# THE LARGE SCALE ANISOTROPY OF THE COSMIC MICROWAVE BACKGROUND RADIATION (INFLATION)

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For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. Uni International 300 N. Zeeb Road Ann Arbor, Ml 48106 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8517051 S c h a e f e r , Robert Karl THE LARGE SCALE ANISOTROPY OF THE COSMIC MICROWAVE BACKGROUND RADIATION Brandeis University University Microfilms International Ph.D. 1985 300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1985 by Schaefer, Robert Karl All Rights Reserved R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE LARGE SCALE ANISOTROPY OF THE COSMIC MICROWAVE BACKGROUND RADIATION A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences of Brandeis University Department of Physics In Partial Fulfillment of the Requirements of the Degree Doctor of Philosophy by Robert Karl Schaefer April 1985 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. This dissertation, directed and approved by the candidate's Committee, has been accepted and approved by the Graduate Faculty of Brandeis University in Partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Iteaff, Graduate School/ E/Arts and Science*^ MAY 1 9 1985 Dissertation Committee Laurence F. Abbott (Chairman) % H A M i ugh/N. Pendleton, if David H. Roberts Steve Rosenberg (Mathematic^) ii R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Copyright by Robert Karl Schaefer 1985 iii R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. ABSTRACT The Large Scale Anisotropy of the Cosmic Microwave Background Radiation (A Dissertation Presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts.) by Robert Karl Schaefer Although the predicted moments of the distribution of the cosmic microwave background radiation in an inflationary universe have previ ously been calculated, the sensitivity of these moments to small changes in the conditions required by inflation had not been explored. In this thesis, formulae are presented for calculating the moments of the background radiation due to an arbitrary perturbation in the gravitation al field of the early universe. It is found that only scalar (energy density) and tensor (gravitational wave) perturbations are important for this effect. Then numerical calculations are performed; first, to confirm the previous inflationary calculations which require a critical cosmological density and a Harrison-Zel'dovich scale invariant spectrum of perturbations, and second, to compute the values of the moments in universes with non-critical energy densities and different spectra (k* )• In all cases we keep the assumption that the perturbations are caused by some random process which follows a Gaussian distribution, as this seems to be a reasonable feature of inflation to keep even in non-inflationary universes. The previously predicted values of the inflation induced moments are confirmed, which imply that £.H< 4 * 1 0 _t and the value of the quadrupole to dipole ratio must be two orders of magnitude smaller than the current upper bound, unless gravitational waves are present. If this latter pos sibility is true then the moments higher than dipole must follow the 1-dependence predicted for gravitational wave perturbations. The values of the moments were found to be sensitive enough to distinguish a k+l or a k“* spectrum from a Harrison-Zel'dovich spectrum in a critical density universe, (but not in a universe which had only a few tenths of the crit ical density). The moments higher than dipole for a Harrison-Zel'dovich spectrum were not very sensitive to differences in the energy density but the quadrupole-to-dipole ratio was. We find that the values of the mulipole moments are indeed a good test of the inflationary model. In the event that an alternative model is found to inflation, the equations pre sented here are general enough to be used to calculate moments from this alternative model. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Acknowledgement I am greatly indebted to Larry Abbott first for suggesting a prob lem which would have interesting results no matter what they were, and second, for his personal kindness and patience throughout ray graduate career. I am also thankful to my family for the unfailing support they gave to me for the duration of my seemingly endless stay in institu tions of higher learning. I am also grateful to the Physics Department secretaries, especially Mimi, for her patience and help in getting me to this stage despite her ongoing balancing act in the Physics office. iv R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. TABLE OF CONTENTS Page 1. Introduction 2. 1 The Sachs-Wolfe Effect and Moments of the Distribution of the Cosmic Background Radiation 5 1. The Generalized Sachs-WolfeEffect 5 2. Gaussian Random Variables andMultipole Predictions A. 8 Ta/T0 12 B. Multipole Moments and Their Expectation Values 12 3. Time Evolution and Radial Dependence of thePerturbations 1. Scalar Perturbations A. 15 17 8T0 /To 17 B. Evolution of € . ( T ) 21 C. The Harmonic Functions Q (x) 22 2. Vector Perturbations 25 A. 8T0 /T0 25 B. Evolution of 26 C. The Harmonic Functions O' Q (x) 3. Tensor Perturbations 27 28 A. 8T0/T0 28 B. Evolution of H^?(Y ) 28 C. The Harmonic Functions Qj- (sD 4. Predictions of Moments 1. 11 30 31 Dipole and Quadrupole Bounds 31 A. eu and the Dipole Moment 33 B. The Quadrupole to Dipole Ratio 35 v R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. Table of Contents - continued Page 2. Higher Multipole Moments 37 A. The Harrison-Zel'dovich Spectrum 37 B. £ H= X ^ tl 38 C. Gravitational Wave Perturbations 5. Conclusions 39 41 APPENDICES A. Spatial Harmonic Functions B. Growth of Density Perturbations During the Radiation Dominated Era 44 64 REFERENCES 70 BIBLIOGRAPHY 73 FIGURE CAPTIONS 76 FIGURES 1-25 78 vi R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CHAPTER 1 INTRODUCTION In the past couple of decades the marriage of the fields of elemen tary particle physics and cosmology has proved to be a fruitful union. Cosmology has provided some constraints on particle physics models unat tainable through other methods (e.g., the limit on the number of neutrino species.1) As particle physics looks to a unification of the fundamental forces of nature at energy scales far beyond the reach of particle accelerators, physicists are increasingly appealing to cosmology for direction. An unexpected benefit of this research occurred when particle physics returned the favor by providing a possible solution of many cosmological puzzles with the invention of the inflationary cosmology.2-1* This model of the very early universe solves in one clean sweep the horizon, flatness, and monopole problems and also provides a natural origin of the density perturbations necessary to form galaxies, and eventually, us. With all its theoretical successes, this model of the universe is as yet observationally untested. It is the purpose of this thesis to explore a possible observational test of this theory through observations of anisotropy in the cosmic microwave background radiation. The inflationary cosmology leads to two predicted properties of the universe which we will examine. First, that the power spectrum of energy density perturbations (and gravitational waves) is scale-invariant.5-8 This is known to astrophysicists as a Harrison-Zel'dovich spectrum after R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 2 the authors who pointed out the need for such a spectrum long before the invention of the inflationary model.9>10 Second, the energy density of the universe must be equal to the critical energy density which is the value critically between an open and closed Robertson-Walker universe. We will use the standard variable for characterizing the energy density fl = p/ptr (energy density)/(critical energy density), which equals 1 for an .infla tionary universe. Attempts have been made to measure £ \ directly which provide a narrow range of values for IT , 0.1 <£l< 4. worth noting here: Two points are first, /I = 1 cannot be completely accounted for with normal baryonic matter. In fact another result from the crossfertilization of particle physics and cosmology is that the total energy due to baryons must be less than the critical energy density by a factor of 1/5-1/711 in order for primordial nucleosynthesis to produce the observed light element abundances. Thus the inflationary cosmology (or any cosmology with jfl > 0.2) requires that the energy density of the universe be dominated by some non-baryonic component. This assumption is not unfounded as some type of "dark matter” is needed to explain observed properties of galaxies.12 The best candidates for this non-baryonic matter are those which fall in the category of cold dark matter (e.g., axions,13-15 photinos,15*17 etc.). The second point worth mentioning is that measure ments of/I from clusters leads to /I £ 0.5 in clusters18-20; hence, although we will not use this assumption explicitly, we point out that for values of fl > 0.5 we assume galaxies are not good tracers of the mass distribution of the universe. This assumption has some supporting evidence in that it seems to help explain the velocity correlations of galaxies in clusters.21 In the inflationary cosmology perturbations are produced in the early R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. universe on all scales according to the Harrison-Zel'dovich spectrum. The long wavelength density and gravitational wave fluctuations22-25 will produce gravitational redshifts of photons in the background radiation leading to anisotropies in the temperature distribution of these photons. The values of multipole moments of the temperature of the photon dis tribution have been previously calculated and thus provide a test of inflation.22-27 The question is, how good a test is this? words, can we tell that f\ = 1 and not say 0.2? In other Can we tell if the spectrum is really a Harrison-Zel'dovich spectrum and not something close to it? In chapter 4 of this thesis we will try to answer these questions by comparing the predictions of the inflationary cosmology with those for other spectra and for open and closed universes. One key element of the inflationary cosmology will be kept in all cases considered— the assumption that the source of the perturbations is a Gaussian random process. is separately testable.28*29 It is interesting to note that this hypothesis We include in our analysis the effects of this uncertainty on the final moment predictions to see under what con ditions the inflationary predictions can be distinguished from those of different spectra and non-critical energy densities. This element of our analysis is unique and has not been used in previous anisotropy calcula tions for universes with spatial curvature.30-32 We will explain the implementation of this assumption in chapter 2. Also in chapter 2 we present the derivation of the general formula for finding fluctuations in the temperature of photons from an arbitrary perturbation in the metric of a Robertson-Walker universe (the Sachs-Wolfe formula). Any arbitrary perturbation in the metric (and hence the gravitational field) can be decomposed into 3 types according to their behavior under R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. 