Small-Scale Secondary Anisotropics in the Cosmic Microwave Background

by

Jonathan Dudley

M.Sc.

Department of Physics

McGill University

Montreal, Quebec

August 2007

A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science in Physics

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While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada ACKNOWLEDGEMENTS

First and foremost I wish to acknowledge the the efforts and contributions of my supervisor Gil Holder, whose guidance and discussions have proved invaluable to my graduate experience. I also wish to thank Olivier Dore for valued contributions to this work as well as the use of the Legacy Archive for Microwave Background

Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science.

I would also like to acknowledge all the educators whose dedication and interest in physics helped spark my own. Amongst them there are two that deserve special mention: Mr. Robert Miller and Prof. Roman Koniuk. Their seemingly endless knowledge and genuine passion for teaching are the two principle reasons that I be­ came, and remain, a student of physics.

Also deserving of mention are the denizens of ERP-225 who provided the insightful and often esoteric discussion that is so necessary for a productive workspace.

Finally I wish to acknowledge the support of family and friends who provided excel­ lent avenues of procrastination while also keeping me apprised of affairs in the "real world".

n ABSTRACT

One of the main harbingers of the modern age of precision cosmology, the Cosmic

Microwave Background (CMB) has proven itself to be a veritable trove of cosmolog-

ical information. With the aid of experiments such as the WMAP satellite, precision

measurements of the CMB anisotropy spectra are now being made.

This work will first explore the physics of the Vishniac effect, a small-scale CMB

temperature anisotropy created by the Compton scattering of CMB photons by free electrons caught in a line-of-sight bulk flow. The Vishniac effect arises due to a den­

sity enhancement in the electrons caused by gravitational potentials in both the lin­ ear and nonlinear regimes and contributes significant power to the CMB temperature

anisotropy power spectrum on small scales. This effect is strongly dependent upon

cosmology and as such this dependence is investigated for all experimentally-allowed values of the fundamental cosmological parameters. This analysis is performed for both the linear Vishniac effect as well as its nonlinear extension. Following this anal­ ysis a fitting function, capable of predicting the power generated by the Vishniac effect over a range of scales for any allowed cosmology, is investigated. This function proves to be an accurate and efficient way of computing the Vishniac effect for any input cosmology.

The next small-scale phenomenon to be explored is the small-scale CMB polariza­ tion anisotropies generated by Thomson scattering of the local photon quadrupole

iii anisotropy during reionization. The underlying physics behind this effect are stud­ ied along with the observational information it potentially contains. Observational data of the remote quadrupole is capable of improving constraints in the CMB tem­ perature anisotropy spectrum on scales of I ~ 11 as well as offering information concerning the reconstruction of the primordial density perturbations on gigaparsec scales in our local universe.

IV ABREGE

Etant la principale prediction de la cosmologie moderne, le fond diffus cos- mologique s'est revele etre un outil indespensable de la recherche actuelle en cos­ mologie. A l'aide de projets tel que le Satellite WMAP, des mesures precises des anisotropics du rayonnement fossile sont desormais disponibles.

Premierement, ce travail explore la physique de l'effet Vishniac. Cette anisotropie de temperature a petite echelle du fond diffus est creee par la diffusion Compton de photons par des electrons libres en mouvement dans la ligne d'observation du flux principal. L'effet Vishniac nait d'une surdensite locale d'electrons due des potentiels gravitationels dans les regimes lineaires et non-lineaires. De plus, l'effet

Vishniac contribue significativement aux anisotropics de temperature du fond diffus cosmologique a petite echelle. Cet effet depend fortement de la cosmologie choisie et c'est pourquoi cette dependence est sujette de nombreuses investigations avec toutes les valeurs possibles experimentalement des parametres cosmologiques. L'analyse du regime lineaire de l'effet Vishniac et sa contrepartie non-lineaire ont ete effectue dans le but de produire une fontion d'ajustement de courbe capable de predire la puis­ sance generee par cet effet differentes echelles et ce pour n'importe quelle cosmologie autorisee. Cette fonction d'ajustement est une methode precise et efHcace pour de­ terminer l'effet Vishniac.

Le second effet a petite echelle etudie est la polarisation des anisotropics dues la

v diffusion Thomson du photon local du quadrupole d'anisotropie au moment de la re-ionisation. Les lois de la physique impliquees par cet effet sont etudiees ainsi que de les informations d'observations potentiellement contenues. Les donnees ob­ servers du quadrupole non local sont capables d'ameliorer les contraintes posees sur les anisotropies de temperature du rayonnement fossile a des echelle de £ ~ 11. De plus, les observations offrent de nombreuses informations sur la reconstruction des perturbations primordiales de densite sur plusieurs gigaparsecs dans notre univers local.

VI TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii ABSTRACT iii ABREGE v LIST OF TABLES ix

LIST OF FIGURES x

1 Introduction 1

1.1 Overview 1 1.2 Physical Cosmology 3 1.3 Preliminaries 6 1.4 Early Times 8 1.5 Inflation and Growth of Perturbations 10

2 Cosmic Microwave Background 16

2.1 Recombination 16 2.2 Fluctuations on the Sky 21 2.3 CMB Anisotropics 24 2.4 Reionization 27 3 Vishniac Effect 31

3.1 CMB Temperature Anisotropy Studies 31 3.2 Reionization and Secondary Anisotropics 33 3.3 Vishniac Effect 35 3.4 Contribution to the CMB spectrum 42 3.5 Modelling the Vishniac Effect 49 3.6 Summary and Future Work 56

vn 4 CMB Remote Quadrupole Studies 58

4.1 Intoduction 58 4.2 CMB Polarization 59 4.3 Polarization from a Quadrupole Anisotropy 61 4.4 Polarization from Reionization 64 4.5 Studying the Remote Quadrupole through Secondary Polarization Anisotropies 67 4.6 Constrained Modes 71 4.7 Summary and Future Work 76

5 Summary 79

References 82

vin LIST OF TABLES

Table Page 3-1 Parameters, values and descriptions for the standard ACDM cosmology. All values are taken from [67]. Note that in a flat universe the dark energy density is defined as tt\ = l — flm. 44 3-2 Best fit parameters for the linear Vishniac effect 51 3—3 Best fit parameters for the maximum Vishniac effect 52

IX LIST OF FIGURES

Figure Page

3-1 Comparison of the linear Vishniac effect, the approximation of the linear Vishniac effect and the maximum effect from both the linear and nonlinear density power spectra. ... 41 3-2 The spectrum of primary CMB anisotropics is exponentially damped at small scales (large values of £). This allows for secondary anisotropics like the Vishniac effect to dominate on sub-arcminute scales. Primary CMB spectra were produced by the CAMB online implementation of the CMBFAST [65] code for a fiducial model where only zre is allowed to vary and n = 1.0, fif, = 0.04, Q,m = 0.27, ftA = 0.73 and h = 0.72 43

3-3 Histograms of the linear Vishniac effect power for £=1000 (solid red), 5000 (long-dashed green) and 10000 (short- dashed blue). Results from 9187 allowed cosmologies binned in 0.02/Ltk2 intervals 46

3-4 Histograms of the maximum Vishniac effect power for £=1000 (solid red), 5000 (long-dashed green) and 10000 (short-dashed blue). Results from 9187 allowed cosmolo­ gies binned in 0.02/xk2 intervals 47

3-5 Plot of rms power for ten values of £ for the linear (left) and maximum (right) Vishniac effects. The shaded region represents the 68% confidence contour while the dotted lines represent the spectra for three randomly-chosen cosmologies 48

x 6 Linear (left) and Maximum (right) Vishniac Effect Fitting Residuals. Deviations in the data versus the model for all 9187 representative cosmologies where a value of 1 indicates agreement 53

7 Comparison of fitting function with actual data at t = 4000 for the linear effect (red crosses) and the maximum Vishniac effect (blue squares). The line x = y is shown dashed 55 1 Linear polarization created by Thomson scattering of ra­ diation with a quadrupole anisotropy where solid blue (dashed red) represents cold (hot) incoming radiation. Figure adapted from [40] 60

2 Last scattering surfaces for a local observer and one located at z = 11. Comoving distances to the remote observer and local last scattering surfaces are labelled as well as their redshift equivalences. Figure adapted from [58] 69 3 Principle components of the Fisher matrix 73

4 First four eigenvectors of the real-space polarization analysis. 74

5 Normalized amplitudes of position-space eigenvalues 75

6 Sum of first ten position-space eigenvectors 76

XI Chapter 1

Introduction

1.1 Overview The primary focus of this work is the theory and analysis of small-scale secondary anisotropies in the spectra of Cosmic Microwave Background, hereafter denoted as the CMB. While the CMB and its primary anisotropies were formed during the era of recombination (z ~ 1100), secondary anisotropies in the CMB were created much later via scattering processes during the era of reionization [z « 11). As will be seen in Chapters 3 and 4 these secondary anisotropies will provide strong constraints upon cosmology and structure formation in the universe.

In order to properly describe these anisotropies a brief summary of modern cos- mological theory is presented in Chapter 1. The concept and metric of an expanding universe are explored followed by an overview of the early-time evolution of the uni­ verse. Also presented are the origins and growth rates of density perturbations in the universe as well as review of inflationary theory.

1 2

In Chapter 2 the underlying physics of the CMB and its anisotropies are outlined.

Here the perturbations present in the baryon-photon fluid prior to recombination are reviewed as are the details of how these perturbations manifest themselves on the sky. Chapter 2 also provides an analysis and summary of the creation of primary anisotropies in the CMB as well as the late-time reionization of the universe. This discussion will prove vital in motivating and deriving the Vishniac effect, a secondary temperature anisotropy explored in Chapter 3, as well as the secondary polarization anisotropies created during reionization as presented in Chapter 4.

Chapter 3 will examine the late-time scattering of CMB photons in the reionized universe, specifically focusing upon the Vishniac effect. As will be shown, the Vish­ niac effect is a second order Doppler effect wherein CMB photons are scattered by density-enhanced and energetic free electrons caught in a bulk flow. This effect will be derived for density perturbations in both the linear and nonlinear regimes and solutions will be evaluated for both the full calculation as well as an approximation.

Following this initial calculation the Vishniac will be computed for the full range of allowed cosmologies and limits will be placed upon its contribution to the CMB temperature anisotropy spectrum on small scales. Finally a fitting function will be created to model the effect and permit much faster computations. This function will prove to be both efficient and accurate across the entire allowed cosmological parameter space.

In Chapter 4 the polarization anisotropy created by inhomogeneous reionization is examined. This late-time effect will depend upon the primary photon anisotropies discussed in Chapter 2 and will be found to contribute to the CMB polarization 3 anisotropy spectra on small scales. The physics responsible for this polarization are examined including the temperature quadrupole anisotropy that induces it. Sub­ sequent to this an analysis of the theoretical constraints provided by the potential observation of this signal is performed. It is discovered that under ideal observing conditions this polarization signal has the capacity to provide information pertaining to the primordial perturbations within the local horizon on gigaparsec scales and be­ low. It is these primordial perturbations that seeded the evolved structure observed today in the local universe.

1.2 Physical Cosmology Long considered to be the domain of philosophers and theologians, physical cosmology entered the realm of quantifiable and precision physics in the early 20 century. Einstein's theory of general relativity brought about not only the concept of curved space-time but also demonstrated that the universe must be either in a state of expansion or contraction depending upon its total energy content. This result contrasted with the widely held belief that the universe was static and prompted the introduction of a "cosmological constant" designed to balance the possible con­ traction of the universe thus preserving its assumed static nature. It was quickly realized that this was an unstable and artificial solution and a more natural expla­ nation was sought. One proposed solution, presented by Georges Lemaitre, was that the universe began as a sort of "primeval atom" and expanded in size to its current observed state [56]. This theory was quite controversial and later earned the derisive moniker of the "big bang" theory. Observational support for this theory came about through Edwin Hubble's observations that galaxies in every direction appear to be 4 receding and that the more distant a galaxy was the more rapidly it was receding.

Observations at that time were, unfortunately, dominated by systematic errors and the analysis of these recessional velocities led to the untenable prediction that the universe was younger than the planet Earth. Due to these and other difficulties the steady state theory of the universe continued to compete for acceptance with big bang theory for much of the 20 century [47, 56].

This situation persisted until the discovery of the Cosmic Microwave Background in 1965 [47, 56]. A direct prediction of big bang theory, the CMB is a pervasive and isotropic field of radiation produced by atomic recombination in the early uni­ verse. This discovery as well as more accurate observational techniques lead to the widespread adoption of the theory of a big bang universe.

