Cosmological Fluctuations in Standard and Conformal Gravity

University of Connecticut OpenCommons@UConn

Doctoral Dissertations University of Connecticut Graduate School

6-5-2020

Matthew Phelps University of Connecticut - Storrs, [email protected]

Follow this and additional works at: https://opencommons.uconn.edu/dissertations

Recommended Citation Phelps, Matthew, "Cosmological Fluctuations in Standard and Conformal Gravity" (2020). Doctoral Dissertations. 2532. https://opencommons.uconn.edu/dissertations/2532 Cosmological Fluctuations in Standard and

Conformal Gravity

Matthew Phelps, Ph.D.

University of Connecticut, 2020

In the theory of cosmological perturbations, methods of simplifying the equations of motion, eliminating non-physical gauge modes, and constructing the solutions broadly fall into one of two approaches. In the first approach, one first fixes the gauge freedom by imposing coordinate constraints and then solves the fluctuation equations directly in terms of the metric perturbation. In the context of conformal gravity, we implement this method by constructing a gauge condition that is invariant under conformal transformations, allowing the full set of exact solutions to be obtained within any background geometry that is conformal to flat (thus covering all cosmologically relevant geometries). With this construction, we show that in a radiation era Robertson-Walker cosmology, conformal gravity fluctuations grow as t4. The other approach, standard within modern cosmology, is to decompose the metric perturbation into a basis of scalars, vectors, and tensors defined according to their transformation behavior under three-dimensional rota- Matthew Phelps – University of Connecticut, 2020 tions. With this basis automatically facilitating gauge invariance, it is asserted that the scalar, vector, and tensor sectors decouple within the fluctuation equations themselves, àla the SVT decomposition theorem. Here we investigate the status of the decomposition theorem by solving the fluctuation equations exactly within standard cosmologies as applied to both Einstein gravity and conformal gravity, finding that in general the various SVT gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the

fluctuation equations themselves one must additionally impose spatial boundary conditions, with an exact form depending on the given background geometry. To match the underlying transformation group of GR and thus provide a manifestly covariant formalism, we introduce an alternate scalar, vector, tensor basis with components defined according to general four-dimensional coordinate transformations. In this basis, the fluctuation equations greatly simplify, and while one can again decouple them into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves. Cosmological Fluctuations in Standard and

Conformal Gravity

Matthew Phelps

B.S., University of Colorado Colorado Springs, 2014

A Dissertation

Submitted in Partial Fulﬁllment of the

Requirements for the Degree of

Doctor of Philosophy

at the

University of Connecticut

2020 Copyright by

Matthew Phelps

2020

i APPROVAL PAGE

Doctor of Philosophy Dissertation

Cosmological Fluctuations in Standard and

Conformal Gravity

Presented by

Matthew Phelps, B.S.

Major Advisor

Philip Mannheim

Associate Advisor

Alex Kovner

Associate Advisor

Vasili Kharschenko

University of Connecticut

2020

ii To Aaron M. Phelps and Renee S. Phelps, lost but never forgotten

iii ACKNOWLEDGEMENTS

To Philip Mannheim - Working under you will always be one of the great honors of my life. Your brilliance is outshined only by your love and driven curiosity of the ﬁeld, which have served as a continual source of inspiration throughout. To my family, your continual encouragement has meant a great deal to me. To Ray

Retherford - Not of my blood but forever my brother true and through. You were there every time I needed you. To Michaela Poppick - Your unwavering support is nothing short of admirable and I am indebted to you for leading me to become a better person. To H. Perry Hatchﬁeld, for helping me not lose touch of the most important aspects of myself amongst such a transformational time. Your taste in music may only be rivaled by the gods themselves. To Candost Akkaya, for injecting high doses of humor into an otherwise mechanical environment, and for so many parallel journeys we have shared together. To Jason Hancock - Thank you for your faith in my abilities and treating me like an equal. Working with you was an unexpectedly great pleasure. To Lukasz Kuna - Our foray into the world of cryptocurrency could never be forgotten. To my past advisors, Robert

Camley and Karen Livesey, for mastering the balance of challenging my mind while building my intellectual conﬁdence. To all other faculty and friends, I thank you for helping me through this entire journey.

iv TABLE OF CONTENTS

1. Introduction ...... 1

1.1 Overview...... 6

2. Formalism ...... 9

2.1 Einstein Gravity...... 10

2.1.1 Fluctuations Around Flat in the Harmonic Gauge..... 14

2.2 Conformal Gravity...... 14

2.2.1 Conformal Invariance...... 17

2.2.2 Fluctuations Around Flat in the Transverse Gauge.... 23

2.2.3 On the Energy Momentum Tensor...... 25

2.3 Cosmological Geometries...... 30

3. Scalar, Vector, Tensor (SVT) Decomposition ...... 33

3.1 SVT3...... 35

3.1.1 SVT3 in Terms of hµν in a Conformal Flat Background.. 36

3.1.2 SVT3 in Terms of the Traceless kµν in a Conformal Flat

Background...... 38

3.1.3 Gauge Structure and Asymptotic Behavior...... 39

3.2 SVTD...... 48

3.2.1 Gauge Structure (D = 4)...... 53

3.3 Relating SVT3 to SVT4...... 55

v 3.4 Decomposition Theorem and Boundary Conditions...... 58

3.4.1 SVT3...... 58

3.4.2 SVT4...... 64

4. Construction and Solution of SVT Fluctuation Equations ... 66

4.1 SVT3...... 71

4.1.1 dS4 ...... 71

4.1.2 Robertson Walker k = 0 Radiation Era...... 76

4.1.3 General Roberston Walker...... 86

4.1.4 Robertson Walker k = −1...... 105

4.1.5 δWµν Conformal to Flat...... 124

4.2 SVT4...... 130

4.2.1 Minkowski...... 131

4.2.2 dS4 ...... 134

4.2.3 General Robertson Walker...... 157

4.2.4 δWµν Conformal to Flat...... 160

4.3 Gauge Invariants, SVT Mixing, and the Decomposition Theorem. 162

4.3.1 SVT3 Fluctuations Around an Anti de Sitter Background. 163

4.3.2 SVT3 Fluctuations Around a General Conformal to Flat

Background...... 165

4.3.3 Imposition of SVT Gauge Conditions...... 170

vi 5. Construction and Imposition of Gauge Conditions ...... 172

5.1 The Conformal Gauge and General Solutions in Conformal Gravity 174

5.1.1 The Conformal Gauge...... 174

5.1.2 δWµν in an Arbitrary Background...... 178

5.1.3 δWµν in a Conformal to Flat Minkowski Background... 185

5.1.4 Calculation Summary...... 188

5.2 Imposing Speciﬁc Gauges within δGµν ...... 189

5.3 Robertson Walker Radiation Era Conformal Gravity Solution... 193

6. Conclusions ...... 200

Bibliography ...... 204

A. SVT by Projection ...... 209

A.1 The 3 + 1 Decomposition...... 209

A.2 Vector Fields...... 210

A.3 Transverse and Longitudinal Projection Operators for Flat Space-

time Tensor Fields...... 211

A.4 Transverse-Traceless Projection Operators for Flat Spacetime Ten-

sor Fields...... 213

A.5 Transverse and Longitudinal Projection Operators for Curved Space-

time Tensor Fields...... 215

vii A.6 D-dimensional SVTD Transverse-Traceless Projection Operators

for Curved Spacetime Tensor Fields...... 216

B. Conformal to Flat Cosmological Geometries ...... 219

B.1 Robertson-Walker k = 0...... 219

B.2 Robertson-Walker k > 0...... 219

B.3 Robertson-Walker k < 0...... 220

B.4 dS4 and AdS4 Background Solutions...... 221

B.5 dS4 and AdS4 Background Solutions - Radiation Era...... 222

C. Computational Methods ...... 223

C.1 GitHub Repository...... 223

C.2 Mathematica and xAct ...... 223

C.3 Detailed Calculation Example...... 225

C.3.1 Conformal Transformation and Perturbation of Gµν .... 225

C.3.2 Background Equations of Motion...... 226

C.3.3 δGµν and δTµν Fluctuations...... 227

C.3.4 Field Equations...... 229

C.3.5 Field Equations (Gauge Invariant Form)...... 229

C.3.6 Conservation...... 230

C.3.7 Higher Derivative Relations...... 233

viii Chapter 1

Introduction

In the theory of cosmological fluctuations, the perturbative equations of motion have historically been acknowledged as forming a quite difficult and complex set of coupled partial differential tensor equations. 1 As a result, extensive methods of simplifying the equations of motion and eliminating non-physical gauge modes have been continually developed in order to construct the perturbative solutions, with such methods broadly falling into one of two categories.

The first entails the construction and imposition of a suitable gauge condition that aims to reduce the equations of motion into a simplified form such that a solution is readily obtainable. Here, no further decomposition upon the fluctuations is performed, with solutions being expressed directly in terms of the metric tensor fluctuation and the energy momentum tensor components. While use of such gauges are quite effective at facilitating solutions within Minkowski background geometries, their efficacy begins to diminish when faced with backgrounds

1 For instance, within Weinberg’s cosmology book [1], he notes that even after performing additional simpliﬁcations, solving the cosmological perturbative equations is still “fearsomely complicated”.

1 2 associated with a non-vanishing curvature tensor and hence backgrounds that are most relevant to the study of cosmology. However, we note that there are some interesting exceptions regarding a gauge’s reductive power that can occur if one works within gravitational theories that possess additional symmetries in comparison to standard Einstein gravity; we will in fact explore such an alternative theory, namely conformal gravity, extensively throughout this work.

The second method of simpliﬁcation of the equations of motion consists of decomposing the metric and matter ﬂuctuations into a basis of scalars, vectors, and tensors. First developed by Lifshitz in 1946 [2], the scalar, vector, tensor

(SVT) basis is characterized according to how the ﬂuctuations transform under three-dimensional rotations, and has become the de facto approach to modern treatments of cosmology [1,3,4,5,6,7] (as we later develop an SVT formalism appropriate for higher dimensions, we will refer to the canonical SVT basis as

SVT3 when the three-dimensional nature warrants emphasis). In fact, such an

SVT3 decomposition is deeply intertwined within the calculation of large scale observables in the universe such as the contributions of SVT3 scalar fields which dominate the cosmic microwave background power spectrum [1,8]. One of the primary features afforded by implementing the SVT3 basis, first developed in [3], is that one may readily construct particular combinations of scalars, vectors, and tensors that are invariant under gauge transformations. By collecting the set of six total gauge invariant quantities, one may then express the evolution equations 3

entirely in terms of the gauge invariants, thus eﬀectively identifying and removing

non-physical modes from the prescription.

Within literature [3,5,9], the utilization of the SVT3 basis actually goes well

beyond just a means of decomposing the perturbations and constructing gauge

invariants, with one of its primary eﬃcacies rather coming from the assertion that

within the equations of motion, the scalar, vector, and tensor sectors are to com-

pletely decouple. Such assertion is referred to as the cosmological decomposition

theorem.

As the analysis of the cosmological decomposition theorem forms one of the

primary focal points of this work, we digress and brieﬂy motivate an investiga-

tion into the applicability of the theorem by considering a representative example

of a scalar-vector decomposition of two vectors ﬁelds into their transverse and

longitudinal components via

Ai + ∂iA = Bi + ∂iB. (1.1)

i Here a transverse Ai obeys ∂ Ai = 0 and B and Bi are to represent functions given

i by the evolution equations with Bi also transverse and thus obeying ∂iB = 0.

According to the decomposition theorem, the vectors Ai and Bi are taken to decouple and evolve independent of the scalars, with an analogous statement holding for the scalars. Hence, the theorem implies

Ai = Bi, ∂iA = ∂iB. (1.2) 4

It is clear, however, that (1.2) does not directly follow from (1.1). Rather, if we

i ijk apply ∂ and ∂j to (1.1) we obtain

i ijk ∂ ∂i(A − B) = 0, ∂j(Ak − Bk) = 0. (1.3)

Now from (1.3), one may only conclude that A and B can be deﬁned up to an

i arbitrary scalar C obeying ∂ ∂iC = 0. Similarly, for the transverse vectors, Ak and

ijk Bk can only diﬀer by any function Ck that obeys ∂jCk = 0, i.e. an irrotational

ﬁeld Ck expressed as the gradient of a scalar. Thus, while we have achieved a separation of scalars and vectors in (1.3) by virtue of applying higher derivatives, we conclude that (1.3) can only imply (1.2) if some additional information is provided. In the context of the full cosmological ﬂuctuation equations, we shall investigate the requisite constraints that force an analogous (1.2) to follow from an analogous (1.3).

Returning to the discussion of the two broad methods used in solving the

fluctuation equations, within this work we explore both in detail. In regard first to constructing and imposing appropriate gauge conditions, we find particularly useful implementations in the context of conformal gravity that permit the fluctuations to be expressed in a remarkably straightforward and simple form with an immediately obtainable solution. Within a maximally symmetric background

(e.g. de Sitter), we also evaluate a broad range of gauges as applied to standard

Einstein gravity.

As for the SVT3 basis, the construction of the decomposition itself can be 5 carried out by means of the transverse and longitudinal projector formalism which we have developed in AppendixA. In effect, each SVT3 scalar, vector, or tensor is defined in terms of non-local spatial integrals of the metric perturbation. With the very SVT3 definitions being intrinsically non-local, a number of considerations must be addressed in great detail pertaining to the interplay of spatially asymptotic behavior, the decomposition theorem, and gauge invariance.

To better match the underlying dimensionality of the spacetime geometry, one may also consider an SVTD decomposition into scalars, vector, and tensors according to their transformation behavior in D dimensions. Specifically, we find implementation of an SVT4 formalism provides meaningful simplifications not shared within SVT3, with the fluctuation equations being able to be expressed in more compact and readily solvable forms. However, with an additional dimension, the role of asymptotic behavior within the issues of gauge invariance and the decomposition theorem must be revisited and carefully analyzed separately from the SVT3 considerations.

As previously mentioned, the treatment of cosmological perturbations may proceed diﬀerently in comparison to standard Einstein gravitation if one works in a theory of gravitation that possesses additional symmetries. To this end, we carry out an analogous analysis of cosmological perturbations in the context of a conformally invariant (at the level of the action) theory of gravitation. Conformal gravity has been advanced as a candidate alternative to standard Einstein gravity 6 and reviews of its status at both the classical and quantum levels may be found in [10, 11, 12]. While our treatment of conformal gravity in this text is purely classical, the establishment of unitarity and the positivity of its inner product at the quantum level may be found in [13, 14, 15, 16]. 2 Additional astrophysical and cosmological support for conformal gravity is motivated by the excellent agreement between ﬁts to galactic rotation curves of 138 spiral galaxies presented in [11,

15, 29] and to ﬁts of the accelerating universe Hubble plot data presented in

[10, 12]. With all cosmologically relevant backgrounds of interest being able to be expressed as conformal to flat geometries (cf. AppendixB), we shall find that despite the initially expansive form of the fourth order fluctuation equations, the invariance properties of the Bach tensor under conformal transformations enables remarkable reduction and simplification in both the SVT formalism as well as under the imposition of a specific gauge. Throughout this thesis, all constructions, solutions, and analysis of cosmological perturbations in standard Einstein gravity will be frequently contrasted with the analogous treatment in conformal gravity.

1.1 Overview

In Ch.2, we ﬁrst introduce the formalism within the theory of cosmological perturbations, starting with the gravitational action and ending with the ﬁrst order perturbative equations of motion. A discussion of gauge transformations

2 Additional studies of conformal gravity and of higher derivative gravity theories in general can be found in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. 7 and imposition of gauge conditions in the context of the simplest background geometry (e.g. Minkowski) is performed. In Sec. 2.2, we introduce the theory of conformal gravity, with an important development pertaining to the unique conformal transformation properties contained within Sec. 2.2.1.

Next, the SVT formalism is developed in detail within Ch.3 in both three

(SVT3) and D dimensions (SVTD). Here we discuss correlations between gauge invariance and spatially asymptotic behavior, as well as provide a deeper exploration of the cosmological decomposition theorem. In addition, by referencing the gravitational tensor associated with conformal gravity, we eﬀectively relate the

SVT3 basis to the SVT4 basis.

With the SVT basis developed, in Ch.4 we proceed to apply the decomposition to a variety of cosmologically relevant backgrounds in both the SVT3 and

SVT4 basis. As one cannot generically prove the validity of the decomposition theorem within an arbitrary background, we must assess its applicibilty by solving the equations of motion exactly in each cosmological background separately, and we do so here for de Sitter and Robertson Walker geometries with arbitrary

3-curvature. In each case, a number of important issues are addressed related to gauge invariance, boundary conditions, and constraints needed to recover the decomposition theorem. In addition, we explore the mixing of the separate SVT components within gauge invariants around particular geometries and provide means to remedy such intertwining. 8

Finally, in Ch.5, we explore the implementation of specific gauge conditions with an aim towards simplifying the fluctuations equations to a solvable form. In the conformal theory, we construct a gauge condition that retains its form under conformal transformation, whereby usage of such a gauge leads to a significant simplification of the equations of motion, leading to an equation of motion comprised of just a single term. Here we succinctly solve the fluctuation equations exactly within the large class of conformal to flat backgrounds, which automatically covers all cosmological geometries of interest. Lastly, we form a generalized gauge condition where upon variation of its various coefficients, we effectively reduce the Einstein equations of motion within a de Sitter background to a set of small compact expressions that can then be solved exactly. We note that in Einstein gravity, the components of the metric fluctuation cannot be decoupled entirely, with such an issue not arising in conformal gravity due to its unique trace properties arising from conformal invariance. Chapter 2

Formalism

Before we can enter the discussion of the technical methods used to decompose and simplify the cosmological fluctuation equations, we must first introduce the necessary formalism describing the interaction of gravitation and matter. The general procedure, repeated for both standard and conformal gravity, consists of varying a classical gravitational action (a general coordinate scalar) with respect to the metric, with stationary solutions yielding the equations of motion. The metric is then decomposed into zeroth and first order contributions where we obtain the background and perturbed fluctuation equations, respectively. Serv- ing as a prototypical example of what is to come, we illustrate the form of the

ﬂuctuation equations in their simplest conﬁguration, namely within a source-less

Minkowski background geometry. Following convention [30], we impose a standard gauge condition (e.g, the harmonic or transverse gauge), allowing us to solve the equations of motion exactly.

In the case of conformal gravity, there are particular properties not shared within Einstein gravity [11] that deserve special attention which are also explored

9 10 here. Namely, the additional symmetry contained within conformal gravity per- mits extremely useful transformation properties and directly leads to very compact equations of motion (with one less degree of freedom) if the background metric itself can be shown the exhibit the same underlying symmetry properties.

In addition, for matter actions relevant to conformal gravity (actions necessarily possessing conformal invariance) we explore two non-trivial geometries [31, 32, 33] in which the background energy momentum tensor may be shown to vanish. We contrast the separation of gauge invariance within each sector (i.e. the gravitational and matter sector) with Einstein gravity, as applied to the equations of motion within the presence of non-trivial vacuum geometries.

Finally, we provide an overview of the spacetime geometries studied in cosmology and their underlying motivations. Via coordinate transformations, each of the cosmological geometries of interest can be cast into a conformal to ﬂat form, a detail whose importance cannot be understated and serves a crucial role in the development and solution of the ﬂuctuation equations throughout this work.

2.1 Einstein Gravity

The formulation of the Einstein ﬁeld equations ﬁrst begins by introducing the

Einstein-Hilbert action [30]

1 Z I = − d4x(−g)1/2gµνR . (2.1) EH 16πG µν 11

Variation of this action with respect to gµν yields the Einstein tensor

16πG δIEH µν µν 1 µν α 1/2 = G = R − g R α. (2.2) (−g) δgµν 2

Upon speciﬁcation of a matter action, IM, an energy momentum tensor may like-

wise be constructed by variation with respect to the metric,

2 δIM 1/2 = Tµν. (2.3) (−g) δgµν

Requiring the total gravitational + matter action IEH + IM to be stationary under

variation of gµν then yields the Einstein equations of motion

1 Rµν − gµνRα = −8πGT µν. (2.4) 2 α

The Einstein tensor itself is covariantly conserved via the Bianchi identities,

1 ∇ Rµν = ∇νRµ =⇒ ∇ Gµν = 0. (2.5) µ 2 µ µ

As a first step towards describing fluctuations in the universe, we may decompose the metric gµν(x) into a background metric and a first order perturbation according to

(0) µν gµν(x) = gµν (x) + hµν(x), g hµν ≡ h, (2.6)

whereby Gµν can be decomposed as

(0) Gµν = Gµν(gµν ) + δGµν(hµν). (2.7)

By virtue of the Bianchi identities, the 10 components of the symmetric Gµν can

be reduced to 6 independent components in total. In terms of perturbations of 12

the curvature tensors, the decomposition of Gµν takes the form

1 G(0) = R(0) − g(0)R(0)α (2.8) µν µν 2 µν α 1 1 δG = δR − h R(0)α − g δRα . (2.9) µν µν 2 µν α 2 µν α

Under a coordinate transformation of the form xµ → xµ − µ(x), with µ of O(h), the perturbed metric transforms as

hµν → hµν + ∇µµ + ∇µν. (2.10)

0 For every solution hµν to δGµν +8πGδTµν, a transformed hµν = hµν +∇µµ +∇µν will also serve as a solution - hence the set of four functions µ serve to deﬁne the gauge freedom under coordinate transformation. If the gauge is ﬁxed, the initial

10 components of the symmetric hµν are reduced to 6 individual components.

As will be discussed later, one can also construct gauge invariant quantities as functions of the hµν and express the ﬁeld equations entirely in terms of 6 gauge invariant functions (see Ch.3).

As regards the perturbed gravitational and energy momentum tensors, under xµ → xµ − µ(x) they transform as

(0) λ (0) λ (0) λ δGµν → δGµν + G µ∇νλ + G ν∇µµ + ∇λGµν

(0) λ (0) λ (0) λ δTµν → δTµν + T µ∇νλ + T ν∇µµ + ∇λTµν . (2.11)

(0) If Gµν = 0, then δGµν will be separately gauge invariant. On the other hand, if

(0) Gµν 6= 0, then it is only δGµν + 8πGδTµν that is gauge invariant by virtue of the 13 background equations of motion. (The transformation behavior of tensors under the gauge transformation as given in (2.11), otherwise known as the Lie derivative, is in fact generic to all tensors deﬁned on the manifold. One can check that it readily holds for (2.10)).

With aim towards a description of ﬂuctuations in the universe, let us perturb the metric around an arbitrary background according to

2 µ ν (0) 2 µ ν ds = gµνdx dx = (gµν + hµν + O(h ) + ...)dx dx . (2.12)

(0) The zeroth order Gµν is given as (2.8) and the ﬁrst order ﬂuctuation evaluates to

1 1 αβ 1 α 1 α δGµν = − 2 hµνR + 2 gµνh Rαβ + 2 ∇α∇ hµν − 2 gµν∇α∇ h

1 α 1 α 1 αβ 1 − 2 ∇α∇µhν − 2 ∇α∇νhµ + 2 gµν∇β∇αh + 2 ∇µ∇νh. (2.13)

(Here the covariant derivatives ∇ are deﬁned with respect to the background

(0) gµν and all curvature tensors are taken as zeroth order). For the purpose of illustrating gauge ﬁxing and, later, the SVT decomposition, we evaluate (2.13) within a Minkowski background viz.

2 µ ν (0) ds = (ηµν + hµν)dx dx , ηµν = diag(−1, 1, 1, 1),Gµν = 0

1 α 1 α 1 α 1 α δGµν = 2 ∇α∇ hµν − 2 gµν∇α∇ h − 2 ∇α∇µhν − 2 ∇α∇νhµ

1 αβ 1 + 2 gµν∇β∇αh + 2 ∇µ∇νh. (2.14)

In then taking δTµν = 0, the equations of motion describing the gravitational

ﬂuctuations in an empty universe (vacuum) are given by

1 α 1 α 1 αβ 1 α 0 = 2 ∇α∇ hµν − 2 gµν∇α∇ h + 2 gµν∇β∇αh − 2 ∇µ∇αhν 14

1 α 1 − 2 ∇ν∇αhµ + 2 ∇ν∇µh. (2.15)

2.1.1 Fluctuations Around Flat in the Harmonic Gauge

In order to solve (2.15), we have two general approaches: a). Use the coordinate

freedom in hµν to impose a speciﬁc gauge, typically one in which the equations of

motion are most simpliﬁed b). Determine gauge invariant quantities as functions

of hµν and express the equation of motion entirely in terms of the gauge invariants.

As an example of using method a) to solve (2.15), we may select coordinates

that satisfy the harmonic gauge condition [30]

µ 1 ∇ hµν − 2 ∇νh = 0, (2.16)

whereby (2.15) reduces to

1 α 1 0 = 2 ∇α∇ hµν − 2 gµνh . (2.17)

α The trace deﬁnes a solution for h whereafter hµν can be solved as ∇α∇ hµν = 0.

We will see that method b) is facilitated by the use of the scalar, vector, tensor decomposition as discussed in detail within Ch.3.