4 spatial transformations in the background space-time: scalar, vector, and tensor. In chapter 3 we use the elegant gauge-invariant formalism of Bardeen33 for treating each specific class of perturbation in the Sachs-Wolfe formula to find the effects these fluctuations have on the microwave background radiation. Finally we will draw a few conclusions from the small catalogue of moment predictions displayed in chapter 4. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 5 CHAPTER 2 THE SACHS-WOLFE EFFECT AND MOMENTS OF THE DISTRIBUTION OF THE COSMIC MICROWAVE BACKGROUND RADIATION The main effect of small energy-density perturbations on the cosmic microwave background radiation is the gravitational redshift they produce. This redshift has become known as the Sachs-Wolfe effect after their original work for a flat Robertson-Walker universe.31* Since this is the main source of anisotropy in the cosmic microwave background we present the full derivation of the generalized Sachs-Wolfe formula in section 2.1 for all Robertson-Walker-Friedmann universes. In section 2.2 we show how we define multipole moments from the Sachs-Wolfe effect. We also explain how we implement the assumption that the fluctuation amplitudes are gaussian distributed. 2.1) The Generalized Sachs-ffolfe Effect A well known consequence of General Relativity is that the distribution of matter-energy in the universe determines the geometry of space-time. Different distributions will be associated with different metrics describing their manifolds, and thus their respective geodesics will also be different. If we consider small perturbations in the matter- energy density of the universe we must expect small variations in the geodesics as well. This fact implies that photons reaching us from different directions in a perturbed expanding universe may have redshifts slightly different from the standard redshifts resulting from these (null) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. geodesic variations. This phenomenon is known as the Sachs-Wolfe effect. The original work of Sachs and Wolfe considered perturbations only in a "flat" Robertson-Walker universe. Their work was extended by Anile and Motta35 to include open (K = -1) and closed (K = +1) Robertson-Walker metrics, although they worked in a different gauge than Sachs and Wolfe. Of course, the final formulae will be gauge invariant and it doesn't really matter which gauge is chosen, provided one is careful enough to completely specify the gauge. For completeness we will present the derivation of the Sachs-Wolfe formula for all three cases (open, closed, and flat) in the Sachs-Wolfe gauge. The choice of gauge will become irrelevant when we rewrite the formula in terms of Bardeen's33 gauge invariant variables. This will also eliminate any worries of unphysical "gauge mode" solutions36-39 appearing as solutions to the perturbation equations. We will find it to be convenient to use the conformal time 'f defined by where R(t) is just the usual Robertson-Walker scale factor which describes the general expansion of the universe with time. This gives us the (unperturbed) Robertson-Walker metric (2 .1 .1) where = (1 + Kjjj*), x° = 'X , and: K= +1 K= 0 K= -1 closed universe 4 ^ flat universe open universe. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 7 At this point we also note that latin indices are summed from 1 to 3 and greek indices are summed from 0 to 3 and that c = 1 in our units. We now wish to perturb the metric so that and we remember that /)£='£, not the usual time, t. We will assume that the universe is a perfect fluid with a four velocity U 01 . We will choose our coordinates so that the fluid is at rest (U *■ = 0) and also choose h oo = 0. These four conditions constitute our ( and Sachs' and Wolfe's ) choice of gauge. The restriction U* U,* = -1 now means that C /s e n We must now find the null geodesics in our perturbed space. These perturbed null geodesics will give rise to an anisotropy in the cosmic background radiation. To make the task simpler, we only need to find the null geodesics of d s l , where J U * = S \ r ) d A K because the geodesics of two conformally related spaces are parallel. The affine parameters describing geodesics in these two spaces do not nec essarily coincide,35 but they are related. p* So, for example, the vectors tangent to the null geodesic in the full space are simply related to the tangent vectors "p*4 in the barred space (specified by dl*' ) by J3 = b f J (2.1.3) Using y as our affine parameter along the null geodesic in the barred space, the tangent vectors "p * are just R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. The equation for this null geodesic is = <ty, O After performing the variation and doing an integration by parts we get the Euler-Lagrange type differential equation which we will solve perturbatively. We start by breaking up the tangent vector pT* into the tangent vector pT* for the Robertson-Walker background and a small perturbation ‘p < : — ■f A') (2.1.4) -4- JF)* - f J0>) + f m To zeroeth order the geodesic equation is fw jr x 3 ^ f&> f '« or rewriting dq + a t** I f w + tv,*)fl ft4 = 0 Instead of finding the general solution to this equation we will solve only for radial geodesics. If we put ourselves at the origin of our coordinate system (r = 0), then the only photons which we can observe from external R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. sources must travel along radial geodesics.1+0 The tangent vector to these unperturbed radial geodesics is easily found to be ( - M s) where e <2-1,5a) is the unit vector in the radial direction "e = r*/|?|. We then get _ dr , <h _ Jbo v If we arrange y=0 to be the origin at the point of observation ( /t'o >0»0>0)> then t= r. - r K=-| C2.taMl(%) < 2 - 1 ' 5 b ) K= o k = +i /L = ) (_ a t « / > v ( H O Note that because we are considering photons received at the origin dr/d is negative. MW We can use the zeroeth order solution p, . to obtain a first order equation for p*4 : CO or: - JP,!’ ± l R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. <2.1.6) 10 We will only need the zeroeth component pc° [ dr A d f^) If we define "e^( o)= 0, the solution of (2.1.6) is <?M0 o a 3r Z.JL Q JT\P (2.1.7) f< » - f® We will use this solution to find the redshift Z from the relation1*1 (2 .1 .8 ) Quantities to be evaluated at the time of emission % will be denoted by an e subscript and quantities to be evaluated at observation time 7*o will be denoted with an o subscript. Using eqs. (2.1.2), (2.L.3), and (2.1.4) in eq. (2.1.8), +z =4* !\S 'e is -Se — f c o Also using eqs. (2.1.5) and (2.1.7), -%'X r l+Z (2.1.9) l+*i This is our generalized Sachs-Wolfe formula. It tells us how the redshift is related to perturbations in the metric, which in turn are related to the state of the matter-energy density of the universe through Einstein's field equations. Our next task will be to rewrite equation (2.1.9) in R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 11 terms of Bardeen's gauge invariant variables. We can then use this generalized Sachs-Wolfe formula in the determination of the temperature distribution of the cosmic microwave background radiation. We will do this by predicting the expected values of multipole moments of the distribution of the background radiation. We will also assume the perturbations are caused by some Gaussian process. This assumption of gaussian fluctuations will have to be put into the boundary conditions of the perturbation equations, so we will first examine the implementation of the gaussian assumption and how 1l will affect the prediction of multipole moments, then we can fully recast equation (2.1.9) in terms of Bardeen's variables with the proper boundary conditions already enforced. 2.2) Gaussian Random Variables and Multipole Moment Predictions A key element in the analysis presented here is the assumption of a Gaussian distribution of the perturbation amplitudes. The inflationary cosmology in fact does not predict the amplitudes of the fluctuations, but rather gives the width of a Gaussian probability distribution for the perturbations. We will include this Gaussian probabilistic description in our non-inflationary predictions as well, because there is evidence to support this assumption in the observation of the correlations of rich galactic clusters.28-29 The uncertainty in our predictions will be realized by making the fluctuation amplitudes proportional to a Gaussian distributed random variable. We will examine this in more detail once we explain how the moments are derived. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 12 2.2A) 8Te /Te We would like to predict the observationally relevant quantity S T6 /T0 , the fluctuation in the observed temperature of the photon distribution. These variations can come from two sources: a) the Sachs- Wolfe effect and b) fluctuations in the density of matter when photons are last emitted (i.e., recombination, or photon decoupling time). Any perturbation in the density of matter which exists just before the photons decouple from matter will be subsequently have its image "frozen" in to the photon distribution at decoupling, which for long wavelengths can be taken to be an instantaneous process.1*2 perturbation recombination. STg This causes a in the average temperature of photons Te This initial temperature Tg + 8Te at distribution gets further distorted by the gravitational Sachs-Wolfe effect. Thus the temperature observed today is (2 .2 .1 ) where 1+Z is given by eq. (2.1.9). Equation (2.2.1) is our master equation for predicting fluctuations in the temperature of the photons today. 2.2B) Multipole Moments and Their Expectation Values We can manipulate eq. (2.2.1) to get an expression for 8 T& /T0 . Once this is done, we can expand STj, /T0 in terms of the spherical harmonics to get the amplitudes of the various moments (2 .2 .2 ) The Gaussian nature of the perturbations is taken care of by making the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 13 coefficients a. Xrrr> proportional to the random variable unfortunate choice of letter, but this the literature.)The/a^/m(k) sl. JLrm (k) (an is the notationappearing parametrize the amplitude individual fluctuation mode is excited. in to which each The expectation value of satisfies f k = -| ( 2 - 2 - 3) $ where k = k=+i - (R+1)K, R is the rank of the type of perturbation under consideration (scalar, vector, or tensor), and p > 0 /3 = K = 0 , - 1 K = + 1 Our final result will be to predict the rotationally invariant quantities I (2.2.4) rn\- ~x which we can now do using eq. (2.2.3). o the expectation value of a^ Jim We can think of eq. (2.2.4) as averaged over an ensemble of universes. We can determine a probability distribution for the values a^ , namely p f. \_ f i r !