first term in (5.3) that will generate polarization is second-order in the velocity of the electrons. After dividing out the frequency dependence of thermal fluctuations given by (1.32), and expressing it in terms of the spherical harmonics, the coefficient of the β2 term becomes

x x 1 r4π x x 1 √  √ coth µ2 + = 2 coth Y + 4πY + πY . (5.4) 2 2 2 5 2 2 3 20 00 00

Note that the coordinate system has been set by taking the z-axis to align with the electron velocity. From this, we see that in temperature units, the ‘kinematic’ quadrupole due to the velocity of the electrons is given by r − 2 4π ae = g(x)T β2, (5.5) 20 3 5 CMB

This term has a nonthermal frequency dependence,

x x g(x) = coth , (5.6) 2 2

For reference, g is plotted as a function of frequency in Fig. 5–1.

If the direction of the electron velocity is given by (θv, φv) in some coor- dinate system, the quadrupole term in (5.3) becomes

2 1 2 8π X µ2 − = P (ˆv · ˆn) = Y ∗ (θ , φ )Y (ˆn), (5.7) 3 3 2 15 2m v v 2m m=−2

69 5

4

3 ) ν ( g 2

1

0 0 100 200 300 400 500 ν (GHz)

Figure 5–1: The non-thermal frequency dependence of the kinematic quadrupole. and so we can rewrite (5.5) more generically as

− 8π ae = g(x)T β2Y ∗ (θ , φ ), (5.8) 2m 15 CMB 2m v v giving the quadrupole moments of the incoming radiation field as seen by the moving electrons. From equation (3.39), the outgoing polarization along the z-axis will be generated by a2,±2 moments in the incoming radiation. Since we are interested in calculating the polarization signature on the entire sky, we will need to determine the polarization produced in an arbitrary direction. To do this, we will follow a similar procedure to that used in section 3.3 and rotate the incoming field through the Euler angles (θe, φe) to obtain the moments of a

70 e− new multipole expansiona ˜`m: m Z ! e− X ` ∗ ∗ X e− e− a˜`m = Dm0m(θe, φe, 0) dΩ Y`m0 a`00m00 Y`00m00 . (5.9) m0=−m m00

` where Dm0m is the Wigner rotation symbol. The polarization emitted along

e− the outgoing direction (θe, φe) will come froma ˜2,±2:

e− X ±2 ∗ e− a˜2,±2 = Dm2 (θe, φe, 0) a2m. (5.10) m Since the initial radiation field (5.8) is not necessarily azimuthally symmetric,

e− unlike the case discussed in section 3.3, it was necessary to rotate a2m through an additional Euler angle θe. As a result, (5.10) is the generalization of (3.41) for non-azimuthally symmetric radiation fields. Since a Wigner rotation is related to the spin-weighted spherical harmon- ics, the polarization-generating multipoles may be more explicitly written as r − 4π X − a˜e = Y ∗ (θ , φ ) ae . (5.11) 2,±2 5 ∓2 2m e e 2m m

From (3.39), the polarization emitted in the direction (θe, φe) by a single Thom- son scattering off an electron moving in the direction (θv, φv) becomes r 3 X − Q(θ , φ ) ± iU(θ , φ ) = σ Y ∗ (θ , φ ) ae , (5.12) e e e e T 50 ±2 2m e e 2m m

e− e− where a2m = a2m(θv, φv) is given by (5.8). To calculate the polarization from this scattering as measured by an ob- server, note that the emission angles φe, θe are related to the angles φ, θ seen by the observer by θe = π − θ and φe = φ + π (see Fig. 5–2). Making this substitution, we find that

∗ m ±2Y2m(θe, φe) = (−1) ±2Y2,−m(θ, φ), (5.13)