2.2 Conformal Gravity

Conformal gravity is a candidate theory of gravitation based on a pure metric action that is not only invariant under local Lorentz transformations (to thus possess the properties of general coordinate invariance and adherenace to the equivalence principle as found in Einstein gravity) but also invariant under local 15

2α(x) conformal transformations (i.e. transformations of the form gµν(x) → e gµν(x)

with α(x) arbitrary). Such a metric action that obeys these symmetries is uniquely prescribed and is given by a polynomial function of the Riemann tensor [10]

Z 4 1/2 λµνκ IW = −αg d x (−g) CλµνκC Z 1 ≡ −2α d4x (−g)1/2 R Rµκ − (Rα )2 , (2.18) g µκ 3 α

where αg is a dimensionless gravitational coupling constant, and

1 C = R − (g R − g R − g R + g R ) λµνκ λµνκ 2 λν µκ λκ µν µν λκ µκ λν 1 + Rα (g g − g g ) (2.19) 6 α λν µκ λκ µν

is the conformal Weyl tensor [34]. Under conformal transformation gµν(x) →

2α(x) λ λ e gµν(x), the Weyl tensor transforms as C µνκ → C µνκ. As the tracless component of the Riemann tensor, Cλµνκ vanishes in geometries that are conformal to ﬂat.

As described in [11], conformal invariance requires that there be no intrinsic mass scales at the level of the Lagrangian; rather, mass scales must come from the vacuum via spontaneous symmetry breaking. In such a process, particles may localize and bind into inhomogeneities comprising astrophysical objects of interest

(e.g. stars and galaxies). With inhomogeneities violating the conformal symmetry in the spacetime geometry, the transition from a cosmological background geometry to the cosmological fluctuations associated with inhomogeneities corresponds to a shift from conformal to flat geometries to non-conformal flat geometries. 16

However, when decomposed into a background and ﬁrst order contribution, the

underlying conformal symmetry contained within the background allows one to

tame the complexity of the ﬂuctuations due to the transformation properties of

the Weyl tensor.

Variation of the Weyl action (2.18) with respect to the metric gµν(x) yields the fourth-order derivative gravitational equations of motion [10][35]

2 δIW µν µλνκ µλνκ − 1/2 = 4αgW = 4αg 2∇κ∇λC − RκλC (−g) δgµν 1 = 4α W µν − W µν = T µν, (2.20) g (2) 3 (1)

µν µν where tensors W(1) and W(2) are given by

1 W µν = 2gµν∇ ∇βRα − 2∇ν∇µRα − 2Rα Rµν + gµν(Rα )2, (1) β α α α 2 α 1 W µν = gµν∇ ∇βRα + ∇ ∇βRµν − ∇ ∇νRµβ − ∇ ∇µRνβ (2) 2 β α β β β 1 − 2RµβRν + gµνR Rαβ. (2.21) β 2 αβ

Here T µν is the conformal invariant matter source energy-momentum tensor. With

µν IW being both general coordinate invariant and conformal invariant, W is auto-

µν µν matically covariantly conserved and traceless and obeys ∇νW = 0, gµνW = 0

(i.e. without needing to impose any equation of motion or stationarity, thus hold-

ing on every variational path).

Upon ﬁrst glance, the fourth order (2.21) takes a considerably more complex

form in comparison to the relatively terse second order Einstein equations

1 1 − Rµν − gµνRα = T µν. (2.22) 8πG 2 α 17

However, in solving the vacuum equations associated with (2.21), two types of solutions arise: those associated with a vanishing Weyl tensor and those associated with a vanishing Ricci tensor. As a consequence of the former, since all cosmological relevant geometries of interest can be expressed in the conformal to

ﬂat form,

2 2 µ ν 2 2 2 2 2 ds = −Ω (t, x, y, z)ηµνx x = Ω (t, x, y, z)[dt − dx − dy − dz ], (2.23) for appropriate choices of the conformal factor Ω(t, x, y, z) they serve as vacuum solutions. Regarding the latter, solutions with vanishing Ricci tensor necessarily encompass all vacuum solutions to Einstein gravity. In this sense, the set of solutions to the vacuum equations in conformal gravity forms a superset of all such vacuum equations in Einstein gravity. As a particular example, both gravitational theories admit the Schwarzschild solution exterior to a static, spherically symmetric source [31], with the Schwarzschild solution geometry not expressible in conformal to ﬂat form.

2.2.1 Conformal Invariance

With the Weyl action (2.18) being locally conformal invariant, W µν(x) possesses the transformation property that upon a conformal transformation of the form

2 µν −2 µν µν gµν(x) → Ω (x)gµν(x) =g ¯µν(x), g (x) → Ω (x)g (x) =g ¯ (x), (2.24)

µν W (x) and Wµν(x) transform as

W µν(x) → Ω−6(x)W µν(x) = W¯ µν(x), 18

−2 ¯ Wµν(x) → Ω (x)Wµν(x) = Wµν(x), (2.25)

¯ where the functional dependence of Wµν(x) ong ¯µν(x) is equivalent to that of

Wµν(x) on gµν(x). To be noted is that (2.25) holds regardless of whether or not the metric gµν(x) is conformal to ﬂat. If we further decompose gµν(x) andg ¯µν(x) into a background metric and a ﬁrst order perturbation according to

2 (0) µ ν (0) ds = −[gµν + hµν]dx dx , gµν(x) = gµν (x) + hµν(x),

µν µν µν (0) ¯ g (x) = g(0)(x) − h (x), g¯µν(x) =g ¯µν (x) + hµν(x),

µν µν ¯µν g¯ (x) =g ¯(0)(x) − h (x), (2.26)

¯ then Wµν(x) and Wµν(x) will decompose as

(0) (0) Wµν(gµν) = Wµν (gµν ) + δWµν(hµν),

¯ ¯ (0) (0) ¯ ¯ Wµν(¯gµν) = Wµν (¯gµν ) + δWµν(hµν). (2.27)

To clarify, within (2.27) Wµν(hµν) is evaluated in a background geometry with

(0) ¯ ¯ metric gµν (x), whereas Wµν(hµν) is evaluated in a background geometry with

(0) metricg ¯µν (x).

Since the gravitational sector Wµν is conformal invariant, the matter sector

−2 ¯ Tµν must necessarily also transform as Ω (x)Tµν(x) = Tµν(x). Repeating a decomposition into background and ﬁrst order components, we obtain for the energy momentum tensor

(0) (0) ¯ ¯(0) (0) ¯ ¯ Tµν(gµν) = Tµν (gµν ) + δTµν(hµν), Tµν(¯gµν) = Tµν (¯gµν ) + δTµν(hµν). (2.28) 19

The utility of the conformal transformation properties described allow us to ﬁnd solutions around conformally transformed geometries using only knowledge of the form of the solution around the original geometry. Speciﬁcally, if we know how to

(0) (0) solve for ﬂuctuations hµν(x) around a background gµν (x), (that is, if gµν (x) is such a geometry that solutions to δWµν(hµν) = δTµν(hµν)/4αg may be found apriori)

¯ ¯ ¯ ¯ then we can ﬁnd obtain solutions to δWµν(hµν) = δTµν(hµν)/4αg for ﬂuctuations

¯ (0) hµν(x) around a background metricg ¯µν (x) by deﬁning

¯ 2 ¯ ¯ −2 hµν(x) = Ω (x)hµν(x), δWµν(hµν) = Ω (x)δWµν(hµν). (2.29)

Implementing conformal gravity solutions found in past work (e.g. [10]), one can use the determined structure of the fluctuations around a flat background to construct the fluctuations around any background that is conformal to flat by virtue of (2.29). As mentioned, since all cosmologically relevant background geometries can be cast into the conformal to flat form, the conformal transformation properties give rise to an extremely convenient and powerful methodology to solving that fluctuation equations, despite the fourth-order nature and expansive form of the gravitational equations of motion.

We can continue to use conformal invariance (i.e. under a conformal trans-

2 formation of general metric gµν → Ω (x)gµν the Bach tensor Wµν transforms

−2 as Wµν → Ω (x)Wµν) to determine the trace depedendent properties of Wµν.

Taking h as a ﬁrst order perturbation in the metric and using the conformal 20

transformation properties, we ﬁnd up to ﬁrst order

h h h W g(0) + g(0) = W 1 + g(0) = W (0)(g(0)) + δW g(0) µν µν 4 µν µν 4 µν µν µν µν 4 µν h = 1 − W (g(0)), 4 µν µν

and hence

h h − W (g(0)) = δW g(0) , (2.30) 4 µν µν µν 4 µν

or, in its full form

h (0) 1 1 2 1 αβ 2 αβ δWµν( 4 gµν ) = − 4 h(− 6 gµνR + 2 gµνRαβR + 3 RRµν − 2R Rµανβ

1 α α α − 6 gµν∇α∇ R + ∇α∇ Rµν − ∇µ∇ Rνα

α 2 − ∇ν∇ Rµα + 3 ∇ν∇µR)

h (0) = − 4 Wµν(gµν ). (2.31)

To take make full use of the dependence of δWµν on h we introduce the quantity

Kµν(x) deﬁned as

1 K (x) = h (x) − g(0)(x)gαβh , (2.32) µν µν 4 µν (0) αβ

µν with Kµν being traceless with respect to the background metric g(0). Substituting

(2.32) into δWµν(hµν) we obtain

h h δW (h ) = δW K + g(0) = δW (K ) + δW g(0) . (2.33) µν µν µν µν 4 µν µν µν µν 4 µν

If we work in a background that is conformal to ﬂat, then (2.30) will vanish which

implies from (2.33) that

δWµν(hµν) = δWµν(Kµν). (2.34) 21

Hence in a conformal to ﬂat geometry, the trace of hµν not only decouples but also vanishes, with the ﬂuctuation equations being able to be entirely expressed in terms of the nine component Kµν.

We may also ﬁnd a relationship between hµν and the trace of the ﬂuctuation

δWµν itself. First, we note that the tracelessness of Wµν implies

µν (0)µν µν (0) g Wµν(gµν) = g − h Wµν + δWµν = 0. (2.35)

To ﬁrst order this equates to,

µν (0) (0)µν −h Wµν + g δWµν = 0 (2.36)

and thus we obtain

(0)µν µν (0) g δWµν(hµν) = h Wµν(gµν ). (2.37)

For reference, we state the full form of the above as

(0)µν µν 1 2 1 αβ 2 αβ g δWµν = h (− 6 gµνR + 2 gµνRαβR + 3 RRµν − 2R Rµανβ

1 α α α α − 6 gµν∇α∇ R + ∇α∇ Rµν − ∇µ∇ Rνα − ∇ν∇ Rµα

2 + 3 ∇ν∇µR)

µν (0) = h Wµν(gµν ) (2.38)

Consequently, in a conformal to ﬂat background, the trace of the ﬂuctuation tensor

itself will vanish. 22

Owing to this decoupling of the trace, for conformal to ﬂat backgrounds one may substitute the usage of (2.27) instead by the usage of

(0) (0) Wµν(gµν) = Wµν (gµν ) + δWµν(Kµν),

¯ ¯ (0) (0) ¯ ¯ Wµν(¯gµν) = Wµν (¯gµν ) + δWµν(Kµν), (2.39)

where

(0) 2 (0) g¯µν (x) = Ω (x)gµν (x), (2.40)

¯ 2 Kµν(x) = Ω (x)Kµν(x). (2.41)

Consequently, in the context of conformal gravity, when constructing ﬂuctuations

(0) (0) in ag ¯µν background from the ﬂuctuations in a gµν background that is conformal

to ﬂat, we here on utilize (2.41) rather than (2.29).

Summarizing the conformal properties of conformal gravity, we have shown

that for ﬂuctuations around a conformal to ﬂat background, we can reduce the

equations to a dependence on the traceless Kµν without needing to make any

reference to the actual detailed form of the ﬂuctuation equations at all. Given that

one also possesses the freedom to make four general coordinate transformations,

one can further reduce the nine-component Kµν to ﬁve independent components,

all without needing to make any reference to the ﬂuctuation equations. Any

further reduction in the number of independent components of Kµν can only

be achieved through use of residual gauge invariances or the structure of the 23

ﬂuctuation equations themselves. Within Ch.5 we make use of the coordinate

freedom and implement a particular gauge condition (motivated by its conformal

transformation properties) that yields ﬂuctuation equations in which there is no

mixing of any of the components of Kµν with each other.

2.2.2 Fluctuations Around Flat in the Transverse Gauge

To illustrate the use of gauge conditions within conformal gravity, we investiage

ﬂuctuations around a Minkowski background. In such a background it was found in [10], that δWµν takes the form, prior to imposing any gauge conditions

1 δW = (ηρ ∂α∂ − ∂ρ∂ )(ησ ∂β∂ − ∂σ∂ )K µν 2 µ α µ ν β ν ρσ 1 − (η ∂γ∂ − ∂ ∂ )(ηρσ∂δ∂ − ∂ρ∂σ)K . (2.42) 6 µν γ µ ν δ ρσ

µν µν µν Within a ﬂat background, the Lie derivative of K leads to ∂νK → ∂νK −

ν µ µ ν µν µν µ ν ∂ν∂ − ∂ ∂ν /2 and ∂µ∂νK → ∂µ∂νK − 3∂µ∂ ∂ν /2. Hence, in order to

µν µ µ ν construct a gauge condition ∂νK = f for arbitrary f , we can solve for ∂ν

µ µν µ µ and then for in order to set ∂νK = f . If we elect to select an f such

µν that ∂µK = 0 (i.e. the transverse gauge condition), then (2.42) reduces to the remarkably compact and simple form

1 δW = ησρηαβ∂ ∂ ∂ ∂ K . (2.43) µν 2 σ ρ α β µν

Note that the tensor components of Kµν that were coupled in (2.42) are now completely decoupled in (2.43). (This may be constrasted with conformal to ﬂat

ﬂuctuations in Einstein gravity where, as we as far as we aware, there is no gauge 24

in which such a complete decoupling occurs. In Sec. 5.2 we present a selection

of gauges that yield maximal simpliﬁcation and decoupling, with it being evident

that a complete decoupling is prevented only by the presence of the trace h of the

metric ﬂuctuation).

To solve 4αgδWµν = δTµν with its associated (2.43), we deﬁne the fourth- order derivative Green’s function which obeys

α β 0 4 0 ∂α∂ ∂β∂ D(x − x ) = δ (x − x ), (2.44)

in which the solution (in the transverse gauge) is given by

Z 1 4 0 0 0 Kµν(x) = d x D(x − x )δTµν(x ). (2.45) 2αg

The retarded Green’s function solution to (2.44)[36] is given by

1 D(FO)(x − x0) = θ(t − t0 − |x − x0|), (2.46) 8π

with θ(t − t0 − |x − x0|) vanishing outside the light cone.

α β The solutions to the fourth order wave equation ∂α∂ ∂β∂ Kµν = 0 may be solved in terms of momentum eiginstates, given by [11, 22]

ik·x ik·x ∗ −ik·x ∗ −ik·x Kµν = Aµνe + (n · x)Bµνe + Aµνe + (n · x)Bµνe , (2.47)

2 2 µ where k0 = k , where Aµν and Bµν are polarization tensors, and where n =

(1, 0, 0, 0) is a unit timelike vector. With n·x = t, we see that ﬂuctuations around

a ﬂat background grow linearly in time. In total, given δTµν,(2.45) can be solved 25

completely, and for a localized δTµν the asymptotic solution for Kµν is given by

(2.47). (In Sec. 5.3, we analogously construct the eigenstate solutions to the wave

equation within a curved Robertson Walker radiation era (k = −1) background

to ﬁnd solutions with leading time behavior of t4.)

2.2.3 On the Energy Momentum Tensor

The matter ﬁeld T µν in conformal gravity is behaves in a diﬀerent nature in

comparison to standard Einstein gravity. The root of the diﬀerence of the en-

ergy momentum tensor between the two theories stems from the statement that

µν µν µν 4αgW = T must be conformally invariant, from which it follows that T

must transform in the same manner under conformal transformations as W µν.

Consequently, in conformal to ﬂat cosmological backgrounds where W µν vanishes,

T µν must also vanish. However, in the literature two ways in which it could vanish

non-trivially have been identiﬁed, one involving a conformally coupled scalar ﬁeld

[33], and the other involving a conformal perfect ﬂuid [37].

In the case of a conformally coupled scalar ﬁeld S(x) we deﬁne the matter

action as

Z 1 1 I = − d4x(−g)1/2 ∇ S∇µS − S2Rµ + λ S4 (2.48) S 2 µ 12 µ S

where λS is a dimensionless coupling constant. As written, the IS action is the most general curved space matter action for the S(x) ﬁeld that is invariant under both general coordinate transformations and local conformal transformations of 26

−α(x) 2α(x) the form S(x) → e S(x), gµν(x) → e gµν(x)[33]. Variation of IS with respect to S(x) yields the scalar ﬁeld equation of motion

1 ∇ ∇µS + SRµ − 4λ S3 = 0, (2.49) µ 6 µ S

while variation with respect to the metric yields the matter ﬁeld energy-momentum

tensor

2 1 1 T µν = ∇µ∇νS − gµν∇ S∇αS − S∇µ∇νS S 3 6 α 3 1 1 1 + gµνS∇ ∇αS − S2 Rµν − gµνRα − gµνλ S4. (2.50) 3 α 6 2 α S

By using (2.49) within (2.50), the energy-momentum tensor obeys the expected traceless condition

µν gµνTS = 0. (2.51)

If we take the scalar ﬁeld as the spontaneously broken non-zero constant expec-

tation value S0, the scalar ﬁeld wave equation and the energy-momentum tensor

reduce to the form

α 2 R α = 24λSS0 , 1 1 T µν = − S2 Rµν − gµνRα − gµνλ S4 S 6 0 2 α S 0 1 1 = − S2 Rµν − gµνRα . (2.52) 6 0 4 α

Now, in a de Sitter (dS4) geometry deﬁned by

Rλµσν = K[gµσgλν − gµνgλσ],Rµν = −3Kgµν 27

α µν µν α R α = −12K,R = (1/4)g R α, (2.53)

µν µν since W vanishes identically, TS will also vanish identically in the same ge-

2 ometry. And with curvature constant K being taken as K = −2λSS0 we ﬁnd

that though W µν and T µν both vanish identically, as noted in [33], the conformal

µν µν cosmology governed by 4αgW = T admits of a non-trivial de Sitter geometry

2 solution, with a non-vanishing four-curvature K = −2λSS0 .

To discuss another avenue in which T µν can vanish non-trivially [37], we set

λS = 0 within IS (an operation which still preserves the conformal invariance of

IS), and we evaluate the scalar ﬁeld wave equation in a generic Robertson-Walker

geometry deﬁned as

dr2 ds2 = dt2 − a2(t) + r2dθ2 + r2 sin2 θdφ2 = dt2 − a2(t)γ dxidxj.(2.54) 1 − kr2 ij

As discussed in [37], solutions to the scalar ﬁeld wave equation (2.49) within the

Roberston Walker geometry obey

1 d2f 1 + kf(p) = γ−1/2∂ [γ1/2γij∂ g(r, θ, φ)] = −λ2, (2.55) f(p) dp2 g(r, θ, φ) i j where p = R dt/a(t), S = f(p)g(r, θ, φ)/a(t), γij is the metric of the spatial part of the Robertson-Walker metric, and λ2 is a separation constant. Inspection of

(2.55) reveals that f(p) obeys a harmonic oscillator equation with characteristic frequencies ω2 = λ2 + k. In addition, we may look for separable solutions in g(r, θ, φ) viz.

` m g(r, θ, φ) = gλ(r)Y` (θ, φ), (2.56) 28

` with gλ(r) necessarily obeying

∂2 (2 − 3kr2) ∂ `(` + 1) (1 − kr2) + − + λ2 g` (r) = 0. (2.57) ∂r2 r ∂r r2 λ

From here, we proceed with an interesting step and perform an incoherent averaging over all allowed spatial modes associated with a given ω. Upon calculating the sum over all modes, for each ω we obtain [37]

ω2(gµν + 4U µU ν) (λ2 + k2)(gµν + 4U µU ν) T µν = = , (2.58) S 6π2a4(t) 6π2a4(t) where U µ is a unit timelike vector. With (2.58) being traceless, the incoherent averaging over the spatial modes has yielded an energy momentum tensor of the perfect ﬂuid form, namely

µν µ ν µν TS = (ρ + p)U U + pg , ρ = 3p, (2.59)

2 µν for appropriate values of ρ and p. Inspecting (2.58), we see that if ω = 0, TS = 0

2 2 µν and if ω = λ +k, we can satisfy TS = 0 non-trivially if and only if k is negative.

00 Thus, we proceed with k negative. In performing an incoherent averaging for TS

(recalling that we are taking ω = 0 here), we obtain [37]

" 3 # 1 X X T 00 = γii|∂ (g` Y m(θ, φ))|2 + k|g` Y m(θ, φ)|2 . (2.60) S 6 i (−k)1/2 ` (−k)1/2 ` `,m i=1

It has been shown in [37] that the sum in (2.60) in fact vanishes identically.

µν With scalar ﬁeld modes providing a positive contribution to TS , the negative contributions of the gravitational ﬁeld from its negative spatial curvature serve 29

00 to cancel the scalar modes identically, resulting in a vanishing TS . As regards the solutions to (2.57), with negative k these are determined to be associated

µν Legendre functions. Despite TS vanishing non-trivially, (2.57) still contains an inﬁnite number of solutions, each labelled with a diﬀerent ` and m. Hence, we

µν shown that TS admits of a non-trival vacuum solution that can be obtained by taking an incoherent average over the spatial modes associated with the solutions of the scalar ﬁeld.

While the choice of negative k may warrant concern in the standard treatment of gravitation and cosmology, where the universe geometry is phenomenologically taken as k = 0, in conformal gravity it poses no such restriction as evidenced in past work [29, 31, 32, 38, 39, 40]. In applications of conformal gravity to astrophysical and cosmological data it has been found that phenomenologically k should be negative. Specifically, in previous works within conformal cosmology negative k fits to the accelerating universe Hubble plot data have been presented in [10, 12], along with negative k conformal gravity fits to galactic rotation curves presented in [10, 12].

A last aspect worth mentioning in regards to the diﬀerence between the matter source in conformal and Einstein gravity concerns the interplay of gauge invariance. While a background T µν may vanish, this does not necessarily imply that its perturbation will also vanish (with its vanishing dependent upon the vanishing of δW µν, which in all cosmological applications is necessarily non-zero). 30

Now, in standard Einstein gravity with a non-zero background T µν, neither the

ﬂuctuation in the background Einstein tensor or the ﬂuctuation in the background

T µν will separately be gauge invariant. It is only the perturbation of the entire

µν µν α µν R − g R α/2 + 8πGT that is gauge invariant. Namely, one must impose the background equations of motion to the ﬂuctuation equations to ensure gauge

µν invariance. Moreover, there are no nontrivial background solutions to G(0) = 0

- all solutions demand a vanishing curvature tensor. However, within conformal gravity, any background that is conformal to ﬂat will cause the background ﬂuc-

µν tuations to vanish and we have identiﬁed two scenarios in which the TS itself vanishes non-trivially. Consequently, the background equations of motion serve

µν µν µν no role in enforcing gauge invariance within 4αgδW = δT , and thus δW and δT µν are separately gauge invariant. Specifically, we shall find that in any background that is conformal to flat, δW µν can be expressed entirely in terms of the gauge invariant components of the metric. Through the following chapters, we will illustrate the role of gauge invariance explicitly in both standard and conformal gravity using a Scalar, Vector, Tensor formulation as well as through the imposition of gauge conditions.

2.3 Cosmological Geometries

The cosmological principle asserts that on a large enough scale, the structure of spacetime is homogeneous and isotropic. Allowing for expansion or contraction of the universe over time, the generic metric that satisﬁes these criteria is the Fried- 31 mann–Lemaˆıtre–Robertson–Walker (FLRW, commonly cited as RW) [9] metric

dr2 ds2 = −dt2 + a(t)2 + r2dθ2 + r2 sin2 θdθ2 . (2.61) 1 − kr2

Here the scale factor a(t) describes the expansion of space in the universe and k ∈

{−1, 0, 1} describes the global geometry of the universe, being spatially hyperbolic,

ﬂat, or spherical respectively.

In a universe dominated by a cosmological constant (as relevant to inﬂation), one may solve the Einstein equations Gµν = −8πGΛgµν to determine the requisite metric. For Λ > 0, the solution is the deSitter geometry (and Λ < 0 the anti deSitter geometry), which describes a maximally symmetric space with curvature tensors of the form

Rλµνκ = Λ(gλνgνκ − gλκgµν),Rµκ = −3Λgµκ,R = −12Λ. (2.62)

With deSitter space possessing higher symmetry than the most general FLRW space (i.e. more Killing vectors), it is in fact a special case of Roberston Walker as can be seen in the choice of coordinates

ds2 = −dt2 + e2Λt(dr2 + r2dθ2 + r2 sin2 θdφ2), (2.63) which corresponds to k = 0, a(t) = e2Λt within (2.61) and further discussed within

Appendix B.4 and Appendix B.5.