__________ { Q a L 1 * 3 -S’— ( a l W ) ( ( £ ) ^ +l C R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (2.2.5) 14 where 0" r JL (2.2.6) QJL+1 Using this probability distribution we can determine "error bars” corresponding to one standard deviation which we will present in the results section. This is an important feature in that the error bars will tell you whether a given set of multipole moments for one model can theoretically be distinguished from another set, regardless of how small observational errors can be made. We see that we are really expanding the fluctuations in terms of the modes £^^(k) We have now shown how to incorporate the Gaussian assumption into predictions of the multipole moments of the temperature variations in the cosmic microwave background radiation. This leaves only the radial dependence and the (conformal) time dependence which will be discussed in the next chapter. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 15 CHAPTER 3 TIME EVOLUTION AND RADIAL DEPENDENCE OF THE PERTURBATIONS The most elegant method for treating cosmological perturbations is the gauge invariant formulation of Bardeen.33 In this chapter we apply this formalism to solve for the time evolution of these perturbations. Bardeen's approach uses an expansion in spatial harmonic functions to describe the purely spatial dependence. In curved space this is the equivalent of a Fourier transformation of the Einstein equations, which then yields an ordinary differential equation to solve for the time dependence. The solutions of the time evolution equations will be presented here, but we only show the pieces of the spatial harmonic functions which are necessary for our calculations. The complete spatial harmonic functions along with some of their properties will be presented in gory detail in Appendix A. At various points in the derivation we will also need to know how the Robertson-Walker scale factor depends on the conformal time 'Y. The Friedmann Equations in terms of 'Y become (3.1) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. 16 where /O and P are the background energy density and pressure, and a dot indicates a derivative with respect to '7'. Since we will be concerned with times later than the time of recombination, which is later than the time of matter domination, we will need the solution to these equations for P = 0, which is (3.2) AX/r^ where ^ and S 0 are the values of the present energy density and scale factor. Perturbations in various quantities can be classified as scalars, vectors, and tensors, according to how they transform under spatial coordinate transformations in the background spacetime. The spatial and temporal evolution of the three types will be governed by a separate set of equations for each type. Any completely general perturbation of the gravitational field can be written as a linear combination of the three types, with each type evolving independently of the others. Therefore, our analysis will be broken into three sections, one for each class of perturbation. It is also worth mentioning that the equations given here are valid only for adiabatic perturbations. In each of the three sections we will first rewrite the expression for /T0 the Bardeen variables and the appropriate spatial harmonics. in terms of Then we will describe the functional form of the evolution of the Bardeen variables and the spatial harmonics. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 17 3.1) Scalar Perturbations 3.1A) STb /T0 Scalar perturbations are characterized by the scalar spatial harmonic functions Q (1?) which satisfy the Helmholtz equation D^Gt + l l Q - 0 where D 3. = D t D' (3.1.D and D ^ is the three dimensional covariant derivative in the spaces of (2.1.1). We will write Q (3) in the form Q(#)=r ( 3 - i - 2) j-)/*i and so we will label individual modes by k, 1, and m. For each mode we define perturbations using Bardeen's notation X, ■ - B W Q iU ) - (3.1.3) aHLWQ(?)f., +aHT(T)Q,I») and the perturbations in a (perfect fluid) stress energy tensor with adiabatic perturbations are given by ST>S j° = (pto+ p with R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. <3.i.4) 18 (3.1.5a) (3.1.5b) where 6 ^ is the three space metric in eq. (2.1.1). variables A, B, HL , HT , and 8 In terms of the Bardeen defines the gauge-invariant combinations: 0l A = m HL+ iH T+i. (l)^'je(f)H t o -1- u £ M = = 5+ Vs = V " (3.1.6) )N -b) ±Hr Our goal here is to rewrite eqs. (2.2.1) and (2.1.9) in terras of these gauge-invariant variables. We first note that the temperature fluctuation in the emitting plasma previously mentioned in section 2.2A can be written as (3.1.7) where ^ 0 ^ = since is proportional to T* Using this and the definitions of eq. (3.1.5) in eq. (2.2.1) we find R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 19 S L c r\ I - 1 - fV* 0 (3.1.8) where the perturbations are evaluated at values of T terms of y by eq. (2.1.5b). and x given in Using the identity for an arbitrary vector V; jf J y ' L 0 ' vl (3.1.9) along a radial null geodesic and the fact that 4 = - & + jrei^ from eq. (2.1.5b), we can rewrite eq. (3.1.8) as ( ¥ ■ “i q p - Ai Mf I r R \ ~ W -T S -T e . + (3.1.11) "i- -V7c L '0 3 H X t A i'TJ -r ^t n q 1/ The first term in (3.1.11) only contributes to the raonopole moment of S Te /Tp so we will absorb it into our definition of T0 . The next step is to re-express (3.1.11) in terms of the gauge-invariant variables of eq. (3.1.6). Here a slight complication arises. energy density fluctuation 8^ appears. gauge invariant variables and v| In (3.1.11) the photon Therefore we must introduce in analogy with (3.1.6),^** R eproduced with perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission. 20 P (3.1.12) v > s= v „ ' i h t Eq. (3.1.11) is valid in the gauge A = in Bardeen's notation. =0, the Sachs Wolfe gauge From this we can rewrite ST0 /T0 using gauge- invariant variables as ST. /I £a+ T ' a ( ^ +(j)v/))q 11 X *i(v,wv,vaOQi’:r -f - (3.1.13) i k We now use the evolution equations for the gauge-invariant variables1*1^ 5 which come from energy and momentum conservation and from the Einstein equations. For a perfect fluid in a matter dominated universe these give <|a = - | H (3.1.14) i l u s A v x = t [ (dk K)/(X-3K)] (e - 1e A - s K ^ This leads to an expression for /T0 involving £ y as well as the total energy-density fluctuation variable £ . However, for the long wavelength modes which dominate the multipole moments we will evaluate, € y is proportional to £ , £ y = 4/3£ . This can be verified using the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 21 multifluid extension of Bardeen's formalism1*1**115 and is a result of the fact that these long wavelength fluctuations are more massive than the maximum Jeans mass. Therefore, we can write (3.1.11) purely in terms of £ as I^ = r e-r0 Tc ~ J C - 3 K 'I I + Q '^=0 (3.1.15) 4-3 To obtain our final results we must sum and/or integrate over the mode variables k, 1, and m and project out the appropriate moment. We do this numerically and present the results in chapter 4. 3.IB) Evolution of £ (T) The gauge invariant variable £ Is € - obeys the evolution equation33 0 (3.1.16) This can be most easily solved by writing (3.1.16) as a function of S/S which in turn is a function of T • Defining W= f (3.1.17) eq. (3.1.16) has the solution ( r ) < K ^ M = V / ( w a+ K ) which is a shrinking mode and can be ignored. (3.1.18) The growing mode is R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 22 J v y 3’ | w ( w a- i ) £ u d M i A ( w ) k i(w)- < w ^ l+ I = - i |^= 0 | W(w^-t) Oa A a u We will characterize the amplitude of £ of horizon crossing (i.e., when S/S = k). (3.1.19) K~+l (vJt) by its value at the time Thus we write (3.1.20) where aA/m is the Gaussian random variable satisfying (2.2.3). of a The role now becomes clear; while the Einstein equations specify how each fluctuation mode must evolve in time, they say nothing about the initial amplitudes of the modes. The a ^ are a way to enforce that each mode is initially excited to an amplitude which follows a Gaussian random probability distribution, whose width scales as €^. Since each mode independently follows this distribution there can be no correlation between modes. This is the information of eq. (2.2.3). ATI' is introduced so that The factor of will agree with the definition commonly used in inflationary cosmology.5>2 7 3 In the case of inflation is a constant but for non scale invariant spectra it is a function of k. The expression (3.1.20) is to be substituted into eq. (3.1.13) to obtain 3.1C) The harmonic functions Q(x) The harmonic functions QO?) appearing in (3.1.11) have been expressed R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 23 .1 .2 1 ) J{/m The radial functions are given by the following expressions. For K = 0 <jvV) = jjiiM) (3, 1 .22 ) where j^(kr) is a spherical Bessel function which satisfies the orthonormality relation (no sum on 1) { jx ( M p (Jm The radial functions ) = § (A - M ) «• 1.23) (j)^ for K ^ 0 have been given by Harrison.1*6 We have changed the normalization slightly to make their orthogonality properties more like those of j^(kr). — For K = -1 --- ' r - v - W (3. 1.24) where Pj£ is an associated Legendre function, A y 1.25) and P A /n = o In analogy with (3.1.23) these functions satisfy the orthonormality R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 24 condition (no sum on 1) ,-£ & ). * V ) 4 » ,V ) = g i 5 ( ^ ) Further properties of c .1.26) (j)^ are given in appendix A. For K = +1 (tr*r / ^ V ‘— z6 D c .1.27) QjcA-f^ -J- + /S with = = W&F c .1.28) /s = 3 ^ ^ - - and n?= i if-*') These functions satisfy the orthogonality relation (no sum on 1) . co f i f e *> ) ^ v Further properties are also given in appendix A. .1.29) Note that in any case k and ^3 are related by .1.30) The orthonormality condition can then be summarized as y£Jtb- . J (I+Kff ^ Of space _ Tc, f K=0,-1 ^ W -2/3° .1.31) ^ K=-H R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 25 3.2) Vector Fluctuations 3.2A) &T0 /Te We now repeat the analysis of section 3.1 for vector fluctuations. These are the least physically interesting as they can only shrink with time. They are presented for completeness. The perturbations are characterized by a vector harmonic satisfying p a Q(p + j w = o and D 1Q f = o Defining Q i f - su* (Pi Q f + Dj Q i ' ) (3-2-2) we can write the fluctuation hyuy in the form Ju= o (3.2.3) X , = Jiij = - B 0)( r ) 2 Q .‘ < W H?(r) Q?j & ) and perturbations in the stress energy tensor as 5 V = 51/= o (3.2.4) &T-“=if’ b>))q(;Y^) We next rewrite eqs. (2.1.9) and (2.2.1) in terras of these variables to obtain R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 26 STa To0 0 ” <4[i Q'i’e‘e3-*B<0Q"el] Integrating by parts and using the identity (3.1.9) this becomes a ,ji $ [ta r- (3.2.6) For vector harmonics Bardeen33 defines the gauge-invariant variables i(I) _L |/- BU )-j 1 /j CO Hr (3.2.7) Vc - v l° - B0> and However in the gauge we have chosen (V^ =0), so eq. (3.2.6) can be rewritten in terms of the gauge invariant variables as -j lo 3.2B) (3.2.8) Evolution of Perturbations For a pressureless, perfect fluid the gauge invariant variables andy V^. satisfy the evolution equations 1 tf-a-K. y = p\Jc s (3.2.9) and V.. = ' j Vc The second equation can be solved immediately to give only a shrinking mode R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 27 Vc °< (3.2.10) ys There is no growing mode for the vector perturbations. There is also no growing mode in a radiation dominated universe. 