71 Figure 5–2: The geometry of a single Thomson scattering off a moving electron as seen by an observer at the origin. and so the observer will measure polarization in the direction (θ, φ) resulting from one Thomson scattering according to: r 3 X − Q(θ, φ) ± iU(θ, φ) = σ (−1)mae Y (θ, φ). (5.14) T 50 2m ±2 2,−m m The total polarization resulting from many Thomson scatterings can then be calculated by integrating the contribution due to one electron along the line of sight, so that

r 3 8π Z Q(θ, φ) ± iU(θ, φ) =g(x)T dr a(r)σ n (θ, φ, r)β2(θ, φ, r) CMB 50 15 T e X × Y2m(θv(r), φv(r)) ±2Y2m(θ, φ). (5.15) m In general, this will depend strongly on the distributions of electrons and their velocities, and will be a complicated signal, producing both E and B modes, as well as multipoles on many scales. A similar idea is pursued by Baumann

72 et al. [74], where the small-scale polarization produced by random peculiar velocity fields and electron density from haloes is calculated statistically. However, if we neglect the difficulties associated with the distributions of ionized electrons and their velocities and adopt a toy model of a region of con- stant electron density with a bulk flow of constant direction, the polarization signal simplifies considerably. In the simplest case of an observer at the center of a sphere of constant-directional bulk flow and constant electron density, the signal can be expressed as a purely E mode quadrupole, since

1 aE = − ( a + a ) (5.16) 2m 2 2 2m −2 2m 8π r 3 Z = −g(x)T dr a(r)σ n β2(r)Y (θ (r), φ (r)) (5.17) CMB 15 50 T e 2m v v and

i aB = − ( a − a ) = 0. (5.18) 2m 2 2 2m −2 2m

Taking the even simpler case where the velocity of the bulk flow is constant out to the edge of the sphere, the signal simplifies further, so that

8π r 3 aE = −g(x)T τ(z )β2 Y (θ , φ ). (5.19) 2m CMB r 15 50 2m v v where τ(zr) gives the optical depth to the edge of the sphere or, in the case of a ‘dark flow’ extending to the horizon, the optical depth to reionization. The signal in (5.19) is the result of a single quadrupole moment–when the direction of the velocity aligns with the z-axis, it becomes

2r2π aE = − g(x)T τ(z )β2, (5.20) 20 5 15 CMB r

In writing (5.19) as (5.20), we have effectively undone the step from (5.5) to (5.8), so that the initial radiation field is once again azimuthally symmetric.

73 In summary, the polarization quadrupole moment of (5.19) or (5.20) is the geometric result of the Thomson scattering of the Doppler-shifted CMB temperature monopole by free electrons. The resulting signal is a quadrupole in the simplest case of a ‘bubble’ of constant velocity within a region of con- stant electron density. If the distributions of electrons and velocities are fully considered, the signal will gain higher-order terms and a B mode component. 5.2 Polarization Signatures of Bulk Flow Measurements

Let us examine the current and future prospects of measuring the large- scale local bulk flow using this polarization signature. Since τ is particularly sensitive to electrons at high redshifts (see equation 4.2), this technique is particularly well suited to detecting coherent bulk flows on extremely large scales, and very poor at detecting small-scale flows, unlike most methods used for measuring bulk flows. Using (5.20), we can calculate the magnitude of the polarization signal expected for some of the recent bulk flow measurements presented in section 2.4 or predicted ΛCDM bulk flows on various scales, as derived in section 2.3. Several examples, calculated at a frequency of 150 GHz are shown in Table 5–1. We have emphasized larger, more measurable bulk flows, and include a possible ΛCDM bulk flow, the measurements of Feldman et al. [37], the measurements of Kashlinsky et al. [45], and a ‘dark flow’ with the same velocity as the Kashlinsky et al. measurements. For this last calculation, we take

τ(zr) = τ = 0.088, since we would expect a signal to be generated by electrons

E across the entire post-reionization universe. This signal, a20 = −1000 nK, is considerably stronger than the others due to the sensitivity of τ(zr) to higher redshifts; similarly, the expected ΛCDM signals are considerably weaker. We would like to use the current WMAP large-scale polarization data to put an upper limit on the possible ‘dark flow’. Since the magnitude of the bulk