A remarkable aspect about the Roberston Walker geometry (and dS4 or

AdS4 by extension) is that in each global geometry (hyperbolic, ﬂat, spherical) 32

the space can be written in conformal to ﬂat form. As a simple example, if we

take the k = 0 (ﬂat 3-space) metric of (2.61) according to

ds2 = −dt2 + a(t)2 dr2 + r2dθ2 + r2 sin2 θdθ2 , (2.64)

R dt then in transforming the time coordinate via τ = a(t) , the geometry may be written in the form

ds2 = a(τ 2)(−dτ 2 + dr2 + r2dθ2 + r2 sin2 θdφ2). (2.65)

If we are interested in the k = −1/L2 hyperbolic space, a proper choice of coor-

dinates allows the Roberston Walker to be expressed as

4L2a2(p0, r0) ds2 = −dp02 + dr02 + r02dθ2 + r02 sin2 θdφ2 (2.66) [1 − (p0 + r0)2][1 − (p0 − r0)2]

whereas for the k = 1/L2 spherical 3-space (2.61) takes the form

4L2a2(p0, r0) ds2 = −dp02 + dr02 + r02dθ2 + r02 sin2 θdφ2 .(2.67) [1 + (p0 + r0)2][1 + (p0 − r0)2]

The coordinate transformations necessary to bring the co-moving Roberston Walker

forms into the conformal to ﬂat basis are given in detail within AppendixB, in-

cluding the conformal factors relevant to the radation era.

As mentioned at the end of Sec. 2.2.3, since all the cosmological geometries

of interest possess a coordinate expression where the space is conformal to ﬂat,

(0) within such geometries the background Bach tensor vanishes Wµν = 0 to thus

render δWµν to independently be a gauge invariant tensor, i.e. without reference

to the equation of motion Chapter 3

Scalar, Vector, Tensor (SVT) Decomposition

In the ﬁeld of perturbative cosmology, it is standard to ﬁrst introduce a 3+1

decomposition of the metric perturbation followed by a decomposition into scalars,

vectors, and tensors according to the underlying background 3-space. A la the

SVT decomposition [3,4,5,6], referred to as SVT3 with this work.

In this chapter, we ﬁrst develop the SVT3 formalism by separately form-

ing relations between SVT3 components and integrals over hµν, as well as their

inversions (i.e. higher derivative relations between hµν and SVT3 quantities.) In

utilizing the SVT3 decomposition in conformal gravity, we also form the analogous

relations between the traceless Kµν and the requisite SVT3 components.

We then investigate the behavior of the SVT3 quantities under gauge transformations, and along with reference to the ﬂat space gauge invariant Einstein tensor δGµν, we construct the set of gauge invariant quantities. (In fact, such a gauge invariant construction lies behind the core utility of implementing the

SVT3 decomposition in the ﬁrst place). In forming the gauge invariants, we highlight their dependence upon underlying assumptions of being asymptotically

33 34 well-behaved and we investigate the consequence of gauge invariance absent of any deﬁned asymptotic behavior.

To provide a manifestly covariant decomposition, we introduce the general

SVTD decomposition for scalars, vectors, and tensors within arbitrary dimension

D [41]. Analgous to the SVT3 discussion, we provide integral and derivative relations between SVT4 components and hµν and ﬁnd that the cosmological ﬂuc- tuation equations take a considerably more compact and simple form in the new

SVT4 formalism. Being simpler, these equations are thus more straightforward to solve. The gauge invariants are then constructed and the role of asymptotic behavior within the establishment of the SVT4 decomposition itself is analyzed.

With help from the structure of the conformal gravity ﬂuctuation δWµν, we are able to determine the relationship between the SVT3 and SVT4 components, with particular emphasis on the gauge invariant quantities.

Finally, we introduce and analyze a core theorem within this work called the cosmological decomposition theorem. This theorem asserts that SVT3 scalars, vectors, and tensors decouple within the ﬂuctuation equations themselves, to thus evolve independently. We carefully inspect the dependence of the decomposition upon underlying constraints related to boundary conditions, determining that the theorem requires ﬂuctuations to be well behaved asymptotically for such an equation decoupling to occur. The analysis is repeated with respect to the SVT4 decomposition, where the validity of the decomposition does not hold generally 35

but rather must be determined on a case by case basis according the speciﬁc

background geometry.

3.1 SVT3

The discussion of the three dimensional SVT expansion begins by taking a ﬂat

2 2 i j background geometry of the form ds = dt − δijdx dx where δij represents

a generic ﬂat 3-space metric (equating to the Kronecker delta for a Minkowski

background). Upon introducing a metric ﬂuctuation hµν and performing a 3+1

decomposition, the geometry may be written as 1

2 µ ν ds = (−ηµν − hµν)dx dx

2 ˜ i ˜ ˜ = (1 + 2φ)dt − 2(∇iB + Bi)dtdx − [(1 − 2ψ)δij + 2∇i∇jE

˜ ˜ i j +∇iEj + ∇jEi + 2Eij]dx dx , (3.2)

˜ i ˜ i ij ˜ where ∇i = ∂/∂x and ∇ = δ ∇j (with Latin indices) are deﬁned with respect to

the background three-space metric δij. In addition, the SVT3 components within

1 In application to cosmological backgrounds, we will find it convenient to decompose the fluctuation around a conformal to flat background by incorporating an explicit factor of Ω2(x), with the perturbed geometry taking the form 2 2 2 i ds = Ω (x) (1 + 2φ)dt − 2(∇˜ iB + Bi)dtdx − [(1 − 2ψ)δij + 2∇˜ i∇˜ jE i j +∇˜ iEj + ∇˜ jEi + 2Eij]dx dx . (3.1)

i Here Ω(x) is an arbitrary function of the coordinates, where ∇˜ i = ∂/∂x (with Latin index) i ij −2 ij and ∇˜ = δ ∇˜ j (not Ω δ ∇˜ j) are deﬁned with respect to the background 3-space metric δij. SVT3 elements obey the same relations as in (3.3), i.e. transverse and traceless with respect to the background 3-space metric. 36

(3.2) are required to obey

ij ˜ ij ˜ jk ˜ ij δ ∇jBi = 0, δ ∇jEi = 0,Eij = Eji, δ ∇kEij = 0, δ Eij = 0. (3.3)

As written, (3.2) contains ten elements, whose transformations are deﬁned with

respect to the background spatial sector as four 3-dimensional scalars (φ, B, ψ, E),

two transverse 3-dimensional vectors (Bi, Ei) each with two independent degrees of freedom, and one symmetric 3-dimensional transverse-traceless tensor (Eij) with

two degrees of freedom. A la, the scalar, vector, tensor (SVT) decomposition.

To implement the decomposition of hµν to the SVT3 form in (3.2), we utilize transverse and transverse-traceless projection operators as applied to tensor and vector components to yield a decomposition into scalars, vectors, and tensors. Both the 3+1 decomposition and projection operators have been derived in developed in detail within AppendixA.

3.1.1 SVT3 in Terms of hµν in a Conformal Flat Background

Following [41, 42] and making use of the projection operators in AppendixA, we

express the ten degrees of freedom of the SVT3 components in a conformal to

ﬂat background in terms of the original ﬂuctuations hµν. First we introduce the

3-dimensional Green’s function obeying

ij ˜ ˜ (3) 3 δ ∇i∇jD (x − y) = δ (x − y). (3.4)

2 Upon setting hµν = Ω (x)fµν, the line element of (3.1) takes the form

2 2 α β ds = −[Ω (x)ηαβ + hαβ]dx dx 37

2 α β = −Ω (x)[ηαβ + fαβ]dx dx

2 2 i j 2 i i j = Ω (x) dt − δijdx dx − f00dt − 2f0idtdx − fijdx dx ,

ij ˜ ˜ i ˜ j ˜ ˜ ˜ ˜ k ˜ ˜ k δ fij = −6ψ + 2∇i∇ E, ∇ fij = −2∇iψ + 2∇i∇k∇ E + ∇k∇ Ei,

˜ i ˜ j ˜ ˜ k ˜ ˜ k ˜ ˜ ` ∇ ∇ fij = −2∇k∇ ψ + 2∇k∇ ∇`∇ E 4 1 = ∇˜ ∇˜ k∇˜ ∇˜ `E + ∇˜ ∇˜ kδijf 3 k ` 3 k ij ˜ ˜ k ˜ ˜ k ij = 4∇k∇ ψ + ∇k∇ (δ fij), Z 3 (3) ˜ i ˜ 2φ = −f00,B = d yD (x − y)∇yf0i,Bi = f0i − ∇iB, 1 Z 1 ψ = d3yD(3)(x − y)∇˜ k∇˜ ` f − δk`f , 4 y y k` 4 k` Z 3 Z 1 E = d3yD(3)(x − y) d3zD(3)(y − z)∇˜ k∇˜ ` f − δk`f , 4 z z k` 4 k` Z Z 3 (3) ˜ j ˜ y 3 (3) ˜ k ˜ ` Ei = d yD (x − y) ∇yfij − ∇i d zD (y − z)∇z ∇zfk` ,

˜ ˜ ˜ ˜ 2Eij = fij + 2ψδij − 2∇i∇jE − ∇iEj − ∇jEi. (3.5)

One may readily check that Bi, Ei, and Eij are indeed transverse by applying

appropriate derivatives, thus conﬁrming their obeying (3.3). 2 The integral form

of the inversions of the SVT3 components is unique up to integration by parts,

which plays a role in the analysis of asymptotic behavior, discussed in detail within

Sec. 3.1.3.

We)where here and throughout we use the notation given in [30]

2 ˜ i In (3.5) a symbol such as ∇y, y indicates that the derivative is taken with respect to the y coordinate and likewise for other latin coordinates. 38

3.1.2 SVT3 in Terms of the Traceless kµν in a Conformal Flat Background

We have shown in Sec. 2.2 that in conformal to ﬂat backgrounds, the perturbed

Bach tensor δWµν may be expressed entirely in terms of the traceless Kµν. As such, it will prove useful to be able to express the SVT components in terms of

2 the traceless part of fµν. Deﬁning Kµν = Ω kµν, we have

2 −2 αβ αβ Kµν = hµν − (1/4)Ω ηµνΩ η hαβ = hµν − (1/4)ηµνη hαβ, (3.6)

whereby we factor out the conformal factor to form the traceless kµν as

ij kµν = fµν − (1/4)ηµν[−f00 + δ fij]. (3.7)

Substituting fµν in terms of this kµν, we obtain from (3.5) the following integral relations for the SVT components:

3 1 1 1 k = f + δk`f , k = f , k = f + δ f − δ δk`f , 00 4 00 4 k` 0i 0i ij ij 4 ij 00 4 ij k` 1 Z φ = − f ,B = d3yD(3)(x − y)∇˜ i k ,B = k − ∇˜ B, 2 00 y 0i i 0i i 1 Z 3 1 ψ = d3yD(3)(x − y)∇˜ k∇˜ ` k − k + f , 4 y y k` 4 00 2 00 Z 3 Z 1 E = d3yD(3)(x − y) d3zD(3)(y − z)∇˜ k∇˜ ` k − k , 4 z z k` 4 00 Z Z 3 (3) ˜ j ˜ y 3 (3) ˜ k ˜ ` Ei = d yD (x − y) ∇ykij − ∇i d zD (y − z)∇z ∇zkk` , 1 2E + 2∇˜ ∇˜ E + ∇˜ E + ∇˜ E = k − δ k ij i j i j j i ij 2 ij 00 1 Z + δ d3yD(3)(x − y)∇˜ k∇˜ ` k . (3.8) 2 ij y y k`

Here can see that all SVT3 components can be expressed in terms of kµν along

−2 with a single component of fµν = Ω (x)hµν, namely f00. Recalling that δWµν can 39

only depend on kµν, we note that the combination φ+ψ is independent of f00 and

only depends on kµν. Hence, we expect this coupled combination to be associated

with the scalar SVT component of conformal gravity. Indeed, we conﬁrm such a

relation later in Sec. 4.1.5.

3.1.3 Gauge Structure and Asymptotic Behavior

As given in (3.2) and its integral form in (3.5), we have shown the form of the

SVT3 decomposition of hµν comprising 10 independent components of scalars,

vectors, and tensors. Due to the coordinate freedom, it must hold that linear

combinations of the SVT quantities form precisely six gauge invariant quantities

(a reduction from ten initial degrees of freedom minus four coordinate transfor-

mations). Consequently, we seek to determine the coeﬃcient combinations of the

SVT quantities that form the gauge invariants. In general, this may be accom-

plished by manipulating the relations between the SVT components and the com-

ponents of hµν in a general background. This procedure is carried out in (3.12) in a ﬂat background and in (4.118) within a general Roberston Walker background.

Before discussing these results, it is informative to ﬁrst analyze the structure of the gauge invariants within Einstein gravity in a source-free ﬂat background. With the background Tµν = 0 vanishing, the perturbed Einstein tensor δGµν itself is a

completely gauge invariant tensor. As a function only of the metric, inspection of

the components of the Einstein tensor will thus reveal the appropriate ﬂat space 40

gauge invariant combinations. The Einstein ﬂuctuation takes the form,

ab ˜ ˜ δG00 = −2δ ∇b∇aψ,

˜ ˙ 1 ab ˜ ˜ ˙ δG0i = −2∇iψ + 2 δ ∇b∇a(Bi − Ei),

¨ ab ˜ ˜ ˙ ¨ ab ˜ ˜ δGij = −2δijψ − δ δij∇b∇a(φ + B − E) + δ δij∇b∇aψ

˜ ˜ ˙ ¨ ˜ ˜ 1 ˜ ˙ ¨ 1 ˜ ˙ ¨ +∇j∇i(φ + B − E) − ∇j∇iψ + 2 ∇i(Bj − Ej) + 2 ∇j(Bi − Ei)

¨ ab ˜ ˜ −Eij + δ ∇b∇aEij,

µν ij ab ˜ ˜ ¨ ab ˜ ˜ ˙ ¨ g δGµν = −δG00 + δ δGij = 4δ ∇b∇aψ − 6ψ − 2δ ∇b∇a(φ + B − E),

(3.9) where the dot denotes the time derivative ∂/∂x0. As mentioned, while the generic metric ﬂuctuation hµν has ten components, because of the freedom to make four gauge transformations on the coordinates (i.e hµν → hµν − ∂µν − ∂νµ), δGµν can only depend on a total of six of them. Looking at the individual components

˙ ¨ of δGµν, we see that these are proportional to the combinations ψ, φ + B − E,

˙ Bi − Ei, and Eij.

However, with these identiﬁcations, there still remains a degree of ambiguity

as to whether the combinations listed form the gauge invariants, or whether it is

in fact derivative combinations that are truly gauge invariant. Here one must

proceed carefully, as the gauge invariance of δGµν entails that only when taken

with the various derivatives that appear in (3.9) will these combinations be gauge

invariant. For example, we may only state deﬁnitively that δG00 is gauge invariant 41

ab ˜ ˜ (hence δ ∇b∇aψ). The gauge invariance of ψ itself cannot be assumed through

the analysis on δGµν alone.

To further investigate gauge invariance issues, we express each of the various SVT3 components in terms of combinations of the original components of hµν. Such a procedure has been derived in [42] and is to be contrasted with the integral formulations in (3.5). Speciﬁcally, in (3.5) we mentioned uniqueness up to integration by parts, whereas here such issues are avoided as we merely apply sequences of derivatives to hµν to form the requisite gauge invariant structure, and

afterward analyze which SVT combinations are formed as a result. The gauge

invariants take the following form: using the deﬁnition

˜ 2φ = −h00,Bi + ∇iB = h0i,

˜ ˜ ˜ ˜ hij = −2ψδij + 2∇i∇jE + ∇iEj + ∇jEi + 2Eij, (3.10)

we apply derivatives to obtain the relations

ij ˜ ˜ i ˜ j ˜ ˜ ˜ ˜ k ˜ ˜ k δ hij = −6ψ + 2∇i∇ E, ∇ hij = −2∇iψ + 2∇i∇k∇ E + ∇k∇ Ei,

˜ i ˜ j ˜ ˜ k ˜ ˜ k ˜ ˜ ` ∇ ∇ hij = −2∇k∇ ψ + 2∇k∇ ∇`∇ E, (3.11)

which then allow us to form the gauge invariants, taking the form

1 h i ∇˜ ∇˜ kψ = ∇˜ i∇˜ jh − ∇˜ ∇˜ k(δijh ) , k 4 ij k ij 3 1 ∇˜ ∇˜ k∇˜ ∇˜ `E = ∇˜ i∇˜ jh − ∇˜ ∇˜ k(δijh ), k ` 4 ij 4 k ij ˜ ˜ k ˜ k ∇k∇ B = ∇ h0k, 42

˜ ˜ k ˜ ˜ k ˜ ˜ k ∇k∇ Bi = ∇k∇ h0i − ∇i∇ h0k,

˜ ˜ k ˜ ˜ ` ˜ ˜ k j ˜ k ˜ ` ∇k∇ ∇`∇ Ei = ∇k∇ ∇ hij − ∇i∇ ∇ hk`, 1 ∇˜ ∇˜ kE = ∇˜ ∇˜ kh − ∇˜ ∇˜ kh − ∇˜ ∇˜ kh k ij 2 k ij i kj j ki ˜ ˜ k` ˜ ˜ k +∇i∇j(δ hk`) + δij∇k∇ ψ

˜ ˜ +∇i∇jψ, 1 ∇˜ ∇˜ `∇˜ ∇˜ kE = ∇˜ ∇˜ `∇˜ ∇˜ kh − ∇˜ ∇˜ kh − ∇˜ ∇˜ kh ` k ij 2 ` k ij i kj j ki 1 h i +∇˜ ∇˜ (δk`h ) + δ ∇˜ ∇˜ ` + ∇˜ ∇˜ × i j k` 4 ij ` i j h i ˜ m ˜ n ˜ ˜ k mn ∇ ∇ hmn − ∇k∇ (δ hmn) ,

˜ ˜ ` ˜ ˜ k ˙ ˜ ˜ ` ˜ ˜ k ˜ ˜ ` ˜ ˜ k ˜ ˜ ` ˜ j ∇`∇ ∇k∇ (Bi − Ei) = ∇`∇ ∇k∇ h0i − ∇`∇ ∇i∇ h0k − ∂0∇`∇ ∇ hij

˜ ˜ k ˜ ` +∂0∇i∇ ∇ hk`,

˜ ˜ k ˜ ˜ ` ˙ ¨ 1 ˜ ˜ k ˜ ˜ ` ˜ ˜ ` ˜ k 3 2 ˜ i ˜ j ∇k∇ ∇`∇ (φ + B − E) = − 2 ∇k∇ ∇`∇ h00 + ∇`∇ ∂0∇ h0k − 4 ∂0 ∇ ∇ hij

1 2 ˜ ˜ k ij + 4 ∂0 ∇k∇ (δ hij). (3.12)

Given (3.12) one can readily check that under a gauge transformation hµν →

˜ ˜ k ˜ ˜ ` ˜ ˜ k ˜ ˜ ` ˜ ˜ k ˙ hµν − ∂µν − ∂νµ the combinations ∇k∇ ψ, ∇`∇ ∇k∇ Eij, ∇`∇ ∇k∇ (Bi − Ei)

˜ ˜ k ˜ ˜ ` ˙ ¨ and ∇k∇ ∇`∇ (φ + B − E) are gauge invariant. We see here that it was in fact necessary to apply higher order derivatives than found in δGµν in order to express

each of the SVT3 components entirely in terms of combinations of components

˙ of the hµν. Hence, we repeat that it is not the quantities ψ, Eij, Bi − Ei and

φ + B˙ − E¨ themselves that are necessarily gauge invariant; rather, it is their

derivatives that are gauge invariant. In comparing (3.12) to (3.9) we see that 43

˜ ˜ k it is the quantity ∇k∇ ψ that appears in δG00 and that it is the combination

˜ ˜ k ˜ ˜ k ˜ ˜ ∇k∇ Eij − δij∇k∇ ψ − ∇i∇jψ that appears in δGij. Thus these combinations are automatically gauge invariant.

To touch basis with (3.5), we could proceed to integrate the relevant equa-

˙ ¨ ˙ tions in (3.12) in order to check gauge invariance for ψ, φ + B − E, Bi − Ei and

Eij themselves, since we can set

1 Z h i ψ = d3yD(3)(x − y) ∇˜ k∇˜ ` h − ∇˜ y ∇˜ m(δk`h ) , 4 y y k` m y k` 1 Z φ + B˙ − E¨ = − h + ∂ d3yD(3)(x − y)∇˜ kh 2 00 0 y 0k Z Z 2 3 (3) 3 (3) − ∂0 d yD (x − y) d zD (y − z) × 3 1 ∇˜ i∇˜ jh − ∇˜ ∇˜ k(δijh ) 4 ij 4 k ij Z Z 1 ˜ ˜ ` ˜ ˜ k 3 (3) 3 (3) = − 2 ∇`∇ ∇k∇ d yD (x − y) d zD (y − z)h00 Z Z ˜ ˜ ` 3 (3) 3 (3) k + ∂0∇`∇ d yD (x − y) d zD (y − z)∇z h0k Z Z 2 3 (3) 3 (3) − ∂0 d yD (x − y) d zD (y − z) × 3 1 ∇˜ i∇˜ jh − ∇˜ ∇˜ k(δijh ) , (3.13) 4 ij 4 k ij and

Z h i ˙ 3 (3) ˜ k ˜ y ˜ y ˜ k Bi − Ei = d yD (x − y) ∇y∇kh0i − ∇i ∇yh0k Z Z h i 3 (3) 3 (3) ˜ k ˜ z ˜ j ˜ z ˜ k ˜ ` − ∂0 d yD (x − y) d zD (y − z) ∇z ∇k∇zhij − ∇i ∇z ∇zhk` Z Z h i ˜ ˜ ` 3 (3) 3 (3) ˜ k ˜ z ˜ z ˜ k = ∇`∇ d yD (x − y) d zD (y − z) ∇z ∇kh0i − ∇i ∇z h0k 44

Z Z h i 3 (3) 3 (3) ˜ k ˜ z ˜ j ˜ z ˜ k ˜ ` − ∂0 d yD (x − y) d zD (y − z) ∇z ∇k∇zhij − ∇i ∇z ∇zhk` , 1 Z h i E = d3yD(3)(x − y) ∇˜ y∇˜ kh − ∇˜ y∇˜ kh − ∇˜ y∇˜ kh + ∇˜ y∇˜ y(δk`h ) ij 2 k y ij i y kj j y ki i j k` 1 Z h i Z + d3yD(3)(x − y) δ ∇˜ y∇˜ ` + ∇˜ y∇˜ y d3zD(3)(y − z) × 4 ij ` y i j h i ˜ m ˜ n ˜ z ˜ k mn ∇z ∇z hmn − ∇k∇z (δ hmn) , (3.14) where we make use of the ﬂat space Green’s function D(3)(x − y) obeying

1 δij∇˜ ∇˜ D(3)(x − y) = δ3(x − y),D(3)(x − y) = − , i j 4π|x − y| Z eiq·x d3yeiq·yD(3)(x − y) = − . (3.15) q2

2 ij ˜ i (Here q = δ qiqj, and in ∇y the y indicates that the derivative is taken with

respect to the y coordinate, and likewise for other coordinate choices.)

As eluded to below (3.5), there remains however an issue within (3.13) and

(3.14). Speciﬁcally, since ψ and Eij are manifestly gauge invariant as is (they are

ij expressed in terms of the gauge-invariant ﬂat 3-space δRij and δ δRij), in order

˙ ¨ ˙ to show the gauge invariance of φ + B − E and Bi − Ei we would need to be able

˙ ¨ ˜ ˜ ` ˜ ˜ k to integrate by parts (i.e., for φ + B − E we would need to bring ∇`∇ ∇k∇ and

˜ ˜ ` ˙ ˜ ˜ ` ∇`∇ inside the double integral, while for Bi − Ei we would need to bring ∇`∇

˙ inside, and similarly to show tranverness for Bi − Ei and Eij we need to be able

to integrate by parts.) Consequently, we are then forced to one of three scenarios.

Either a) we must put constraints on how hµν is to behave asymptotically, or b)

˜ ˜ ` ˜ ˜ k restrict to requiring in the Eij sector that only ∇`∇ ∇k∇ Eij be gauge invariant

˜ ˜ ` ˜ ˜ k and that only ∇`∇ ∇k∇ Eij be transverse or c) in the Eij plus ψ sector restrict 45

˜ ˜ k ˜ ˜ k ˜ ˜ to requiring that only ∇k∇ Eij − δij∇k∇ ψ − ∇i∇jψ be gauge invariant and that

˜ ˜ k only ∇k∇ Eij be transverse.