3.2C) Vector Harmonics Because the unit vector ~e appearing in eq. (3.2.8), which points from the observer out to the source, is in the radial direction we only need the radial component of the vector harmonic 7$^ . are given in appendix A. The radial component Other components is given by (3.2.11) where (3.2.12) The functions (j)^ and the variable Y are as defined in section 3.1C, with P> now given by (3.2.13) for K = 0, ±1. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 28 3.3) Tensor Perturbations 3.3A) 8T0 /T0 caO Tensor perturbations are characterized by tensor harmonics Q;. 1 satisfying + o Qf] = Q'tl D 1q = o - Q ?1 ^ For tensor perturbations the fluctuation variable of relevance here is Ht as A.. - A . i - o and st: (a) - o is already gauge invariant so we immediately obtain a gauge invariant form for CT Q lo C \ Ye ~ ~ J 3.3B) H WO &TC /Te . n ci^, from eqs. (2.2.1) and (2.1.9), I iCT) /*\ Ctt Ht i ! Q4 e ei C3.3.3) Evolution of HL* satisfies the equation (for a perfect fluid) + ( i) f i r + (a i i + H r r ° For a matter dominated universe we have a shrinking mode solution R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.3.4) 29 Ht *• ^ (2 ) which can be ignored, and a growing-mode solution (3.3.5) K N y ^ f ^a|&(Z) where (j>1 are the scalar radial functions given in section 3.1C, > a= (3*-3K (3.3.6) and Z is given by K = - 1 2/ a Zr c <h l K ~ (3.3.7) o £ As in section 3.IB we characterize the size of the fluctuations at the horizon crossing time by dividing out the value of S/S = k. We will also use an arbitrary constant H h at such that to get the form H„/^S (3.3.8) where (3.3.9) Ail) = (zM T such iliod §■-jfz. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 30 3.3C) Tensor Harmonics As in the case of the vector harmonics, we will only need the radial component of the tensor harmonics since "e-in eq. (3.3.3) points in the radial direction. This component can be written as (3.3.10) where K=-) (3.3.11) K= o N?= K=+ l and <j>* is as given in section 3.1C where now /S is K~= (3.3.12) /3a - 3 K R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 31 CHAPTER 4 PREDICTIONS OF MOMENTS 4.1) Dipole and Quadrupole Bounds In the preceeding sections we have derived formulae for calculating the 1 = 1 , 1 = 2 , and higher moments of the distribution of the cosmic microwave background radiation by using the appropriate functions in eq. (2-2.1) for 8Te /Te . We would like to examine some physically interesting cases and see what information we can first get out of the predicted dipole and quadrupole alone. Remember that the amplitude of our density fluctuations was proportional to £ H (see eq. 3.1.20). If inflation provides the correct picture of the early universe, we could predict the value of €.H if only we knew the correct unified field theory of elementary particle physics. Unfortunately we cannot say what the correct unified field theory is, so we do not know a priori what £ h what limit the dipole imposes on is. Abbott and Wise1*3 have calculated in a purely inflationary universe. The dipole has also been considered by others1*7~l *9 in a similar manner. Here we would like to consider the more general cases in which we have initial conditions similar to those naturally provided by Inflation, but are produced by some now unknown mechanism. A Harrison-Zel'dovich spec trum (or a spectrum very close to it) of Gaussian density perturbations is required for successful galaxy formation whether or not inflation is correct.12 If we could extract Information about the value of the energy R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. 32 density jfL (= or about deviations from a Harrison Zel'dovich spectrum of perturbations we could observationally test the inflationary universe, perhaps excluding inflation from models of the early universe. We believe such is possible and we will discuss this more after first presenting bounds on . For a fixed value of £ H , the predictions for the size of the of the dipole and quadrupole moments vary with SI . We can thus use the moments to give us information about as a function of X I . In order to make our results even more general, we will consider small deviations from a Harrison-Zel'dovich Spectrum by parametrizing as where n = 0 gives the Harrison-Zel'dovich spectrum and X is a constant to be evaluated when the horizon size is equal to the wavelength of the mode under consideration. (The letter X is used to signify that it is the amplitude when the fluctuation crosses or "Xs" the horizon.) In non-scale-invariant spectra the value of X becomes proportional to some scale length to the power n/2, X sionless. (size) ^ , so X is no longer dimen- But if, as these spectrum deviations require, inflation is not the correct model of the universe we don't know how density pertur bations can form in the first place, and so the power law spectrum for k is just a guess that requires minimal information, and also has been used in models of galaxy formation. We will now look at the bounds on X imposed by the measured anisotropy of the microwave background radiation. We can obtain a weak bound on X from the present limit on the quadrupole moment50*51 < R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 33 This gives a 90% confidence bound on X ) X ^ (fl/euj X | 0 for -1 < n < 1 and 0.05 < XI < 2 (see figure 1). However, we can get much stronger bounds by considering the dipole moment. This is not as straightforward as imposing the quadrupole bound and we will discuss the complications in the next section. 4.1A) and the Dipole Moment Unlike the other moments of the background radiation distribution, the dipole is sensitive to short wavelength fluctuations. This is most easily seen through the fact that the value of the dipole depends on the velocity of the observer. In fact the bulk of the observed dipole5°>51 seems to be due to infall velocity of the Milky Way galaxy.52 Thus we cannot simply set our predicted dipole equal to the observed value of the dipole and determine X, because the short wavelength fluctuations have grown too big to be treated by linear perturbation theory. Calculating the behavior of these short wavelength modes is quite complicated and model dependent, so we have adopted a different strategy. We have cal culated the contribution of only the linear modes to the dipole moment. Since the modes are uncorrelated the total dipole moment must be larger than just the contribution from linear modes. Thus we can demand that the value of X must be small enough to make the linear contribution at least as small as the total observed dipole moment. Of course, the possibility exists that the linear part of the dipole is anti-aligned so the two effects could cancel, but the probability of this occurring is very small, much smaller than the probability that the dipole is outside our 90% confidence range. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 34 We must cut off our integral over modes of different wavenumber at some ky^ in order to prevent including non-linear modes. The largest scale of structure in the universe seems to be the supercluster scale which is the order of 60 Mpc today, so we will pick our kmax correspond ing to a wavelength of 60 Mpc today. bound on X. This will give us our best possible This kp,^ corresponds to a value of 1 for the expectation value of the function of Peebles53 for a flat universe (K = 0) and a Harrison-Zel'dovich spectrum. As an added complication we find that a fluctuation of wavenumber kmox crosses the horizon (wavelength = horizon size) while the universe is still dominated by radiation (not matter). The growth of perturbations during this phase is very different from that of the matter dominated era A check of the equations for multiple uncoupled perfect fluids4*1* in non flat Robertson-Walker universes reveals that we areconsldering values of XI for which we can use the equations for K = 0 to an accuracy of 1%. Thus we can use the treatment of Abbott and Wise1*3 to describe the behavior of 6. in section 3.1. The technique used by Abbott and Wise, along with a check of the accuracy of using the (K = 0) equations is presented in Appendix B. Using our 60 Mpc cutoff then we get the bounds on X shown graphically in figures 2, 3, and 4. As can be seen from figures 2, 3, and 4 the dipole moment provides quite a strong limit on X. n = 0, we see that X For example, in the inflationary case,_Q. = 1, must be less than 3.6 x 10 is just barely big enough to allow galaxy formation. the dipole c< k mftx, so our upper bound on € ^ of inflationary models, the value of £ h (see figure 2), which Note that forXI~l, In the context is significant in that it is not easy to arrange for a potential in a particle theory which can both pro R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. 35 vide for a successful inflationary phase and predict such a small value of Another popular set of parameters is n = 0 and Xh = 0.2, for which must be less than 5.4 * 10 allowed by inflation. , a value about twice as big as that If one wants to make galaxies in his/her favorite early universe scenario, figures 2, 3, and 4 provide information about the initial amplitudes of the density perturbations. While we have provided some constraints on the conditions of the early universe, we have not learned anything about the value of n or n. Next we will consider the value of the ratio of the quadrupole to dipole moments which may in fact give us such information. 4.IB) The Quadrupole to Dipole Ratio An annoying feature of the previous discussion of results is that the predictions of the moments are proportional to an unknown constant X. We would like to get information from the cosmic background radiation which is independent of the value of X. By dividing the quadrupole by the dipole we can cancel the factors of X in each individual moment. By the same reasoning used in the previous section, we see that our predicted value of the quadrupole to dipole ratio (Q/D) must be larger than the observed value of Q/D because we have calculated only part of the dipole moment. Since we have at present an upper bound on Q we have only an upper bound on Q/D (see previous discussion of quadrupole bound on X). Thus all we can say now is R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 36 As the measurement of the quadrupole gets more precise, the value of (Q/D)|obs will decrease. When the quadrupole measurement is accurate enough to yield a positive result, we will be able to rule out all theories with a predicted (Q/D)^ < (Q/D)^ . The results of our calculations are presented in figure 5. The curves represent the 90% confidence upper bounds on the values of these ratios. When the quadrupole is found the observed value of Q/D will be represented by a horizontal line, which will eliminate all theories with values of Q/D below that line. For convenience we have multiplied the predicted Q/D ratio by the observed value of the dipole so that the value of the quadrupole can be read directly from the vertical scale on the right side of figure 5. Looking at figure 2 we can see that in order for inflation to survive, the observed value of Q ( = <(aa)a'> ) must be over 2 orders of magnitude smaller than the present upper bound, not a pleasant thought for observers. If the quadrupole is found very near to its present upper bound, however, this could spell death to both the inflationary models and the Harrison-Zel'dovich spectrum. The only way for Inflation to evade this bound would be if copius amounts of gravitational waves are produced by the inflationary era. This possibility and observation al constraints on gravitational wave influences on the microwave back ground will be discussed at the end of the next section. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 37 4.2) Higher Multipole Moments In the previous section we gave some hope that we may get strong constraints on our models of the early universe from the quadrupole to dipole ratio. However, if the quadrupole is found to be very small compared to its present bound, the Q/D ratio will not give much infor mation about modelling the early universe. If this is indeed the case, we would like to discover if different models give distinguishably different sets of values for the higher moments. Again, if we knew what was and whether it depended on k, we could tell which model was correct on the basis of the quadrupole moment alone. Since we do not know £ h a priori, we must use the quadrupole moment to fix the value of X, (the constant amplitude of the fluctuations at the time they cross the horizon). The question then becomes, "Once the quadrupole is measured, how many higher moments (octupole, etc.) will need to be measured before we distinguish one model from another. ing cases in the following sections. We will consider a few interest In all cases we will plot the predicted values along with their theoretical error bars. These error bars arise from the fact that the physical perturbing process is assumed to be Gaussian. Thus, we can determine upper and lower limits corre sponding to 1 O' error bars (see reference [43]). Because of this feature, it may be impossible to distinguish two sets of multipole moments regardless of how small observational errors can be made to be, because of the intrinsically random nature of the underlying physical process. 4.2A) The Harrison-Zel'dovich spectrum We present the values of the first few moments up to 1 = 9, calculated using the Harrison-Zel'dovich scale invariant spectrum of R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 38 perturbations in the matter density of the universe. In figure 6 we show the values of the moments for 3 values of the parameter XL. In the next section we will show how these moment predictions change with small deviations from the Harrison-Zel'dovich spectrum for a fixed XL. In the the three spectra under consideration, the behavior of the moments as a function of XI is similar. As we can see from figure 6, the distribution of moments for XL = 1 and XL = 0.2 are nearly identical, even if the moments are measured up to 1 = 9. Thus we do not realistically expect that higher moments will be able to help us distinguish between the important cases XL = 1 and XI < 1 for a Harrison-Zel'dovich spectrum. If XT. really is less than 1, we will have to rely on the hope that the Q/D ratio test will tell us this. On the positive side, these moments may in fact give us our best upper bound on X I , as it can be seen that XT. = 2 is quite distinguishable from XI = 1. The moments for a variety of other values of XT. ( 0.05, 0.5, 0.75, and 1.5) are presented in figures 7 through 10. 4.2b ) eH= XX f We would also like to test whether the Harrison-Zel'dovich spec trum is discemable from some weak deviations from it. we are considering the two cases = X k+l and with the Harrison-Zel'dovich spectrum, parisons for n = X. Specifically, = X k 1 contrasted We present these com = 0.2, i, and 2 in figures 11, 12, and 13 respectively. Figure 11 shows us that with X L = 0.2, it is not easy to separate the moments for different powers of k. Only if the observational errors are much less than the theoretical Gaussian fluctuations can we hope to distinguish the different powers of k dependence, and then only after many moments are measured. If XT. = 1 (or even XT. = 2) The observers R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 39 should have an easier job as the distributions of moments for the differ ent powers of k separate after only a few moments, as we see in figures 12 and 13. In figures 14 through 17 we present moments for other values of /I ( 0.05, 0.5, 0.75, and 1.5). From the figures we can see that the values of the higher moments (high 1) decrease relative to the lower moments as we decrease the exponent of k in the spectrum. This is because when we decrease the exponent of k we put more power into the long wavelength fluctuations to which the lower moments are more sensi tive. 4.2C) Gravitational Wave Perturbations If the measured value of the quadrupole moment turns out to be too large to allow Inflationary models (according to figure 6), there is still a chance to save the inflationary picture. It has been shown elsewhere22-25 that an inflationary epoch in the early universe will produce long wavelength gravitational waves in a Harrison-Zel'dovich spectrum. Because of the tensor character of the gravitational waves, no dipole anisotropy is produced. Thus, even if the quadrupole moment is found to be close to its present upper bound, it could simply mean that the quadrupole is dominated by the gravitational wave contribution and inflation would still be allowed. If the gravitational waves dominate the quadrupole moment, they will in fact dominate all the moments, because the 1 dependence of the moments caused by density perturbations and gravitational waves is similar. So for inflation to escape our Q/D ratio test requires the moments higher than the dipole to behave as if they were caused by a universe filled with gravitational waves. If this is indeed the case we would like to know if it is possible to tell this through the higher moments. For example, can we now R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 40 distinguish our inflationary (.0- = 1) density perturbation moments from those of produced by inflationary long wavelength gravitational waves? To see if this is feasable, we have plotted the values of this case in figure 18, where once again we have normalized the values of the quadrupoles to 1. We see that the values of the moments in the two cases are indistinguishable, so the distribution of higher moments in the inflationary universe is basically the same whether they are caused by density perturbations or gravitational waves. If the value of the Q/D ratioallows fl figure 19 shows that the values of = 0.2 and forbids .d. = 1, the higher moments (if measured accurately enough) can make or break inflation. For completeness, we present the values of the moments in the cosmic background radiation due purely to gravitational waves in open, flat, and closed Robertson-Walker-Friedman universes in figures 20 through 24. Harrison-Zel'dovich spectrum is assumed in all cases. A As before, we have used the values of fi. = 0.05, 0.2, 0.5, 0.75, 1.5, and 2. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 41 CHAPTER 5 CONCLUSIONS Formulae have been derived for calculating the moments of the anisotropy of the cosmic microwave background radiation for an arbitrary perturbaton of the gravitational field in all three types of RobertsonWalker universes (open, flat, and closed). It has been found that the amplitudes of perturbations which transform like vectors can only shrink with time and thus are not important for the Sachs-Wolfe effect on the microwave background. We have also explained how to implement the assumption that the process which causes these perturbations is random with a Gaussian distribution. Actual computations of the moments due to density and gravitational wave perturbations have been performed for the inflationary universe as well as a range of relevant alternatives to an inflationary cosmology for comparison leading to the following conclusions: 1) Confirming previous work, references [25,27,43], the microwave background radiation anisotropy provides some tough constraints for the inflationary cosmology. A) An upper bound on ( € M is related to the underlying elementary particle physics theory of inflation and also equals the amplitude of density perturbations when their wavelengths are equal to the horizon size). We find that with 90% confidence R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 42 r B) ^ n I wq ve/engTh of cutoff j — T * I GO I The value of the observed quadrupole must be at least two orders of magnitude smaller that the current upper bound if the anisot ropy is caused by density perturbations. C) If the quadrupole moment Is much larger than the value pre sented in figure 5, then the quadrupole moment could be signalling the presence of gravitational waves. The higher moments must then follow the 1 dependence presented in figure 18. 2) The multipole moments provide a useful test of the Harrison- Zel'dovich spectrum as we can distinguish between it and some small deviations from it in a critical density universe. 3) If the Harrison-Zel'dovich spectrum proves correct then the multipole moments will probably not give us any information about the value of the energy density. If inflation flunks the quadrupole test then we must first rule out the presence of gravitational waves in order to say that the energy density is not the critical value. If inflation does pass the quadrupole test, the values of the higher moments are not much help in distinguishing values in the range 0.2 <.f"L< 1, where the true value is believed to be anyway. 4) As distasteful to theorists as it may be these multipole moments can provide ways of killing the inflationary model. This could happen in two ways: A) as much as k**. The spectrum deviates from a Harrison-Zel'dovich by at least This can be discovered through the quadrupole-to-dipole ratio in combination with the higher moments. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 43 B) If I"! is outside the range 0.2 to 1.3 with a Harrison- Zel'dovich spectrum then inflation can be ruled out. If inflation is killed by either of these possibilities, then the Q/D ratio and the 1 dependence of the higher moments presented in chapter 4 can be used as a diagnostic tool for pointing the way towards the correct model of the universe. If a better model is found, the equations in chapter 3 are general enough to be used to calculating moments to test this new model. In any event, it is quite amazing to consider the fact that, as Larry Abbott has pointed out to me, we are considering measurements of phenomena on the largest scales of the universe possible which ultimately will give us information about phenomena about physics on the smallest scales being considered by particle physics. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 44 APPENDIX A SPATIAL HARMONIC FUNCTIONS We present the derivation and explicit forms of the spatial har monic functions which are solutions of a generalized Helmholtz equation. We will separate the discussion into a section for each of the three types of harmonics: scalar, vector, and tensor. We will present the derivation of the scalar harmonics <*&) in detail since they are the basis for the vector and tensor harmonics as well. The treatment of the scalar functions follows that of Harrison1*6 and the treatment of the vector and tensor functions follows that of Tomita.51* Al) Scalar Harmonic Functions We want to find the eigenfunctions Q(o0 of the covariant Laplacian in a general Robertson-Walker space. We need to solve the Helmholtz equation where D *5. = D I D t and D is a spatial covariant derivative of our space defined by the spatial part of our metric, <JU? = with S ( T ) [ ~ d r :L 4- (Al.2) = 1 + K — . With this metric we can see that only the radial dependence of the m ) stant K. Tp-Jfc'd!#] will change with the value of the curvature con In flat space (K = 0) eq. (Al.l) becomes R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 45 ( V a- + A ’-jC H a ) = O <a i .3) which is easily solved by Q (x) = exp(ik*lc). We can express the Q (?) in terms of spherical coordinates by remembering the relation J!)fyn with the defined by This form of the Q (?) is more useful for our purposes because we want to expand our formulae in terms of the to get the multipole moments. It is worthwhile to note that the equations for Q (1?) and for the pertur bation amplitudes depend only on the magnitude of k and not on its direction. The directional part of k here is introduced purely as a convenience so that we can write the solutions as exp(i^.?), a function which is simple and easy to think about. However, we can dispense with the directional parts of k along with the accompanying factors of i and 4If to write Q (?) as Q ( * ) = Y jL m which is perfectly acceptable. w i.4 ) Thus the general form for our solutions will be where (p^~ = CQ j (kr) for K = 0, = k 3- + K, and C q is a normali- R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. 46 zation constant. As previously stated, only the radial equations differ in the curved spaces and the are eigenfunctions of the angular part of the covariant Laplacian for any value of K. The radial equation be comes 0 (Al.6) For K=0 we see that this reduces to The spherical Bessel equation for the argument kr. We will next consider separately the equations for K = +1 and K = -1. For K = +1, let a s & jT L * (A1.7) /l£ = This y is the same y which is used as the affine parameter along the null geodesics described in section 2.1. The radial equation (A1.6) then becomes j g + i ) [ x - j With C^- the equation for A d AUr* fij. J o o (A 1' 8) J T - 0 (A1.9) = is X ( \ + 0 - 17 J with For K = -1, let - A A n Jb /k R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. (Al.10) 47 and just as in K = +1 we note that y is the parameter describing the null geodesics of section 2.1. The radial equation is now (Al.11) In this case define • The equation for TT(<i^) is AlrtJn* then ^ "TT-0 [x(x+iV ^ If we define the variable Cj = K (ai. 12) we can turn both radial equations into a single equation i d JL ■ where now \ ( \ + 1) = Kk1^ X x Since V* - O (Al.13) (cos( ^ )) with has two solutions = - £ ± ( , + k ^ = P(^ , or P ^ (Al.14) = P^^we will use V = X * integer but JU. is not an integer we can use P^ solutions instead of P^*- and the origin. it , and now has solutions which are associ ated Legendre functions P^(cos(^)) and and V-= \. (&$£ Also sinceyU ± V is an and P^ . We will demand that as independent be regular at This boundary condition eliminates P J vl+ ’A^as a solution and "Jb-'h leaves us with P^ as the only solution we need to consider. When talking about these eigenfunctions and their eigenvalues, it is more convenient to redefine k in terras of the variable . For K = +1 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 48 we define (A1.15) r = f - Then the radial function becomes <f>p = C+| In order that (fp^ A PyJ fi (CXL^) be single valued, (A1-16) must be an integer, i.e., It can be seen that the values of ft = 1 and ft = 2 correspond to modes which are pure gauge terms.33*36 Thus our spectrum of eigenvalues for K = +1 are J N /3* - 1 ,3=3^ S y .. j3 > £ wi-w For K = -1, we will define |3 such that Jl = fi* 4 | (A1.19) so the eigenfunctions are Because space is open there are no periodic boundary conditions to satisfy, so ft can take on any positive real value. Thus the spectrum for K = -1 is 'k I N ~~ /3a-H / 3^> 0~ R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (Al.21) 49 We have yet to discuss our normaliztion conditions so we can fix the values of the constants C0 , CL| , and C4, . Our condition is that rr js (g ^ L (q (*)),,- W (A1-22) all1 space. where y9 " = X + K and 8 (f^/3') is the delta "function" with respect to the measure ^(k) 5T/3,/2.^)= ^(p') ]Sjx(K) (Al.23) The measure JU(k) is defined differently for different curvatures in Table 1. Table 1. The k-space measure in curved spaces Mm ') K -1 - Dirac delta function 0 ~ Dirac delta function +1 T- ft- (5=3 1 & , P><2> - Kronecker delta where we will use the definition (A1.24) for the rest of the section on scalars. We have now specified the nor malization in which our functions now take the form (with their normali zation constants evaluated) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 50 ir* ~ _^_^ A>.—^ ) 7 / ?T 2.|3 AX/Kj /\^_ i^ K= + l where N l' X - v r "1=° i f + Jt 1 ^ = fr n (f 1n 2. /n- ^ o ' r f ) (A1.26) - r^)/ after Harrison1*5 and y is defined as (A1.7) for K = +1, (A1.10) for K = -1, and y = r for K = 0. While equation (A1.25) provides a neat well known form for the functions > it is of little value for numerical computation. will present some properties of the (p^ We useful in calculating which are derivable from the tabulated properties55-57 of the Bessel and Legendre functions. First of all we can rewrite the derivative of the radial functions which appears in the scalar formula for Ve ‘ d£ Q(2) - 1)1^ L ^ 8T0 /Tp eq. (3.1.15) as ^Tn. Since we will need to calculate the functions <A1-27) anyway, it is useful to know that we can express this derivative in terms of them. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 51 X cotky $ - o3 (j)"' K= -1 i £+1 eh dn'P K- o a y t ^ tpf - J p - c x + n * <j)f This leaves us with the task of calculating the (Al.28) |<- + 1 (j)^-s. We can do this recursively using the relations f (it-1)jr $'* - fr (j)Jl-x %s K =r o <A1-29) K= + 1 C and exact formulae for two of the c±p^ s, e.g., the 1=0 and 1=1 modes K=-l l/nr K = 0 (Al.30) K -+1 and K= -I *;= ^ x< Kj p P T [ o - p t to - K= o p K -+1 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 52 A closed form for any other (p^ can be found from the generating functions K--I ( - i ^ 1 i+i 4£= )<rO (A1.31) ( - 0 v. K=+l /syTTf For small values of ky it is better to use the Taylor series for (M- since the recursion relations would then involve subtractions of A* numbers very close in magnitude, resulting in the loss of significant figures. r The Taylor series are — ~~ \H ' ‘ (j1 (M+Oi! c (_|) (.|y K=-i 1 *rr2\~1 I fA _ X _ (A1.32) K ^ o P (ax+i)!l ^ m -1 1s ' - * - ” ® K - +1 where v(/3a+a+j)1'i k--i am ft K=0 D„ = T T - (!•+-> +!t) r-1 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 53 In this form it is now clear that at small distances and short wave lengths (small y and large k), we recover the flat space limit & 4 ^ = (ai-33) as we must since space is locally flat. A2) Vector Harmonic Functions We now want to find the solution of the vector Helmholtz equation (A2.1) where t is divergenceless D‘Qt>= o <42'2) As in the scalar case we will decompose the Q ^ into modes defined by k, 1, and m Ot\ Q?(Z)=L (Q'c /if* (A2.3) ^ 1 Counting the degrees of freedom we see we have two degrees (3 components - 1 constraint =2). In flat Euclidean space we usually think of this as two choices of polarization vector in the transverse plane. coordinates the two solutions are modes with parity (-1)^ In spherical and (-l)^+l. Thus any arbitrary transverse vector field can be expressed as a linear combination of even parity and odd parity modes. In deriving these solutions for the curved spaces (K = ±1), it is easiest to use the R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission. 54 coordinates described by the three space line element "k + <A2- 4 > where K ~ K h. \ " A>> y Notice that we can recover the flat space limit These coordinates are useful because we can treat both closed and open universes simultaneously and the connections take on a simple form. We present only the non-zero coefficients. ~ - A C n ^ c o o . 1^ Q r % - 1 I'i. r 3" - co* *ai = & 5 r = '*» r 4= t o * E^ M'i where x 1 = ^ , xa Q 1 r s= '-S'JL = (? , and x 3 = (J) . We now present separately the solutions of the even and odd parity modes, a) Solutions of even parity When we use eq. (A2.2) in (A2.1) we find we get an equation which involves solely the radial component 1 (3). We can expand Qt(* 0t) in i terms of spherical harmonics (A2-6) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. 55 to get the equation for the radial function of (A2.7) and taking the limit K-*0, cos£-> 1, and hV * $ V + 0 ? - K^sin£-*' y XJ^ h l‘ » Equation (A2.7) has the solution v . [ K = i l '‘ " U f where and (A2.8) K= ° (p-* are the solutions of the scalar equation defined in (A1.25) TP is defined here as / 3 and Ne * = X + a K <A2- 9> is a normalization constant. We get only one solution to this second order differential equation because we enforce the same boundary conditions as in the scalar case. and (A2 =2), Using (A2.6) and (A2.8) in £4 3• (A2.1) we can find the other components of . We present the solutions in terms of the variable y which is more useful than ^ $Te /T formulae R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. for our 56 where M I) " y3> " and r /UndhsAs^ AJLnr* (^s)filLw Mt+f) Vlib 3<t> K*-i X (A2.ll) ‘t’") with (j)/^ K- ~ I (A2.12) vl(b‘ = < K - 0 /*» A £ K - + I They obey the normalization condition R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 57 (A2.13) f s(M-) all Space. where £(^(2,0 is the appropriate Dirac delta or Kronecker delta as defined in eq. (A1.20). b) The solutions of odd parity The radial equation (A2.7) does not admit a solution of odd parity and thus S® =0. The remaining components are then solved for using eqs. (A2.1) and (A2.2). The solution is as follows Q? = I N. (A2.14) cm j£,/m ' where N o = a* i )J%p and & ) * „ A / 1 v 3 (^> AX tk O- — Y IJim (A2.15) s > ° ) /him with K = -I nf K = >IAV K - + o R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (A2.16) 58 The odd parity harmonics have the same normalization as the even harmonics a/1 Space A3) Tensor Harmonic Functions We now want to find the solutions of the generalized tensor Helmholtz equation ( D* + X ) Q u = 0 (A3-U d subject to the constraints DlQ1?. = O i (A3.2) QSj = Qfc and Q*u = o \&) C i As in the vector case there are two degrees of freedom which correspond to even and odd parity modes. Any transverse traceless symmetric tensor can be expressed as a linear combination of even and odd parity modes, a) Solutions of even parity Using conditions (A3.2) in eq. (A3.1) we find that the equation for the radial-radial component decouples from the other components an can be expressed as Q» =X)(tv\ L'XU%)XJW) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 59 resulting in an equation for the radial function “Xft, (using the coordi nates of eq. (A2.4)) (A3.4) Of- ^ + )-A=° K=O These are solved by K4 o (A3.5) y;= \ where the K= o % are the radial part of the scalar harmonic functions defined in eq. (A2.25), N& is a normalization constant, and is now defined by /6*= X +•3 K (A3.6) Once this component is known we can use eqs. (A3.1) and (A3.2) to find relationships between the radial-radial component and the other compo nents. We present the results Q n) = Z Ne (Ge(A3.7) ki<.