74 Table 5–1: Polarization signature for large-scale bulk flows at ν = 150 GHz E Velocity (km/s) Maximum redshift (zr) −a20 (nK) 100 0.03 0.006 400 0.03 0.1 1000 0.25 6 1000 & 10.5 1000

2 flow, β , and its extent, τ(zr), have indistinguishable effects on the resulting polarization, we define a single parameter,

2 Aflow = τ(zr)β , (5.21) to include contributions from both quantities. We then reparameterize (5.19) in terms of Aflow, so that

8π r 3 aE = −A g(x)T Y (θ , φ ). (5.22) 2m flow CMB 15 50 2m v v

2 Finally, we perform a χ fit for Aflow, using (5.22) with data from the WMAP7 foreground-reduced V-band1 [16]. We extract the quadrupoles present in the WMAP data using HEALPIX2 [75], and compare them to the theoretical predictions with freely varying Aflow and the average direction of velocity from Kashlinsky et al. [45]: (l, b) = (288◦, 30◦). We choose this direction since their proposed ‘dark flow’ is the only signal large enough to be possibly constrained by the WMAP data. The variance in the χ2 fit is given by

theory C2 = C2 + N2, (5.23)

1 available at http://lambda.gsfc.nasa.gov/product/map/dr4/maps_ band_forered_r9_iqu_7yr_get.cfm. 2 http://healpix.jpl.nasa.gov

75 14

12

10

8 2 χ 6

4

2

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 6 Aflow × 10

2 Figure 5–3: The results of the χ fit for Aflow.

theory where C2 is the theoretical value of EE power spectrum quadrupole calcu- lated by CAMB and N2 is the experimental noise of the E mode quadrupole, which we calculate by averaging the nearby WMAP BB C`’s in the same map. Since no B mode signal is expected to be measured by WMAP, the measured values of the BB power spectrum should provide an accurate estimate of the E mode noise on the same scales. The result is plotted in Fig. 5–3; it is consistent with zero with an upper

−6 limit of Aflow = 3 × 10 . This value of Aflow would correspond to a ‘dark flow’ extending out to reionization of magnitude ∼ 2000 km/s. Although this does not rule out any of the predictions from recent measurements, it suggests that future polarization data might be able to provide new constraints on large-scale bulk flows.

76 We expect this improvement to come from future experiments with more sensitivity to large-scale polarization. In particular, at ` = 2 Planck is ex- pected to have a sensitivity of 15 nK [18], PIXIE a sensitivity of 2 nK [17], and proposals for CMBPol have < 1.5 nK [19, 18]. Although no experiment currently proposed is sensitive enough to measure a ΛCDM bulk flow, they should be powerful enough to provide constraints on large-scale bulk flows. In particular, Planck data may be able to test for large dark flows, and a PIXIE or a CMBPol type experiment could potentially measure non-ΛCDM flows even if they do not represent a ‘dark flow’ on larger scales. The crucial property that will allow the extraction of this small signal from the much larger polarization signal from the primordial CMB temperature quadrupole is the nonthermal frequency dependence g(x) of the kinetic signal. Since the polarization signal due to the primordial quadrupole will have a thermal frequency dependence, using measurements across several frequency bands, the primordial signal can be ‘integrated down’. Experiments such as Planck and especially PIXIE will provide data in many different frequency bands, which should allow this to be done accurately. In the derivation of an upper limit on the size of Aflow above, we have not attempted to subtract away the primordial signal. In this chapter we have shown that a large-scale bulk flow of ionized elec- trons produces a characteristic quadrupolar polarization signal with a nonther- mal frequency dependence. This signal provides a new approach for testing for large-scale bulk flows using measurements of the E mode quadrupole at several frequencies. For signals as large as have been claimed by recent mea- surements, we can expect detection by future experiments. Otherwise, future experiments can be expected to place upper limits on the magnitude of the local coherent bulk flow.