Beyond its role as a confirmation of big bang theory, the CMB was also pre­ dicted to contain a wealth of information concerning the fundamental properties of the universe. The fantastic potential of the CMB spurred the development of nu­ merous precision experiments. The first experiment to measure the anisotropy in

CMB beyond the dipole due to the Earth's motion relative to the CMB was the

Cosmic Background Explorer (COBE) [6, 27, 50]. Launched in 1989, the COBE satellite was designed to measure both the blackbody temperature of the CMB as well as the first twenty multipoles of the CMB temperature anisotropy spectrum [6].

Following the success of COBE, balloon-borne experiments like BOOMERANG [60] and MAXIMA [59] as well as ground-based experiments like DASI [44] and CBI [17] began collecting data in the late 1990's. These experiments were able to observe the

CMB temperature anisotropy spectrum on smaller scales than the COBE satellite 5

and provided new constraints on cosmological parameters. Some early results are

documented in [66] and indicate a universe that is 13.7 billion years in age, spa­

tially flat with a baryon energy density of only a few percent. The next advance

came courtesy of the Wilkinson Microwave Anisotropy Probe (WMAP) in 2001 [7].

WMAP was not only much more sensitive than COBE but was also able to probe

much smaller scales. The scientific contributions produced by WMAP have been of

monumental importance in the field of cosmology and include significant refinements

on previous cosmological constraints as well as highly accurate observations of the

CMB temperature anisotropy spectrum on scales far smaller than COBE's earlier

limits [7, 33, 67, 70].

These results brought about not only new information and constraints but also

new questions as well as modifications to the theoretical framework of cosmology.

With the scientific content of the CMB not nearly exhausted, future experimental frontiers entail higher sensitivity on smaller (arcminute and lower) scales as well as better sensitivity to polarization. Part of the struggle now lies in analysing and mod­ elling small-scale CMB anisotropics as well estimating their contribution to the CMB spectra. Understanding small-scale temperature anisotropics such as the Vishniac effect is an important facet in the analysis of future CMB data and may also help in constraining cosmological parameters. Observations of polarization anisotropies such as those created during the era of reionization have the potential to improve upon fundamental errors in current cosmological measurements and may contain informa­ tion pertaining to primordial density fluctuations inside the local cosmic horizon. 6 1.3 Preliminaries As previously mentioned Hubble observed that the more distant a galaxy ap­ peared the more rapidly it receded, thus providing evidence for an expanding uni­ verse. Based upon isotropy and homogeneity in the universe, the two pillars of cosmology, it was possible to describe this expanding universe via the Friedman-

Robertson-Lemaitre-Walker metric, defined in many modern cosmology textbooks

(such as [12, 20, 47, 56]) as

ds2 = -dt2 + a(t)2 (^3^ + r2d02 + r2sinW) , (1-1) where a(t) is the scale factor of the universe, related to redshift z such that a =

l (1 + z)~ and normalized such that today a(t0) = 1. In this equation K corresponds to the curvature of the universe such that K < 0 signifies an open universe, K > 0 signifies a closed universe and K — 0 is a flat universe. When combined with Einstein's equation, R^ = 87rG(TM„ — ^g^T), it yields the so-called Friedman equations

87rG K

a 4TTG . „ . ,„ „. - = -^-(/o + 3p), (1.2) where p(a) is a measure of the total energy density of the universe (including matter, radiation and dark energy) and H = ^ is a measure of the rate of expansion of the universe often expressed in terms of the Hubble constant, Ho — 100/t km sec-1 Mpc-1. The equation of state relating pressure to energy density can then be denned as

P = wp, (1.3) where the energy density of a species is

3{1+w) P(xa- . (1.4)

For matter w = 0 and thus pM oc a-3, for radiation w = | and pn oc a-4 and for vacuum energy w = — 1 and p\ oc a0. Thus at early times the scale factor (a(t) oc t1^2) was very small and energy density due to radiation would have dominated followed by a period of matter domination (a(t) oc i2//3) and finally at late times vacuum energy domination (a(t) oc eHt).

From equation 1.2 it is also possible to define an energy density parameter for a species i, normalized by the critical density required for a flat universe as

a = —, (i.5) Pcrit where

PerU = g^. (1.6)

This sum of energy densities from all species is the total energy density of the universe and will determine the overall geometry of the universe.

It is now useful to derive some of the key measures of cosmology. From equation

1.1 one can express the distance that light has travelled since the big bang as the the conformal time /"* dt' ,-. ^ Jo a(*) 8

As a corollary the sound horizon is defined to be the distance a wave can travel prior to a time rj

rs(V)= [ drics{rj), (1.8) Jo where the sound speed of a fluid is defined as

Cosmological distances are often expressed in terms of the comoving distance to a redshift z *{z)=L'w)d* (LIO) where

1 2 H(z) = tf0[a»(l + zf + nK(l + zf + QK] ' (1.11)

Similarly the angular diameter distance can be expressed as

1/2 /2 DA = ^ sinh(l4 X), (1.12) where DA —• X as &K —> 0. 1.4 Early Times

Following the initial successes of big bang theory there arose some worrying inconsistencies as documented in [8, 12]. One of these concerns was known as the

"Horizon Problem" and had to do with the observed isotropy and homogeneity in the visible universe. Under standard Big Bang theory widely separated areas of the universe could not have ever been in causal contact and would thus not be expected to be in equilibrium. Because of this one would not expect the universe 9 to be homogeneous and isotropic on very large scales. This problem was especially evident in the CMB which maintained a very nearly perfect blackbody temperature across scales that under standard big bang theory could never have been in causal contact.

The second problem was known as the "Flatness Problem" and concerned the observed value of the total energy density parameter Q. It was found that the total energy density in the universe was very close to the critical density required for a geometrically flat universe. This presented a fine-tuning problem since as one extrapolated to the distant past the curvature of the universe would have to have been extremely close to zero as any initial deviation would grow with time, making a late-time flat universe extremely unlikely.

A resolution to both of these problems was found in the theory of inflation, a brief period of time during which the universe would grow at an exponential rate. During this period the Hubble parameter would remain constant resulting in a scale factor that increases exponentially, a(t) = em. Sub-horizon causally-connected regions could be stretched to super-horizon scales, resulting in the homogeneity and isotropy observed in both the CMB and the large scale universe. This brief inflationary period

(lasting approximately 10~33 seconds) would also have the effect of diluting any initial curvature. This alleviates the fine-tuning problem in that even a universe that was not flat initially would be "stretched" by inflation towards Q = 1 resulting in the observed flatness today. In this way inflationary theory became a central component of the modern big bang theory. 10

Following the inflationary period would be the era of radiation domination. As the universe expanded it would continue to cool; eventually the temperature of the universe would decrease sufficiently for nuclei to form in a process known as big bang nucleosynthesis [20, 47, 56]. This process would result in hydrogen comprising 75% of all nuclei with helium comprising approximately 24% and all other light elements comprising the remainder.

Following nucleosynthesis the next important stage in the evolution of the uni­ verse would be the end of the radiation dominated era and the beginning of matter domination. By solving for matter-radiation equality it is possible to find the redshift at which this occurred, from [20]

-E^ = £HL (1.13) Pcrit Pcrit 5 fir 4.15 x 1Q- = QI1 a4 h?aA a3 l ' } 4-15 x 10-5 a- = sue (L15) 4 2 1 + Zeg = 2.4 x 10 nmh . (1.16)

After this time matter energy density will be the dominant form of energy den­ sity in the universe and the scale factor will increase more quickly, a(t) oc t2/3 in the matter-dominated era. Matter domination would continue until vacuum energy density became dominant, as occurs at late times.

1.5 Inflation and Growth of Perturbations Inflationary theory predicts that soon after the big bang the universe entered a brief phase of exponential expansion. During this time the non-zero temperature of 11 the vacuum in the accelerating universe would produce fluctuations in the inflaton field 4> (following [12])

|A^|fc - TGH, (1.17) where TQH = ^ is the Gibbons-Hawking temperature of a universe dominated by potential energy and k is the wavenumber of the perturbation. These led to fluc­ tuations in the energy density (again assuming a universe which is dominated by potential energy)

Sp = Vi(4>)5. (1.18)

The density fluctuations are often expressed as

(^ YES U]{k)d{\nk), (1.19) \ r rvas/ •> where

A2(fc) = ^T"- (1-20) is the dimensionless representation of the power spectrum of the perturbation and

*s^/e"*"7*x (L21) is the fractional density perturbation in Fourier space [12]. These fluctuations were then "stretched" to super-horizon scales by the rapid expansion of space, resulting in a nearly scale-independent spectrum of primordial density fluctuations.

Current inflationary theories favour a "slow roll" condition which requires that the potential of the scalar field responsible for inflation (denoted as Vj) be very nearly 12 flat and very slowly changing as defined by the following parameters (from [12]) «. - \™l (I)' (1-22) V m = mP^J> (L23)

19 2 where mp is the Planck mass (1.2xl0 GeV/c ) and ej,T7j << 1. The slow roll parameters 77^ and e* also modify the scale-invariance of the spectrum of initial fluc­ tuations via the "spectral index"

n = 1 - 6e* + 2Vi, (1.24) where n — 1 corresponds to a scale-invariant spectrum.

The power spectrum of the gravitational potential $ can then be written in terms of the spectral index n as

2 k <5$(fc)| ,„_! , . * = —2?r2 ' ( ' where a value of n = 1 would correspond to a scale-invariant spectrum [9, 56]. The density (equation 1.20) and gravitational potential power spectra are related via the cosmological Poisson equation

4 Al = l(JLj nm(i + zy&i (i.26)

where Ho is the Hubble parameter and Q,m is the total matter density expressed in

29 -3 units of the critical density (pcrit — 3HQ/8TTG = 1.879 x 10~ h? g cm ). Equation 1.26 can be used in conjunction with equation 1.25 to find the scale dependence of 13 the density power spectrum

A? - kn+3 (1.27) and consequently

|4|2 ~ kn. (1.28)

The relations in equations 1.27 and 1.25 will be explored further in Chapter 3.

It is now possible to investigate how these perturbations evolve in time. To do so requires the continuity equation, the Euler equation and the Poisson equation for gravity (as defined in [9])

p + V-(pv) = 0 (1.29)

v + (v- V)v + -Vp + V$ = 0 (1.30)

V2$ = A-nGp. (1.31)

The perturbations of the energy density, velocity and pressure can be written in terms of the physical coordinate x as

p(t,x) = pQ(t)(l + 6t(t,x)) (1.32)

v(t,x) = v0(t,x) + <5v(i,x) (1.33)

p(t,x) = Po(t) + 6p(t,x), (1.34)

where again 5e is the real space fractional energy density perturbation and 5p =

(?s5p is derived from the sound speed (?s (equation 1.9). By switching to comoving coordinates such that

x(t) = a(t)x'(t) (1.35) 14 it is possible to solve the three previous hydrodynamical equations and obtain

2 2 I + 2-je - %V x,5e - 47rGpcA = 0. (1.36) a a2 x From this equation one can find the Jeans length

2 fcj = (W)" . (1.37)

For physical wavelengths shorter than the Jeans length (kp > kj) the solution is in the form of a power-law damped oscillator

Sk(t) ~ a-l/2(£)e[±-sfc/dtVi(t\ (L38)

For fluctuations whose physical wavelength is longer than the the Jeans length

2 3 (kp « kj) the general solution in the matter dominated era (a(t) oc £ / ) is

2/3 Sk(t) = cxi + c2r\ (1.39)

Thus there are two possible solutions with two constants; one solution corresponds to a growing mode while the other corresponds to a decaying mode. As noted in [47] at late times only the projection onto the growing mode is important. One can then derive, as done by Heath in [32] and later in [13, 47, 56], the functional form of the growth rate of density perturbations in an expanding universe DW=^4rl dz{1+z)[w)) • (1-40) 15

While this growth rate does not depend upon the spatial scale of the perturbation it is sensitive to the cosmology of the universe, specifically the matter and dark energy densities Qm and Cl\. Chapter 2

Cosmic Microwave Background

2.1 Recombination The primary focus of this work is the analysis of small scale anisotropies pro­ duced in CMB spectra at late times. However, much of this analysis is based upon what has been learned of the universe as measured by CMB anisotropies generated when the universe was only 300,000 years old. Much of this information comes from observations of temperature anisotropies in the photon distribution performed by experiments such as the WMAP satellite [7, 33, 67, 70]. These deviations from a uniform blackbody spectrum provide important constraints on cosmological param­ eters and theories.