«_ lu+xtii-t-1)jl a- o e '/ We remember j3 is defined as in eq. (A3.6). R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 60 The G & are given by K=-i I ( ‘*4 K- o K = l&e \ ^xx 30 Y 'lim = t + ' \ 2(i> -Y \&-J/bJL/m d(J> ‘i/rr, (A3.8) I£ Gt v>. I 35* YIjL /m C G, 43 = T, L - “ t o >» X s fi 2 <^ \G i I X + 1 o Y. je*Y3 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 61 and r x aj I d t:=JLU+OdyW1 jU+OYfi* ' i ±_ rh* K-° Kr+i Cot v. r ( y^AAL/K AA 1 d T * •5p. = K- -1 l J (~ n (x*i)u+iu(£-r) Co-d. <tiKi (A3.9) I cC- i - X V^ /4-^kA,7- ’)/y(UmAZ K= K -& U K- i- 1 a +1 '*O X i v ^ K~~l T Hp* '= JC+k-X J— (cH1,4 V V ^lAv1 ; " '1 (ft ((&"+'" K-0 K-M V g and the G^- s satisfy the normalization condition o (A3.10) o V e ' / V> lVlUo*\ kfLW^t U ^ all spoxc where is the appropriate Dirac or Kronecker delta as defined in eq. (A1.20). R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 62 b) The solutions of odd parity There is no odd parity solution to the radial-radial component equation (A3.4), so G* = 0, just as there is no odd parity solution of the vector radial component. The odd parity solutions are then pSL(rC\ (A3.11) „y^-i)o±ji_u±3i where G° =o ^ l\ / fn° y,„\ (A3.12) l&. ItaY - -c*c &xJLm J L r( \ . (j.a'b tp (>9.m M ™ ' 0r3 e ' - X ™ * ^ ( - S e h X ^ c*t°r*'h) with R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. I 63 16/a K -r'A I d , 1 (A3.13) K - o U-f K =+ l and have the same normalization as the even parity modes, r dfy (A3.14) ctU space. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 64 APPENDIX B GROWTH OF DENSITY PERTURBATIONS DURING THE RADIATION DOMINATED ERA To get the dipole bound in chapter 4 we had to cutoff our integral (or sum) over /3 modes at a wavelength which corresponds to 60 Mpc today. This wavelength fluctuation crosses the horizon while the density of the universe is still dominated by radiation. Fluctuations which cross the horizon during this time grow at a different rate than that of the matter dominated era presented in section 3.1. To treat the behavior of perturbations during the radiation dominated phase, we have to solve the equations for £(d^ and insure that the function £(V)goes smoothly over from the radiation to matter dominated eras. This has already been done for critical density universes in reference [43]. We will first show that their results can be used here for our range of non-critical density universes, and then briefly present their results for completeness. In the radiation dominated era we need to consider how the matter fluctuations behave in the presence of a dominant photon component. To do this we need the multifluid generalization of Bardeen's formalism which is provided in references [44,45]. We will first solve for the evolution of perturbations in the photon density and then find out how they influence the matter fluctuations. To begin with we cannot use the form of S(/y) provided by eq. (3.2) because we can no longer neglect the photon density and pressure in the R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 65 Friedmann equations ir= •-** sv p -> - k (B.l) (!)'= • f c where .« < - l f d s ) / l r - 3 / U I . and are the densities of matter and radiation and we have used the equation of state for photons -p^. = these equations are for . The solutions of T 1<C /^n-v r K- -I r y, 81r & ?>(r) n *6 K -0 r u o (B.2) K-+1 y U A v T V and for ^ An A u r d . 'l ( 1 x ^ ) K * - l Ka a^ o ) z( ) K— l R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (B.3) 66 where is the time when matter begins to dominate and is defined by # With these definitions we find that S and S are continuous at • <v t '■y Notice that the argument in eq. (B.3) is — and not simply -j£ as presented in eq. (3.2). concerned with 'Y » This is because in section 3 we are only and so 'T'+ The equation governing — "Y in eq. (3.2). in a radiation dominated universe can be found from Bardeen,33 0 (B.4) Following reference [44] we recast this equation in terms of the dimensionless parameter x which is the ratio of the sound horizon length to Tf times the fluctua tion wavelength. Equation (B.4) becomes a £ti+ X R + 1 e Y = o (b.5) where a prime denotes an x derivative, and K = k (|P (b-6) R tells us whether or not we can neglect curvature effects and has the physical interpretation R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 67 ^ = ^fachus S ^ ( t ') ~ ( U o n z o n of curvature s iz e ') *\ R - ,X k / f ] ‘ As long as R « = / V- S i StZg. I ra d iu s erf — Y curvature-J cur 1 we can ignore curvature effects and just use the equations for K = 0 (R = 0). For our range of jfl (10 < £"1 < 0.01) the largest value of R while the universe is radiation dominated is R — 0.002 (for X L = 0.01), so we are accurate to at least of order < 1% by neglect ing the curvature terms. £ v = f ” A , d ( The growing mode solution to this equation is - a * * ) ik ) <b- 7 > where A^a£j is just a constant. We next need the equation1*5 which gravitationally couples the photon density perturbations to the matter density perturbations during the _ 5^ radiation dominated era. We will also use the approximation k « K, which is valid for any fluctuation which crosses the horizon during the radiation dominated epoch. (This approximation is equivalent to setting R = 0.) c + (|)e + a.(f)(l As before, rewriting this in terms of our parameter x R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 68 €' (B.9) Ignoring the R terra we get £ " + ^ - [ i + a t i + ^ ^ ' ^ e ' (B.10) % i (,4 i 4 ' U - ¥ ) X % 2- ^ - 6 C !T £(")£) is then solved for by numerically integating eq. (B.10). have to match the solutions of € We then for the matter and radiation dominated eras at the time of equal densities ( A good numerical fit is provided by using an £(T) modified to include the short wavelength effects of the radiation dominated epoch. modified to emod(T) I \ include rad. dom.J = /e defined Ame(T) [ by eq. 3.1.20 j (B.11) 0.0005-JC^] (B.12) era effects and A m is given by1*3 [l 4 and Xm = X(Tm) = ' Jz 0 . 0 4 X ^ + and k ^ is the wavenumber which corresponds to the wavelength which just crosses the horizon at Y . This fit is good up to about k = 13 k ^ . Notice that for fluctuations which cross the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 69 horizon during the matter dominated era A^Xl. To get our dipole bound we just substitute our modified Gfnoi (ecl* (B.ll)) wherever G appeared in our equation for £T0 /T0 (3.1.15). R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 70 REFERENCES 1) Steigraan, G., Schramm, D. M., and Gunn, J., Phys. Lett. B66 (1977), 202 . 2) Guth, A., Phys. Rev. D23 (1981), 347. 3) Linde, A., Phys. Lett. B108 (1982), 389. 4) Albrecht, A. and Steinhardt, P., Phys. Rev. Lett. _48 (1982), 1220. 5) Guth, A. and Pi, S.-Y., Phys. Rev. Lett. 49 (1982), 1110. 6) Bardeen, J., Steinhardt, P., and Turner, M. S., Phys. Rev. D28 (1983), 679. 7) Staroblnskii, A., Phys. Lett. B117 (1982), 175. 8) Hawking, S., Phys. Lett. B115 (1982), 295. 9) Harrison, E. R., Phys. Rev. Dl_ (1970), 2726. 10) Zel'dovich, Ya. B., Monthly Not. Roy. Astronom. Soc. 160 (1972), IP. 11) Yang., J., Turner, M. S., Stelgman, G., Schramm, D. N., and Olive, K. A., Astrophys. J. 281 (1984), 493. 12) Blumenthal, G., Faber, S., Primack, J., and Rees, M., Nature 311 (1984), 517. 13) Preskill, J., Wise, M., and Wilczek, F., Phys. Lett. B120 (1983), 127. 14) Abbott, L. F. and Sikivie, P., Phys. Lett. B120 (1983), 133. 15) Dine, M. and Fischler, W., Phys. Lett. B120, (1983), 137. 16) Cabibo, N., Farrar, G., and Maiani, L., Phys. Lett. B105, (1983), 155. 17) Ellis, J., Hagelin, J. S., Nanopoulos, D. V., Olive, K. A., and Srednicki, M., Nucl. Phys. B238 (1984), 453. 18) Davis, M., Geller, M., and Huchra, J., Astrophys. J. 221 (1978), 1. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 71 19) Peebles, P. J. E., Astron. J. 84 (1979), 730. 20) Press, W. A. and Davis, M., Astrophys. J. 259 (1982), 449. 21) Davis, M., Efstathiou, G., Frenk, C., and White, S. D. M., Astrophys. J. (in press). 22) Starobinskii, A., JETP Lett. 30 (1979), 683. 23) Rubakov, V., Sazhira, M., and Veryaskin, A., Phys. Lett. B115, (1982), 189. 24) Fabbri, R. and Pollock, M., Phys. Lett. B125 (1983), 445. 25) Abbott, L. F. and Wise, M., Nucl. Phys. B244 (1984), 541. 26) Peebles, P. J. E., Astrophys. J. (Letters) 263 (1982), LI. 27) Abbott, L. F. and Wise, M., Phys. Lett. B135 28) Kaiser, N., Astrophys. J. (Letters) 284 (1984), L9. 29) Politzer, H. D., and Wise, M., Astrophys. J. (Letters) 285 (1984), LI. 30) Fabbri, R., Guidi, I., and Natale, V., Astrophys. J. 257 (1982), 17. (1984), 279. 31) Tomita, K. and Kenji, K., Prog. Theor. Phys. 69_(1983), 32) Wilson, M., Astrophys. J. 273 (1983), 2. 828. 33) Bardeen, J., Phys. Rev. D22 (1980), 1882. 34) Sachs, R. and Wolfe, A., Astrophys. J. 147 (1967), 73. 35) Anile, A. and Motta, S., Astrophys. J. 207 (1976), 685. 36) Lifshitz, E. M. and Khalatnikov, I. M., Adv. Phys. 12_ (1963), 185. 37) Field, G. B. and Shepley, L. C., Astrophys. Space Scl. 1 (1968), 309. 38) Sakai, K., Prog. Theor. Phys. 4_1 (1969), 1461. 39) Press, W. H. and Vishniac, E. T., Astrophys. J. 239 (1980), 1. 40) McVittie, G. C., General Relativity and Cosmology, Univ. of Illinois Press, Urbana, 1965. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 72 41) Schrodinger, E., Expanding Universes, Cambridge Univ. Press, Cambridge, 1956. 42) Silk, J., in Cosmology and Particles (Proc. of the XVI Recontre de Moriond - Astrophysics Meeting) Les Ars, Savoie, 1981. 43) Abbott, L. F. and Wise, M., Astrophys. J. (Letters) 282 (1984), L47. 44) Abbott, L. F. and Wise, M., Nucl. Phys. B237 (1984), 226. 45) Wise, M., private communication. 46) Harrison, E. R., Revs. Mod. Phys. ^9 (1967), 862. 47) Kaiser, N., Astrophys. J. (Letters) 273 (1983), L17. 48) Silk, J. and Wilson, M., Astrophys. J. 243 (1981), 14. 49) Silk, J. and Wilson, M., Astrophys. J. (Letters) 244 (1981), L37. 50) Fixsen, D., Cheng, E., and Wilkinson, D., Phys. Rev. Lett. _5£ (1983), 620. 51) Lubin, P., Epstein, G., and Smoot, G., Phys. Rev. Lett. 50, (1983), 616. 52) Hogan, C. J., Kaiser, N., and Rees, M., in The Big Bang and Element Creation, The Royal Society, London, 1982. 53) Peebles, P. J. E., The Large Scale Structure of the Universe. Princeton Univ. Press, Princeton, 1980. 54) Tomita, K., Prog. Theor. Phys. 68^ (1982), 310. 55) Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966. 56) Erdelyi, A., Higher Transcendental Functions, vol. 1 of The Bateman Manuscript Project, McGraw-Hill, New York, 1953. 57) Arfken, G., Mathematical Methods for Physicists. Academic Press, New York, 1966. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 73 BIBLIOGRAPHY Abbott, L. F. and Sikivie, P., Phys. Lett. B120 (1983), 133. Abbott, L. F. and Wise, M., Phys. Lett. B135 (1984), 279. Abbott, L. F. and Wise, M., Astrophys. J. (Letters) 282 (1984), L47. Abbott, L. F. and Wise, M., Nucl. Phys. B244 (1984), 541. Abbott, L. F. and Wise, M., Nucl. Phys. B237 (1984), 226. Albrecht, A. and Steinhardt, P., Phys. Rev. Lett. 4j3 (1982),1220. Anile, A. and Motta, S., Astrophys. J. 207 (1976), 685. Arfken, G., Mathematical Methods for Physicists, Academic Press, New York, 1966. Bardeen, J., Phys. Rev. D22 (1980), 1882. Bardeen, J., Steinhardt, P., and Turner, M. S., Phys. Rev. D28 (1983), 679. Bluraenthal, G., Faber, S., Primack, J., and Rees, M., Nature 311 (1984), 517. Cabibo, N., Farrar, G., and Maiani, L., Phys. Lett. B105 (1983), 155. Davis, M., Geller, M., and Huchra, J., Astrophys. J. 221 (1978),1. Davis, M., Efstathiou, G., Frenk, C., and White, S. D. M., Astrophys. J. (in press). Dine, M. and Fischler, Ellis, J., W., Phys. Lett. B120 (1983), 137. Hagelin, J. S., Nanopoulos, D. V., Olive, K. A., and Srednicki, M., Nucl. Phys. B238 (1984), 453. Erdelyi, A., Higher Transcendental Functions, vol. 1 of The Bateman Manuscript Project, McGraw-Hill, New York, 1953. Fabbri, R., Guidi, I., and Natale, V., Astrophys. J. 257 (1982), 17. Fabbri, R. and Pollock, M., Phys. Lett. B125 (1983), 445. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 74 Field, G. B. and Shepley, L. C., Astrophys. Space Sci. _1_ (1968), 309. Fixsen, D., Cheng, E., and Wilkinson, D., Phys. Rev. Lett. f>0 (1983), 620. Guth, A., Phys. Rev. D23 (1981), 347. Guth, A. and Pi, S.-Y., Phys. Rev. Lett. 49 (1982), 1110. Harrison, E. R., Revs. Mod. Phys. 39^ (1967), 862. Harrison, E. R., Phys. Rev. D_l_ (1970), 2726. Hawking, S., Phys. Lett. B115 (1982), 295. Hogan, C. J., Kaiser, N., and Rees, M., in The Big Bang and Element Creation. The Royal Society, London, 1982. Kaiser, N., Astrophys. J. (Letters) 273 (1983), L17. Kaiser, N., Astrophys. J. (Letters) 284 (1984), L9. Lifshitz, E. M. and Khalatnikov, I. M., Adv. Phys. 12 (1963), 185. Linde, A., Phys. Lett. B108 (1982), 389. Lubin, P., Epstein, G., and Smoot, G., Phys. Rev. Lett. 5(3(1983), 616. Magnus,W., Oberhettinger, F., and Soni, R. P.. Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966. McVittie, G. C., General Relativity and Cosmology, Univ. of Illinois Press, Urbana, 1965. Peebles, P. J. E., Astron. J. j54 (1979), 730. Peebles, P. J. E., The Large Scale Structure of the Universe. Princeton Univ. Press, Princeton, 1980. Peebles, P. J. E., Astrophys. J. (Letters) 263 (1982), LI. Politzer, H. D., and Wise, M., Astrophys. J. (Letters) 285 (1984), LI. Preskill, J., Wise, M., and Wilczek, F., Phys. Lett. B120 (1983), 127. Press, W. A. and Davis, M., Astrophys. J. 259 (1982), 449. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 75 Press, W. H. and Vishniac, E. T., Astrophys. J. 239 (1980), 1. Rubakov, V., Sahim, M., and Veryaskin, A., Phys. Lett. B115 (1982), 189. Sachs, R. and Wolfe, A., Astrophys. J. 147 (1967), 73. Sakai, K., Prog. Theor. Phys. 41 (1969), 1461. Schrodinger, E., Expanding Universes, Cambridge Univ. Press, Cambridge, 1956. Silk, J., in Cosmology and Particles (Proc. of the XVI Recontre de Moriond - Astrophysics Meeting) Les Ars, Savoie, 1981. Silk, J. and Wilson, M., Astrophys. J. 243 (1981), 14. Silk, J. and Wilson, M., Astrophys. J. (Letters) 244 (1981), L37. Starobinskii. A., JETP Lett. 30 (1979), 683, Staroblnskii, A., Phys. Lett. B117, (1982), 175. Steigman, G., Schramm, D.M., and Gunn, J., Phys. Lett. B66 (1977), 202. Tomlta, K., Prog. Theor. Phys. 68^ (1982), 310. Tomita, K. and Kenji, K.,Prog. Theor. Phys. Wilson, M., Astrophys. J. (1983), 828. 273 (1983), 2. Wise, M., private communication. Yang., J., Turner, M. A., Astrophys. J. S., Steigman, G., Schramm, D. N., and Olive, K. 281 (1984), 493. Zel'dovich, Ya. B., Monthly Not. Roy. Astronom. Soc. 160 (1972), IP. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 76 FIGURE CAPTIONS 1. Bounds on X from the upper limit on the quadrupole moment* With 90% confidence, the allowed values of X (as a function of/I) are below the lines for each of the three spectra as shown. 2-4. Dipole bounds on X. With 90% confidence, the allowed values of X (as a function of-0.) are below the lines in the figures. Figures 2, 3, and 4 show the upper bound lines for Harrison-Zel'dovich, k’* , and k4 1 spectra respectively. 5. The quadrupole to dipole ratio. Shown here are the 90% confi dence bounds on the linear contribution to the quadrupole to dipole ratio. For convenience, the values of the ratio have been multiplied by the observed dipole moment in the left hand scale. Thus the quadrupole moment must be found below the value shown for a particular spectrum and XL value to be a viable model. The right hand scale shows the unaltered values of the predicted linear quadrupole to dipole ratio. 6. A comparison of moment predictions with 1 CT error bars for a Harrison-Zel’dovich spectrum of density perturbations for 3 different values of XI : 0.2, 1.0, and 2.0. 7-10. The 1-dependence of higher moments from a Harrison-Zel'dovich spectrum of density perturbations, again with 1CT error bars. Figures 7, 8, 9, and 10 show the moments for ft = 0.05, 0.5, 0.75, and 1.5 respectively. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 77 11-13. Comparison of 3 different spectra: k 0 , k+l , and k”* for a given value of XL* Figures 11, 12, and 13 show the 1-dependence for .0 - = 0.2, 1.0, and 2.0. 14-17. The 1-dependence of the higher moments for the non-scale-v. *1 o invariant spectra « h = k X _ for particular values of XL . Figures 14, 15, 16, and 17 show the moments for XL values 0.05, 0.5, 0.75, and 1.5. 18. Inflationary predictions for the 1-dependence of the higher moments for density fluctuations vs. gravitational waves. This comparison can only be made in the event that the quadrupole is found to be small enough to pass the quadrupole-dipole ratio test. 19. Comparison of the inflationary gravitational wave moments and non-inflationary (fl = 0.2) Harrison-Zel'dovich density perturbation moments. 20. A comparison of the 1-dependence of moments from a scaleinvariant spectrum of gravitational waves for the /I values: 0.2, 1.0, and 2.0. 21-24. Predictions of the moments due to a Harrison-Zel'dovich spectrum of gravitational waves for certain values of £ L . Figures 21, 22, 23, and 24 show predicted moments for X L = 0.05, 0.5, 0.75, and 1.5 respectively. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FIGURE 1 Quadripole 90% confidence upper bound on X. Values of X are allowed below curves. i 7 -S ' XIo s o.o% o.oS" 0.07 a. T R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. FIGURE 2 Dipole 90/S confidence upper bounds on X, for a Harrison Zel’dovich spectrum *H= X . Values of X are allowed below the curve. 100 a.o -c. O.i 1-0 5 SX- R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. FIGURE 3 Dipole 90$ confidence upper bounds on X for a k' spectrum. Values of X are allowed below the curve -S' *|0 a. 5 M - R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FIGURE 4 Dipole 90% confidence upper bounds on X for a k +l( = £ h/)0') spectrum. Values of X are allowed below the curve. loo x{07 2 jr JO- R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. FIGURE 5 Quadrupole to dipole ratio. bound o n (0-/Q )»bs X 10 X 10 X /0 xlo X X I- R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 1---- »---- 1 <» 4 ^ 4 ________________________________ • * < | -g | f a Harrison-Zel'dovich ’’ . ■----- 1 H *------ 1 I <1 *- to FIGURE I to «- of moments I for it 6. Comparison H spectrum d ci - 6 i i i it ii i i c—. ^ Ii » < i t i i i i I i i i i fi i i I Ii i i i *0 O I — /\,S^ | *0 —■ o r< * W> rt ' Ii I q (i 1 >dy R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. to I— •---- 1 FIGURE 7. Harrlson-Zel'dovich spectrum for fl = 0.05 h-*- I— - CO lo 0 6 t 1 I cC / R eproduced with perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission. h -— I - <r~ do in o o |— «--1 -p O (D Q m. JG 0 T> "8 I—I <D tSI 1 lo g CO ----- 1 1 w CO I *- cO C3 M ci*' lo d Uj 6 I i I Or \ R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. ■' h - — t So in c*— U o <+H •p o <D CU CO £ o •H > O I .----- 1 T) (u t-i I C o CQ •H **) w CTv Of o E v> o o t T R eproduced with perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. Zel'dovich spectrum for A = 1.5 OO FIGURE 10. Harrison cO o or 4c*c> a ox/ I N/ 4 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. G>- I t © ---- 1 I <D60 ~ < i? t5 e xt* ^ I d>- y) u) Vjj 0.2 © 0 © I of Spectra'for o- -Q- I---- © ------- 1 -<D- F <*) Qr~ cv ■^r i i i t—i-i I i , i i I i i i i I i i i i I i t i i in b 0 O nr 1 I /\ 'd 1 t / v l y 4 R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission. 11. & ------- FIGURE i Comparison i----- © — i i 1 q i— to ©-----1 I i ©- f f ©- for e- I----- <D~ I-----© --------1 I---- © -------- 1 i <D*4 M 7 if v r.i «** m Sl> Ui -ffi- ® c-7 <£> <D \------ 0 - i------ &1..1 I I I 1.1 M I I--I E) l l I I I I O < I I lA I I l I f n O. rit I I I I I I U I I I f J. .t- o i /n ./\ V * - cr* s/|v R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Comparison ©- 12. I < & r- of spectra 1-------- FIGURE i <D----- 1 A.= -© © ----- 1 |-o— | I— ®— I MB— I I— <P— I I— ©— i -4 r— & — i <s> © 0 <D l— ©— i for p. = CM of spectra I— a---1 |— <&---1 I— <D- | ©- ©— 13. 1 1 Comparison t-Q 1 ■ I ■ w> j FIGURE t— *--©- I . . 1 1 1 . , . , I ■ n 1 1 n i ■ I . ■ 1 1 c< i 1 1 1 1 1 1 . i > ■ 1111 I ■ I w i n 1 1 1 1 1 Ixi I ~s R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. X FIGURE 14. Comparison of non-scale-invaiant spectra for fl = 0.05 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. a JL X FIGURE 15. 3 4 7 c. 7 Conparison of non-scale-invariant spectra for XI = 0.5 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. -1___________ 1___________I___________I___________i___________I___________ i___________ L 3- 2> tf S' (. 7 % SL <o2>) <£>' -l i s X FIGUHE 16. Comparison of non-s cale-invariant r spectra f o r A = 0.75 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. o A/tv a jg) -3 3- c 3 Jt FIGURE 17. 7 2 r Comparison of non-scale-invariant spectra for /I = 1.5 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. i— I— *— o— I s- a 3 2 4- I W o’ a. £ .o I— • --------- 1 ■fc •* 5 o ---- 1 I— $- +* > 5 u Lo of Inflationary I— H cO X<Y 1 cO and density r- 18. Comparison w9 C gravitational waves OQ perturbations I FIGURE — I R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •'JVM <^ro.v d ’a + iowU - lln -O A 0 I ■Tur cl A -a s FIGURE 19. A (. 8 Comparison of inflationary gravitational waves with fl- = 0.2 density perturbations UJOVtfi I— ■— I I— «— I OO 1 --- 1 wave moments *----- *- 1 I— *- d I! - i I' II 3 <i d *- i T I /'v/' <* Is* 1 cMcr c0 - 1 FIGURE 20. 1 cO of gravitational I — 1 Comparison i * for a Harrison-Zel'dovich -9-- 1 v ' v 4 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. I— -— 1 - <r~ i rD ) OT<S il. 4 R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. FIGURE 21. <T\ Gravitational wave moments for A = 0.05 1--- -----1 “i <r~ <Je? m o n c M O <4-4 c a) i 6 a) § 5s <9 a) C o *H 4J OS •H 4J § M a eg eg O c~f (O 4 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 60 LO r^. u o 00 a 0) a o a <u 4J 5 s cd 4J •H § M O C O CM ci D O H P*4 4 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. m r4 II a u o CO u c 0) 0 § <D § CtJ CS O (0 u •H 5 o 8 8 w c« 4 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.

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