77 CHAPTER 6 Conclusions In this work, we have presented two effects which lead to distinctive sig- natures in the large-scale cosmic microwave background polarization power spectrum. Essentially all polarization on scales of ` . 20 is created during the reionization of the universe. The specific shape of the power spectrum on these scales is not well-determined since it depends on the details of reioniza- tion history. This presents an opportunity–we can use the polarization power signal along with other experimental measurements to place constraints on the processes leading to reionization. The first effect we investigate is the dependence of the ‘reionization bump’ on a dark-star-driven reionization. The models depend strongly on the number of dark stars, their lifetimes, and the escape fraction of ionization photons and the star formation efficiency. These parameters serve primarily to shift the time of reionization, although they do not do so independently. In particular, an increase in the fraction and lifetime of the dark stars can significantly reduce the effects of the astrophysical parameters. If dark stars are present, they allow a wider range of other astrophysical parameters. The second effect presented here, the quadrupolar polarization signature, is subdominant to the reionization bump, as the magnitude of the primor- dial CMB temperature quadrupole (which leads to the reionization signa- ture) is much larger than the quadrupole due to Doppler-shifting of the CMB monopole for any reasonable velocity. This signal is a prediction for the form of specific quadrupole moments, so that, unlike the reionization polarization, the kinematic polarization is not subject to cosmic variance. It will be difficult

78 to measure since it has a very small magnitude, although due to its nonther- mal frequency dependence, it can in principle be measured by any sufficiently sensitive instrument with multiple frequency bands. Although we have limited discussion to a ΛCDM universe with a specific set of parameters, the results presented in chapters4 and5 are relatively robust to the exact formulation of the model. In particular, both effects will be present for any remotely similar set of ΛCDM parameters, and are only strongly affected by the value of τ. The magnitude of the large-scale signal created during the epoch of reion- ization varies strongly with the optical depth—this is the primary information we use to constrain the reionization history. A small amount of information about the reionization history is also present in the shape and location of the reionization bump. It is only mildly sensitive to the choice of ΛCDM param- eters other than τ: notably, varying values of Ωb and to a lesser extent, H0 can cause a shift in the location of the reionization bump by affecting the calculated value of zreion [69]. This behavior can be seen from the dependence on other parameters of (4.2). The kinetic polarization quadrupole, on the other hand, is only really affected by the value of τ, through its dependence on the electron density.

Nonetheless, a different choice of the other parameters (in particular Ωm, on which the velocity factor strongly depends) could lead to a different period of structure growth, and so perhaps greater or lesser velocities of the bulk flows. This would affect the magnitude of the signal we might expect to see, but in general this difference is unimportant, since the signal due to ΛCDM is con- siderably smaller than expected to be measurable by upcoming experiments, and the constraints on Ωm are already tight.

79 For sensible non-ΛCDM models containing e.g. curvature, time-varying dark energy, or other more exotic components, we expect the overall behavior of the universe to be fairly similar to that predicted by ΛCDM. Under these circumstances, we would expect to see signals in both cases, although the de- tails may be a bit different. For the existence of either effect it is necessary only to have a period of reionization, and in the case of the kinematic polarization, large-scale bulk flows. After the recent cosmological successes from measuring the temperature anisotropies of the CMB, and with more sensitive experiments, the cosmo- logical community is beginning to look to the polarization of the CMB as a new source of cosmological information. The basic theoretical polarization signals from recombination have long been known and are expected to pro- vide important complementary information. The large-scale polarization of the CMB will provide new information about a later epoch of the universe. With the advent of very sensitive large-scale polarization experiments, mea- surements of the large-scale CMB polarization have the potential to provide new astrophysical bounds and increase our knowledge of cosmology.

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