Approximately 300,000 years after the big bang, long after nucleosynthesis, the temperature in the universe would have cooled to ~ 1 eV. At this temperature it became possible for the recombination of atoms to commence. Ionized species such as hydrogen and helium could capture free electrons, form atoms and emit photons in the process. Although the ionization energy of hydrogen (the dominant atomic

16 17 species in the universe) is 13.6 eV the high photon to baryon ratio in the universe would have reduced the recombination rate significantly [20, 56]. Because of this recombination occurred after the temperature in the universe had dropped to ~ 1 eV, at a redshift of approximately z* ~ 1100. The photons emitted at this time would bear an imprint of the density perturbations in this early universe. In the neutral, post-recombination universe CMB photons would not interact significantly with free electrons and would thus undergo "free-streaming", maintaining their characteristic temperature and polarization spectra. It is these photons that are observed today and whose anisotropy spectra provide some of the most sensitive probes of fundamental cosmological parameters in the universe.

In the period prior to recombination the baryons and photons in the universe would still be coupled via Compton scattering processes and would behave as a single baryon-photon fluid. The spectrum of scale-free perturbations that emerged from inflation would be imprinted upon this fluid. During this period the sound speed of the baryon-photon fluid would be cs ~ -%= [20, 47, 56]. The comoving Jeans length at this time would be of the order of a few hundred Mpc, approximately as large as the horizon at the time (in comoving coordinates); modes larger than the Jeans length would be growing and smaller modes would be undergoing damped oscillations. Thus there would exist fluctuations on all scales undergoing gravitational contractions due to the combined fluctuations of the dark matter, baryons and photons while also undergoing rarefactions due to pressure feedback in the coupled fluid. The compressional heating and rarefaction of these oscillations as well as an effective 18

Doppler shift will be imprinted on the temperature spectrum outgoing photons as recombination takes place.

Prior to recombination it is possible to describe the perturbations to the photon distribution that led to temperature fluctuations (working in units such that G = c = 1); the monopole of the photon distribution may be written as

@o(v,k) = ~(v,k). (2.1)

These perturbations can be expressed in terms of a driven, damped harmonic oscil­ lator, following Hu [37]

a l -\- a where F is a gravitational forcing term, R = j0&/4p7 is the baryon to photon ratio (R ~ 0.6 at recombination [24]) and overdots represent derivatives with respect to conformal time r? (equation 1.7). The sound speed can be expressed as a function of the baryon to photon ratio R such that it is clear that higher baryon densities result in lower sound speeds in the baryon-photon fluid

2 _ dp _ 1 (2 3) °s~ dp~ 3(1 + R) [Z'6) and in the limit of a pure radiation field the sound speed is cs = -7=. In describing the FRLW metric in equation 1.1 the universe was assumed to be smooth. In order to describe the perturbed universe, as performed in [9, 20], small perturbations ^ and $ are introduced into the metric such that

ds2 = {1 + 2W(x, t))dt2 - a{t)2{l + 2$(x, t))(dx2 + dy2 + dz2) (2.4) 19

where $ represents the Newtonian potential and $ corresponds to the gravitational potential which are related via \I/ ~ — . The gravitational forcing term can then be expressed as

^•'^•-STTJi*-*- (2'5) In equation 2.5 the effect of gravitational infall is described by ^, $ represents the

dilation effect and $ is the modification to the damping caused by the expansion of

the universe. By utilising equations 2.2 and 2.5 one can describe the overall equation

governing the acoustic oscillations in the baryon-photon fluid as

1 + + |e + 1 = $-* (2.6) ^ ir55 ^ - * T 1 + R

The solutions to this equation will describe the fluctuations in the baryon-photon

fluid on intermediate and small scales.

On the largest scales one can characterise the perturbation to the photon dis­

tribution as follows (as done in [20])

e.t*,*) = ?5J|a). (2.7)

Since the photons at recombination had to travel out of both positive and negative

potentials their wavelengths were stretched (\l> < 0 ) or compressed (^ > 0) and thus the effective temperature observed today is actually 0o + ^f. Setting \& ~ —$

one may write

(e0 + *)(M*) = ^(M*)- (2-8)

By following [20] this expression can also be represented in terms of the dark matter density fluctuation 5 by integrating the large scale evolution equation 6 = —3$ with 20

the initial condition that 5 = |<1> such that

S(V*) = 1*P - 3 [<%*) - *p] = 2*(^), (2.9)

where <&p refers to the primordial potential set during inflation and can be expressed

as $ rsj ^$p assuming that recombination occurs long after matter-radiation equal­

ity. Thus the temperature anisotropy can be expressed in terms of the dark matter

density fluctuation

(90 + tt)(M*) = -£<%•)• (2-10)

This expression states that large, overdense regions will produce cold spots on the

observed sky due to the large potential that the photons must climb out of, whereas

underdense regions will appear as hot spots on very large scales. It also gives a

proportionality of \ between observed temperature fluctuations and initial density

fluctuations, an important constraint to many theories of structure formation.

On the smallest scales the acoustic oscillations in the baryon-photon fluid be­

come washed out by diffusion damping. At scales far below the sound horizon rs (equation 1.8) the coupling of photons to baryons breaks down and oscillations at the photon diffusion scale are exponentially damped. In general the damping factor

(as in [37]) can be defined as

P(?7,fc) = e(-fc/fc^2, (2.11) where the damping scale is

/:-2 1 /"J 1^ + 4(1 +JZ)/5 21

m Here TT(J)) = j dr] neaTd is the optical depth , ne and a? are the electron number density and Thomson cross-section respectively, TT = ^p = —neaTa is the scattering rate and % refers to the conformal time today. In the limit of small baryon ratios the damping length is approximately Af> ~ / d-q/tr which corresponds to a random walk process with a Compton mean free path of Ac = l/fy. Thus all small-scale oscillations in the photons and the baryons will be damped, resulting in a power spectrum that decreases exponentially for k > 0.05Mpc_1. As will be seen in Chap­ ter 3, on small scales this damping causes the primary CMB anisotropy spectra to become subdominant to secondary scattering anisotropics like the Vishniac effect.

The aforementioned fluctuations will persist in the baryon-photon fluid until recombination at z* ~ 1100 at which point hydrogen can combine with free electrons and form neutral atoms. As this occurs the free electron density drops quickly over a small time scale. The photon distribution along with its temperature perturbations is able to free stream as it is no longer coupled to the baryons. In this newly neutral universe photons would be able to travel distances of many Hubble lengths without interaction, thus preserving their distinct temperature spectrum. It is the anisotropies in this temperature spectrum, caused by the initial oscillations in the baryon-photon fluid, that allow for strict constraints on fundamental cosmological constants like the scalar spectral index ns as well as the baryonic and dark matter energy densities, fi& and £ldm, among others.

2.2 Fluctuations on the Sky Having derived the fluctuations that were present in the baryon-photon fluid prior to recombination it is important to describe how those fluctuations manifest 22 themselves on the sky. Since the signal of the photon temperature anisotropy presents itself on a sphere it is useful to decompose it into spherical harmonics (as performed in [20, 37])

— (p) = @(x,p,r)) = 53 S ae™(x,v)Yem(p), (2.13) e=i m=-t where (returning to standard vector notation) x is the position of observation, p is the incoming direction of photons and £, m are the spherical harmonic conjugates to the real space unit vector p. Using the properties of spherical harmonics it is possible to express the temperature perturbation in terms of the observable quantity, the a^m's n aem&v) = fj^J '*f dnY;m(p)e(k,p,v). (2.14)

The mean value of all the a£TO's is zero but the variance of the a^TO's, Ce, is nonzero

(aem) = 0 (2.15)

(Ct. (2.16)

C/ is a measure of the width of the distribution of the a^m's. Since the number of measurable a^m's for any £ is 2£ + 1 there is clearly a limit to how well Ce can be measured. This limit is known as cosmic variance

[&Ce \ °* / cosmic variance v ^ < L and is most pronounced at small values of £, corresponding to the largest scales in the universe thus placing an absolute bound on the amount of information available at these scales in the primary CMB signal. 23

It is now possible to formulate an expression for Ci in terms of the dark matter overdensity 8 and the Legendre moments of the photon distribution for each A;-mode;

Qe is defined in [20] as

0/(fc, 7?0) ^ [ 90(fc, V*) + *(fc, V*)]Jt[k(Vo ~ V* (£+l)jt[k(rio-v*)} + 36i(fc,77*) ( je-i[k(j]0 - ??*)]

drje-^ \y(k,r]) - $(M)J j^k^o - rf)], (2.18) where ji is the spherical Bessel function, 0o is the photon distribution monopole and

©i is the photon distribution dipole. Ct can then be expressed as

2 f°° 2 e (k) C = - dkk P(k) e (2.19) t n Jo S(k) here P(k) refers to the power spectrum of the dark matter density fluctuations

S(k)5*(k')) = (27rf<53(fc - fc')P(fc). (2.20)

A more common expression is the dimensionless power per logarithmic interval

k3 / k \3+n A^)U = ^Ps(k) = 81 \JLJ T\k), (2.21) where n is again the scalar spectral index of the power spectrum, 8H is the amplitude of density fluctuations at the present Hubble scale and T(k) is known as the matter transfer function defined in [23] as

8(k,z = 0) 8(0,z = oo) T{k)^ (2.22) 8{k,z = oo) 6(0, z = 0) 24 where it is assumed that in the post-recombination era that the transfer function of the baryons (now essentially pressureless) approaches the shape of the dark matter transfer function. During radiation-domination photon streaming prevents potentials from growing thus density perturbations evolve differently depending upon whether the universe is matter or radiation dominated. This imprints a characteristic scale on the matter transfer function defined by the size of the particle horizon at matter- radiation equality

/2 _2 2 -1 keq = (2ttmH%zeqy = 7.46 x 10 fimft e^ Mpc , (2.23) where

4 2 zeq = 2.50 x 10 Qmh e^ (2.24) and 02.7 is the ratio of the CMB temperature to 2.7 K. The transfer function has a value of 1 for k < keq and its amplitude is suppressed on scales smaller than keq. On intermediate and small scales the transfer function is affected by the baryon oscil­ lations, formed prior to recombination while baryons were still coupled to photons.

On the smallest scales the baryon transfer function is damped and no oscillations are present.

2.3 CMB Anisotropies In this section the generation of the large scale anisotropies in the CMB photon distribution will be reviewed. These anisotropies will prove integral not only in the generation of secondary polarization as explored in Chapter 4 but also in constraining 25 cosmological parameters such as those required in calculating secondary anisotropies like the Vishniac effect as presented in Chapter 3.

On large scales one of the primary sources of anisotropy is the Sachs-Wolfe effect, caused by the gravitational redshift of CMB photons as they exit potentials at the time of last scattering. This effect will depend upon the growth function, as described in equation 1.40, during the matter-dominated era such that D{a) oc a.

Motivated by equation 2.8 the anisotropy spectrum due to the Sachs-Wolfe effect is derived in [20] to be

2 Cr -

In this expression it is clear that if one plots the anisotropy as £(£+l)Ce then the

Sachs-Wolfe effect presents itself as a plateau in the spectrum at low £. At late times there is also the integrated Sachs-Wolfe (ISW) effect which arises from the influence of dark energy. Generally when a photon traverses a potential it experiences no net redshift; dark energy removes this symmetry by altering gravitational potentials at late times thereby introducing anisotropies into the photon spectrum. This is known as the integrated Sachs-Wolf effect and is explored more thoroughly in [18, 57, 62].

These large scale anisotropies, specifically the quadrupole temperature anisotropy, will be integral in Chapter 4 in creating polarization anisotropies during reionization. 26

On scales smaller than the sound horizon at last scattering the compressions

and rarefactions of the baryon-photon fluid imprint a distinctive pattern of peaks

and troughs onto the temperature anisotropy power spectrum. It is these peaks that

provide precise, model-independent constraints on important cosmological quantities.

Prior to recombination the baryon-photon fluid would be undergoing oscillations on

all scales; these scales correspond to different wavelengths of the perturbation mode

k, each of which would be "frozen in" during a different phase in its oscillation as

recombination proceeds. Since the photons in the fluid will have some bulk motion

associated with the phase of their oscillation there will also be a Doppler shift (gen­

erally expressed as ^ = ^f) imprinted upon the photon spectrum. The angular

positions of the peaks helps to constrain the geometry of the universe; following [37]

one can define the positions of the peaks as

kprefa) = jm, (2.27)

where again rs(rj*) is the sound horizon (equation 1.8) at last scattering. This can be approximated in angular units as

£P = kpDA(r]*), (2.28) where DA{T]*), as defined in equation 1.12, is the angular diameter distance to the last scattering surface. In this way the location of the first peak is a measure of the sound horizon at last scattering and thus also a measure of the curvature of the universe. Equally important are the relative heights of the peaks as the amplitude of the oscillations in the photon-baryon fluid will depend strongly upon the baryon 27 content in the universe as they increase the effective mass of the coupled fluid. Dark matter will also be influential as it allows for gravitational potentials to grow during the era of matter domination and thus increases the amplitude of the second peak relative to the first. These parameter constraints are integral towards estimating the power contributed by secondary temperature anisotropies like the Vishniac effect as explored in Chapter 3 as well as secondary polarization anisotropies as explored in

Chapter 4.