To see how method a), constraining the asymptotic behavior of hµν, may

resolve the issues of integration by parts, we shall take hµν to be localized in space

iq·x−iω(q)t and oscillating in time. Specifically, for each mode we will set hij = ij(q)e with ω(q) as yet undefined (and thus not necessarily equal to q), and where ij(q) serves as the polarization tensor. As a localized packet, we constrain the form of the polarization tensor by excluding any functional dependence of the form δ(q) or δ(q)/q. Thus, referring to (3.13) and (3.14), for spatially localized fluctuations comprising a single mode, the quantities ψ and Eij given in (3.13) and (3.14) evaluate to

[qkq` (q) − q2δk` (q)] ψ = eiq·x−iω(q)t k` k` , 4q2

3 In a similar manner, we may also integrate the remaining SVT3 components, obtaining Z 3 (3) ˜ i 2φ = −h00,B = d yD (x − y)∇yh0i, Z ˜ 3 (3) ˜ i Bi = h0i − ∇i d yD (x − y)∇yh0i, 1 Z Z h i E = d3yD(3)(x − y) d3zD(3)(y − z) 3∇˜ k∇˜ ` h − ∇˜ z ∇˜ k(δk`h ) , 4 z z k` k z k` Z Z 3 (3) 3 (3) h ˜ z ˜ k j z ˜ k ˜ ` i Ei = d yD (x − y) d zD (y − z) ∇k∇z ∇zhij − ∇i ∇z ∇zhk` (3.16)

i i As constructed, we see that ∇˜ Bi = 0. However to show ∇ Ei = 0, we need to be able to k ` integrate by parts. Using (3.10) and (3.12) directly, we can then show that both ∇˜ k∇˜ ∇˜ `∇˜ (φ+ k ` B˙ − E¨) and ∇˜ k∇˜ ∇˜ `∇˜ (Bi − E˙ i) are gauge invariant, with the gauge invariance of φ + B˙ − E¨ and Bi − E˙ i themselves then following when defining B, Bi, E and Ei according to (3.16). Hence, granted the freedom to integrate by parts, we can show that for fluctuations around flat spacetime all of the six ψ, Eij, φ + B˙ − E¨ and Bi − E˙ i quantities that appear in δGµν as given in (3.9) are gauge invariant. 46

[q2 (q) − q qk (q) − q qk (q) + q q δk` (q)] E = eiq·x−iω(q)t ij i kj j ki i j k` ij 2q2 (δ q2 + q q )[qkq` (q) − q2δk` (q)] + ij i j k` k` . (3.17) 4q4

˜ j ˜ j With application of ∇ , one may conﬁrm the transverse relation ∇ Eij = 0. To

P iq·x−iω(q)t construct a wave packet, we sum over all modes viz. hij = q aqij(q)e ,

to then obtain

k ` 2 k` X [q q k`(q) − q δ k`(q)] ψ = a eiq·x−iω(q)t , q 4q2 q 2 k k k` X [q ij(q) − qiq kj(q) − qjq ki(q) + qiqjδ k`(q)] E = a eiq·x−iω(q)t ij q 2q2 q (δ q2 + q q )[qkq` (q) − q2δk` (q)] + ij i j k` k` , (3.18) 4q4

˜ j iq·x−iω(q)t where again ∇ Eij = 0. Since the set of all e plane waves is complete for ﬂuctuations around ﬂat, any mode can be expanded as a general sum hij =

P iq·x−iω(q)t q aqij(q)e , with it following that (3.18) then holds for the complete plane wave basis. Hence, by constructing the ψ and Eij in a localized plane-wave

˜ j basis, we conﬁrm the transverse relation ∇ Eij = 0 without encountering issues related to integration by parts.

While we have demonstrated the role asymptotic behavior plays within tradeoff of transverse behavior vs. gauge invariance, it is also of importance to consider under conditions the SVT3 decomposition of hµν may be afforded in the first place. We revisit the SVT3 derivation constructed in [42] with an eye towards boundary conditions and asymptotic behavior.

Let us suppose that we are given a general vector Ai and we desire to extract 47

out its transverse and longitudinal components, to thereby construct a relation

i i Ai = Vi + ∂iL where ∂iV = 0. Applying ∂ , it follows that

i i ∂i∂ L = ∂iA . (3.19)

Recalling the Green’s identity

i i i i A∂i∂ B − B∂i∂ A = ∂i(A∂ B − B∂ A), (3.20) and introducing the Green’s function

i 3 ∂i∂ D(x − y) = δ (x − y), (3.21) the general solution to (3.19) is of the form

Z 3 (3) y j L(x) = d yD (x − y)∂j A (y) (3.22)

Z i y (3) (3) y + dSy L(y)∂i D (x − y) − D (x − y)∂i L(y) . (3.23)

Now utilizing Ai = Vi + ∂iL, it follows that

Z x x 3 (3) y j Ai(x) = Vi(x) + ∂i L = Vi(x) + ∂i d yD (x − y)∂j A (y) Z x i y (3) (3) y + ∂i dSy L(y)∂i D (x − y) − D (x − y)∂i L(y) . (3.24)

i Upon applying ∂x to (3.24), we obtain

Z i x i y (3) (3) y ∂x∂i dSy L(y)∂i D (x − y) − D (x − y)∂i L(y) = 0, (3.25) 48

R i (3) (3) to thus establish that ∂i dS (L∂iD − D ∂iL) is transverse. At this point, we appear to have constructed two transverse components - both Vi and

Z i (3) (3) ∂i dS (L∂iD − D ∂iL) (3.26)

itself. To allow Vi(x) to serve as the unique transverse component of Ai(x), we

R i (3) (3) must then require that ∂i dS (L∂iD − D ∂iL) vanish. Consequently, we see that we must inadvertently impose a constraint on Ai, namely that Ai be spatially asymptotically well-behaved. Thus, the unique decomposition of Ai into longitudinal and traverse components necessarily entails an assumption regarding the asymptotic behavior of Ai.

3.2 SVTD

While the SVT3 formalism presented thus far has demonstrated that the gauge invariant SVT3 quantities are covariant, one may note that these quantities have been deﬁned with respect to the three-dimensional subspace of the full four- dimensional spacetime. With the general hµν behaving under coordinate transformation as

∂x0α ∂x0β h → h0 = h , (3.27) µν µν ∂xµ ∂xν αβ

we see that a quantity such as h00 = −2φ may transform into a scalar involving

vector components. To see this, let us form the four vector h0µ, which in terms of 49

SVT3 components takes the form   −2φ       B + ∇˜ B  1 1  h0µ =   . (3.28)   B + ∇˜ B  2 2     ˜  B3 + ∇3B Now, with the full four-dimensional coordinate transformation mixing each of the four components of h0µ, we see that the transformation of φ may induce a contribution due to a vector Bi.

Thus to decompose the hµν into a set of scalars, vectors, and tensors that remain closed under the Poincare group, we must develop a formalism that matches the underlying space-time dimensionality; namely an SVT4 formalism. We proceed to do so here in a ﬂat spacetime, following the series of steps given within

[41]. It is no additional overhead to generalize this to D dimensions here, with even further generalization to arbitrary curved spacetimes found in detail within

AppendixA.

We deﬁned Greek indices to range over the full D-dimensional space and begin with the construction of a symmetric rank two tensor Fµν, taken to be transverse and traceless in the full D-dimensional space. 4 Accounting for the dimensionality and the transverse traceless constraints, Fµν tensor will have D(D+

1)/2−D −1 = (D +1)(D −2)/2 components. To facilitate the decomposition, we introduce a D-dimensional vector Wµ. In terms of this Wµ and h, and motivated

4 The previously introduced Eij was only transverse and traceless in a 3-dimensional subspace. 50

by [43], we define the general hµν fluctuation around a flat D-dimensional space to be of the form

2 − D Z h = 2F + ∇ W + ∇ W + ∇ ∇ dDyD(D)(x − y)∇αW µν µν ν µ µ ν D − 1 µ ν α g ∇ ∇ Z − µν (∇αW − h) − µ ν dDyD(D)(x − y)h, (3.29) D − 1 α D − 1

where the ﬂat spacetime D(D)(x − y) obeys

µν (D) (D) g ∇µ∇νD (x − y) = δ (x − y). (3.30)

Parallel to the SVT3 asymptotic behavior discussed in Sec. 3.1.3, it is implied

α in the form of hµν that the D-dimensional integrals exist, speciﬁcally with ∇ Wα

being sufficiently well-behaved at infinity. To make the Fµν that is defined by

(3.29) be transverse and traceless requires D+1 conditions, D to be supplied by

Wµ and one to be supplied by h. Given the deﬁned (3.29), one may take the trace

and conﬁrm that Fµν is traceless as written. To assess whether Fµν is transverse,

we apply ∇ν to (3.29) yielding

ν α ∇ hνµ = ∇α∇ Wµ. (3.31)

Thus (3.31) serves to deﬁne the as yet determined D components of Wµ in terms

α of the hµν. Questions on the asymptotic behavior of ∇ Wα are thus directly

linked the asymptotic behavior of hµν. Hence, for a Wµ that obeys (3.31) and

is suﬃciently bounded, the D-dimensional rank two tensor Fµν is transverse and

traceless. 51

α Uponj applying ∇α∇ to (3.29) we obtain

2 − D ∇ ∇αh = 2∇ ∇αF + ∇ ∇αh + ∇ ∇αh + ∇ ∇ ∇αW α µν α µν ν αµ µ αν D − 1 µ ν α g ∇ ∇ − µν (∇α∇βh − ∇ ∇αh) − µ ν h, (3.32) D − 1 αβ α D − 1 and on rearranging we obtain

α α α ∇α∇ hµν − ∇ν∇ hαµ − ∇µ∇ hαν + ∇µ∇νh 2 − D = 2∇ ∇αF + ∇ ∇ [∇αW − h] α µν D − 1 µ ν α g − µν (∇α∇βh − ∇ ∇αh). (3.33) D − 1 αβ α

Now in ﬂat spacetime, we note that the perturbed curvature tensors δR and δRµν are independelty gauge invariant. Thus, armed with the SVTD formalism, we use these to determine the gauge invariants

α α α 2δRµν = ∇α∇ hµν − ∇ν∇ hαµ − ∇µ∇ hαν + ∇µ∇νh (3.34)

α β α −δR = ∇ ∇ hαβ − ∇α∇ h. (3.35)

Making use of

β α α β α ∇β∇ [∇ Wα − h] = ∇ ∇ hαβ − ∇α∇ h, (3.36) we deﬁne

Z α D (D) α β α ∇ Wα − h = d yD (x − y)[∇ ∇ hαβ − ∇α∇ h], (3.37) 52

α and with this solution and reference to (3.34) and (3.35) we see that ∇α∇ Fµν is thus gauge invariant. However, we note again that parallel to the discussion the

α SVT3 tensor Eij, to show that ∇α∇ Fµν is transverse requires that we one can justify an integrate by parts.

For the remaining SVTD components, we make the following deﬁnitions

1 1 Z 2χ = [∇αW − h], 2F = dDyD(D)(x − y)[D∇αW − h], D − 1 α D − 1 α Z D (D) α Fµ = Wµ − ∇µ d yD (x − y)∇ Wα. (3.38)

µ As constructed, within (3.38) Fµ is deﬁned such that ∇ Fµ = 0. As for χ, we note with (3.37) we ﬁnd that χ is the integral of a gauge-invariant function so that

α ∇α∇ χ is also automatically gauge invariant. Finally, with (3.38) we can express

(3.29) in terms of the SVTD quantities as

hµν = −2gµνχ + 2∇µ∇νF + ∇µFν + ∇νFµ + 2Fµν, (3.39)

to thus complete the SVTD basis decomposition of hµν. In a general D-dimensional basis Fµν has (D + 1)(D − 2)/2 components, the transverse Fµ has D − 1 components, the two scalars χ and F each have one component, and together they comprise the D(D + 1)/2 components of a general hµν. If we elect to take D = 3,

(i.e. a decomposition of hij) we can then recognize (3.39) as the spatial piece of

SVT3 given in (3.2), providing a check on our results. 53

3.2.1 Gauge Structure (D = 4)

Counting the degrees of freedom of the ﬂuctuation δGµν around ﬂat D-dimensional we arrive at a total of D(D +1)/2−D = D(D −1)/2 independent gauge-invariant combinations. For the SVTD components, Fµν has (D + 1)(D − 2)/2 components and χ has one. That is, the combination of χ and Fµν precisely forms a total of

D(D − 1)/2. Moreoever with the derivatives of both being gauge invariant, we deduce that δGµν can only depend on Fµν and χ. Applying appropriate derivatives to the quantities given (3.38) and (3.39), we indeed form the following gauge invariant relations

1 2∇ ∇αχ = ∇α∇βh − ∇ ∇αh , α D − 1 αβ α

α β β α α α 2∇α∇ ∇β∇ Fµν = ∇β∇ [∇α∇ hµν − ∇ν∇ hαµ − ∇µ∇ hαν + ∇µ∇νh] 1 + [(D − 2)∇ ∇ + g ∇ ∇γ][∇α∇βh − ∇ ∇αh], D − 1 µ ν µν γ αβ α 1 δR = [2∇ ∇αF + 2(2 − D)∇ ∇ χ − 2g ∇ ∇αχ], µν 2 α µν µ ν µν α

α δR = 2(1 − D)∇α∇ χ, 1 δG = δR − g gαβδR = ∇ ∇αF µν µν 2 µν αβ α µν

α +(D − 2)(gµν∇α∇ − ∇µ∇ν)χ,

µν α g δGµν = (D − 2)(D − 1)∇α∇ χ. (3.40)

As written, we conﬁrm that δGµν indeed depends upon only Fµν and χ, with one

ν readily being able to check that δGµν obeys conservation ∇ δGµν = 0. Inspection

µν α of the components of δGµν reveals that only infer that from g δGµν that ∇α∇ χ 54

α is alone gauge invariant. However, by applying ∇α∇ to the δGµν equation we can

α β then additionally deduce that ∇α∇ ∇β∇ Fµν is gauge invariant. Importantly, we

α note that Fµν nor ∇α∇ Fµν may be determined to gauge invariant without con- sideration issues related to its asymptotic behavior. We will continue a discussion of the role of gauge invariance and boundedness in the following section. In addition, further detail of the SVTD construction, formalism, and gauge invariance is discussed within AppendexA.

Demonstrated in (3.40), we also observe that when written in the SVT4 basis the ﬂuctuation equations take a considerably simpler form than when written according to the SVT3 basis (a result that will continue to carry over when we study the SVT4 ﬂuctuation equations in curved background within Ch.4).

Hence, by implementing the covariant formalism we have eﬀectively replaced the

˙ ¨ ˙ six gauge-invariant combinations ψ, Eij, φ + B − E and Bi − Ei of SVT3 by the six gauge-invariant combinations Fµν and χ of SVT4. Consequently, the SVTD approach has provided a more compact set of gauge-invariant combinations and equations within δGµν and will prove to be simpler to solve, hence justfying the beneﬁcial utility of matching the fundamental transformation group associated with GR.

For geometries more general than flat spacetime, since the gauge-invariant equations must contain a total of six gauge-invariant degrees of freedom, they must be comprised of the five-component Fµν and some combination of the five 55

other components that appear in (3.39).

3.3 Relating SVT3 to SVT4

Thus far we have presented both and SVT3 and SVT4 decomposition of the metric

ﬂuctuation hµν. In the SVT3 development, we have determined the individual

SVT3 components in terms of hµν in both integral form (3.8) and in terms of higher order derivative relations (3.12). In addition, we have determined the

SVT3 gauge invariants, using the gauge invariant δGµν as reference. Likewise,

we have repeated the analysis for SVT4. Given that both formalisms present

valid decompositions with their structure only diﬀering the dimensionality of the

underlying space-time slicing, it remains to show the relationship between SVT3

and SVT4, which we develop here.

For ﬂuctuations around a ﬂat background, we have seen that the SVT4

transverse traceless tensor Fµν is comprised of ﬁve independent components. Count-

ing the degrees of freedom, to compose the Fµν in terms of SVT3 quantities, we must include the two-component transverse-traceless three-space rank two tensor, a two-component transverse three-space vector and a one-component three-space scalar. In addition, we have determined that if Fµν is asymptotically well behaved to thus permit integration by parts, Fµν itself is then gauge invariant and thus its associated combinations of SVT3 quantities must be gauge invariant. Such a requirement demands that the SVT3 vector component of Fµν must be propor-

˙ tional to Bi − Ei (with Eij being the only SVT3 tensor, we have already account 56

for such). To lastly pin down the last remaining gauge invariant associated with

the SVT3 scalar presents a diﬃculty, as we would appear to not have any means

˙ ¨ to signify whether it is ψ or φ + B − E that should comprise Fµν. One may try to use the relation for χ with δGµν to ﬁx the remaining degree of freedom, but comparing the SVT3 and SVT4 expansions of δGµν as given in (3.9) and (3.40) does in fact not enable us to uniquely specify the needed scalar combination.

Consequently, we thus need to construct the ﬂuctuation equations associated with a pure metric gravitational tensor other than the Einstein δGµν, with the

requirement that such a tensor also be independently gauge invariant in the ﬂat

background. It would also be beneﬁcial if such a gravitational tensor were able

to form a separation between Fµν and χ. Earlier we have determined that the perturbed curvature tensors themselves are gauge invariant, and thus we make appeal to any remaining curvature tensors not yet considered. Speciﬁcally, we have found such a tensor that meets the requisite properties, namely the Bach tensor associated with conformal gravity. As a pure metric tensor obtained by variation of the conformal gravity action of (2.18), we restate its form here [10]

1 δW = (ηρ ∂α∂ − ∂ρ∂ )(ησ ∂β∂ − ∂σ∂ )K µν 2 µ α µ ν β ν ρσ 1 − (η ∂γ∂ − ∂ ∂ )(ηρσ∂δ∂ − ∂ρ∂σ)K , (3.41) 6 µν γ µ ν δ ρσ

(0) With Wµν vanishing in conformal to flat background, and recalling that δWµν is traceless, the perturbed Bach tensor comprises five independent gauge invariant components. In the SVT4 basis, it must then solely be a function of the five 57

component gauge invariant Fµν. Naturally, we proceed to evaluate (3.41) in the

SVT3 basis to determine its form, given as

2 δW = − δmnδ`k∇˜ ∇˜ ∇˜ ∇˜ (φ + ψ + B˙ − E¨), 00 3 m n ` k 2 1 δW = − δmn∇˜ ∇˜ ∇˜ ∂ (φ + ψ + B˙ − E¨) + δmnδ`k∇˜ ∇˜ ∇˜ ∇˜ (B − E˙ ) 0i 3 i m n 0 2 m n ` k i i `k ˜ ˜ 2 ˙ −δ ∇`∇k∂0 (Bi − Ei) , 1 δW = δ δ`k∇˜ ∇˜ ∂2(φ + ψ + B˙ − E¨) + δ`k∇˜ ∇˜ ∇˜ ∇˜ (φ + ψ + B˙ − E¨) ij 3 ij ` k 0 ` k i j mn `k ˜ ˜ ˜ ˜ ˙ ¨ ˜ ˜ 2 ˙ ¨ −δijδ δ ∇m∇n∇`∇k(φ + ψ + B − E) − 3∇i∇j∂0 (φ + ψ + B − E) 1 + δ`k∇˜ ∇˜ ∇˜ ∂ (B − E˙ ) + δ`k∇˜ ∇˜ ∇˜ ∂ (B − E˙ ) − ∇˜ ∂3(B − E˙ ) 2 ` k i 0 j j ` k j 0 i i i 0 j j h i2 ˜ 3 ˙ mn ˜ ˜ 2 −∇j∂0 (Bi − Ei) + δ ∇m∇n − ∂0 Eij, (3.42)

where Kµν = hµν −(1/4)gµνh. Likewise, evaluating (3.41) in the SVT4 basis given in (3.39) yields

α β δWµν = ∇α∇ ∇β∇ Fµν. (3.43)

As an aside, we note the extremely compact and simple structure of the SVT4

ﬂuctuation equation in conformal gravity, with its exact solution being readily obtainable (particularly compared to δGµν within (3.40)).

Inspection of the SVT3 structure reveals that φ+ψ +B˙ −E¨ is to unambigu- ously serve as the three-dimensional scalar piece of Fµν. In addition, from (3.40) we

α ij ˜ ˜ ˙ ¨ ij ˜ ˜ ¨ can identify χ according to 3∇α∇ χ = −δ ∇i∇j(φ+ψ+B−E)+3δ ∇i∇jψ−3ψ.

Thus we have determined the underlying relationship between SVT3 and SVT4, 58

with ﬂuctuations around ﬂat spacetime quantity Fµν necessarily containing Eij,

˙ ˙ ¨ Bi − Ei and φ + ψ + B − E.

3.4 Decomposition Theorem and Boundary Conditions

In attempts to solve the cosmological ﬂuctuation equations that have been presented in the literature, appeal is commonly made to the decomposition theorem.

The theorem entails the assertation that within the fluctuation equations themselves the scalar, vector, and tensor sectors decouple and evolve independent. As mentioned in the introduction, the investigation of the decomposition theorem forms a core component of this work. We proceed by inspecting the interplay of asymptotic behavior and boundary conditions in establishing the validity of theorem below, as applied to both SVT3 and SVT4, taking a flat background and reduced fluctuation equation forms as illustrative examples. Within Ch.4, we touch basis again with the decomposition theorem as applied to specific cosmological geometries in order to determine the underlying assumptions that need to hold for such a theory to be valid.

3.4.1 SVT3

We begin with a schematic example of an SVT3 representation of two vector

ﬁelds which have been decomposed into their traverse vector and longitudinal scalar components

Bi + ∂iB = Ci + ∂iC, (3.44) 59

i where the B and Bi are given by (3.2), obeying ∂ Bi = 0 and where the C and Ci are to represent functions given by the evolution equations with Ci also obeying

i ∂iC = 0. Now, according to the decomposition theorem, the vectors Bi and Ci are taken to decouple and evolve independent of the scalars, with an analogous statement holding for the scalars. Thus, we are to take

Bi = Ci, ∂iB = ∂iC. (3.45)

It is clear, however, that in such an action, (3.45) does not follow from (3.44). For

i ijk if we apply ∂ and ∂j to (3.44) we obtain

i ijk ∂ ∂i(B − C) = 0, ∂j(Bk − Ck) = 0, (3.46) and from this we can only conclude that B and C are deﬁned up to an arbitrary

i scalar D obeying ∂ ∂iD = 0. For the vector sector analogously, Bk and Ck can

ijk only diﬀer by any function Dk that obeys ∂jDk = 0, i.e. an irrotational ﬁeld Dk expressed as the gradient of a scalar. In composing (3.46) we have appropriately separated the scalar and vector components within(3.44), obtaining a decomposition for the components that does not follow by proceeding from (3.46) to (3.45).

Speciﬁcally, without providing some additional information or constraints, one cannot directly proceed from (3.46) to (3.45). However, given the imposition of such constraints, such a decomposition may be possible, with the constraints in fact needing to be in form of spatially asymptotic boundary conditions. Alter- natively, if one were to elect to form a decomposition without the imposition of 60 boundary condition or constraints, we see from (3.46) that we necessarily need to go to higher derivatives in order to establish the decomposition. Thus we will continue to investigate both branches of applying said decomposition theorem.

Continuing with the example of (3.46), we ﬁrst proceed from (3.46) to (3.45) by imposing asymptotic boundary conditions. As we have done within (3.17), we take a complete basis of three-dimensional plane waves (serving as a complete

i basis for ∂i∂ ) and we express the general solution for B − C = D in the form

X ik·x D = ake , (3.47) k where the ak are constrained to obey

2 k ak = 0. (3.48)

Inspecting (3.48) itself carefully, we observe that (3.48) does not lead to ak = 0 directly. In fact since k2δ(k) = 0, k2δ(k)/k = 0 we can take other forms for the ak, with the general form of

ak = αkδ(kx)δ(ky)δ(kz) δ(kx)δ(ky)δ(kz) δ(kx)δ(ky)δ(kz) δ(kx)δ(ky)δ(kz) +βk + + , (3.49) kx ky kz

with constants αk and βk. Substituting αk as given into (3.47) would serve to eliminate the x dependence from D and consequently provide a constant D that does not vanish at spatial inﬁnity. Alternatively, if we substitute the βk term into

(3.47) we would construct a scalar D that grows linearly in x, to therefore also 61

not vanish at spatial inﬁnity. Hence by imposing asymptotic conditions, we can

i exclude solutions containing a non-zero D obeying ∂i∂ D = 0, to thus yield the requisite decomposition theorem. Consequently, we have connected validity of the decomposition theorem as being hinged upon the very existence of the SVT3 basis in the ﬁrst place as both require asymptotic boundary conditions.

We can further our understanding the role of boundary conditions by in-

i specting the behavior of ∂i∂ D = 0 in coordinate space. Recalling the Green’s identity

i i i i A∂i∂ B − B∂i∂ A = ∂i(A∂ B − B∂ A), (3.50)

i we take a general scalar A to obey ∂i∂ A = 0 and take B to be the Green’s function D(3)(x − y) which obeys

i (3) 3 ∂i∂ D (x − y) = δ (x − y). (3.51)

As a result, we can now express A as an asymptotic surface term of the form

Z i y (3) (3) y A(x) = dSy A(y)∂i D (x − y) − D (x − y)∂i A(y) , (3.52) with dS representing the integration over a closed surface S. If the asymptotic surface term vanishes, then A will also vanish identically. Hence we arrive at two

i non-trivial solutions to ∂i∂ A = 0, a) with A constant or b) of the form n · x with n a normal vector. However, both of these must be explicitly excluded if they are to be well behaved asymptotically. Hence, requiring that the asymptotic surface 62

i term in (3.52) vanish consequently forces the remaining solution to ∂i∂ A = 0 to be A = 0.

Using a polar coordinate basis, the formulation is adapted by using a polar

(ﬂat) metric γij and replacing (3.52) by

Z ∂D(3)(x, y) ∂A(y) A(x) = dS A(y) − D(3)(x, y) , (3.53) ∂n ∂n

with ∂/∂n is the normal derivative to the surface S, and with the Green’s function

obeying

i (3) −1/2 3 ∇i∇ D (x, y) = γ δ (x − y). (3.54)

Taking D(3)(x, y) = −1/4π|x − y|, the surface integral then becomes

1 Z 1 ∂A(y) ∂ 1 A(x) = dS − A(y) . (3.55) 4π |x − y| ∂n ∂n |x − y|

As written, the asymptotic surface term will vanish conditioned on A(y) behaving

as 1/rn for n > 0.