2.4 Reionization Following recombination the universe existed in a mainly neutral state of primar­ ily hydrogen and helium atoms. Without the pressure support of coupled photons, density perturbations would be able to grow. During this early period there would not yet be any stars or other sources of ionizing radiation in the universe. Precur­ sors for star and galaxy formation in the form of dark matter "halos" would exist, attracting baryons into growing gravitational wells. On small scales the condition for hydrostatic balance would apply

dp = _GpM^ dr r1 where Menc denotes the total enclosed mass of the object. Expressed in terms of a redshift-dependent baryonic mass the Jeans mass as described by Ferrara and Marri in [25] is

S/2 3/2 M^6x^(l±i)~ (^) ntMs, (2.30) 28

where Tgas is the gas temperature, M0 is a solar mass and masses larger than Mj will collapse under their own gravity. Gas clouds with masses larger than Mj should

collapse provided that the cooling time of the gas is shorter than the dynamical

time. This constraint is easily met in metal rich gases as there exist many available

cooling modes; the primordial gas however would be extremely metal poor, comprised

primarily of the hydrogen and helium created during big bang nucleosynthesis and

thus would have very few viable cooling modes [14, 25, 29]. Furthermore, virial

temperatures for these large objects are thought to be typically less than 8000.ftT

which is below the 104K at which cooling via Lycc transitions in hydrogen becomes

efficient [29, 31].

The primary cooling path for smaller objects would be through the small amounts

of molecular hydrogen, Hi, produced in protogalactic gas clouds following recombi­

nation [28, 29]. The abundance and density of H2 would increase during the collapse, allowing for the formation of early stars. As suggested in [25] in order for sufficient

cooling to occur the initial gas clouds required to form stars at a reshift of z ~ 30

6 would have to have been quite large, on the order of (M m 10 MQ). As they cooled

2 these large clouds could fragment into smaller objects (M* « 1O M0). These first stars (commonly referred to as Pop III objects) would be massive, extremely hot, short-lived objects that would inject their surroundings with ionizing radiation dur­

ing their brief lives as well as providing metals and explosive energy in their demises

as supernovas [4, 14, 25, 28, 29]. The net effect of these early stars is the slow but

steady reionization of the universe such that by z ~ 6.5 the universe is almost entirely reionized. 29

Observational evidence for the ionization state of the universe comes from the absorption of light by neutral hydrogen, a phenomenon first described by Gunn and

Peterson in [30]. The Lyman-a line of neutral hydrogen at 1216A absorbs the emitted light from distant quasars, leaving a distinct feature in the object's spectrum. Since neutral hydrogen along the observer's line of sight would absorb the feature in its own frame the net effect is a "trough" in the spectrum corresponding to the redshifted absorptions of all the hydrogen along the line of sight. Thus from observing distant quasars it is possible to deduce that the universe has been nearly fully ionized since z ~ 6.5 [4, 29].

In this era of reionization, CMB photons would again be in contact with free electrons and could again undergo scattering. This scattering would have two distinct effects on the CMB spectrum. The first effect would be a suppression of the primary anisotropies corresponding to the optical depth of the reionized region and the second effect would be the creation of secondary anisotropies in the CMB spectra. If one expresses the background temperature as T and the perturbation to this temperature as 6 the initial temperature of a photon can be written as T[l + 0]. Reionization, as described in [12, 20], will suppress this radiation by a factor of e~TT while also contributing a fraction (1 — e~TT) of photons at the background temperature T. The observed photon temperature then becomes

T[l + e]e_7T + T(l -e~rr)= T[l + Ge'^), (2.31) thus the fractional anisotropy observed today is Qe~TT where TT ~ 0.09. Reionization has suppressed the initial anisotropy 6 by a factor of e~TT on all scales smaller than 30 the horizon at reionization. This damping can only affect modes that are within the horizon at the time of reionization and thus larger modes {£ < r]Q/rjreian) would not be affected by this suppression. As stated the second effect of reionization would be the creation of secondary anisotropics in the CMB spectra. These anisotropics would be created via interactions with free electrons and would affect both the temperature and polarization spectra of the CMB as will be seen in Chapters 3 and 4.

On large scales the polarization created during reionization provides constraints upon the integrated optical depth to reionization, TT [40, 51, 54]. The angular size of this signal will depend upon the horizon size at reionization with late-time reionization scenarios corresponding to larger (lower £) scales. In the event of a

"patchy" reionization period, a polarization anisotropy can also be created on small scales via the interaction of the photon quadrupole anisotropy with pockets of free electrons [22, 58]. This signal will contain information concerning primordial density perturbations inside the local horizon and will be investigated in Chapter 4. A separate small scale CMB anisotropy occurs through the Compton scattering of

CMB photons in ionized regions moving with some bulk velocity along the line of sight. This secondary temperature anisotropy is known as the Vishniac effect [53, 71] and will be examined in Chapter 3. Chapter 3

Vishniac Effect

3.1 CMB Temperature Anisotropy Studies One of the earliest and most important measurements made of the CMB was that of its precise blackbody temperature, as completed by the COBE satellite in

1990 [6, 27, 50]. Here it was found that the temperature of the CMB was very nearly constant across the sky with a level of anisotropy of approximately 1 in 105.

These temperature anisotropics and specifically the acoustic peaks in the temper­ ature anisotropy power spectrum are valuable sources of cosmological information.

The first few of these peaks were measured by the WMAP satellite launched in 2001

[7, 67]. From these measurements a plethora of cosmological information was gleaned including important constraints on parameters such as the matter density as well as the overall geometry of the universe; however many cosmological parameters remain relatively poorly constrained. With future experiments promising not only improved

31 32 precision but also exploring smaller angular scales it has become increasingly impor­ tant to understand the mechanisms that contribute to the small-scale temperature anisotropy spectrum as well as their sensitivities to cosmological parameters.

An important mechanism for generating small-scale temperature anisotropies is the Sunyaev-Zeldovich (SZ) effect, explored in [36, 38, 42, 68] and many others. This late-time effect arises from the Compton scattering of cold CMB photons by hot

(~ 107K) intra-cluster free electrons. The high energies of these electrons coupled with the non-negligible integrated optical depths created by large clusters causes this effect to be a dominant contributor to the CMB temperature anisotropy spectrum on arcminute scales. This effect also has the feature that it has a unique frequency dependence with a null point at 217GHz. This dependence allows the thermal SZ effect to be effectively subtracted from the primary CMB anisotropy spectrum via sensitive multi-frequency measurements.

A similar effect occurs at late times whereupon CMB photons scatter in a re­ gion of enhanced electron density that is moving with a net bulk velocity along the line of sight. This creates a distortion in the anisotropy spectrum and contributes power on small scales. When the gravitational perturbation causing the local density enhancement of free electrons is in the linear regime this is known as the Vishniac effect [21, 38, 39, 43, 53, 71]. When this scattering is caused by nonlinear collapsed structures like galaxy clusters it is known as the kinetic SZ effect or equivalently the nonlinear extension of the Vishniac effect. These effects do not share the unique spectral signature of the thermal SZ; in fact they have the same spectral signatures 33

as the primary anisotropies and so are impossible to subtract from the primary tem­

perature anisotropy spectrum via multi-frequency measurements. Accordingly it is

extremely important to understand the scale at which these effects act as well as

their amplitudes. The Vishniac effect, as will be seen, also proves to be an effec­

tive constraint upon cosmological parameters. Depending strongly upon parameters

such as the amplitude of mass fluctuations as well as the physics of reionization the

Vishniac effect is important not only as a dominant contributor to the primary CMB

temperature anisotropy but also as a test of cosmology in its own right.

3.2 Reionization and Secondary Anisotropies The CMB and its primary anisotropies were formed at the time of last scat­

tering, z* ~ 1100. This spectrum would have remained relatively unchanged until

reionization {zreion ~ 11) at which time the primary anisotropies would experience a damping corresponding to the optical depth r^ — 0.09 and secondary anisotropies would be created in the photon distribution. At late times, line of sight velocities in baryon distributions can create anisotropies via the Doppler effect

eD(h) = JdXg(x)n-Mx), (3-i) where g = TTGTTT is the visibility function, overdots correspond to derivatives with respect to the comoving distance xiz)> ** *s the direction on the sky and V& is the baryon velocity field.

In linear theory gravitationally induced flows are irrotational and thus curl-free, simply depending upon the gradient of the scalar gravitational potential as seen from

equation 1.30. Because of this baryon velocities will orient themselves parallel to the 34 wavevector k of the perturbation. In general a photon will experience a Doppler shift corresponding to the baryon velocity. However, for modes smaller than the horizon there will be an equal number of photon scatterings on both sides of the perturbation and thus the total Doppler effect will be a coherent sum of shifts with alternating signs, resulting in severe cancellations. Thus, excluding very large scales, one expects the anisotropy due to the first order Doppler effect to be negligible.

At first order all modes along the line of sight cancel over many wavelengths of the perturbation; perpendicular modes, while not suffering cancellations, do not contribute to the Doppler effect. Thus, at first order there is no signal and second order Doppler effects become important. The Doppler effect depends not only on the baryon velocity field vj, but on the velocity field modulated by the probability of scattering TrVb (since g — fre"77'). Clearly regions with higher concentrations of free electrons will cause more scattering. As performed in [37] it is thus possible to define an effective baryon velocity that depends upon the electron number density ne as well as the ionization fraction xe

q(x) = [1 + <5ne(x)/ne] [1 + 5xe(-x)/xe) vfc(x)

= [l + <5b(x)][l + fee(x)/xe]vb(x). (3.2)

These second order effects are capable of escaping the cancellations suffered by the first order effects thus permitting a contribution to the CMB temperature anisotropy power spectrum. The following sections will assume a constant ionization fraction xe while allowing for the electron number density ne to vary. In Chapter 4 a polarization anisotropy that arises from variations in xe will be discussed. 35

3.3 Vishniac Effect

The Vishniac effect, first explored by Ostriker and Vishniac in [53, 71] and later in [21, 38, 39, 43] among others, is a secondary temperature anisotropy, generated at z « 1100, that occurs when baryon density perturbations, Sj,, are still in the linear regime. Here a spatial variation in the density of baryons (assuming no variation in the ionization fraction xe) modulates the visibility ^ = ^ = Sb, creating an effective velocity

vb(l + 8g/g) = vb + vg, (3.3) which allows for the possibility of an additive Doppler effect.

Following closely the derivation by Hu in [38] it is first useful to express the baryon velocity field in terms of its components such that v&(k) = (vx,vy, vz) where z || k

v6(k) = ^°>z+£^>^. (3.4) m=±m- l v The projection of v&(k) is then described as

• I fi-v (k) = -i^5]t;Jm)y (m)(n). (3.5) 6 3 1 m=—1

The projections of only the vector modes of the velocity field v5 are then written as

„0*>(k) = £±p.v,. (3.6)

By reexpressing the velocity field in terms of a convolution of the linear velocity field and the baryonic density field as

vff(k) = y^v6(k1)5b(fc2), (3.7) 36 where k2 = k — ki, the projection of the vector modes becomes

(3.8)

1 Expressing the m = 1 modes as a power spectrum and using the relation F1 (k2) =

Ip-Y^ki) one finds the logarithmic power spectrum of vg 2 A^/f^l^kOI A^(^A|(^-|A^t(fcl)A^(fc2) (3.9)

Expanding upon the approximation of equation 1.25, the gravitational potential power spectrum is written as

1 *i = l(-^Y~ ^(1+zfDHink), (3.10) which, in conjunction with equation 1.26, leads to an expression for the density power spectrum / h \ "+3 (3.11)

In order to describe the projection of 3-dimensional two-point statistics on the

2-dimensional sky the Limber equation may be used [38, 45, 46].

x Cf = 2^Jd xDAAl (3.12) where X is any general field that is expressed as the line of sight integral of another field S and DA is the angular diameter distance. Then the angular power spectrum for the Vishniac effect becomes e 2 A m) ^ = ^/^/ ^£ 2 > (3.13) m=±l 37 where AVg is evaluated at the wavenumber that projects onto the angular scale I at x(z)

k = H0-£-. (3.14)

Utilising the fact that the m = ±1 modes contribute equally and by deriving the velocity power spectrums using equations 3.11 and 1.29

D^ I Af, (3.15) 'v \D k the angular power spectrum can then be reexpressed as Cf = ^jixD\{sfJ^Jy. (3.16)

Here Iy is a mode coupling integral described as Iv = r iV1 f d„(i-m-^ww, (3,7) Jo J-i ViVl Agfc(fc) A|(fc) with

fi = k • kj

yi = ki/k

y2 = k2/k = ^/l - 2/xj/i + y\. (3.18)

By assuming that the largest contribution to the mode coupling integral come from the limit y\ « 1 the integral can be approximated (as done by Hu in [38]) as

i2 ^c) 4 f°°dy1Al(ky1,ri)

3j0 y\ Af6(M) 38

" no Ho) A- (3-19) where

vrms = JfAlb. (3.20)

While accurate for very small scales where the density and velocity fields have de­ coupled this mode coupling approximation proves insufficient on scales larger than

A; ~ 0.5/iMpc-1. The discrepancy between the two calculations can be seen in Fig.