With asympyotic conditions having been shown to yield the decomposition

theorem, we now focus the discussion on the higher derivative relations 5 necessary

for said decomposition. Speciﬁcally, we apply sequences of derivatives to the ﬂat

space ﬂuctuation of the Einstein tensor (which we recall is gauge invariant entirely

on its own in this background), to obtain

ab ˜ ˜ 0 = δ ∇b∇aψ, 5 As with (3.12) we note that we need to go to fourth-order derivatives to establish decomposition. 63

ab ˜ ˜ cd ˜ ˜ ˙ ¨ 0 = δ ∇b∇aδ ∇c∇d(φ + B − E),

ab ˜ ˜ cd ˜ ˜ ˙ 0 = δ ∇b∇aδ ∇c∇d(Bi − Ei),

ab ˜ ˜ cd ˜ ˜ ¨ ef ˜ ˜ 0 = δ ∇b∇aδ ∇c∇d(−Eij + δ ∇e∇f Eij), (3.56)

Inspection of (3.9) shows that a decomposition theorem requires

ab ˜ ˜ 0 = −2δ ∇b∇aψ,

˜ ˙ 0 = −2∇iψ,

1 ab ˜ ˜ ˙ 0 = 2 δ ∇b∇a(Bi − Ei),

¨ ab ˜ ˜ ˙ ¨ ab ˜ ˜ 0 = −2δijψ − δ δij∇b∇a(φ + B − E) + δ δij∇b∇aψ

˜ ˜ ˙ ¨ ˜ ˜ +∇j∇i(φ + B − E) − ∇j∇iψ,

1 ˜ ˙ ¨ 1 ˜ ˙ ¨ 0 = 2 ∇i(Bj − Ej) + 2 ∇j(Bi − Ei),

¨ ab ˜ ˜ 0 = −Eij + δ ∇b∇aEij. (3.57)

Within (3.57), for any SVT quantity D that obeys

ab ˜ ˜ ab ˜ ˜ cd ˜ ˜ δ ∇a∇bD = 0 or δ ∇a∇bδ ∇c∇dD = 0 (3.58)

ab ˜ ˜ if impose spatial boundary conditions such that D or δ ∇a∇bD vanishes, the decomposition theorem will then follow for the ﬂuctuation δGµν. Thus, while a decoupling of the SVT representations can be achieved by going to higher derivatives, it is still necessary to constrain the behavior of such quantities according to

(3.58) in order to recover the decomposition theorem. 64

3.4.2 SVT4

Similar to our treatment of the SVT3 decomposition theorem, we now investi-

gation the status of the theorem in the SVT4 basis. We being with the four-

dimensional analog of (3.44):

Fµ + ∂µF = Cµ + ∂µC, (3.59)

where the F and Fµ are given by (3.38), and where the C and Cµ are representative

µ of the ﬂuctuation equations, with Cµ obeying ∂µC = 0. For the decomposition

theorem to hold one must take

Fµ = Cµ, ∂µF = ∂µC. (3.60)

Just as with the SVT3 case, (3.60) does not follow from (3.59), since on applying

ν σ ∂µ and µνστ n ∂ we obtain

µ ν σ τ τ ∂µ∂ (F − C) = 0, µνστ n ∂ (F − C ) = 0. (3.61)

By applying derivatives, we have indeed successfully decomposed the components.

However, such is determined only up to a function D = F −C, which need only to

µ obey ∂µ∂ D = 0. We proceed to use a complete basis of plane waves to represent

µ D (now four-dimensional plane waves, covering the ∂µ∂ operator), to obtain

X ik·x−ikt D = ake , (3.62) k

i where k = |k|. Importantly, in contrast to the ak in (3.47) which obey kik ak = 0,

µ µ here we have no such constraint on ak, as the ak obey kµk ak = 0 where kµk = 65

k2 − k2 is identically zero. In addition we can continue retaining the form of

µ 2 ∂µ∂ D = 0 by taking ak = exp(−a k · k). Upon taking the real part of D, we

obtain

3 Z d k 2 2 Re[D] = Re e−a k +ik·x−ikt (2π)3 1 r + t 2 2 r − t 2 2 = e−(r+t) /4a + e−(r−t) /4a . (3.63) 16π3/2a3 r r

We see that (3.63) is in fact not constrained by a spatially asymptotic condition since as r → ∞, Re[D] falls oﬀ as exp(−r2), both for ﬁxed t and for points on the light cone where r = ±t. In fact, such a D is even being well-behaved at r = 0. Thus because of the metric signature (i.e. opposing sign of space and time

µ coordinates), the quantity kµk = 0 leads to a form of D that cannot generally

µ satisfy the decomposition theorem, even in presence of the constraint ∂µ∂ D =

0. The investigation of whether a decomposition theorem can be satisﬁed at all must be performed on a case by case basis according to the diﬀering background geometries, of which we explore within Ch.4. Chapter 4

Construction and Solution of SVT Fluctuation Equations

We now apply the SVT3 and SVT4 decompositions of hµν developed in detail within Ch.3 to the fluctuation equations themselves and proceed to find exact solutions within backgrounds relevant to cosmology. Of particular importance, for each background geometry we analyze the conditions necessary for the decomposition theorem (cf. Sec. 3.4) to hold. Repeatedly, we find that a separation of the scalar, vector, and tensor sectors of the fluctuations may be generically achieved

(i.e. without any assumptions) by application of higher derivatives which effectively project out the transverse and longitudinal components. Hence, for each geometry we may achieve an SVT separation at the equation level in terms of fourth or higher order fluctuation equations. Once this achieved, we may then analyze precisely which conditions may be required in order to recover the naive decomposition one obtains when immediately separating scalars, vectors, and tensors at the outset. Since one cannot validate the decomposition theorem in general for arbitrary backgrounds, we evaluate the SVT3 and SVT4 fluctuations for a number of specific geometries.

66 67

As for the speciﬁc geometries, we ﬁrst evaluate SVT3 decompositions of

∆µν = δGµν+δTµν = 0 (with 8πG = 1), starting with a de Sitter background. Here we solve the ﬂuctuation equations ∆µν = 0 exactly, ﬁnding that the decomposition theorem can be recovered by imposing spatially asymptotic boundary conditions upon the SVT3 components.

With de Sitter serving as the simplest geometry, we then proceed to solve the ﬂuctuations in a Robertson Walker geometry with k = 0 in the radiation era.

Here the gauge invariants necessarily include contributions from both the matter sector δTµν, in this case a traceless perfect ﬂuid, and the SVT functions of hµν within the gravitational sector δGµν. Once again, the decomposition theorem is maintained upon the imposition of good asymptotic spatial behavior. In this particular background, we recover the familiar [1] solution of the transverse traceless

2 1/2 sector τ Eij being proportional to comoving time t .

To generalize the Roberston Walker case, we tackle a more complex set of equations corresponding to a Roberston Walker background with arbitrary

3-curvature and a perfect fluid form with unspecified equation of state. To simplify the fluctuations, we make heavy use of various curvature identities involving appropriate commutations of covariant derivatives. From the structure of the fluctuations, a series of gauge invariants are readily identified; however, to determine the full set, recourse must be made to inversions of the SVT3 quantities in terms of higher derivatives of hµν, which is developed in detail. In addition, the rela- 68

tionship between the background density ρ and pressure p is investigated. In the

1 high temperature limit radiation era, we have p = 3 ρ and in the low temperature limit matter dominated era we have p = 0. While the equation of state at the limits may obey p = wρ for w constant, in the transition between the two eras (as appropriate for recombination, for example) we show that the equation of state does not take such a simple form and is altogether more complicated. However, to assess the validity of the decomposition theorem, the relationship between ρ

and p need not be explicitly assigned.

A concrete application of the developed general SVT3 Roberston Walker case is constructed in the k = −1 cosmology. Here it can be shown that one can ﬁnd realizations of the evolution equations in the scalar, vector, and tensor sectors that would not lead to a decomposition theorem in those respective sectors.

However, if we force boundary conditions such that vector and tensor modes vanish at χ = ∞ instead of limiting to a constant value (and in addition are well behaved at the origin), then it follows that decomposition theorem holds.

In addition to standard Einstein gravity, we construct and solve the SVT3

ﬂuctuations within conformal gravity, determining that the imposition of boundary conditions leads to very simple forms for the evolution equations in a conformal to Minkowski background. Interestingly, one can also invert ﬁve of the SVT3 relations in terms of δWµν, to thus serve as a convenient alternative SVT3 formulation comprised of integral functions of a curvature invariant (i.e. the Bach tensor). 69

The discussion is then steered towards the SVT4 Einstein fluctuation equations as applied to de Sitter and Robertson Walker backgrounds, where an analogous procedure is carried through, namely solving the evolution equations exactly and determining the necessary conditions for the decomposition to hold. In contrast to the SVT3 fluctuations, here we find sets of solutions that are well behaved at the both the origin and infinity but nonetheless violate the decomposition theorem. To satisfy such a decomposition theorem, one would have to make a judicious choice of initial conditions at an initial time. Absent any compelling reason to do so, we conclude the decomposition theorem does in fact not hold in the SVT4 formulation.

In evaluating the conformal gravity cosmological ﬂuctuations in a de Sitter background, we ﬁnd δWµν takes a conveniently simple form, to only depend on the transverse traceless Fµν, with the decomposition theorem being automatic.

Additionally, one ﬁnds that δWµν may be related directly to the standard Einstein transverse traceless sector viz.

α 2 T θ δWµν = (∇α∇ − 4H )(δGµν + δTµν)

α 2 α 2 = (∇α∇ − 4H )(∇α∇ − 2H )Fµν,

a˙ with H = a . In continuing to evaluate conformal gravity SVT4 ﬂuctuations in

a general conformal to ﬂat background, we show that despite the presence of a

non-vanshing spatial 3-curvature, the evolution equations may still be expressed 70

in the extremely compact and readily solvable form

−2 σ τ δWµν = Ω ∂σ∂ ∂τ ∂ Fµν, (4.1) a form which has a direct corresponce to the solution (5.32) as developed in

Sec. 5.1.3 where the conformal gauge has been used to reduce the traceless Kµν to ﬁve independent degrees of freedom. Once again, in conformal gravity the decomposition theorem automatically holds without any auxiliary assumptions concerning asymptotic behavior.

In analyzing the SVT4 fluctuations in a de Sitter geometry with a conformal factor (Sec. 4.2.2), we find that the SVT4 scalar gauge invariant necessarily involves a contribution from the vector sector. In maintaining gauge invariance, one therefore must conclude that the decomposition can only apply to gauge invariant quantities themselves, not the separation of independent scalars, vectors, and tensors. In fact, such a construction is not novel to SVT4 itself and is shown to apply to SVT3 as exemplified in an anti de Sitter or conformal to flat backgrounds with a spatially dependent conformal factor, (i.e. Ω(τ, xi)). For instance, in such a conformal to flat background the scalar gauge invariant is given by

−1 ˙ ˙ −1 ˜ i ˜ η = ψ − Ω Ω(B − E) + Ω ∇ Ω(Ei + ∇iE), (4.2)

with the presence of the Ei vector illustrating such mixing. However, the intertwining of scalars and vectors can be avoided if one opts to ﬁx the gauge freedom by selecting a speciﬁc choice of coordinate constraints corresponding to 71

the conformal-Newtonian gauge [4].

4.1 SVT3

4.1.1 dS4

In the SVT3 formulation within a de Sitter background, the background and

ﬂuctuation metric can be expressed in the conformal to ﬂat form

1 ds2 = (1 + 2φ)dτ 2 − 2(∇˜ B + B )dτdxi − [(1 − 2ψ)δ + 2∇˜ ∇˜ E H2τ 2 i i ij i j ˜ ˜ i j +∇iEj + ∇jEi + 2Eij]dx dx , (4.3)

˜ i j where the ∇i denote derivatives with respect to the ﬂat 3-space δijdx dx metric.

In terms of the SVT3 form for the ﬂuctuations the components of the perturbed

δGµν are given by [42]

6 2 δG = − ψ˙ − ∇˜ 2(τψ + B − E˙ ), 00 τ τ 1 1 3 δG = ∇˜ 2(B − E˙ ) + ∇˜ (3B − 2τ 2ψ˙ + 2τφ) + B , 0i 2 i i τ 2 i τ 2 i δ δG = ij − 2τ 2ψ¨ + 2τφ˙ + 4τψ˙ − 6φ − 6ψ ij τ 2 +∇˜ 2 2τB − τ 2B˙ + τ 2E¨ − 2τE˙ − τ 2φ + τ 2ψ 1 h i + ∇˜ ∇˜ −2τB + τ 2B˙ − τ 2E¨ + 2τE˙ + 6E + τ 2φ − τ 2ψ τ 2 i j 1 h i + ∇˜ −2τB + 2τE˙ + τ 2B˙ − τ 2E¨ + 6E 2τ 2 i j j j j j 1 h i + ∇˜ −2τB + 2τE˙ + τ 2B˙ − τ 2E¨ + 6E 2τ 2 j i i i i i 6 2 −E¨ + E + E˙ + ∇˜ 2E , (4.4) ij τ 2 ij τ ij ij 72

where the dot denotes the derivative with respect to the conformal time τ and

˜ 2 ij ˜ ˜ ∇ = δ ∇i∇j. For the de Sitter SVT3 metric the gauge-invariant metric combinations are (see e.g. [42])

˙ ¨ ˙ ˙ α = φ + ψ + B − E, β = τψ + B − E,Bi − Ei,Eij. (4.5)

(For a generic SVT3 metric with a general conformal factor Ω(τ) the quantity

−(Ω/Ω)˙ ψ + B − E˙ is gauge invariant, to thus become β when Ω(τ) = 1/Hτ, with

the other gauge invariants being independent of Ω(τ).) In terms of the gauge-

invariant combinations the ﬂuctuation equations ∆µν = δGµν + δTµν = 0 take the form

6 2 ∆ = − (β˙ − α) − ∇˜ 2β = 0, (4.6) 00 τ 2 τ

1 2 ∆ = ∇˜ 2(B − E˙ ) − ∇˜ (β˙ − α) = 0, (4.7) 0i 2 i i τ i

δ h i 1 ∆ = ij −2τ(β¨ − α˙ ) + 6(β˙ − α) + τ∇˜ 2(2β − τα) + ∇˜ ∇˜ (−2β + τα) ij τ 2 τ i j 1 1 + ∇˜ [−2(B − E˙ ) + τ(B˙ − E¨ )] + ∇˜ [−2(B − E˙ ) + τ(B˙ − E¨ )] 2τ i j j j j 2τ j i i i i 2 − E¨ + E˙ + ∇˜ 2E = 0, (4.8) ij τ ij ij

µν 2 ¨ ˙ ˜ 2 2 ˜ 2 g ∆µν = H [−6τ(β − α˙ ) + 24(β − α) + 6τ∇ β − 2τ ∇ α] = 0, (4.9)

to thus be manifestly gauge invariant.

If there is to be a decomposition theorem the S, V and T components of

∆µν will satisfy ∆µν = 0 independently, to thus be required to obey

6 2 1 2 − (β˙ − α) − ∇˜ 2β = 0, ∇˜ 2(B − E˙ ) = 0, ∇˜ (β˙ − α) = 0, τ 2 τ 2 i i τ i 73

δ h i 1 ij −2τ(β¨ − α˙ ) + 6(β˙ − α) + τ∇˜ 2(2β − τα) + ∇˜ ∇˜ (−2τβ + τ 2α) = 0, τ 2 τ 2 i j 1 ∇˜ [−2τ(B − E˙ ) + τ 2(B˙ − E¨ )] 2τ 2 i j j j j 1 + ∇˜ [−2τ(B − E˙ ) + τ 2(B˙ − E¨ )] = 0, 2τ 2 j i i i i 2 −E¨ + E˙ + ∇˜ 2E = 0. (4.10) ij τ ij ij

To determine whether or not these conditions might hold we need to solve the ﬂuctuation equations ∆µν = 0 directly, to see what the structure of the solu-

2 tions might look like. To this end we ﬁrst apply τ∂τ − 1 to −τ ∆00/2, to obtain

τ 2∇˜ 2β˙ + 3τ(β¨ − α˙ ) − 3(β˙ − α) = 0, (4.11)

2 µν 2 and then add 3τ ∆00 to g ∆µν/H to obtain

τ 2∇˜ 2α + 3τ(β¨ − α˙ ) − 3(β˙ − α) = 0. (4.12)

Combining these equations and using ∆00 = 0 we thus obtain

∇˜ 2(α − β˙) = 0, ∇˜ 2β = 0, (4.13) and

τ 2∇˜ 2(α + β˙) + 6τ(β¨ − α˙ ) − 6(β˙ − α) = 0. (4.14)

Applying ∇˜ 2 then gives

∇˜ 4(α + β˙) = 0, ∇˜ 4(α − β˙) = 0. (4.15)

˜ 2 Applying ∇ to ∆0i in turn then gives

˜ 4 ˙ ∇ (Bi − Ei) = 0, (4.16) 74

ijk ˜ while applying ∇j to ∆0k gives

1 ijk∇˜ ∇˜ 2(B − E˙ ) = 0. (4.17) 2 j k k

˜ 4 Finally, to obtain an equation that only involves Eij we apply ∇ to ∆ij, to obtain

2 ∇˜ 4 −E¨ + E˙ + ∇˜ 2E = 0. (4.18) ij τ ij ij

As we see, we can isolate all the individual S, V and T gauge-invariant combinations, to thus give decomposition for the individual SVT3 components. However, the relations we obtain look nothing like the relations that a decomposition theorem would require, and thus without some further input we do not obtain a decomposition theorem.

To provide some further input we impose some asymptotic boundary conditions. To this end we recall from Sec. 3.4 that for any spatially asymptotically bounded function A that obeys ∇˜ 2A = 0, the only solution is A = 0. If A obeys

∇˜ 4A = 0, we must first set ∇˜ 2A = C, so that ∇˜ 2C = 0. Imposing boundary conditions for C enables us to set C=0. In such a case we can then set ∇˜ 2A = 0, and with sufficient asymptotic convergence can then set A = 0. Now a function could obey ∇˜ 2A = 0 trivially by being independent of the spatial coordinates altogether, and only depend on τ. However, it then would not vanish at spatial infinity, and we can thus exclude this possibility. With such spatial convergence for all of the S, V and T components we can then set

˙ ˙ ¨ ˙ ˜ 2 α = 0, β = 0, β = 0,Bi − Ei = 0, −τEij + 2Eij + τ∇ Eij = 0. (4.19) 75

Since this solution coincides with the solution that would be obtained to (4.10) un-

der the same boundary conditions, we see that under these asymptotic boundary

conditions we have a decomposition theorem.

In this solution all components of the SVT3 decomposition vanish identically

2 except the rank two tensor Eij. Taking Eij to behave as ijτ f(τ)g(x) where ij is a polarization tensor, we ﬁnd that the solution obeys

τ 2f¨+ 2τf˙ − 2f ∇˜ 2g = = −k2, (4.20) τ 2f g

2 where k is a separation constant. Consequently Eij is given as

2 ik·x Eij = ij(k)τ [a1(k)j1(kτ) + b1(k)y1(kτ)]e , (4.21)

2 where k · k = k , j1 and y1 are spherical Bessel functions, and a1(k) and b1(k)

are spacetime independent constants. For Eij to obey the transverse and traceless

ij ˜ j ij conditions δ Eij = 0, ∇ Eij = 0 the polarization tensor ij(k) must obey δ ij =

j 0, k ij(k) = 0. Then, by taking a family of separation constants we can form a

transverse-traceless wave packet

X 2 ik·x Eij = ij(k)τ [a1(k)j1(kτ) + b1(k)y1(kτ)]e k X sin(kτ) τ cos(kτ) = (k) a (k) − ij 1 k2 k k cos(kτ) τ sin(kτ) +b (k) + , (4.22) 1 k2 k

and can choose the a1(k) and b1(k) coeﬃcients to make the packet be as well-

behaved at spatial inﬁnity as desired. Finally, since according to (4.3) the full 76

2 2 −Ht ﬂuctuation is given not by Eij but by 2Eij/H τ , then with τ = e /H, through

2 2 2Ht the cos(kτ)/k term we ﬁnd that at large comoving time Eij/τ behaves as e ,

viz. the standard de Sitter ﬂuctuation exponential growth.

4.1.2 Robertson Walker k = 0 Radiation Era

In comoving coordinates a spatially ﬂat Robertson-Walker background metric

2 2 2 i j takes the form ds = dt − a (t)δijdx dx . In the radiation era where a perfect

ﬂuid pressure p and energy density ρ are related by ρ = 3p, the background

energy-momentum tensor is given by the traceless

Tµν = p(4UµUν + gµν), (4.23)

µν 0 i where g UµUν = −1, U = 1, U0 = −1, U = 0, Ui = 0. With this source

the background Einstein equations Gµν = −Tµν with 8πG = 1 ﬁx a(t) to be

a(t) = t1/2. In conformal to ﬂat coordinates we set τ = R dt/t1/2 = 2t1/2, with

the conformal factor being given by Ω(τ) = τ/2. In conformal to ﬂat coordinates

4 0 the background pressure is of the form p = 4/τ while U = 2/τ, U0 = −τ/2. In

this coordinate system the SVT3 ﬂuctuation metric as written with an explicit

conformal factor is of the form

τ 2 ds2 = (1 + 2φ)dτ 2 − 2(∇˜ B + B )dτdxi − [(1 − 2ψ)δ + 2∇˜ ∇˜ E 4 i i ij i j ˜ ˜ i j +∇iEj + ∇jEi + 2Eij]dx dx , (4.24) and the ﬂuctuation energy-momentum tensor is of the form

δTµν = δp(4UµUν + gµν) + p(4δUµUν + 4UµδUν + hµν). (4.25) 77

As written, we might initially expect there to be ﬁve ﬂuctuation variables in

the ﬂuctuation energy-momentum tensor: p and the four components of δUµ.

µν However, varying g UµUν = −1 gives

00 00 δg U0U0 + 2g U0δU0 = 0, (4.26)

i.e.

1 τφ δU = − (g00)−1(−g00g00δg )U = − . (4.27) 0 2 00 0 2

Thus δU0 is not an independent of the metric ﬂuctuations, and we need the ﬂuctu-

ation equations to only ﬁx six (viz. ten minus four) independent gauge-invariant

metric ﬂuctuations and the four δp and δUi (counting all components). With ten

∆µν = 0 equations, we can nicely determine all of them.

To this end we evaluate δGµν, to obtain

6 2 δG = ψ˙ + ∇˜ 2(−τψ + B − E˙ ), 00 τ τ 1 1 1 δG = ∇˜ 2(B − E˙ ) + ∇˜ (−B − 2τ 2ψ˙ − 2τφ) − B , 0i 2 i i τ 2 i τ 2 i δ δG = ij − 2τ 2ψ¨ − 2τφ˙ − 4τψ˙ + 2φ + 2ψ ij τ 2 +∇˜ 2 −2τB − τ 2B˙ + τ 2E¨ + 2τE˙ − τ 2φ + τ 2ψ 1 h i + ∇˜ ∇˜ 2τB + τ 2B˙ − τ 2E¨ − 2τE˙ − 2E + τ 2φ − τ 2ψ τ 2 i j 1 h i + ∇˜ 2τB − 2τE˙ + τ 2B˙ − τ 2E¨ − 2E 2τ 2 i j j j j j 1 h i + ∇˜ 2τB − 2τE˙ + τ 2B˙ − τ 2E¨ − 2E 2τ 2 j i i i i i 2 2 −E¨ − E − E˙ + ∇˜ 2E , (4.28) ij τ 2 ij τ ij ij 78

where the dot denotes ∂τ . For a spatially ﬂat Robertson-Walker metric the gauge- invariant metric combinations are (see e.g. [42])

˙ ¨ ˙ ˙ α = φ + ψ + B − E, γ = −τψ + B − E,Bi − Ei,Eij. (4.29)

(For a generic (4.24) with conformal factor Ω(τ), the quantity −(Ω/Ω)˙ ψ + B − E˙ becomes γ when Ω(τ) = τ/2, with the other gauge invariants being independent of Ω(τ).) In terms of the gauge-invariant combinations the ﬂuctuation equations

∆µν = δGµν + δTµν = 0 take the form

16 8 3τ 2 16 6 2 ∆ = − δU − φ + δp − ψ + (α − γ˙ ) + ∇˜ 2γ = 0, (4.30) 00 τ 3 0 τ 2 4 τ 4 τ 2 τ

8 4 1 2 ∆ = − δU + ∇˜ ψ + ∇˜ 2(B − E˙ ) − ∇˜ (α − γ˙ ) = 0, (4.31) 0i τ 3 i τ i 2 i i τ i

δ h i ∆ = ij τ 4δp − 16ψ − 8τ(α ˙ − γ¨) + 8(α − γ˙ ) − 4τ∇˜ 2(τα + 2γ) ij 4τ 2 1 1 + ∇˜ ∇˜ (τα + 2γ) + ∇˜ [2(B − E˙ ) + τ(B˙ − E¨ )] τ i j 2τ i j j j j 1 2 + ∇˜ [2(B − E˙ ) + τ(B˙ − E¨ )] − E¨ − E˙ + ∇˜ 2E = 0,(4.32) 2τ j i i i i ij τ ij ij

64 32 24 8 24 gµν∆ = δU + φ − (α ˙ − γ¨) − ∇˜ 2α − ∇˜ 2γ = 0. (4.33) µν τ 5 0 τ 4 τ 3 τ 2 τ 3

Since ∆µν is gauge invariant, we see that it is not δU0, δUi and δp themselves

4 that are gauge invariant. Rather it is the combinations δU0 + τφ/2, δp − 16ψ/τ ,

2 ˜ and δUi − τ ∇iψ/2 that are gauge invariant instead. Since we have shown above

that δU0 + τφ/2 is actually zero, conﬁrming that it is equal to a gauge-invariant 79

quantity provides a nice check on our calculation. However, since δU0 + τφ/2 is

µν zero, we can replace the ∆00 and g ∆µν equations by

3τ 2 16 6 2 ∆ = δp − ψ + (α − γ˙ ) + ∇˜ 2γ = 0, (4.34) 00 4 τ 4 τ 2 τ

24 8 24 gµν∆ = − (α ˙ − γ¨) − ∇˜ 2α − ∇˜ 2γ = 0. (4.35) µν τ 3 τ 2 τ 3

To put δUi in a more convenient form we decompose it into transverse and

˜ longitudinal components as δUi = Vi + ∇iV where

Z ˜ i ˜ 2 ˜ i 3 (3) ˜ i ∇ Vi = 0, ∇ V = ∇ δUi,V (x, τ) = d yD (x − y)∇yδUi(y, τ). (4.36)

In terms of these components the ∆0i = 0 equation takes the form

8 1 8 4 2 ∆ = − V + ∇˜ 2(B − E˙ ) − ∇˜ V + ∇˜ ψ − ∇˜ (α − γ˙ ) = 0. (4.37) 0i τ 3 i 2 i i τ 3 i τ i τ i

˜ i ˜ 2 Applying ∇ to ∆0i and then ∇ in turn yields

8 4 2 ∇˜ i∆ = ∇˜ 2 − V + ψ − (α − γ˙ ) = 0, 0i τ 3 τ τ 8 1 ∇˜ 2 − V + ∇˜ 2(B − E˙ ) = 0, (4.38) τ 3 i 2 i i

ijk ˜ while applying ∇j to ∆0k yields

8 1 ijk∇˜ ∆ = ijk∇˜ − V + ∇˜ 2(B − E˙ ) = 0. (4.39) j 0k j τ 3 k 2 k k

2 We thus identify V − τ ψ/2 and Vi as being gauge invariant.

To determine more of the structure of the solutions to the ﬂuctuation equa-

˜ i ˜ j tions we apply ∇ ∇ to the ∆ij = 0 equation and then use the ∆00 = 0 equation 80

4 to eliminate δp − 16ψ/τ from ∆ij, to obtain

τ 2 4 2 2 ∇˜ 2 δp − ψ − (α ˙ − γ¨) + (α − γ˙ ) 4 τ 2 τ τ 2 2 h i = − ∇˜ 4γ + 3∇˜ 2(α ˙ − γ¨) = 0. (4.40) 3τ

˜ 2 µν With the application of ∇ to the g ∆µν = 0 equation yielding

3∇˜ 2(α ˙ − γ¨) + τ∇˜ 4α + 3∇˜ 4γ = 0, (4.41) we obtain

∇˜ 4(τα + 2γ) = 0. (4.42)

On applying ∂τ to (4.42) and using (4.42) and (4.40) we obtain

6 3τ∇˜ 4α˙ = −3∇˜ 4α − 6∇˜ 4γ˙ = ∇˜ 4γ − 6∇˜ 4γ˙ = 3τ∇˜ 4γ¨ − τ∇˜ 6γ. (4.43) τ

The parameter γ thus obeys

6 6 ∇˜ 4 ∇˜ 2γ − 3¨γ − γ˙ + γ = 0. (4.44) τ τ 2

We can treat (4.44) as a second-order diﬀerential equation for ∇˜ 4γ. On setting

˜ 4 ∇ γ = fk(τ)gk(x) for a single mode, on separating with a separation constant

−k2 according to

∇˜ 2g 3f¨ + 6f˙ /τ − 6f /τ 2 k = k k k = −k2, (4.45) gk fk √ √ we ﬁnd that the τ dependence of fk(τ) is given by j1(kτ/ 3) and y1(kτ/ 3),

while the spatial dependence is given by plane waves. Since the set of plane waves 81 is complete, the general solution to (4.44) can be written as

X √ √ ik·x γ = [a1(k)j1(kτ/ 3) + b1(k)y1(kτ/ 3)]e + delta function terms, k (4.46) where the delta function terms are solutions to ∇˜ 4γ = 0, solutions that, in analog to (3.49), are generically of the form δ(k), δ(k)/k, δ(k)/k2, δ(k)/k3.

Proceeding the same way for α we obtain

3 τ 3∇˜ 4α˙ = 3∇˜ 4γ¨ − ∇˜ 6γ = − ∇˜ 4(τα¨ + 2α ˙ ) + ∇˜ 6α. (4.47) 2 2

The parameter α thus obeys

12 ∇˜ 4 ∇˜ 2α − 3¨α − α˙ = 0. (4.48) τ

˜ 4 On setting ∇ α = dk(τ)ek(x) for a single mode, we can separate with a separation constant −k2 according to

∇˜ 2e 3d¨ + 12d˙ /τ k = k k = −k2. (4.49) e dk √ √ The τ dependence of dk(τ) is thus given by j1(kτ/ 3)/τ and y1(kτ/ 3)/τ, while the spatial dependence is given by plane waves. The general solution to (4.48) is thus given by

1 X √ √ α = [m (k)j (kτ/ 3) + n (k)y (kτ/ 3)]eik·x τ 1 1 1 1 k +delta function terms, (4.50) 82 where the delta function terms are solutions to ∇˜ 4α = 0. Finally, we recall that

α and γ are related through ∇˜ 4(τα + 2γ) = 0, with the coeﬃcients thus obeying

m1(k) + 2a1(k) = 0, n1(k) + 2b1(k) = 0. (4.51)

Having determined α and γ, we can now determine δp − 16ψ/τ 4 from the

∆00 = 0 equation, and obtain

16 8 8 δp − ψ = − (α − γ˙ ) − ∇˜ 2γ τ 4 τ 4 3τ 3 X 8 2 ∂ 8k2 √ √ = + + [a (k)j (kτ/ 3) + b (k)y (kτ/ 3)] eik·x τ 4 τ ∂τ 3τ 3 1 1 1 1 k +delta function terms

X 8k √ 8k2 √ = a (k) √ j (kτ/ 3) + j (kτ/ 3) 1 τ 4 3 0 3τ 3 1 k X 8k √ 8k2 √ + b (k) √ y (kτ/ 3) + y (kτ/ 3) 1 τ 4 3 0 3τ 3 1 k +delta function terms. (4.52)

˙ ˜ j To determine Bi − Ei we apply ∇ to ∆ij = 0, to obtain

1 1 2 2 ∇˜ 2 (B − E˙ ) + (B˙ − E¨ ) = ∇˜ (α ˙ − γ¨) + ∇˜ 2γ , (4.53) τ i i 2 i i i τ 3τ from which it follows that

1 1 2 2 ∇˜ i∇˜ 2 (B − E˙ ) + (B˙ − E¨ ) = ∇˜ 2 (α ˙ − γ¨) + ∇˜ 2γ = 0. (4.54) τ i i 2 i i τ 3τ

On now applying ∇˜ 2 to (4.53) we obtain

1 1 ∇˜ 4 (B − E˙ ) + (B˙ − E¨ ) = 0, (4.55) τ i i 2 i i 83

just as required for consistency with (4.40). Equation (4.55) can be satisﬁed

through a 1/τ 2 conformal time dependence, and while it could also be satisﬁed

˜ 4 ˙ via a spatial dependence that satisﬁes ∇ (Bi − Ei) = 0, viz. the above delta

function terms. With plane waves being complete we can thus set

1 X B − E˙ = a (k)eik·x + F (τ) × delta function terms, (4.56) i i τ 2 i k

i where the ai(k) are transverse vectors that obeys k ai(k) = 0, and where F (τ) is

an arbitrary function of τ.

After solving (4.53) and (4.54), from the second equation in (4.38) we can

then determine Vi as it obeys

8 1 1 X ∇˜ 2V = ∇˜ 4(B − E˙ ) = k4a (k)eik·x. (4.57) τ 3 i 2 i i 2τ 2 i k

From (4.39) we can infer that

8 1 − V + ∇˜ 2(B − E˙ ) = ∇˜ A, (4.58) τ 3 i 2 i i i

˜ 2 ˜ where A is a scalar function that obeys ∇ A = 0, with ∇iA being curl free. We

recognize A as an integration constant for the integration of (4.57). From the ﬁrst

equation in (4.38) we additionally obtain

8 4 2 ∇˜ 2(− V + ψ) = ∇˜ 2(α − γ˙ ) τ 3 τ τ 2 X √ √ = √ k3[a (k)j (kτ/ 3) + b (k)y (kτ/ 3)]eik·x τ 3 1 0 1 0 k +delta function terms. (4.59) 84

To determine an equation for Eij we note that the δij term in ∆ij can be

2 2 ˜ 2 ˜ 2 written as (δij/4τ )[−8τ(α ˙ − γ¨) − 4τ ∇ α − (32/3)τ∇ γ]. Through use of (4.40),

(4.41) and (4.42), we can show that ∇˜ 2 applied to this term gives zero. Then given (4.55) and (4.42), from (4.32) it follows that Eij obeys

2 ∇˜ 4 −E¨ − E˙ + ∇˜ 2E = 0. (4.60) ij τ ij ij

˜ 4 Setting ∇ Eij = ij(k)fk(τ)gk(x) for a momentum mode, the τ dependence is

given as j0(kτ) and y0(kτ), with the general solution being of the form

X 0 0 ik·x Eij = [aij(k)j0(kτ) + bij(k)y0(kτ)]e + delta function terms. (4.61) k

Since according to (4.24) the full tensor ﬂuctuation is given not by Eij but by

2 1/2 2 τ Eij/2, then with τ = 2t , we ﬁnd that at large comoving time τ Eij behaves

as t1/2. Thus to summarize, we have constructed the exact and general solution

to the SVT3 k = 0 radiation era Robertson-Walker ﬂuctuation equations for all

˙ 4 2 of the dynamical degrees of freedom α, β, Bi − Ei, Eij, δp − 16ψ/τ , V − τ ψ/2

and Vi.

For a decomposition theorem to hold the condition ∆µν = 0 would need to

decompose into

3τ 2 16 6 2 δp − ψ + (α − γ˙ ) + ∇˜ 2γ = 0, (4.62) 4 τ 4 τ 2 τ

8 1 − V + ∇˜ 2(B − E˙ ) = 0, (4.63) τ 3 i 2 i i 85

8 4 2 − ∇˜ V + ∇˜ ψ − ∇˜ (α − γ˙ ) = 0, (4.64) τ 3 i τ i τ i

δ h i ij τ 4δp − 16ψ − 8τ(α ˙ − γ¨) + 8(α − γ˙ ) − 4τ∇˜ 2(τα + 2γ) 4τ 2 1 + ∇˜ ∇˜ (τα + 2γ) = 0, (4.65) τ i j

1 1 ∇˜ [2(B − E˙ ) + τ(B˙ − E¨ )] + ∇˜ [2(B − E˙ ) + τ(B˙ − E¨ )] = 0, (4.66) 2τ i j j j j 2τ j i i i i

2 −E¨ − E˙ + ∇˜ 2E = 0, (4.67) ij τ ij ij

24 8 24 − (α ˙ − γ¨) − ∇˜ 2α − ∇˜ 2γ = 0. (4.68) τ 3 τ 2 τ 3

To determine whether these conditions might hold, we note that in the α,

γ sector the (4.62) and (4.68) equations are the same as in the general ∆µν = 0

case (viz. (4.34) and (4.35)), but (4.65) is diﬀerent. If we use (4.34) to substitute

for δp in (4.65) we obtain

δ 32τ 1 ij −8τ(α ˙ − γ¨) − 4τ 2∇˜ 2α − ∇˜ 2γ + ∇˜ ∇˜ (τα + 2γ) = 0. (4.69) 4τ 2 3 τ i j

˜ ˜ Given the diﬀering behaviors of δij and ∇i∇j it follows that the terms that they multiply in (4.69) must each vanish, and thus we can set

32τ −8τ(α ˙ − γ¨) − 4τ 2∇˜ 2α − ∇˜ 2γ = 0, (4.70) 3

τα + 2γ = 0. (4.71) 86

Combining these equations then gives

3(α ˙ − γ¨) + ∇˜ 2γ = 0. (4.72)

We recognize (4.40) as the ∇˜ 2 derivative of (4.72) and recognize (4.42) as the ∇˜ 4 derivative of (4.71).

Similarly, in the V ,Vi sector we recognize the two equations that appear in

(4.38) as the ∇2 derivative of (4.63) and the ∇˜ i derivative of (4.64), with (4.39)

˙ ˜ j ˜ 2 being the curl of (4.63). In the Bi − Ei sector we recognize (4.55) as the ∇ ∇

˜ 4 derivative of (4.66), and in the Eij sector we recognize (4.60) as the ∇ derivative of

(4.67). Consequently we see that if spatially asymptotic boundary conditions are such that the only solutions to ∆µν = 0 are also solutions to (4.62) to (4.68) (i.e. vanishing of all delta function terms and integration constants that would lead to non-vanishing asymptotics), then the decomposition theorem follows. Otherwise it does not. Finally, we should note that, as constructed, in the matter sector we have found solutions for the gauge-invariant quantities δp−16ψ/τ 4, and V −τ 2ψ/2.

However since ψ is not gauge invariant, by choosing a gauge in which ψ = 0, we

would then have solutions for δp and V alone.

4.1.3 General Roberston Walker Constructing the Fluctuations

Having seen how things work in a particular background Robertson-Walker case

(radiation era with k = 0, Sec. 4.1.2), we now present a general analysis that can

be applied to any background Robertson-Walker geometry with any background 87

perfect ﬂuid equation of state. To characterize a general Robertson-Walker back-

ground there are two straightforward options. One is to write the background

2 2 i 2 i j i metric in a conformal to ﬂat form ds = Ω (τ, x )(dτ − δijdx dx ) with Ω(τ, x )

depending on both the conformal time τ = R dt/a(t) and the spatial coordinates.

The other is to write the background geometry as conformal to a static Robertson-

Walker geometry:

2 2 2 i j ds = Ω (τ)[dτ − γ˜ijdx dx ] dr2 = Ω2(τ) dτ 2 − − r2dθ2 − r2 sin θ2dφ2 , (4.73) 1 − kr2

i j with Ω(τ) depending only on τ, and withγ ˜ijdx dx denoting the spatial sector

of the metric. These two formulations of the background metric are coordinate

equivalent as one can transform one into the other by a general coordinate transfor-

mation without any need to make a conformal transformation on the background

metric (see AppendixB and Sec. 4.1.5). For our purposes in this section we shall

take (4.73) to be the background metric, and shall take the ﬂuctuation metric to

be of the form

2 2 2 ˜ i ˜ ˜ ds = Ω (τ) 2φdτ − 2(∇iB + Bi)dτdx − [−2ψγ˜ij + 2∇i∇jE ˜ ˜ i j +∇iEj + ∇jEi + 2Eij]dx dx . (4.74)

˜ i ˜ i ij ˜ In (4.74) ∇i = ∂/∂x and ∇ =γ ˜ ∇j (with Latin indices) are deﬁned with respect to the background three-space metricγ ˜ij. And with

ij ˜ ij ˜k γ˜ ∇jVi =γ ˜ [∂jVi − ΓijVk] (4.75) 88

˜k for any 3-vector Vi in a 3-space with 3-space connection Γij, the elements of (4.74)

are required to obey

ij ˜ ij ˜ jk ˜ ij γ˜ ∇jBi = 0, γ˜ ∇jEi = 0,Eij = Eji, γ˜ ∇kEij = 0, γ˜ Eij = 0. (4.76)

With the 3-space sector of the background geometry being maximally 3-symmetric,

it is described by a Riemann tensor of the form

˜ Rijk` = k[˜γjkγ˜i` − γ˜ikγ˜j`]. (4.77)

In [42] (and as discussed in (4.115) to (4.119) below), it was shown that for

the ﬂuctuation metric given in (4.74) with Ω(τ) being arbitrary function of τ, the

˙ ¨ ˙ −1 ˙ ˙ gauge-invariant metric combinations are φ+ψ+B −E, −Ω Ωψ+B −E, Bi −Ei, and Eij. As we shall see, the ﬂuctuation equations will explicitly depend on these speciﬁc combinations.

We take the background Tµν to be of the perfect ﬂuid form

Tµν = (ρ + p)UµUν + pgµν, (4.78)

with ﬂuctuation

δTµν = (δρ + δp)UµUν + δpgµν + (ρ + p)(δUµUν + UµδUν) + phµν. (4.79)

µν 0 −1 i Here g UµUν = −1, U = Ω (τ), U0 = −Ω(τ), U = 0, Ui = 0 for the back-

ground, while for the ﬂuctuation we have

00 00 δg U0U0 + 2g U0δU0 = 0, (4.80) 89

i.e.

1 δU = − (g00)−1(−g00g00δg )U = −Ω(τ)φ. (4.81) 0 2 00 0

Thus just as in Sec. 4.1.2 we see that δU0 is not an independent degree of freedom.

˜ ij ˜ ij As in Sec. 4.1.2 we shall set δUi = Vi + ∇iV , where nowγ ˜ ∇jVi =γ ˜ [∂jVi −

˜k ΓijVk] = 0. As constructed, in general we have 11 ﬂuctuation variables, the six

from the metric together with the three δUi, and δρ and δp. But we only have

ten ﬂuctuation equations. Thus to solve the theory when there is both a δρ and

a δp we will need some constraint between δp and δρ, a point we return to below.

For the background Einstein equations we have

h i ˙ 2 −2 ˙ 2 −2 ¨ −1 G00 = −3k − 3Ω Ω ,G0i = 0,Gij =γ ˜ij k − Ω Ω + 2ΩΩ ,

˙ 2 −2 2 G00 + 8πGT00 = −3k − 3Ω Ω + Ω ρ = 0, h i ˙ 2 −2 ¨ −1 2 Gij + 8πGTij =γ ˜ij k − Ω Ω + 2ΩΩ + Ω p = 0,

ρ = 3kΩ−2 + 3Ω˙ 2Ω−4, p = −kΩ−2 + Ω˙ 2Ω−4 − 2ΩΩ¨ −3, 1 Ω p = −ρ − ρ,˙ (4.82) 3 Ω˙

µν (after setting 8πG = 1), with the last relation following from ∇νT = 0, viz.

conservation of the energy-momentum tensor in the full 4-space. To solve these

equations we would need an equation of state that would relate ρ and p. How-

ever we do not need to impose one just yet, as we shall generate the ﬂuctuation

equations as subject to (4.82) but without needing to specify the form of Ω(τ) or

a relation between ρ and p. 90

For δGµν we have

˙ ˙ −1 ˙ −1 ˜ ˜ a ˙ −1 ˜ ˜ a ˙ δG00 = −6kφ − 6kψ + 6ψΩΩ + 2ΩΩ ∇a∇ B − 2ΩΩ ∇a∇ E

˜ ˜ a −2∇a∇ ψ,

.. ˜ ˙ 2 −2 ˜ −1 ˜ ˜ ˙ ˜ ˙ ˙ −1 ˜ δG0i = 3k∇iB − Ω Ω ∇iB + 2ΩΩ ∇iB − 2k∇iE − 2∇iψ − 2ΩΩ ∇iφ

.. ˙ ˙ 2 −2 −1 1 ˜ ˜ a 1 ˜ ˜ a ˙ +2kBi − kEi − BiΩ Ω + 2BiΩΩ + 2 ∇a∇ Bi − 2 ∇a∇ Ei,

.. ˙ 2 −2 ˙ 2 −2 ˙ ˙ −1 ˙ ˙ −1 δGij = −2ψγ˜ij + 2Ω γ˜ijφΩ + 2Ω γ˜ijψΩ − 2φΩ˜γijΩ − 4ψΩ˜γijΩ

.. .. −1 −1 ˙ −1 ˜ ˜ a ˜ ˜ a ˙ −4Ω˜γijφΩ − 4Ω˜γijψΩ − 2Ω˜γijΩ ∇a∇ B − γ˜ij∇a∇ B

.. ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ˜ ˜ a ˜ ˜ a +˜γij∇a∇ E + 2Ω˜γijΩ ∇a∇ E − γ˜ij∇a∇ φ +γ ˜ij∇a∇ ψ

.. ˙ −1 ˜ ˜ ˜ ˜ ˙ ˜ ˜ ˙ −1 ˜ ˜ ˙ ˜ ˜ +2ΩΩ ∇j∇iB + ∇j∇iB − ∇j∇iE − 2ΩΩ ∇j∇iE + 2k∇j∇iE

.. ˙ 2 −2 ˜ ˜ −1 ˜ ˜ ˜ ˜ ˜ ˜ ˙ −1 ˜ −2Ω Ω ∇j∇iE + 4ΩΩ ∇j∇iE + ∇j∇iφ − ∇j∇iψ + ΩΩ ∇iBj

.. 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ ˜ ˙ 2 −2 ˜ + 2 ∇iBj − 2 ∇iEj − ΩΩ ∇iEj + k∇iEj − Ω Ω ∇iEj

.. .. −1 ˜ ˙ −1 ˜ 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ +2ΩΩ ∇iEj + ΩΩ ∇jBi + 2 ∇jBi − 2 ∇jEi − ΩΩ ∇jEi

.. .. ˜ ˙ 2 −2 ˜ −1 ˜ ˙ 2 −2 +k∇jEi − Ω Ω ∇jEi + 2ΩΩ ∇jEi − Eij − 2Ω EijΩ

.. ˙ ˙ −1 −1 ˜ ˜ a −2EijΩΩ + 4ΩEijΩ + ∇a∇ Eij,

.. µν ˙ 2 −4 ˙ 2 −4 ˙ ˙ −3 ˙ ˙ −3 −3 g δGµν = 6Ω φΩ + 6Ω ψΩ − 6φΩΩ − 18ψΩΩ − 12ΩφΩ

.. .. −3 −2 −2 −2 ˙ −3 ˜ ˜ a −12ΩψΩ − 6ψΩ + 6kφΩ + 6kψΩ − 6ΩΩ ∇a∇ B

.. −2 ˜ ˜ a ˙ −2 ˜ ˜ a ˙ −3 ˜ ˜ a ˙ −2Ω ∇a∇ B + 2Ω ∇a∇ E + 6ΩΩ ∇a∇ E

.. ˙ 2 −4 ˜ ˜ a −3 ˜ ˜ a −2 ˜ ˜ a −2Ω Ω ∇a∇ E + 4ΩΩ ∇a∇ E + 2kΩ ∇a∇ E

−2 ˜ ˜ a −2 ˜ ˜ a −2Ω ∇a∇ φ + 4Ω ∇a∇ ψ. (4.83) 91

We introduce

α = φ + ψ + B˙ − E,¨ γ = −Ω˙ −1Ωψ + B − E,˙ Vˆ = V − Ω2Ω˙ −1ψ,

.. Ω δρˆ = δρ − 12Ω˙ 2ψΩ−4 + 6ΩψΩ−3 − 6kψΩ−2 = δρ + ρψ˙ = δρ − 3(ρ + p)ψ, Ω˙ .. ... Ω δpˆ = δp − 4Ω˙ 2ψΩ−4 + 8ΩψΩ−3 + 2kψΩ−2 − 2ΩΩ˙ −1ψΩ−2 = δp + pψ,˙ (4.84) Ω˙

(in (4.84) we used (4.82)), where, as we show below, the functions δρˆ, δpˆ and

Vˆ are gauge invariant. (The γ introduced in (4.29) is a special case of the γ introduced in (4.84).) Given (4.84) we can express the components of ∆µν =

δGµν + 8πGδTµν quite compactly. Speciﬁcally, on using (4.82) for the background but without imposing any relation between the background ρ and p, we obtain evolution equations of the form

˙ 2 −2 2 ˙ −1 ˜ ˜ a ∆00 = 6Ω Ω (α − γ˙ ) + δρˆΩ + 2ΩΩ ∇a∇ γ = 0, (4.85)

.. ˙ −1 ˜ ˜ ˙ 2 −3 −2 −1 ˜ ˆ ∆0i = −2ΩΩ ∇i(α − γ˙ ) + 2k∇iγ + (−4Ω Ω + 2ΩΩ − 2kΩ )∇iV

˙ 1 ˜ ˜ a ˙ ˙ 2 −3 +k(Bi − Ei) + 2 ∇a∇ (Bi − Ei) + (−4Ω Ω

.. −2 −1 +2ΩΩ − 2kΩ )Vi = 0, (4.86)

˙ 2 −2 ˙ −1 ¨ −1 2 ∆ij =γ ˜ij 2Ω Ω (α − γ˙ ) − 2ΩΩ (α ˙ − γ¨) − 4ΩΩ (α − γ˙ ) + Ω δpˆ

˜ ˜ a ˙ −1 ˜ ˜ ˙ −1 ˙ −1 ˜ ˙ −∇a∇ (α + 2ΩΩ γ) + ∇i∇j(α + 2ΩΩ γ) + ΩΩ ∇i(Bj − Ej)

1 ˜ ˙ ¨ ˙ −1 ˜ ˙ 1 ˜ ˙ ¨ + 2 ∇i(Bj − Ej) + ΩΩ ∇j(Bi − Ei) + 2 ∇j(Bi − Ei)

.. ˙ ˙ −1 ˜ ˜ a −Eij − 2kEij − 2EijΩΩ + ∇a∇ Eij = 0, (4.87) 92

ij ˙ 2 −2 ˙ −1 ¨ −1 2 γ˜ ∆ij = 6Ω Ω (α − γ˙ ) − 6ΩΩ (α ˙ − γ¨) − 12ΩΩ (α − γ˙ ) + 3Ω δpˆ

˜ ˜ a ˙ −1 −2∇a∇ (α + 2ΩΩ γ) = 0, (4.88)

.. µν −3 ˙ −3 g ∆µν = 3δpˆ − δρˆ − 12ΩΩ (α − γ˙ ) − 6ΩΩ (α ˙ − γ¨)

−2 ˜ ˜ a ˙ −1 −2Ω ∇a∇ (α + 3ΩΩ γ) = 0. (4.89)

Starting from the general identities

s s ∇k∇nT`m − ∇n∇kT`m = T mR`snk + T` Rmsnk,

s ∇k∇nAm − ∇n∇kAm = A Rmsnk (4.90)

that hold for any rank two tensor or vector in any geometry, for the 3-space

˜ Robertson-Walker geometry where Rmsnk = k(˜γsnγ˜mk − γ˜mnγ˜sk) we obtain

˜ ˜ ˜ a ˜ ˜ a ˜ ˜ a ˜ ˜ ∇i∇a∇ Aj − ∇a∇ ∇iAj = 2kγ˜ij∇aA − 2k(∇iAj + ∇jAi),

˜ j ˜ ˜ a ˜ ˜ a ˜ j ˜ j ˜ ˜ ˜ j ∇ ∇a∇ Aj = (∇a∇ + 2k)∇ Aj, ∇ ∇iAj = ∇i∇ Aj + 2kAi (4.91)

for any 3-vector Ai in a maximally symmetric 3-geometry with 3-curvature k.