3-1 where the approximated Vishniac spectrum is plotted with the exact solution from equations 3.16 and 3.17 using the WMAP-preferred cosmology of n = 0.96, ft0 = 0.24, 0A = 0.76, Qfe = 0.04, h = 0.73, zr = 11.0. It is evident that the approxi­ mation does capture the small-scale limit of the Vishniac effect, specifically at scales of £ > 10000. At larger scales the approximation breaks down, predicting an overall amplitude that is 40% larger than that of the full calculation as well as predicting a peak in the spectrum on slightly larger scales. Thus it was found that although com­ putationally expensive, the calculation of the full mode coupling integral is required in order to produce accurate results for the Vishniac effect on scales of £ < 10000.

In the calculation of the Vishniac effect it was assumed that density fluctuations in the baryons were linear in nature. This restricts the effect to early times before the formation of dense, nonlinear structures such as large galaxy clusters. The anisotropy caused by free electrons in large clusters that are caught in some bulk flow is known as the kinetic Sunyaev-Zeldovich effect (distinct from the thermal SZ effect). It is possible to consider this kinetic SZ effect as a extension of the linear Vishniac effect for very dense structures. Since nonlinear structure formation occurs on scales that are 39

below the coherence scale of the bulk velocity field, the density and velocity fields can

be thought of as decoupled. The total Vishniac effect from both linear and nonlinear

structure can be calculated by replacing the contribution of the linear density power

spectrum with the nonlinear density power spectrum, leaving the contribution from the linear velocity power spectrum and calculating the full mode coupling integral

under linear theory, resulting in

( • \ 2 4) Aj^Al/v. (3-21) This expression is valid so long as the velocity field of the bulk flow is in the linear regime, a fair approximation in standard ACDM theory.

Having derived an expression for both the linear and total Vishniac effects an expression for the nonlinear density power spectrum was sought. In order to simplify this effort it was assumed that baryonic matter simply traces dark matter with no feedback effects. This approximation, as noted in [38], will have the effect of creating

an upper limit for the amplitude of the spectrum as gas pressure feedback and other baryonic effects will in general decrease the overall baryon density on small scales, lowering the amplitude of the signal. In order to model the nonlinear density power spectrum it was first assumed, as done in [38, 55], that nonlinear fluctuations on a scale k are precipitated by linear fluctuations from a larger scale

1 s klin = [l + Al(k)}- / k, (3.22) 40 where A|_ is the cold dark matter density power spectrum. It is then possible to relate the linear and nonlinear power spectra at the two different scales

2(lin),m> Al(k) = fNL[AZ (klin)}, (3.23) where /JVL is a functional that relates the two spectra as defined by Peacock and

Dodds in [55]. The functional is denned as

1 + Bpx + [Ax\ais 1//3 INL = X (3.24) 1 + {[Ax]agl{9)/[Vxl/2}Y where

fi 7 9i(tt) = 2 [ ™ - "A + (1 + On/2)(l + ^A/70)]-1 (3.25) with the following best-fit parameters [55]

a947 A 0.482(1+ ne///3)-

B 0.226(1 + rieff/3)-1-778

0244 a 3.310(1 + ne///3)-

287 & 0.862(1 + neff/3)-°-

423 V 11.55(1 + ne///3)-°' . (3.26)

Here ne// refers to the effective slope of the power spectrum at a given value of k. By using this formulation it is possible to model the nonlinear density power spectrum and in so doing place an upper limit on the total contribution to the Vishniac effect from both linear and nonlinear structures. 41

1 1—I I I I 111 1 1—I I I I III 1 1—I I I I I

. _ linear (approx.)

0 i 111ii i i i i 11111 i i i i 1111 100 1000 10000 100000 multipole £ Figure 3-1: Comparison of the linear Vishniac effect, the approximation of the linear Vishniac effect and the maximum effect from both the linear and nonlinear density power spectra. 42

It is possible to see the effect that the inclusion of nonlinear structures have on the linear Vishniac effect in Fig. 3-1. While nonlinear structure adds power to the spectrum at all scales its effect is most pronounced when £ > 1000. This is to be expected as the nonlinear structures are more dense and smaller in size than their linear counterparts and thus the majority of their contributions occur on scales corresponding to the approximate size of galaxy clusters on the CMB sky. It is also interesting to note that the amplitude at small scales of the maximum Vishniac effect is more than twice that of the linear effect alone, indicating that scattering from nonlinear structures is a significant contributor of anisotropy on small scales.

As previously mentioned the curve for the maximum effect in Fig. 3-1 is an upper limit assuming that the baryons simply trace the dark matter. As such it is to be expected that there will some smoothing on scales of I > 10000 due to the effects of gas pressure in the baryons, this is explored more extensively in [38]. The extent of this smoothing is unclear so the linear and maximum Vishniac effects are presented as limiting cases. Notwithstanding this small-scale smoothing it is evident that the kinetic SZ effect provides a significant boost to the underlying linear Vishniac signal on sub-arcminute scales.

3.4 Contribution to the CMB spectrum Having calculated the Vishniac effect for a single fiducial cosmology it was then important to estimate the effect that this spectrum could have on the total CMB anisotropy spectrum. Due to diffusing damping, the primary CMB anisotropy spec- 43

10000 1—'—'~r\—<-

1000 Primary

100 t- It

< 10 t-

0.1 10 100 1000 10000 100000

Figure 3-2: The spectrum of primary CMB anisotropics is exponentially damped at small scales (large values of £). This allows for secondary anisotropics like the Vishniac effect to dominate on sub-arcminute scales. Primary CMB spectra were produced by the CAMB online implementation of the CMBFAST [65] code for a fiducial model where only zre is allowed to vary and n = 1.0, Q,\> — 0.04, Q,m = 0.27, nA = 0.73 and h = 0.72. 44

ACDM Model: Parameters, Values and Descriptions

Parameter 68% Marginalized Value Description

2 nmh 0.1277^ !oo79 Total matter density of the universe 2 Q,bh 0.02229 ± 0.00073 Total baryon density of the universe h 0.732to:o32 Hubble constant ns 0.958d_0.016 Slope of the scalar perturbation spectrum TT 0.089 ± 0.030 Optical depth a8 0.76llo:o48 Amplitude of fluctuations Table 3-1: Parameters, values and descriptions for the standard ACDM cosmology. All values are taken from [67]. Note that in a flat universe the dark energy density is defined as Q,A — 1 — ^m- trum is severely damped on small scales as seen from the spectra1 in Fig. 3-2.

This permits the bulk of the small-scale CMB anisotropy signal to be comprised of secondary anisotropies like the Vishniac and kinetic SZ effects. These effects depend strongly upon cosmological parameters like the optical depth to reionization TT and

1 the rms mass fluctuation on 8k" Mpc scales, CT8. In order to explore the possible contribution of the maximum Vishniac effect it is important to calculate it not only for a fiducial cosmology but also for all points in the allowed parameter space.

The parameter space to be explored is that of the standard six-parameter ACDM model which will now be briefly reviewed. The parameters of the model and their best-fit values are found in [67] and are listed in Table 3-1. The values for these pa-

1 Primary CMB spectra were produced by the CAMB online interface found at http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm 45 rameters were found using a Markov Chain Monte Carlo (MCMC) approach, utilising the three year data from the WMAP satellite experiment. As explained in [70] the

MCMC is a method of simulating posterior distributions. The posterior distribution of a set of parameters denoted a given an event x can be expressed as

P(alx) = JP{x\a)P(a)da' (3-27) where V{x\a) is the likelihood of an event x given the model parameters a. V(OL) is an expression of the prior probability density. The MCMC is then capable of exploring the six-dimensional ACDM likelihood surface through random sampling of the posterior distribution. A MCMC analysis is significantly more efficient than a grid search analysis particularly for large dimensions. The only caveats of the ap­ proach are that the chains must converge and be well "mixed" in order to accurately represent the entire parameter space; these conditions are explored further in [70].

The WMAP three-year chains are available online through the Legacy Archive for

Microwave Background Data Analysis2 and provide a simple and effective way to accurately sample the allowed parameter space for the six-parameter ACDM model.

The chains initially held information pertaining to nearly 40000 separate cos­ mologies, each weighted according to its respective likelihood. These chains were then randomly sampled to include only 9187 different cosmologies, still a thorough representation of the allowed parameter space.

2 found at http://lambda.gsfc.nasa.gov/ 46

0.045 1 1 1 1 i 0.04 - ; £ = 10000 it 0.035 ;/jk/ = iooo a 0.03 J. 0.025

| 0.02 i K 0.015 0.01 :j ;;y\ £ = 5000

0.005 i 0 •J 1 1 ''^K^fiA^-.rV, ,•- 0 0.5 1 1.5 2 2.5 3

. „, L , -rZ^

Figure 3-3: Histograms of the linear Vishniac effect power for £=1000 (solid red), 5000 (long-dashed green) and 10000 (short-dashed blue). Results from 9187 allowed cosmologies binned in 0.02/^k2 intervals.

The spectra of the linear and maximum Vishniac effects were then discretized into ten equally spaced points in multipole space and calculated for each of the nearly

10000 cosmologies. These ten points ranged from £ = 1000 to £ = 10000 inclusively.

The large range in £-space was chosen in order to accommodate the spectral peaks of both the linear and maximum Vishniac effects as it was demonstrated earlier that the maximum effect will tend to peak at slightly smaller scales than the linear effect.

The calculation was done separately for the linear and maximum Vishniac effect and the results were separated into 500 linearly spaced bins. Fig. 3-3 shows the 47

0.03

0.025 £ = 1000 s 0.02

a 0.015 I = 5000 •8 •§ *• 0.01

0.005 I = 10000 0 J%Wa.?J,.A 0 12 3 4 5 6 AT/OIK2)

Figure 3-4: Histograms of the maximum Vishniac effect power for £=1000 (solid red), 5000 (long-dashed green) and 10000 (short-dashed blue). Results from 9187 allowed cosmologies binned in 0.02//k2 intervals. results for the linear effect while Fig. 3-4 contains the results of the maximum effect. The rms power and la contours are also presented in Fig. 3-5. In the case of the linear Vishniac effect it is clear that the spectrum is peaking near to £ ~ 3000 as expected from the fiducial model. This result seems to be fairly robust across all permitted cosmologies indicating that the linear perturbations that produce the majority of the power have a characteristic size on the sky. The power at the peak of the effect is 0.80 /zK2 which indicates that this effect may produce a measurable contribution to the primary CMB anisotropy which is significantly 48

2.0

linear Vishniac effect max Vishniac effect

^ . .. ^„:n;%m i a) S o Q_

2L 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 I I

Figure 3-5: Plot of rms power for ten values of £ for the linear (left) and maximum (right) Vishniac effects. The shaded region represents the 68% confidence contour while the dotted lines represent the spectra for three randomly-chosen cosmologies. damped on these small scales. Also of interest is the large spread in power that the linear effect exhibits, this is especially visible in Fig. 3-3. Even in the small range of parameter space that is allowed by the WMAP constraints the amplitude of the linear Vishniac effect varies by an order of magnitude. This variation indicates that the effect is strongly dependent upon slight changes in cosmology making it quite difficult to constrain.

Similar results were found for the maximum Vishniac effect. In this case the peak of the spectrum was observed to fall at smaller scales than the linear effect, as predicted. The majority of the power is found at £ ~ 10000 indicating that small-scale, nonlinear structures are contributing a large fraction of the power. As presented in Fig. 3-5 the power at the peak is 1.9 (JK2 but with a large standard deviation of 0.95 fiK2 indicating that the result is again quite sensitive to cosmology. 49

This sensitivity is shown in Fig. 3-4 where it is again evident that the power in the maximum effect spans an order of magnitude.

On the much smaller scales that the maximum Vishniac effect peaks the primary

CMB anisotropy is strongly damped. This allows for the thermal SZ effect (with an rms amplitude of ~ 10/j.K and the maximum Vishniac effect to be dominant contributors of power on these scales, potentially admitting future detection [35].

Both the upcoming Planck satellite and South Pole Telescope are expected to be capable of detecting the thermal SZ signal at a number of frequencies [15, 61, 64].

SPT in particular has the capability, assuming a nearly-perfect subtraction of the thermal SZ signal, to detect the maximum Vishniac effect as well [61]. The Planck satellite is also capable of observing maximum Vishniac effect from galaxy clusters however the errors in associated cluster velocities are expected to be high, on the order of a few hundred km/s [15]. As noted in [41], the upcoming Atacama Cosmology

Telescope [48] is also expected to be capable of detecting the maximum Vishniac effect through precise multi-frequency measurements of the CMB at small scales.

3.5 Modelling the Vishniac Effect

Having performed the computationally intensive task of computing both the linear and maximum Vishniac effects for a huge class of cosmologies a simpler method of calculating these effects was sought. This was accomplished by creating a fitting function that provided an efficient means of calculating both effects for any cosmology in the allowed parameter space. This function was chosen to have a power law 50 dependence in four of the six cosmological parameters, denned as

S /(,,„.,ft,*,rT) - A (ik)' (J^)' (^Y (^) , (3,8) where A, a, (3,7, S are real parameters to be determined and the fitting function is normalized to a fiducial cosmology. The Hubble constant h and scalar spectral index n were omitted due to the relatively tight constraints on these two parameters as well as their minor influence on the Vishniac effect which, if any, could be encompassed by the overall amplitude variable A. Due to having only five free parameters in a relatively confined parameter space a simple grid search algorithm was initiated in lieu of a full MCMC approach. The grid was set up with a coarseness of 0.1 and at every point in the grid the fitting function was computed for each of the 9187 sample cosmologies and then compared with the exact solution for that cosmology resulting

• 2 in a residual of J2i=i ^"potaffl &— ^or eac^ Possible model. In this fashion it was possible to compute a most probable fitting function for each of the ten £ values

{£ =1000, 2000... 10000). The results for the linear Vishniac effect are presented in

Table 3-2 while the results for the maximum effect are presented in Table 3-3.

The first point to note in both tables is that the amplitude parameter A (ex­ pressed in units of /JK2) gives an approximation of expected power as a function of

£, normalized to a common fiducial model. Since the permitted cosmologies only vary slightly from the fiducial model one can expect its power and peak predictions to be fairly robust. This indicates that the linear effect will produce a spectrum that will tend to peak at £ « 2000 — 4000, a value that is in agreement with the predictions of Fig. 3-5. This allows for the possibility of future detection for the 51

Linear Vishniac Effect Fitting Results

£ Residual fi« Q$ aj r5 A (^K2) 1000 2.5 0.6 2.0 2.9 0.5 0.64 2000 1.6 0.6 1.8 3.5 0.5 0.79 3000 1.8 0.6 1.7 3.8 0.5 0.80 4000 2.3 0.6 1.6 4.0 0.5 0.78 5000 2.5 0.6 2.0 3.6 0.6 0.72 6000 2.6 0.6 1.9 3.7 0.6 0.69 7000 2.6 0.6 1.9 3.8 0.6 0.66 8000 2.6 0.6 1.9 3.8 0.6 0.63 9000 2.7 0.6 1.8 4.0 0.6 0.61 10000 2.8 0.6 1.8 4.0 0.6 0.59

Table 3-2: Best fit parameters for the linear Vishniac effect. linear Vishniac effect as the primary CMB anisotropy spectrum will be significantly damped on scales of £ « 4000. In the case of the maximum Vishniac effect it is apparent that the peak of the spectrum is occurring at a larger amplitude and at smaller scales. From Table 3-3 it appears as though the maximum effect is peaking on scales of £ « 10000 and with an amplitude more than twice that of the linear effect (as predicted by Figs. 3-3 and 3-4). This is consistent with earlier estimates and again confirms the possibility of detecting the maximum Vishniac effect on these sub-arcminute scales.

The second point of interest in Tables 3-2 and 3-3 is that the best-fit parameters a, (3,7 and S all carry a positive value. This indicates that an increase in the matter density, baryon density, mass fluctuations or optical depth has a positive effect on the amplitude of the Vishniac effect. Perhaps this is not surprising in that an increase in any of these four variables has the effect of increasing scattering whether by creating 52

Maximum Vishniac Effect Fitting Results

5 2 e Residual s'ro nf *1 T A (pK ) 1000 10.6 1.2 1.5 3.2 0.4 0.96 2000 6.8 0.6 2.0 4.2 0.3 1.34 3000 4.3 0.6 2.1 4.4 0.3 1.51 4000 4.0 0.6 2.2 4.4 0.3 1.60 5000 3.7 0.6 1.7 5.1 0.2 1.77 6000 3.0 0.6 1.7 5.1 0.2 1.82 7000 2.7 0.6 1.8 5.1 0.2 1.84 8000 2.6 0.6 1.8 5.1 0.2 1.88 9000 2.7 0.6 1.8 5.1 0.2 1.90 10000 2.8 0.6 1.8 5.2 0.2 1.92

Table 3-3: Best fit parameters for the maximum Vishniac effect.

more scatterers (higher TT or fifo) or by increasing the density of scatterers (higher Qm or as) both of which increase the probability of scattering upon which the Vishniac effect depends.

Both the linear and maximum Vishniac effects appear to depend similarly upon

Q,m and £V This would indicate that the dependence upon Clm is primarily in the bulk velocity field as it is shared between the two effects. This supposition is supported by the general expression for velocity flows (deviations from Hubble flow) which is described in [73] as

6 where 5m is defined as the linear mass density contrast and the f2^ dependence is clear. Since the bulk velocity flow is assumed to be in the linear regime in both

Vishniac effects (unlike the density perturbations) it is not surprising to observe a 53

Linear Vishniac Effect Maximum Vishniac Effect

13 1.1 1 0.9 0.8

12 •fcfiCOO 1.1 1 09 0.8

1.2 1.1 1 0.9 0.8

12 1.1 1 0,9 0.8

_J I I L. 0.035 0.CA 0.D45 0.05 0.035 0.W 0.045 0.05

Figure 3-6: Linear (left) and Maximum (right) Vishniac Effect Fitting Residuals. Deviations in the data versus the model for all 9187 representative cosmologies where a value of 1 indicates agreement. 54 similar dependence in J7m. The similar dependence upon Clf, indicates that small increases in the baryon density has only the minor effect of increasing the number of scatters equally in both regimes.

Both effects depend quite strongly upon the mass fluctuation parameter a^ in­ dicating that larger density perturbations that allow for earlier and larger structure formation have a significant impact upon the Vishniac effect. This is especially true in the maximum effect where the production of dense, nonlinear structure formation increases greatly with larger density perturbations. There is also a slight dependence on the optical depth TT indicating that earlier reionization scenarios will favour larger amplitudes specifically for the linear effect.

The results of the parameter fitting are presented in Fig. 3-6. In lieu of plotting against all four cosmological parameters Clb was chosen as a representative parameter.

The figure contains panels representing four separate values of £ at which the models were calculated. In the case of the linear Vishniac effect it is clear that the scatter in the fit is quite small for all scales and that the fit is very nearly flat in £Vspace.

The largest outliers occur at smaller scales {£ > 8000) and even then never exceed a

10% deviation from the true value. While Fig. 3-6 provides evidence for the validity of the fitting function of equation 3.28 it is more impressive in light of the results of Figs. 3-3 and 3-7. In these figures the fitting function can be seen to reduce the order of magnitude scatter seen in Fig. 3-3 to the percent-level scatter in Fig. 3-7.

This wide range of usefull applicability is a testament to both the fitting function and the sensitive cosmological dependence of the Vishniac effect. 55

I "I I III,

W

o

log[AT 20iK2)] Data

Figure 3-7: Comparison of fitting function with actual data at £ — 4000 for the linear effect (red crosses) and the maximum Vishniac effect (blue squares). The line x = y is shown dashed

The results of the maximum Vishniac fit are displayed in Fig. 3-6. Here while it is again evident that the scatter is quite low the functional dependence upon the parameters is less obvious. In the fit for the maximum Vishniac effect the largest errors occur at scales I ~ 1000 but again the largest outliers are never more than 10% off from the true values. The erratic functional dependence of the residuals seems to indicate that the fitting function of equation 3.28 is not accurately capturing all of the physics in the maximum Vishniac effect. This appears to be most evident at scales where nonlinear density perturbations would only begin to have an effect, 56 indicating that nonlinear structure begins to dominate at slightly different scales for different cosmologies thus causing more scatter and an unpredictable fit at scales of

I ~ 1000. These effects dissipate at smaller scales and for values near the peak of the maximum Vishniac effect the fitting function appears to be valid and accurate.

Again it is worth noting that with less than a 10% error the fitting function accurately models an effect that ranges in amplitude over two orders of magnitude (in the case of the maximum Vishniac effect) for all permitted cosmologies with a minimum of computation.

3.6 Summary and Future Work The linear Vishniac effect as well as its maximal nonlinear extension were cal­ culated for the full range of allowed cosmologies. The results of this calculation indicated that the Vishniac effect is strongly sensitive to cosmological parameters displaying a peak amplitude that varied by an order of magnitude for the linear ef­ fect and by nearly two orders of magnitude in the case of the maximum effect. This calculation also found the linear Vishniac effect to have a maximum rms power of

0.80/xK2 at I = 3000 while the maximum Vishniac effect peaked with an rms power of 1.9/iK2 at £ = 10000, confirming the Vishniac effect as a small-scale secondary

CMB anisotropy. Following these calculations a fitting function for both the linear and maximal Vishniac effects was created. This five parameter function was able to accurately predict the widely varying power generated by the Vishniac effect across a large range of scales. 57

With future CMB experiments such as the South Pole Telescope [61] and the

Planck [15] satellite poised to explore the small-scale temperature anisotropy spec­ trum with unprecedented sensitivity and range it is of great importance to understand and quantify the effects that dominate at these scales. Both Planck and SPT are capable of detecting the thermal SZ effect and, as noted in [15, 61], may be capable of detecting the maximum Vishniac effect as well. Although lower in amplitude than the thermal SZ effect the Vishniac effect is not spectrally separable from the primary

CMB temperature anisotropy, rendering it an important contributor at small scales

[11]. The amplitude and shape of the Vishniac spectrum is predicted to depend strongly upon the underlying cosmology of the universe. This dependence has been captured quite efficiently in a five parameter fitting function that has been shown to be quite accurate over the entire range of allowed cosmologies. Through these efforts it is possible not only to predict the expected level of power created by the

Vishniac effect but also to potentially constrain key cosmological parameters in light of a future direct detection. Chapter 4

CMB Remote Quadrupole Studies

4.1 Intoduction As discussed in Chapter 3 the epoch of cosmic reionization is capable of generat­ ing secondary anisotropics in the CMB. In addition to the temperature anisotropics explored in the previous chapter, reionization is also capable of creating secondary polarization anisotropies in the CMB spectrum. This polarization, as explored in

[1, 5, 10, 22, 38, 34, 49, 52, 63] and others, is caused by the Thomson scattering of

CMB radiation with a quadrupole anisotropy pattern during a "patchy" reionization process. The primary interest in observing this polarization signature rests in its intriguing potential to offer information pertaining to the temperature quadrupole at the time of reionization as well as data concerning the primordial density fluc­ tuations responsible for evolved structure within the local horizon. Each of these observations offers a rare opportunity to collect data that is not only complemen­ tary to but also partially uncorrelated with the local CMB sky thus permitting the possibility of improving upon the limitations of cosmic variance on large scales.

58 59

Integral to this polarization signal is the inhomogeneity or "patchiness" of reion- ization which is responsible for creating regions or "bubbles" of free electron. While in principle they are very difficult to observe these ionized bubbles can be inferred from "holes" or gaps in the 21-cm radiation produced by the surrounding neutral

HI regions; this strategy is explored in [2, 19, 34, 72] amongst others. As shown in

[34] the 21-cm maps from reionization will be almost perfectly anti-correlated with maps of the Thomson optical depth resulting from free electrons. This allows for the possible reconstruction of the density field at reionization, an important addition to any future polarization data.