Similarly, noting that for any scalar S in any geometry we have

s ∇a∇b∇iS = ∇a∇i∇bS = ∇i∇a∇bS + ∇ SRbsia,

s s ∇`∇k∇n∇mS = ∇n∇m∇`∇kS + ∇n[∇ SRksm`] + ∇ ∇kSRmsn`

s s +∇m∇ SRksn` + ∇`[∇ SRmsnk], (4.92)

in a Robertson-Walker 3-geometry background we obtain

˜ ˜ a ˜ ˜ ˜ ˜ a ˜ ∇a∇ ∇iS = ∇i∇a∇ S + 2k∇iS, 93

˜ ˜ a ˜ ˜ ˜ ˜ ˜ ˜ a ˜ ˜ 1 ˜ ˜ a ∇a∇ ∇i∇jS = ∇i∇j∇a∇ S + 6k(∇i∇j − 3 γ˜ij∇a∇ )S,

˜ ˜ a ˜ ˜ ˜ ˜ ˜ ˜ a ˜ ˜ ˜ ˜ a ∇a∇ ∇i∇jS = ∇i∇j∇a∇ S + 6k∇i∇jS − 2kγ˜ij∇a∇ S. (4.93)

Thus we ﬁnd that

.. ˜ i ˜ ˜ a ˙ −1 ˙ 2 −3 −2 −1 ˆ ∇ ∆0i = ∇a∇ − 2ΩΩ (α − γ˙ ) + 2kγ + (−4Ω Ω + 2ΩΩ − 2kΩ )V

= 0, (4.94) and thus

˜ ˜ k ˜ ˜ k ˙ 1 ˜ ˜ a ˙ (∇k∇ − 2k)∆0i = (∇k∇ − 2k) k(Bi − Ei) + 2 ∇a∇ (Bi − Ei)

.. ˙ 2 −3 −2 −1 +(−4Ω Ω + 2ΩΩ − 2kΩ )Vi = 0. (4.95)

Also we obtain

ij` ˜ ij` ˜ ˙ 1 ˜ ˜ a ˙ ∇j∆0i = ∇j k(Bi − Ei) + 2 ∇a∇ (Bi − Ei)

.. ˙ 2 −3 −2 −1 +(−4Ω Ω + 2ΩΩ − 2kΩ )Vi = 0. (4.96)

P Conferring Sec. 4.2.2, we have noted that in a de Sitter space if a tensor A M is

L P transverse and traceless then so is ∇L∇ A M Since this holds in any maximally

˜ ˜ a symmetric space the quantity ∇a∇ Eij is transverse and traceless too. Thus given

(4.91) we obtain

˜ j ˜ ˙ 2 −2 ˙ −1 ¨ −1 2 ∇ ∆ij = ∇i[2Ω Ω (α − γ˙ ) − 2ΩΩ (α ˙ − γ¨) − 4ΩΩ (α − γ˙ ) + Ω δpˆ

˙ −1 ˜ ˜ a 1 ˙ ¨ ˙ −1 ˙ +2k(α + 2ΩΩ γ)] + [∇a∇ + 2k][ 2 (Bi − Ei) + ΩΩ (Bi − Ei)] = 0,

(4.97) 94

˜ i ˜ j ˜ ˜ a ˙ 2 −2 ˙ −1 ¨ −1 2 ∇ ∇ ∆ij = ∇a∇ [2Ω Ω (α − γ˙ ) − 2ΩΩ (α ˙ − γ¨) − 4ΩΩ (α − γ˙ ) + Ω δpˆ

+2k(α + 2ΩΩ˙ −1γ)] = 0. (4.98)

Thus we obtain

˜ i ˜ j ˜ ˜ a ij ˜ 2 ˜ 2 ˙ −1 3∇ ∇ ∆ij − ∇a∇ (˜γ ∆ij) = 2∇ [∇ + 3k](α + 2ΩΩ γ) = 0, (4.99)

˜ i ˜ j ij ˜ 2 ˙ 2 −2 ˙ −1 ∇ ∇ ∆ij + kγ˜ ∆ij = [∇ + 3k][2Ω Ω (α − γ˙ ) − 2ΩΩ (α ˙ − γ¨)

−4ΩΩ¨ −1(α − γ˙ ) + Ω2δpˆ] = 0. (4.100)

We now deﬁne A = 2Ω˙ 2Ω−2(α − γ˙ ) − 2ΩΩ˙ −1(α ˙ − γ¨) − 4ΩΩ¨ −1(α − γ˙ ) + Ω2δpˆ and

C = α + 2ΩΩ˙ −1γ. And using (4.93) obtain

˜ ˜ a ˜ ˜ ˜ ˜ a (∇a∇ + k)∇i(A + 2kC) = ∇i(∇a∇ + 3k)(A + 2kC), (4.101)

and thus with (4.99) and (4.100) obtain

˜ ˜ a ˜ ˜ a ˜ ˜ ˜ ˜ a ˜ ˜ b (∇a∇ − 2k)(∇a∇ + k)∇i(A + 2kC) = ∇i∇a∇ (∇b∇ + 3k)(A + 2kC)

= 0. (4.102)

Consequently, on comparing with (4.97) we obtain

˜ ˜ a ˜ ˜ b ˜ j ˜ ˜ a ˜ ˜ b ˜ ˜ c (∇a∇ − 2k)(∇b∇ + k)∇ ∆ij = (∇a∇ − 2k)(∇b∇ + k)[∇c∇ + 2k] ×

1 ˙ ¨ ˙ −1 ˙ [ 2 (Bi − Ei) + ΩΩ (Bi − Ei)] = 0, (4.103)

˙ to give a relation that only involves Bi − Ei. 95

To obtain a relation that involves Eij we proceed as follows. We note that sector of ∆ij that contains the above A and C can be written as

˜ ˜ a ˜ ˜ Dij =γ ˜ij(A − ∇a∇ C) + ∇i∇jC. (4.104)

We thus introduce

1 A = D − γ˜ γ˜abD = (∇˜ ∇˜ − 1 γ˜ ∇˜ ∇˜ a)C, ij ij 3 ij ab i j 3 ij a 1 B = ∆ − γ˜ γ˜ab∆ = (∇˜ ∇˜ − 1 γ˜ ∇˜ ∇˜ a)C ij ij 3 ij ab i j 3 ij a ˙ −1 ˜ ˙ 1 ˜ ˙ ¨ ˙ −1 ˜ ˙ 1 ˜ ˙ ¨ + ΩΩ ∇i(Bj − Ej) + 2 ∇i(Bj − Ej) + ΩΩ ∇j(Bi − Ei) + 2 ∇j(Bi − Ei)

.. ˙ ˙ −1 ˜ ˜ a − Eij − 2kEij − 2EijΩΩ + ∇a∇ Eij = 0, (4.105)

with (4.105) deﬁning Aij and Bij, and with A dropping out. Using (4.91) and the

third relation in (4.93) we obtain

˜ ˜ b ˜ ˜ 1 ˜ ˜ a ˜ ˜ b (∇b∇ − 3k)Aij = (∇i∇j − 3 γ˜ij∇a∇ )(∇b∇ + 3k)C, (4.106)

and via (4.93) and (4.94) thus obtain

˜ ˜ a ˜ ˜ b ˜ ˜ 1 ˜ ˜ a ˜ ˜ b ˜ ˜ c (∇a∇ − 6k)(∇b∇ − 3k)Aij = (∇i∇j − 3 γ˜ij∇a∇ )∇b∇ (∇c∇ + 3k)C

= 0. (4.107)

ij Comparing with the structure of ∆ij andγ ˜ ∆ij, we thus obtain

˜ ˜ a ˜ ˜ b ˜ ˜ a ˜ ˜ b (∇a∇ − 6k)(∇b∇ − 3k)[Bij − Aij] = (∇a∇ − 6k)(∇b∇ − 3k)

˙ −1 ˜ ˙ 1 ˜ ˙ ¨ ˙ −1 ˜ ˙ 1 ˜ ˙ ¨ × ΩΩ ∇i(Bj − Ej) + 2 ∇i(Bj − Ej) + ΩΩ ∇j(Bi − Ei) + 2 ∇j(Bi − Ei) 96

.. ˙ −1 ˙ ˜ ˜ a −Eij − 2kEij − 2ΩΩ Eij + ∇a∇ Eij = 0. (4.108)

˜ i We now note that for any vector Ai that obeys ∇ Ai = 0, through repeated use

of the ﬁrst relation in (4.91) we obtain

˜ ˜ b ˜ ˜ ˜ ˜ ˜ b ˜ ˜ ˜ b (∇b∇ − 3k)(∇iAj + ∇jAi) = ∇i(∇b∇ + k)Aj + ∇j(∇b∇ + k)Ai,

˜ ˜ a ˜ ˜ b ˜ ˜ ˜ ˜ ˜ a ˜ ˜ b (∇a∇ − 6k)(∇b∇ − 3k)(∇iAj + ∇jAi) = ∇i(∇a∇ − 2k)(∇b∇ + k)Aj

˜ ˜ ˜ a ˜ ˜ b +∇j(∇a∇ − 2k)(∇b∇ + k)Ai. (4.109)

On using the ﬁrst relation in (4.91) again, it follows that

˜ ˜ c ˜ ˜ a ˜ ˜ b ˜ ˜ (∇c∇ − 2k)(∇a∇ − 6k)(∇b∇ − 3k)(∇iAj + ∇jAi)

˜ ˜ ˜ c ˜ ˜ a ˜ ˜ b = ∇i(∇c∇ + 2k)(∇a∇ − 2k)(∇b∇ + k)Aj

˜ ˜ ˜ c ˜ ˜ a ˜ ˜ b +∇j(∇c∇ + 2k)(∇a∇ − 2k)(∇b∇ + k)Ai. (4.110)

1 ˙ ¨ ˙ −1 ˙ ˜ i On setting Ai = 2 (Bi − Ei) + ΩΩ (Bi − Ei) (so that Ai is such that ∇ Ai = 0),

and recalling (4.103) we obtain

˜ ˜ c ˜ ˜ a ˜ ˜ b (∇c∇ − 2k)(∇a∇ − 6k)(∇b∇ − 3k) ˜ 1 ˙ ¨ ˙ −1 ˙ ˜ 1 ˙ ¨ ˙ −1 ˙ × ∇i[ 2 (Bj − Ej) + ΩΩ (Bj − Ej)] + ∇j[ 2 (Bi − Ei) + ΩΩ (Bi − Ei)]

= 0. (4.111)

Thus ﬁnally from (4.108) we obtain

˜ ˜ c ˜ ˜ a ˜ ˜ b (∇c∇ − 2k)(∇a∇ − 6k)(∇b∇ − 3k) ×

.. ˙ −1 ˙ ˜ ˜ a − Eij − 2kEij − 2ΩΩ Eij + ∇a∇ Eij = 0. (4.112) 97

Thus with (4.99), (4.100), (4.103), (4.112) together with (4.85), (4.94) and (4.95)

we have succeeded in decomposing the ﬂuctuation equations for the components,

with the various components obeying derivative equations that are higher than

second order.

˙ −1 ˙ With (4.99) only involving α+2ΩΩ γ, with (4.103) only involving Bi −Ei, and with (4.112) only involving Eij, and with all components of ∆µν being gauge

˙ −1 ˙ invariant, we recognize C = α+2ΩΩ γ, Bi −Ei and Eij as being gauge invariant.

˙ With Bi − Ei being gauge invariant, from (4.95) we recognize Vi as being gauge

invariant too. While we have identiﬁed some gauge-invariant quantities we note

that by manipulating ∆µν so as to obtain derivative expressions in which each of these quantities appears on its own, we cannot establish the gauge invariance of all 11 of the ﬂuctuation variables this way since ∆µν only has 10 components.

However, just as with fluctuations around flat spacetime, in analog to (3.12) below we shall obtain derivative relations between the SVT3 fluctuations and the hµν

ﬂuctuations by manipulating (4.74). As we show below, this will enable us to establish the gauge invariance of the remaining ﬂuctuation quantities.

Decomposition Theorem Requirements

To get a decomposition theorem for ∆µν = 0 we would require

˙ 2 −2 2 ˙ −1 ˜ ˜ a 0 = 6Ω Ω (α − γ˙ ) + δρˆΩ + 2ΩΩ ∇a∇ γ,

.. ˙ −1 ˜ ˜ ˙ 2 −3 −2 −1 ˜ ˆ 0 = −2ΩΩ ∇i(α − γ˙ ) + 2k∇iγ + (−4Ω Ω + 2ΩΩ − 2kΩ )∇iV, 98

.. ˙ 1 ˜ ˜ a ˙ ˙ 2 −3 −2 −1 0 = k(Bi − Ei) + 2 ∇a∇ (Bi − Ei) + (−4Ω Ω + 2ΩΩ − 2kΩ )Vi,

˙ 2 −2 ˙ −1 ¨ −1 2 0 =γ ˜ij 2Ω Ω (α − γ˙ ) − 2ΩΩ (α ˙ − γ¨) − 4ΩΩ (α − γ˙ ) + Ω δpˆ

˜ ˜ a ˙ −1 ˜ ˜ ˙ −1 −∇a∇ (α + 2ΩΩ γ) + ∇i∇j(α + 2ΩΩ γ),

˙ −1 ˜ ˙ 1 ˜ ˙ ¨ ˙ −1 ˜ ˙ 1 ˜ ˙ ¨ 0 = ΩΩ ∇i(Bj − Ej) + 2 ∇i(Bj − Ej) + ΩΩ ∇j(Bi − Ei) + 2 ∇j(Bi − Ei),

.. ˙ ˙ −1 ˜ ˜ a 0 = −Eij − 2kEij − 2EijΩΩ + ∇a∇ Eij,

0 = 6Ω˙ 2Ω−2(α − γ˙ ) − 6ΩΩ˙ −1(α ˙ − γ¨) − 12ΩΩ¨ −1(α − γ˙ ) + 3Ω2δpˆ

˜ ˜ a ˙ −1 −2∇a∇ (α + 2ΩΩ γ),

.. −3 ˙ −3 −2 ˜ ˜ a ˙ −1 0 = 3δpˆ − δρˆ − 12ΩΩ (α − γ˙ ) − 6ΩΩ (α ˙ − γ¨) − 2Ω ∇a∇ (α + 3ΩΩ γ).

(4.113)

˜ ˜ Withγ ˜ij and ∇i∇j not being equal to each other, we would immediately obtain

2Ω˙ 2Ω−2(α − γ˙ ) − 2ΩΩ˙ −1(α ˙ − γ¨) − 4ΩΩ¨ −1(α − γ˙ ) + Ω2δpˆ = 0,

α + 2ΩΩ˙ −1γ = 0. (4.114)

We recognize the equations for the components of the ﬂuctuations as being deriva-

tives of the relations that are required of the decomposition theorem. We thus

need to see if we can ﬁnd boundary conditions that would force the solutions to

the higher-derivative ﬂuctuation equations to obey (4.113) and (4.114). 99

Gauge Invariance

2 ij ˜ ˜ i Starting with (4.74), setting hµν = Ω (τ)fµν, f =γ ˜ fij = −6ψ + 2∇i∇ E and taking appropriate derivatives, then following quite a bit of algebra we obtain

1 (3k + ∇˜ b∇˜ )∇˜ a∇˜ α = − (3k + ∇˜ b∇˜ )∇˜ i∇˜ f b a 2 b i 00 1 + ∇˜ a∇˜ −2kf − ∇˜ b∇˜ f + ∇˜ m∇˜ nf 4 a b mn 1 +∂ (3k + ∇˜ b∇˜ )∇˜ if − ∂2 3∇˜ m∇˜ nf − ∇˜ a∇˜ f , 0 b 0i 4 0 mn a (4.115)

1 (3k + ∇˜ b∇˜ )∇˜ a∇˜ γ = − ΩΩ˙ −1∇˜ a∇˜ −2kf − ∇˜ b∇˜ f + ∇˜ m∇˜ nf b a 4 a b mn 1 + (3k + ∇˜ b∇˜ )∇˜ if − ∂ 3∇˜ m∇˜ nf b 0i 4 0 mn ˜ a ˜ −∇ ∇af , (4.116)

˜ a ˜ ˜ i ˜ ˙ ˜ i ˜ ˜ a ˜ (∇ ∇a − 2k)(∇ ∇i + 2k)(Bj − Ej) = (∇ ∇i + 2k)(∇ ∇af0j − 2kf0j

˜ ˜ a ˜ a ˜ ˜ i −∇j∇ f0a) − ∂0∇ ∇a∇ fij

˜ ˜ a ˜ b ˜ i +∂0∇j∇ ∇ fab + 2k∂0∇ fij,(4.117)

˜ c ˜ ˜ b ˜ ˜ a ˜ ˜ d ˜ 2(3k + ∇ ∇c)(−3k + ∇ ∇b)(∇ ∇a − 6k)(∇ ∇d − 2k)Eij

˜ c ˜ ˜ b ˜ ˜ a ˜ ˜ d ˜ = (3k + ∇ ∇c)(−3k + ∇ ∇b)(∇ ∇a − 6k)(∇ ∇d − 2k)fij 1 + (−3k + ∇˜ c∇˜ )(∇˜ b∇˜ − 6k)(∇˜ a∇˜ − 2k)(−2kf − ∇˜ d∇˜ f + ∇˜ m∇˜ nf )˜γ 2 c b a d mn ij 100 " 1 −2(∇˜ c∇˜ − 2k) (3k + ∇˜ b∇˜ )∇˜ ∇˜ 3∇˜ m∇˜ nf − ∇˜ a∇˜ f c 4 b i j mn a 3 −kγ˜ ∇˜ ∇˜ d((3k + ∇˜ e∇˜ )f + (−2kf − ∇˜ g∇˜ f + ∇˜ m∇˜ nf )) ij d e 2 g mn # 3 −kγ˜ (−3k + ∇˜ h∇˜ )((3k + ∇˜ m∇˜ )f + (−2kf − ∇˜ n∇˜ f + ∇˜ m∇˜ nf )) ij h m 2 n mn

1 − (−3k + ∇˜ c∇˜ )(3k + ∇˜ b∇˜ ) ∇˜ ∇˜ a∇˜ (3∇˜ df − ∇˜ f) 3 c b i a dj j ˜ ˜ d ˜ ˜ b ˜ ˜ ˜ ˜ b ˜ c ˜ e ˜ +∇j∇ ∇d(3∇ fbi − ∇if) − 2∇i∇j(3∇ ∇ fbc − ∇ ∇ef) ! ˜ ˜ a ˜ ˜ ˜ a ˜ −2k∇i(3∇ faj − ∇jf) − 2k∇j(3∇ fai − ∇if) . (4.118)

Despite its somewhat forbidding appearance (4.118) is actually a derivative of

˜ a ˜ ˜ b ˜ ˜ a ˜ ˜ b ˜ 2(∇ ∇a − 2k)(∇ ∇b − 3k)Eij = (∇ ∇a − 2k)(∇ ∇b − 3k)fij h i 1 ˜ ˜ ˜ a ˜ b ˜ a ˜ ˜ a ˜ ˜ ˜ b ˜ ˜ b + 2 ∇i∇j ∇ ∇ fab + (∇ ∇a + 4k)f − (∇ ∇a − 3k)(∇i∇ fjb + ∇j∇ fib) h i 1 ˜ a ˜ ˜ b ˜ c ˜ ˜ a ˜ ˜ b ˜ a ˜ a 2 + 2 γ˜ij (∇ ∇a − 4k)∇ ∇ fbc − (∇a∇ ∇b∇ − 2k∇ ∇ + 4k )f , (4.119) a relation that itself can be derived from the D = 3 version of (4.198) with H2 = k

˜ a ˜ ˜ a ˜ by application of (∇ ∇a − 2k)(∇ ∇a − 3k) to (4.198). One can check the validity of these relations by inserting

2 2 ˜ h0i = Ω (τ)f0i = Ω (τ)[∇iB + Bi], (4.120)

2 2 ˜ ˜ ˜ ˜ hij = Ω (τ)fij = Ω (τ)[−2ψγ˜ij + 2∇i∇jE + ∇iEj + ∇jEi + 2Eij]

into them. And one can check their gauge invariance by inserting hµν → hµν −

˙ ∇µν −∇µµ into them. We thus establish that the metric ﬂuctuations α, γ, Bi−Ei and Eij are gauge invariant. And from (4.85), (4.94), (4.95) and (4.100) can thus 101

ˆ establish that the matter ﬂuctuations δρˆ, V , Vi and δpˆ are gauge invariant too.

Interestingly, we see that in going from fluctuations around flat to fluctuations

around Robertson-Walker with arbitrary k and arbitrary dependence of Ω(τ) on

˙ τ the gauge-invariant metric ﬂuctuation combinations α, γ, Bi −Ei and Eij remain the same, though γ does depend generically on Ω(τ).

Background Solution

In order to actually solve the fluctuation equations we will need to determine the appropriate background Ω(τ), and we will also need to deal with the fact that, as noted above, the fluctuation equations contain more degrees of freedom (11) than there are evolution equations (10). For the background first we note that no matter what the value of k, from (4.82) we see that if ρ = 3p then ρ = 3/Ω4, as

written in a convenient normalization (one which diﬀers from the one used in Sec.

4.1.2), while if p = 0 we have ρ = 3/Ω3. Once we specify a background equation of state that relates ρ and p we can solve for Ω(τ) and t = R Ω(τ)dτ. We thus

obtain

p = ρ/3, k = 0 : Ω = τ, p = 1/τ 4, ρ = 3/τ 4, t = τ 2/2,

a(t) = Ω(τ) = (2t)1/2,

p = ρ/3, k = −1 : Ω = sinh τ, p = 1/ sinh4 τ, ρ = 3/ sinh4 τ,

t = cosh τ, a(t) = Ω(τ) = (t2 − 1)1/2,

p = ρ/3, k = +1 : Ω = sin τ, p = 1/ sin4 τ, ρ = 3/ sin4 τ, t = − cos τ, 102

a(t) = Ω(τ) = (1 − t2)1/2,

p = 0, k = 0 : Ω = τ 2/4, p = 0, ρ = 192/τ 6, t = τ 3/12,

a(t) = Ω(τ) = (3t/2)2/3,

p = 0, k = −1 : Ω = sinh2(τ/2), p = 0, ρ = 3/ sinh6(τ/2),

1 t = 2 [sinh τ − τ], a(t) = Ω(τ),

p = 0, k = 1 : Ω = sin2(τ/2), p = 0, ρ = 3/ sin6(τ/2),

1 t = 2 [τ − sin τ], a(t) = Ω(τ). (4.121)

For p = 0 and k = ±1 we cannot obtain a(t) in a closed form. Consequently one ordinarily only determines a(t) in parametric form. As we see, the conformal time

τ can serve as the appropriate parameter.