4.2 CMB Polarization The polarization anisotropy of the CMB, like the temperature anisotropy, is an important source of cosmological data. As previously stated, polarization can be created through Thomson scattering of radiation with a quadrupole anisotropy as is demonstrated in Fig. 4-1. Thomson scattering of isotropic radiation or radiation with a dipole anisotropy does not result in a net polarization. Since Thomson scatter­ ing permits the transmission of the transverse component of the electromagnetic field while suppressing the component parallel to the outgoing direction, radiation with a quadrupole anisotropy, in which the hot and cold regions are separated by 7r/2, will result in linear polarization. Due primarily to the finite thickness of the last scatter­ ing surface approximately 10% of CMB photons will be polarized at recombination by the local photon quadrupole. Polarization can also be created post-recombination by the Thomson scattering of the local photon quadrupole but requires the presence of free electrons. 60

•-Hr

Figure 4-1: Linear polarization created by Thomson scattering of radiation with a quadrupole anisotropy where solid blue (dashed red) represents cold (hot) incoming radiation. Figure adapted from [40].

Generally polarization can expressed in terms of the Stokes parameters. Follow­ ing from [16], one can express two orthogonal electromagnetic plane waves at a point z = 0 as

Ex{t) = E0x(t) cos[ujt + (j>x(t)] (4.1)

Ey(t) = Eoy(t)cos[(jjt +

where EQX and Eoy are the amplitudes of the waves, u> is the angular frequency and

(f)X:V are the phases of the waves. The Stokes parameters can then be written as

E / = lx + E0y (4.3)

E E (4.4) Q = 0X ~ ly

u =. 2E0xE0y cos((f)y - (4.5) 61

V = 2E0xE0ysm(cf)y-(f>x). (4.6)

The Stokes / parameter is a measure of the intensity of the radiation and in the limit of monochromatic waves I2 — Q2 + U2 + V2. Q and U are measures of linear polarization while V is a measure of circular polarization. Since circular polarization is not predicted in CMB polarization anisotropics it will not be considered in the following analyses. The polarization angle can be expressed in terms of Q and U as HtarriGD' (4-?) with a corresponding amplitude

P = ^/Q2 + U2. (4.8)

On a sphere where 9 = 0 coincides with the north pole, Q > 0 corresponds to a

N-S polarization component, Q < 0 is an E-W component, U > 0 corresponds to a NE-SW component and U < 0 is a NW-SE component. Polarization can also be expressed in terms of a curl-free E-mode and a divergence-free B-mode as discussed in [40, 52, 54] and others.

4.3 Polarization from a Quadrupole Anisotropy CMB polarization is created via the interaction of radiation with a quadrupole anisotropy and free electrons, namely Thomson scattering. This effect is demon­ strated in Fig. 4-1 in which incoming hot and cold radiation are separated by 7r/2 resulting in outgoing linear polarization. The principle cause of this quadrupole anisotropy in the CMB is the projection of the Sachs-Wolfe gravitational redshift 62 effect (0 = —$/3) [20, 38, 56, 62]. This projection corresponds to a decomposition of a plane wave fluctuation through spherical harmonics as described in [38] and earlier in equation 2.8 as

This results in a quadrupole anisotropy which can be expressed as

{Q) Q = -Vsf j2(^x*), (4.io) where Q^ refers to the rn = 0 component of the quadrupole Q^m\ distinct from the

Stokes parameter Q and x* = Xrec — X- One can then express the quadrupole power spectrum of the as a function of the gravitational potential power spectrum

A2(0) _ 5,J ]A2

= \5%(^y\l + zfD^ljl{^\ (4.11) where the second line comes from the definition of the gravitational potential power spectrum (equation 3.10) on large scales such that the transfer function T(k) ap­ proaches a value of 1

2 2 1 A| ~ ^(1 + zfD 5 H i^X iffc<0.01/iMpc- . (4.12)

Similar to the calculation of the rms velocity in the Vishniac effect (equation 3.20) performed in Chapter 3 it is possible to define the quadrupole rms by integrating 63 over its power spectrum (equation 4.11).

rfl - [ aKdk \W»

l n n 2 = |^(1 + zYtftfm{X*HQ) - J dxx j 2(x)

2 n = ^5%(l + zfD nl(X*Hoy- rsw(n), (4.13) where contributions from the Sachs-Wolfe effect (as examined in section 2.3) are expressed as r (n\ g /-r[(3-n)/2]r[(3 + n)/2] IW(») = 3^r[(4-n)/2]r[(9-n)/2]- <414> It is worth noting that the quadrupole can also be expressed as

QLs = ^CfQ, (4.15)

e where Cf refers to the variance of the a^m's as described in equations 2.16 and 2.19. Thomson scattering from a quadrupole anisotropy, like the one present at re­ combination, has an angular dependence denned in [3] as

da 3<7T e • e (4.16) dtt 8TT where e and e are the incoming and outgoing polarization directions, ar is the

Thomson cross-section and fi refers to the solid angle. For a single incoming wave this scattering results in linearly polarized light that can be represented as

I = ^/(i + cos^) (4.17) 16-7T V ' Q = ^I'sm2e (4.18) 167T U = 0, (4.19) 64 where /, Q and U refer to the Stokes parameters denned in equation 4.6. After integrating over all incoming directions it is possible to express the Stokes parameters in terms of the components of the quadrupole

3&T 4 T ' r- h (4.20) 167T v uu 3:V7raoo + 3-\ V 5-a 2o

Due to causality constraints the polarization signal created at recombination peaks on a scale smaller than the horizon at last scattering. Also, as previously stated, this signal is expected by be only 10% of the amplitude of the CMB temperature anisotropy due to the finite thickness of the last scattering shell [3, 40, 54].

4.4 Polarization from Reionization The creation of polarization via Thomson scattering requires the presence of free electrons and a quadrupole anisotropy in the photon distribution. These conditions were first met during the era of recombination and resulted in the primary CMB po­ larization signal. These conditions were also present during the epoch of reionization and thus permit the creation of a secondary polarization in the CMB, a phenomenon explored in [1, 5, 10, 22, 38, 34, 49, 52, 63] among others. This secondary polar­ ization will depend strongly upon the history of reionization, specifically the degree of inhomogeneity created in the Thomson optical depth due to so called "patchy" reionization. Since the quadrupole is assumed to be slowly varying during the period or reionization (6 < z < 15) fluctuations in the free electron density due to patchy reionization will be the main cause of polarization anisotropies. 65

Following the work of Dore [22] and Portsmouth [58] the polarization at the position x can be expressed as

6 /2 2 P±(x) = Q(x) ± it/(x) = -—J-TTTCMB(T) Yl ±2^m(x)a2ro(x), (4.22)

m=—2 where r represents the conformal lookback time (r = 770—rj), x remains the comoving distance and TT describes the optical depth to Thomson scattering integrated along the line of sight

Mr) = aT f ne{X)dX. (4.23) Jo The spin-2 spherical harmonic ±2^m are related to ordinary spherical harmonics as Y£_2)n1/2 r 92_cot^±_|_(^_cote)^__J_a2 {i + 2)\_ m x Ye (9,(f>). (4.24)

In order to compute the quadrupole components a2m, an expression for the CMB transfer function is required. Following from equation 4.11 and in the limit of a flat

(n = 1) universe, the quadrupole (£ — 2) transfer function can be simplified as a sum of two terms: the first term corresponding to the previously mentioned Sachs-Wolfe effect and a second term arising from the late-time decay of gravitational poten­ tials due to the influence of vacuum energy at late times. In general the blueshift and redshift that a photon experiences as it enters and subsequently exits a grav­ itational potential are equal in magnitude and thus effectively cancel out. At late times, potentials in a vacuum-dominated universe will begin to decay thus altering the aforestated symmetry and resulting in a net shift in a photon's frequency as it traverses potentials; this is known as the integrated Sachs-Wolf effect and is explored 66

more thoroughly in [18, 57, 62]. The transfer function is then written as

3 9 rT~Trec A2(fc, T) = — 32 [k(r - Trec)} + - / dxjtikx)- (4.25) T-X

Since A^(k, r) is independent of k in linear theory the power spectrum can then be

reexpressed in terms of the initial gravitational potential $j as follows [58]

A/(k,r)=A/(fc,r)$j(k). (4.26)

Where $j is an assumed Gaussian field describing the post-inflationary gravitational potential at some arbitrary initial time. Finally the components of the quadrupole are given by

3 ik x a2m(x) = -4TT f d ke ' A2(k, r)Y2*m(k). (4.27)

Thus for a given reionization scenario it is possible, through equation 4.22, to calcu­ late the expected polarization at any point on the sky.

In order to approximate the strength and distribution of the expected polar­ ization signal it is necessary to use simulations of the reionization process. The specifics of these simulations are beyond the scope of this work but since the ob­ served polarization signal is integrated they require reionization to be both sharp and inhomogeneous; this is explored further in [22]. It is worth noting that the simu­ lation models do depend strongly upon cosmological parameters as well as variables such as photon emissivity as a function of stellar mass and the mechanisms of early star formation. The results of these simulations show great variation in the optical depth during the period of reionization and indicate that some regions had optical 67 depths as high as a few percent which is comparable to the integrated optical depth due to free electrons in the centres of large galaxy clusters.

The amplitude of the polarization signal due to patchy reionization is expected to be on the order of 0.01/xK, indicating that the small-scale signal due to patchy reion­ ization at a redshift of z~ll is subdominant to the primary polarization anisotropies

[22, 52, 54]. The hope for the observation of this signal lies in its distinct spatial signature. This polarization anisotropy is expected to have a relatively constant polarization direction over degree scales while its amplitude variations will be on ar- cminute scales. This differs significantly from the primary polarization anisotropies.

Thus although the polarization anisotropy signal due to patchy reionization is ex­ pected to be small in magnitude its unique spatial dependencies offer hope of future detection.

4.5 Studying the Remote Quadrupole through Sec­ ondary Polarization Anisotropies As mentioned in the previous section the polarization that is generated by patchy reionization is dependant upon the local CMB quadrupole temperature anisotropy.

Owing to the expansion of the universe the perturbations that contributed to the quadrupole at reionization (z ~ 11) correspond to a smaller scale on the local CMB sky. Thus the study of secondary polarization created during reionization permits an inference of the CMB temperature quadrupole present at that time.

Another consequence of the expanding universe is that the observed quadrupole anisotropy today (I = 2) corresponds to scales larger than the horizon size at reion­ ization; conversely the quadrupole at reionization corresponds today to smaller scales 68

(£ « 10). On these scales cosmic variance, as denned in equation 2.17, is the domi­ nant source of uncertainty. This cosmic variance arises due to the limitation of having only one sky from which to draw the 2^+1 independent data points that represent the ensemble average prediction of the variance. This limitation is particularly severe for large scales (small values of £) and presents a floor for the accuracy with which the CMB power spectrum can be observed. Thus access to another CMB sky that was created by density perturbations distinct from those that created the local CMB has the possibility of reducing cosmic variance on large scales. Through polarization studies it is possible to observe a remote quadrupole which, by carrying information pertaining to a separate CMB sky, theoretically allows for an improvement to the cosmic variance on large scales in the local CMB.

A second consequence of constraining the remote quadrupole is the ability to reconstruct the initial density perturbations that created it. A remote quadrupole, as evidenced by equations 4.27 and 4.26 depends upon an initial spectrum of per­ turbations that is distinct from that which created the local CMB quadrupole. This occurs because the last scattering surface for an observer located at z = 0 is dissim­ ilar from the last scattering surface seen by an observer at z = 11. This distinction is illustrated in Fig. 4-2 where it is evident that the perturbations that contributed to the CMB as seen by an observer during reionization lie inside the last scattering surface for an observer located at z = 0. In principle this allows for the unique op­ portunity to study the correspondence between the effect of density perturbations on a remote quadrupole and their subsequent evolution into collapsed structure inside the local horizon. It is also clear that the correlation of the remote quadrupole with 69

Figure 4-2: Last scattering surfaces for a local observer and one located at z = 11. Comoving distances to the remote observer and local last scattering surfaces are labelled as well as their redshift equivalences. Figure adapted from [58]. the local quadrupole decreases with increasing comoving distance to the polarization source. This indicates that the polarization anisotropy created during reionization may provide more independent information than the polarization created by large, nearby galaxy clusters as explored in [1, 10, 58].

Prior to obtaining observational polarization data from patchy reionization it is important to first estimate the scales which can be constrained. This is accomplished via the construction of a Fisher matrix. First one must define a \2 function as in

[20] (distinct from x(z) in equation 1.10)

X2({A }) = £ (4.28) a M2 70 where {Xa} are the set of parameters to be constrained, aE is the experimental

2 uncertainty on function Ei and x reaches a minimum when Xa = Xa (the actual value of the parameter). Changes in %2 indicate how well a parameter can be constrained, a steeper slope corresponding to stronger constraints. It is then possible (in a one

2 parameter generalisation) to expand % about its minimum at Xa — Xa

2 2 2 X (A) = X (A) + ^(A-A) . (4.29)

The first term is ignored since x2 is defined to be at a minimum at A and the second term, J7, can be written as

d2ln£ 1 <92x21 T = 2 2 (4.30) OX 2 OX A=A where the likelihood function C is assumed to be Gaussian (or close to it) and the constraint on the parameter A depends upon \jT. By expanding T one finds

d2E (Ei - Efs) i (4.31) M2 \dX dX2 In general the term (Ei — E°bs) will average to zero, resulting in

1 dEi dEi 2 (4.32) ^E (aE) dX dX ' which, when generalised over many parameters, is expressed as the Fisher matrix (as in [20, 26, 69]) ^ 1 dEiFIT?.. dEiFtK. 2 (4.33) ((jE) dXadX^ where Ei is a generalisation of any observable quantity. 71 4.6 Constrained Modes By utilising the expression for the polarization created by patchy reionization from equation 4.22 in concert with the expression for the Fisher matrix in equa­ tion 4.33 it is possible to predict which modes can be constrained by polarization anisotropy observations. The Fisher matrix in equation 4.33 may then be expressed as ~z° i gp Qp Fa0 = ^ ^dM^dM^Y (4'34) where P is the polarization expressed in equation 4.22, $j(kj) corresponds to the gravitational potential as a function of the wavevector kj and aP is the expected error in the experiment. The theoretical data in the experiment corresponds to polarization measurements taken at a position x on the sky; these sample observation points were chosen to lie on a sphere whose radius is defined by the redshift to reionization, zre = 11. For each component of the Fisher matrix, Fal3, the polarization was calculated and summed over these data points (chosen to number 228 in total).

As can be seen in equation 4.27, only the a2m(x) components in the polarization

P depend upon the parameter $i(k), thus its derivative may be expressed as

3 ik x |gj^ = -47r|d ke - A(fc,r)y;m(k)<5(k-ki) (4.35)

ik x = -47re - A(A;i,r)y;ro(ki). (4.36)

Consequently it is possible to express the derivative of the polarization as

= Mj&) f Vf ^TCWT^-A^T) £ ±2y2m(x)y2*m(U (4.37) as required by equation 4.34. 72

Following a procedure similar to that in [22, 58] a 400 x 400 Fisher matrix was created. In the interest of an efficient and representative analysis the k-vectors were chosen to be two-dimensional and ranged in magnitude from |k| = 0 to |k| = 0.0054.

These wavenumbers can be roughly related to the multipole H by the following relation

t = x(z)H^k (4.38)

£ = lO^/Mpc"1 if h = 0.7 and z = 11. (4.39)

The resulting Fisher matrix was created by summing equation 4.34 over the 228 evenly-spaced theoretical polarization observations. In order to obtain the con­ strained modes the Fisher matrix was then inverted and decomposed into its compo­ nent eigenvectors. The eigenvectors were then ordered by their respective eigenvalues in order to find the principle components of the matrix. The results of this decom­ position are presented in Fig. 4-3. From this Figure it is clear that although there are 400 possible components to the Fisher matrix only the first few are significant as even by the tenth eigenvector the amplitude is significantly suppressed and by the fiftieth eigenvector the signal is indistinguishable from the background. Instrumental noise, represented by ap in equation 4.34 has been set to 1, making the overall am­ plitudes difficult to understand. It is also clear from the first few eigenvectors that the most constrained modes have scales of k > 0.001 corresponding to an angular scale of I ~ 10; this is in agreement with the analysis performed by Bunn in [10] as well as with the scale of reionization seen in the "reionization bump" of the polar­ ization power spectrum seen by WMAP and corresponds roughly to the horizon size at z = \\ [54]. 73

10th Eigenvector 50th Eigenvector

1 st Eigenvector 2nd Eigenvector

"•r 8-3. -nr Q.3,

Figure 4-3: Principle components of the Fisher matrix.

Modes with scales of k < 0.001 were found to be very poorly constrained which is expected as the polarization signal from patchy reionization should not contain any information pertaining to scales larger than the horizon at reionization. Fur­ ther analysis of the data revealed constraints on sub-horizon modes. The largest constraints occurred for modes with scales of k = 0.0025 and k = 0.0036. These amplitudes correspond to angular scales of £ — 25 and £ = 36 respectively, indicating the possibility of also constraining smaller subdominant modes.

Following the analysis of the contrained modes in Fourier space the correspond­ ing real space constraints were investigated. This was accomplished following the 74

3rd Eigenvector 4th Eigenvector

0.2 0.15 0.1 0.05 0 ggggSgS?''^ <',Vjj^

iorj y (MP0) 0 200"0,400 0 2000'400. 0 x (Mpc) x(Mpc)

1 st Eigenvector 2nd Eigenvector

2000,400 0 2000'400, 0 x (Mpc) x(Mpc)

Figure 4-4: First four eigenvectors of the real-space polarization analysis. procedure outlined earlier in this section with the following addition

dP(x) dP(x) 3ikr(x-Xi) (4.40) .(x.) L. 9$.

This again resulted in a 400 x 400 Fisher matrix whose eigenvectors, when ordered by their respective eigenvalues would determine the contrained real space directions.

The values of the first 50 eigenvectors are presented in Fig. 4-5. The results of this analysis are presented in Fig. 4-4. As with the Fourier space analysis only the first few eigenvectors contained any information and again by the tenth eigenvector the signal is nearly indistinguishable from the background noise. Also evident in Fig.

4-4 is that the constrained perturbations carry a characteristic size. These sizes are 75

0.1 t

3 £< 0.01 t-

0.001 t-

1e-04 0 50 100 150 200 250 300 350 400 Eigenvalue #

Figure 4-5: Normalized amplitudes of position-space eigenvalues. dictated by the constrained Fourier modes, notably k = {0.001,0.0025,0.0036}, re­ sulting in radii which vary with the inverse of the constrained modes. These results are also evident in the sum of the first ten position space eigenvectors as presented in Fig. 4-6; this again indicates that remote quadrupole surveys have to potential to probe gigaparsec scales. The separation between the "spots" in Figs. 4-4 and 4-6 also appears to be a function of the distance to reionization where late time reioniza- tion resulted in larger separations than early time reionization scenarios. Finally the geometry of the constraints will depend upon the nature of the polarization survey.

As pointed out in [10], limited sky surveys will not be as effective as full sky surveys 76

Figure 4-6: Sum of first ten position-space eigenvectors. in reconstructing these primordial perturbations. This is due primarily to the signif­ icant correlation between last scattering surfaces for proximate observers. Thus the effectiveness of the constraints from a future polarization survey will greatly depend upon the ability to acquire data from as large a fraction of sky as possible.

4.7 Summary and Future Work The secondary polarization created by the interaction of CMB photons with ionized regions in an inhomogeneous reionization scenario has been discussed. This late-time effect is caused by the Thomson scattering of photons with a quadrupole temperature anisotropy, an anisotropy that is unique to the last scattering surface seen at that time. Measurements of this polarization signal have the potential to 77 reveal information pertaining to the primordial density perturbations that seeded structure within the local cosmic horizon. Through a two-dimensional Fisher infor­ mation analysis both the wave-vectors and position-space counterparts of the con­ strained perturbations were calculated. It was found that the modes corresponding to gigaparsec scales were the most constrained. Modes larger than the horizon at reionization would not be in causal contact and thus were found to be unconstrained as were small-scale modes that do not contribute to the quadrupole anisotropy.

The detectability of the polarization created during reionization has been ex­ plored by Dore et al. in [22]. Here it was noted that the polarization signal will have a unique spatial dependence with the direction of the polarization changing over de­ gree scales while the amplitude remains constant on arcminute scales, a pattern that should be distinct from any primary polarization anisotropy. The amplitude of this secondary polarization signal is expected to be only ~ 0.01/xK, a signal that with current detectors would require much more than a year of observing to detect [22].

As such any detection of this signal in the near future is unlikely.

While the sensitivity required for observing the secondary polarization created during reionization is beyond current experimental limits the information it contains remains a tantalising prospect. The possibility of improving upon a limit as funda­ mental as cosmic variance is a lofty but not unachievable goal that is certainly worth further exploration. The intriguing possibility of reconstructing the primordial den­ sity perturbations that seeded structure within the local cosmic horizon offers to not only resolve the "leap of faith" assumption that the density perturbations seen on the CMB are representative of the perturbations that formed structure locally 78

but also offers important insight into the specifics of structure formation. Future work resides in refining and expanding the results outlined in this section as well

as extending calculations to include expectations based upon actual experimental uncertainties. Chapter 5

Summary

In this work small-scale secondary anisotropies in the Cosmic Microwave Back­ ground were explored. Beginning with an introduction to the basic tenets of modern physical cosmology the origin and theoretical properties of the Cosmic Microwave

Background were explored. Following this overview the Vishniac effect was intro­ duced. This small-scale temperature anisotropy was found to arise as a result of scattering between CMB photons and free electrons in the reionized epoch. This effect was divided into two contexts depending upon whether the perturbations re­ sponsible for the density enhancement in the electrons were in the linear or nonlinear regimes. The linear and maximal nonlinear extension of the Vishniac effect were then calculated for 9187 cosmological sets, representative of the entire WMAP-allowed parameter space [67]. From these results it was discovered that the linear Vishniac effect demonstrated a peak power of 0.8^K2 on an angular scale of £ — 3000 while the maximum effect peaked on an angular scale of £ — 10000 with a power of 1.9/^K2.

Subsequent to this calculation was the derivation of a five-parameter fitting function

79 80 for both the linear and maximum Vishniac effects. This function provided a simple and accurate method of predicting the power of the Vishniac effect on a given scale for all allowed cosmologies.

Also investigated was the small-scale polarization anisotropy created by the scattering of the quadrupole photon anisotropy during a "patchy" or inhomogeneous reionization. The origin of this signal as well as its observational consequences were examined. It was determined that this secondary polarization signal had the po­ tential to constrain the remote temperature quadrupole anisotropy corresponding to the time of reionization. This temperature anisotropy would correspond to smaller angular scales today and thus offers complementary information on angular scales of £ ~ 10 in the temperature anisotropy spectrum, possibly improving upon cosmic variance. This signal also has the capability of constraining the spectrum of pri­ mordial perturbations responsible for this quadrupole anisotropy, perturbations that would lie within the sphere defined by the local last scattering surface. The most constrained perturbations were calculated to be < lOOOMpc in scale, potentially of­ fering insights on the origin and evolution of large scale structure within the local horizon.

The experimental future of the CMB lies in the observation of anisotropies on small scales for both the temperature and polarization spectra. On angular scales beyond t ~ 1000 the CMB temperature anisotropy is expected to be dominated by secondary anisotropies like the Vishniac and SZ effects. Unlike the thermal SZ effect the Vishniac effect will be spectrally indistinguishable from the primary CMB 81 anisotropics thus preventing its subtraction by way of precise multi-frequency mea­ surements. Because of this, an accurate prediction of the expected Vishniac spectrum will be required in order to form a more complete understanding of the CMB tem­ perature anisotropy spectrum on small scales.

Following the tremendous successes in observing the CMB temperature anisotropics, the advent of new and accurate polarization data presents many fascinating op­ portunities. Among these opportunities is the possible observation of the remote quadrupole through the small-scale polarization anisotropy it creates during reion- ization. Although this signal is expected to be faint it offers the fascinating pos­ sibility of improving upon cosmic variance constraints for large scale temperature anisotropies as well as offering a glimpse of the primordial density perturbations that seeded structure within the local cosmic horizon. This possibility alone has the intriguing potential to resolve the "leap of faith" that the primordial density pertur­ bations observed in the local CMB are similar to those inside the horizon and thus providing new information concerning the formation and development of structure in the universe.

The observational future of the CMB lies primarily in the acquisition of data with higher precision and on smaller scales. In this work the origin, observability and information contained in two contributors to small scale anisotropy were discussed.

Although in both cases the amplitude of the signal is expected to be subdominant, the cosmological significance of each effect behooves future experimental efforts. References

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