Relation between δρ and δp

To reduce the number of fluctuation variables from 11 to 10 we follow kinetic theory, and first consider a relativistic flat spacetime ideal N particle classical gas of spinless particles each of mass m in a volume V at a temperature T . As discussed for instance in [10], for this system one can use a basis of momentum eigenmodes, with the Helmholtz free energy A(V,T ) being given as

Z e−A(V,T )/NkT = V d3pe−(p2+m2)1/2/kT , (4.122) so that the pressure takes the simple form

∂A NkT p = − = , (4.123) ∂V T V 103

while the internal energy U = ρV evaluates in terms of Bessel functions as

∂A K (m/kT ) U = A − T = 3NkT + Nm 1 . (4.124) ∂T V K2(m/kT )

In the high and low temperature limits (the radiation and matter eras) we then

ﬁnd that the expression for U simpliﬁes to

U 3NkT m ρ = → = 3p, → 0, V V kT U Nm 3NkT Nm 3p Nm m ρ = → + = + ≈ , → ∞. (4.125) V V 2V V 2 V kT

Consequently, while p and ρ are nicely proportional to each other (p = wρ) in the

high temperature radiation and the low temperature matter eras (where w(T →

∞) = 1/3 and w(T → 0) = 0), we also see that in transition region between the two eras their relationship is altogether more complicated. Since such a transition era occurs fairly close to recombination, it is this complicated relation that should be used there. Use of a p = wρ equation of state would at best only be valid at temperatures which are very diﬀerent from those of order m/K, though for

massless particles it would be of course be valid to use p = ρ/3 at all temperatures.

As derived, these expressions only hold in a ﬂat Minkowski spacetime. How-

ever, A(V,T ) only involves an integration over the spatial 3-momentum. Thus for

the spatially ﬂat k = 0 Robertson-Walker metric ds2 = dt2 − a2(t)(dr2 + r2dθ2 +

r2 sin2 θdφ2), all of these kinetic theory relations will continue to hold with T

taken to depend on the comoving time t. (Typically T ∼ 1/a(t).) Suppose we

now perturb the system and obtain a perturbed δT that now depends on both t 104 and r, θ, φ. In the radiation era where p = ρ/3 we would obtain δp = δρ/3 (and thus 3δpˆ = δρˆ). In the matter era where p = 0, from (4.125) we would obtain

δp = 2δρ/3. In the intermediate region the relation would be much more complicated. Nonetheless in all cases we would have reduced the number of independent

ﬂuctuation variables, though we note that not just in the radiation and matter eras but even in the intermediate region, it is standard in cosmological perturbation theory to use δp/δρ = v2 where v2 is taken to be a spacetime-independent constant.

While we can use the above A(V,T ) for spatially ﬂat cosmologies with k = 0, for spatially curved cosmologies with non-zero k we cannot use a mode basis made out of 3-momentum eigenstates at all. One has to adapt the basis to a curved

2 2 1/2 µ ν 3-space by replacing (p + m ) /kT by (dx /dτ)U gµν/kT (see e.g. [10]), while replacing R d3p by a sum over a complete set of basis modes associated with the propagation of a spinless massive particle in the chosen gµν background, and then follow the steps above to see what generalization of (4.125) might then ensue.

While tractable in principle it is not straightforward to do this in practice, and we will not do it here. While one would need to do this in order to obtain a k 6= 0 generalization of the k = 0 δp/δρ = v2 relation, and while such a generalization would be needed in order to solve the k 6= 0 ﬂuctuation equations completely, since our purpose here is only to test for the validity of the decomposition theorem, we will not actually need to ﬁnd a relation between δp and δρ, since as we now see, we 105

will be able to test for the validity of the decomposition theorem without actually

needing to know the speciﬁc form of such a relation at all, or even needing to

specify any particular form for the background Ω(τ) either for that matter.

4.1.4 Robertson Walker k = −1 The Scalar Sector

We have seen that the scalar sector evolution equations (4.94), (4.98), (4.99) and

(4.100) involve derivatives of the form ∇˜ 2, ∇˜ 2 + 3k where the coeﬃcient of k is

either zero or positive, while the vector and tensor sectors equations (4.95), (4.103)

and (4.112) also involve derivatives such as ∇˜ 2 −2k, ∇˜ 2 −3k, and ∇˜ 2 −6k in which

the coeﬃcient of k is negative. As the implications of boundary conditions are

very sensitive to the sign of the coeﬃcient of k, and we will need to monitor both

positive and negative coeﬃcient cases below. In implementing evolution equations

that involve products of derivative operators such as the generic (∇˜ 2 + α)(∇˜ 2 +

β)F = 0 (F denotes scalar, vector or tensor), we can satisfy these relations by

(∇˜ 2 + α)F = 0, by (∇˜ 2 + β)F = 0, or by F = 0. The decomposition theorem will

only follow if boundary conditions prevent us from satisfying (∇˜ 2 + α)F = 0 or

(∇˜ 2 + β)F = 0 with non-zero F , leaving F = 0 as the only remaining possibility.

It is the purpose of this section to explore whether or not boundary conditions do force us to F = 0 in any of the scalar, vector or tensor sectors. While a decomposition theorem would immediately hold if they do, as we will show in

Sec. 4.1.4 the interplay of the vector and tensor sectors in the ∆ij = 0 relation 106

given in (4.87) will still force us to a decomposition theorem even if they do not.

To illustrate what is involved it is suﬃcient to restrict k to k = −1, and to

take the metric to be of the form

ds2 = Ω2(τ) dτ 2 − dχ2 − sinh2 χdθ2 − sinh2 χ sin2 θdφ2 , (4.126)

where r = sinh χ. Since the analysis leading to the structure for ∆µν given in (4.85)

to (4.89) is completely covariant these equations equally hold if we represent the

spatial sector of the metric as given in (4.126). With k = −1 the scalar sector

evolution equations involving the ∇˜ 2 and ∇˜ 2 − 3 operators take the form

˜ ˜ a (∇a∇ + AS)S = 0. (4.127)

(Here S is to denote the full combinations of scalar sector components that appear

in (4.94), (4.98), (4.99) and (4.100).) In (4.127) we have introduced a generic scalar

sector constant AS, whose values in (4.94), (4.98), (4.99) and (4.100) are (0, −3).

m On setting S(χ, θ, φ) = S`(χ)Y` (θ, φ)(4.127) reduces to

d2 cosh χ d `(` + 1) + 2 − + A S = 0. (4.128) dχ2 sinh χ dχ sinh2 χ S `

In the χ → ∞ and χ → 0 limits we take the solution to behave as eλχ (times an

irrelevant polynomial in χ), and as χn, to thus obtain

2 1/2 λ + 2λ + AS = 0, λ = −1 ± (1 − AS) ,

λ(AS = 0) = (−2, 0), λ(AS = −3) = (−3, 1),

n(n − 1) + 2n − `(` + 1) = 0, n = `, −` − 1. (4.129) 107

For each of AS = 0 and AS = −3 one solution converges at χ = ∞ and the other diverges at χ = ∞. Thus we need to see how they match up with the solutions at χ = 0, where one solution is well-behaved and the other is not.

So to this end we look for exact solutions to (4.128). Thus, as discussed for instance in [31, 44], and as appropriately generalized here, (4.128) admits of solutions (known as associated Legendre functions) of the form

1 d `+1 S = sinh` χ f(χ), (4.130) ` sinh χ dχ

where f(χ) obeys

d3 d + ν2 f(χ) = 0, ν2 = A − 1, (4.131) dχ3 dχ S

with f(χ) thus obeying

f(ν2 > 0) = cos νχ, sin νχ, f(ν2 = −µ2 < 0) = cosh µχ, sinh µχ,

f(ν2 = 0) = χ, χ2. (4.132)

For each f(χ) this would lead to solutions of the form

1 df dSˆ Sˆ = , Sˆ = 0 , 0 sinh χ dχ 1 dχ " # " # d Sˆ d Sˆ Sˆ = sinh χ 1 , Sˆ = sinh2 χ 2 , .... (4.133) 2 dχ sinh χ 3 dχ sinh2 χ

However, on evaluating these expressions it can happen that some of these solutions vanish. Thus for AS = 0 for instance where f(χ) = (sinh χ, cosh χ) the two

solutions with ` = 0 are cosh χ/ sinh χ and 1. However this would lead to the two 108

solutions with ` = 1 being 1/ sinh2 χ and 0. To address this point we note that

ˆ suppose we have obtained some non-zero solution S`. Then, a second solution of

ˆ ˆ ˆ ˆ the form f`(χ)S`(χ) may be found by inserting f`(χ)S`(χ) into (4.128), to yield

d2fˆ cosh χ dfˆ dSˆ dfˆ Sˆ ` + 2Sˆ ` + 2 ` ` = 0, (4.134) ` dχ2 ` sinh χ dχ dχ dχ

which integrates to

ˆ Z df` 1 ˆ ˆ ˆ dχ = , f`S` = S` . (4.135) dχ 2 ˆ2 2 ˆ2 sinh χS` sinh χS`

ˆ 2 Thus for ` = 1, from the non-trivial AS = 0 solution S1 = 1/ sinh χ we obtain

ˆ ˆ 2 a second solution of the form f`S` = cosh χ/ sinh χ − χ/ sinh χ. However, once

we have this second solution we can then return to (4.133) and use it to obtain

the subsequent solutions associated with higher ` values, since use of the chain in

(4.133) only requires that at any point the elements in it are solutions regardless

of how they may or may not have been found.

Having the form given in (4.135) is useful for another purpose, as it allows

us to relate the behaviors of the solutions in the χ → ∞ and χ → 0 limits. Thus

ˆ λχ ` ˆ ˆ suppose that S` behaves as e and as χ in these two limits. Then f`S` must

−(λ+2)χ −`−1 ˆ λχ behave as e and χ in the two limits. Alternatively, if S` behaves as e

−`−1 ˆ ˆ −(λ+2)χ ` and as χ in these two limits, then f`S` must behave as e and χ in the

1/2 two limits. Comparing with (4.129), we note that if we set λ = −1 ± (1 − AS)

1/2 then consistently we ﬁnd that −(λ+2) = −1∓(1−AS) . However, this analysis

shows that we cannot directly identify which χ → ∞ behavior is associated with 109

which χ → 0 behavior (the insertion of either χ` or χ−`−1 into (4.135) generates

ˆ ˆ ˆ the other, with both behaviors thus being required in any S`, f`S` pair), and to determine which is which we thus need to construct the asymptotic solutions directly.

2 For AS = 0, ν = i, the relevant f(ν ) given in (4.132) are cosh χ and sinh χ.

(i) ˜ ˜ a Consequently, we ﬁnd the ﬁrst few S` , i = 1, 2 solutions to ∇a∇ S = 0 to be of the form

cosh χ Sˆ(1)(A = 0) = , Sˆ(2)(A = 0) = 1, 0 S sinh χ 0 S 1 cosh χ χ Sˆ(1)(A = 0) = , Sˆ(2)(A = 0) = − , 1 S sinh2 χ 1 S sinh χ sinh2 χ cosh χ 3 3χ cosh χ Sˆ(1)(A = 0) = , Sˆ(2)(A = 0) = 1 + − , 2 S sinh3 χ 2 S sinh2 χ sinh3 χ 4 5 Sˆ(1)(A = 0) = + , 3 S sinh2 χ sinh4 χ 2 cosh χ 15 cosh χ 12χ 15χ Sˆ(2)(A = 0) = + − − . (4.136) 3 S sinh χ sinh3 χ sinh2 χ sinh4 χ

From this pattern we see that the solutions that are bounded at χ = ∞ are badly-behaved at χ = 0, while the solutions that are well-behaved at χ = 0 are

unbounded at χ = ∞. Thus all of these AS = 0 solutions are excluded by a

requirement that solutions be bounded at χ = ∞ and be well-behaved at χ = 0.

2 For AS = −3, ν = 2i, the relevant f(ν ) given in (4.132) are cosh 2χ and

˜ ˜ a sinh 2χ. Consequently, the ﬁrst few solutions to (∇a∇ − 3)S = 0 are of the form

1 Sˆ(1)(A = −3) = cosh χ, Sˆ(2)(A = −3) = 2 sinh χ + , 0 S 0 S sinh χ 110

cosh χ Sˆ(1)(A = −3) = sinh χ, Sˆ(2)(A = −3) = 2 cosh χ − , 1 S 1 S sinh2 χ 3 cosh χ 3χ 1 Sˆ(1)(A = −3) = 2 cosh χ − + , Sˆ(2)(A = −3) = , 2 S sinh2 χ sinh3 χ 2 S sinh3 χ 5 15 15χ cosh χ Sˆ(1)(A = −3) = 2 sinh χ − − + , 3 S sinh χ sinh3 χ sinh4 χ cosh χ Sˆ(2)(A = −3) = . (4.137) 3 S sinh4 χ

From this pattern we again see that the solutions that are bounded at χ = ∞ are badly-behaved at χ = 0, while the solutions that are well-behaved at χ = 0 are unbounded at χ = ∞. Thus all of these AS = −3 solutions are also excluded by a requirement that solutions be bounded at χ = ∞ and be well-behaved at χ = 0.

With all of these AS = 0, AS = −3 solutions being excluded we must realize

(4.94), (4.98), (4.99) and (4.100) by

.. −2ΩΩ˙ −1(α − γ˙ ) + 2kγ + (−4Ω˙ 2Ω−3 + 2ΩΩ−2 − 2kΩ−1)Vˆ = 0, (4.138)

2Ω˙ 2Ω−2(α − γ˙ ) − 2ΩΩ˙ −1(α ˙ − γ¨) − 4ΩΩ¨ −1(α − γ˙ ) + Ω2δpˆ + 2k(α + 2ΩΩ˙ −1γ)]

= 0, (4.139)

α + 2ΩΩ˙ −1γ = 0, (4.140)

2Ω˙ 2Ω−2(α − γ˙ ) − 2ΩΩ˙ −1(α ˙ − γ¨) − 4ΩΩ¨ −1(α − γ˙ ) + Ω2δpˆ = 0. (4.141)

These equations are augmented by (4.85), (4.88) and (4.89)

˙ 2 −2 2 ˙ −1 ˜ ˜ a 6Ω Ω (α − γ˙ ) + δρˆΩ + 2ΩΩ ∇a∇ γ = 0, (4.142) 111

6Ω˙ 2Ω−2(α − γ˙ ) − 6ΩΩ˙ −1(α ˙ − γ¨) − 12ΩΩ¨ −1(α − γ˙ ) + 3Ω2δpˆ

˜ ˜ a ˙ −1 −2∇a∇ (α + 2ΩΩ γ) = 0, (4.143)

.. 3δpˆ − δρˆ − 12ΩΩ−3(α − γ˙ ) − 6ΩΩ˙ −3(α ˙ − γ¨)

−2 ˜ ˜ a ˙ −1 −2Ω ∇a∇ (α + 3ΩΩ γ) = 0. (4.144)

˜ On taking the ∇i derivative of (4.138), we recognize (4.138) to (4.144) as precisely

being the scalar sector ones given in (4.113) and (4.114). We thus establish the

decomposition theorem in the scalar sector.

The Vector Sector

To determine the structure of k = −1 solutions to the vector sector (4.95) and

˜ ˜ a i i (4.103), we ﬁrst need to evaluate the quantity ∇a∇ V , where V obeys the transverse condition

V cos θ 2V cosh χ ∂ V ∂ V ∇˜ V a = 2 + 1 + ∂ V + 2 2 + 3 3 a sin θ sinh2 χ sinh χ 1 1 sinh2 χ sin2 θ sinh2 χ = 0. (4.145)

˜ ˜ a i On implementing this condition, the (χ, θ, φ) ≡ (1, 2, 3) components of ∇a∇ V

take the form

2 4 cosh χ∂ V cos θ∂ V ∂ ∂ V ∇˜ ∇˜ aV 1 = V 2 + + 1 1 + ∂ ∂ V + 2 1 + 2 2 1 a 1 sinh2 χ sinh χ 1 1 1 sin θ sinh2 χ sinh2 χ ∂ ∂ V + 3 3 1 , sin2 θ sinh2 χ 2 1 2 4V cos θ cosh χ ∇˜ ∇˜ aV 2 = V − + − + 1 a 2 sinh4 χ sin2 θ sinh4 χ sinh2 χ sin θ sinh3 χ 112

2 cos θ∂ V ∂ ∂ V 2 cosh χ∂ V 3 cos θ∂ V + 1 1 + 1 1 2 + 2 1 + 2 2 sin θ sinh2 χ sinh2 χ sinh3 χ sin θ sinh4 χ ∂ ∂ V ∂ ∂ V + 2 2 2 + 3 3 2 , sinh4 χ sin2 θ sinh4 χ 2V ∂ ∂ V cos θ∂ V ∂ ∂ V ∇˜ ∇˜ aV 3 = − 3 + 1 1 3 − 2 3 + 2 2 3 a sin2 θ sinh2 χ sin2 θ sinh2 χ sin3 θ sinh4 χ sin2 θ sinh4 χ 2 cosh χ∂ V 2 cos θ∂ V ∂ ∂ V + 3 1 + 3 2 + 3 3 3 . (4.146) sin2 θ sinh3 χ sin3 θ sinh4 χ sin4 θ sinh4 χ

To explore the structure of the k = −1 vector sector we seek solutions to

˜ ˜ a (∇a∇ + AV )Vi = 0. (4.147)

(Here Vi is to denote the full combinations of vector components that appear in

(4.95) and (4.103).) In (4.147) we have introduced a generic vector sector constant

AV , whose values in (4.95) and (4.103) are (2, −1, −2).

Conveniently, we ﬁnd that the equation for V1 involves no mixing with V2

m or V3, and can thus be solved directly. On setting V1(χ, θ, φ) = g1,`(χ)Y` (θ, φ),

the equation for V1 reduces to

d2 cosh χ d 2 `(` + 1) + 4 + 2 + A + − g = 0. (4.148) dχ2 sinh χ dχ V sinh2 χ sinh2 χ 1,`

To check the χ → ∞ and χ → 0 limits, we take the solutions to behave as eλχ

(times an irrelevant polynomial in χ) and χn in these two limits. For (4.148) the limits give

2 !/2 λ + 4λ + 2 + AV = 0, λ = −2 ± (2 − AV ) , λ(AV = 2) = (−2, − 2), √ λ(AV = −1) = −2 ± 3, λ(AV = −2) = (0, −4),

n(n − 1) + 4n + 2 − `(` + 1) = 0, n = ` − 1, −` − 2. (4.149) 113

Thus for AV = 2 and AV = −1 both solutions are bounded at inﬁnity, while for

AV = −2 one solution is bounded at inﬁnity. Moreover, for each value of AV

one of the solutions will be well-behaved as χ → 0 for any ` ≥ 1 while the other

solution will not be. Thus for AV = 2 there will always be one ` ≥ 1 solution that

is bounded at χ = ∞ and well-behaved at χ = 0. To determine whether we can

obtain a solution that is bounded in both limits for AV = −1, AV = −2 we need to explicitly ﬁnd the solutions in closed form.

To this end we need to put (4.148) into the form of a diﬀerential equation whose solutions are known. We thus set g1,` = α`/ sinh χ, to ﬁnd that (4.148)

takes the form

d2 cosh χ d `(` + 1) + 2 − + A − 1 α = 0. (4.150) dχ2 sinh χ dχ sinh2 χ V `

We recognize (4.150) as being in the form given in (4.128), which we discussed

2 above, with ν = AV − 2.

2 2 (i) Thus for AV = 2, viz. ν = 0 in (4.132) and f(ν = 0) = χ, χ , we ﬁnd V` ,

˜ ˜ a i = 1, 2 solutions to (∇a∇ + 2)V1 = 0 of the form

1 χ Vˆ (1)(A = 2) = , Vˆ (2)(A = 2) = , 0 V sinh2 χ 0 V sinh2 χ cosh χ 1 χ cosh χ Vˆ (1)(A = 2) = , Vˆ (2)(A = 2) = − , 1 V sinh3 χ 1 V sinh2 χ sinh3 χ 2 3 Vˆ (1)(A = 2) = + , 2 V sinh2 χ sinh4 χ 3 cosh χ 2χ 3χ Vˆ (2)(A = 2) = − − , 2 V sinh3 χ sinh2 χ sinh4 χ 2 cosh χ 5 cosh χ Vˆ (1)(A = 2) = + , 3 V sinh3 χ sinh5 χ 114

11 15 6χ cosh χ 15χ cosh χ Vˆ (2)(A = 2) = + − − . (4.151) 3 V sinh2 χ sinh4 χ sinh3 χ sinh5 χ

ˆ (2) The just as required by (4.149), the V` (AV = 2) solutions with ` ≥ 1 are bounded at χ = ∞ and well-behaved at χ = 0. Since they can thus not be excluded by boundary conditions at χ = ∞ and χ = 0 (though boundary conditions do

exclude modes with ` = 0), solutions to (4.95) and (4.103) do not become the

vector sector solutions associated with (4.113). Thus if we implement (4.103) by

˜ ˜ a (∇a∇ + 2)Vi = 0, the decomposition theorem will fail in the vector sector for

modes with ` ≥ 1. Thus an equation such as (4.103) will be solved by

h i ˜ ˜ a ˜ ˜ b 1 ˙ ¨ ˙ −1 ˙ (∇a∇ − 1)(∇b∇ − 2) 2 (Bi − Ei) + ΩΩ (Bi − Ei) = Vi, (4.152) and not by

1 ˙ ¨ ˙ −1 ˙ 2 (Bi − Ei) + ΩΩ (Bi − Ei) = 0. (4.153)

˙ Thus (4.103) is solved by the χ dependence of Bi −Ei and not by its τ dependence,

˙ 2 i.e., not by the Bi − Ei = 1/Ω dependence on τ that one would have obtained

from the decomposition-theorem-required (4.153). This then raises the question

of what does ﬁx the τ dependence in the vector sector. We will address this issue

below.

2 2 For AV = −2 we see that ν = −4 and that f(ν ) = cosh 2χ, sinh 2χ.

2 2 However in the scalar case discussed above where ν = AS − 1, ν would also

2 obey ν = −4 if AS = −3. Thus for AV = −2 we can obtain the solutions to

˜ ˜ a (∇a∇ − 2)V1 = 0 directly from (4.137), and after implementing g1,` = α`/ sinh χ 115

we obtain

cosh χ 1 Vˆ (1)(A = −2) = , Vˆ (2)(A = −2) = 2 + , 0 V sinh χ 0 V sinh2 χ cosh χ cosh χ Vˆ (1)(A = −2) = 1, Vˆ (2)(A = −2) = 2 − , 1 V 1 V sinh χ sinh3 χ cosh χ 3 cosh χ 3χ 1 Vˆ (1)(A = −2) = 2 − + , Vˆ (2)(A = −2) = , 2 V sinh χ sinh3 χ sinh4 χ 2 V sinh4 χ 5 15 15χ cosh χ Vˆ (1)(A = −2) = 2 − − + , 3 V sinh2 χ sinh4 χ sinh5 χ cosh χ Vˆ (2)(A = −2) = . (4.154) 3 V sinh5 χ

ˆ (2) ˆ (2) As required by (4.149), the V2 (AV = −2) and V3 (AV = −2) solutions are bounded at χ = ∞. However, they are not well-behaved at χ = 0. Since they thus can be excluded by boundary conditions at χ = ∞ and χ = 0, if we implement

˜ ˜ a (4.103) by (∇a∇ − 2)Vi = 0, the only allowed solution will be Vi = 0, and the decomposition theorem will then follow.

√ √ √ 2 χ 3 −χ 3 Finally, for AV = −1, viz. ν = i 3, f(ν ) = e , e , the solutions to

˜ ˜ a (∇a∇ − 1)V1 = 0 are of the form

√ √ eχ 3 e−χ 3 Vˆ (1)(A = −1) = , Vˆ (2)(A = −1) = , 0 V sinh2 χ 0 V sinh2 χ √ eχ 3 √ Vˆ (1)(A = −1) = [ 3 sinh χ − cosh χ] , 1 V sinh3 χ √ e−χ 3 √ Vˆ (2)(A = −1) = [− 3 sinh χ − cosh χ] , 1 V sinh3 χ √ eχ 3 √ Vˆ (1)(A = −1) = 3 − 3 3 cosh χ sinh χ + 5 sinh2 χ , 2 V sinh4 χ √ e−χ 3 √ Vˆ (2)(A = −1) = 3 + 3 3 cosh χ sinh χ + 5 sinh2 χ , 2 V sinh4 χ √ eχ 3 √ √ Vˆ (1)(A = −1) = 15 3 sinh χ + 14 3 sinh3 χ − 15 cosh χ 3 V sinh5 χ 116

−24 cosh χ sinh2 χ , √ e−χ 3 √ √ Vˆ (2)(A = −1) = − 15 3 sinh χ − 14 3 sinh3 χ − 15 cosh χ 3 V sinh5 χ −24 cosh χ sinh2 χ . (4.155)

ˆ (1) ˆ (2) All of these solutions are bounded at χ = ∞ and all V` (AV = −1) − V` (AV =

−1) with ` ≥ 1 are well-behaved at χ = 0. Thus if we implement (4.103) by

˜ ˜ a (∇a∇ − 1)Vi = 0, we are not forced to Vi = 0, with the decomposition theorem

not then following in this sector.

The Tensor Sector

For k = −1 the transverse-traceless tensor sector modes need to satisfy

T T γ˜abT = T + 22 + 33 = 0, ab 11 sinh2 χ sin2 θ sinh2 χ cosh χT cosh χT cos θT 2 cosh χT ∇˜ T a1 = − 22 − 33 + 12 + 11 + ∂ T a sinh3 χ sin2 θ sinh3 χ sin θ sinh2 χ sinh χ 1 11 ∂ T ∂ T + 2 12 + 3 13 = 0, sinh2 χ sin2 θ sinh2 χ cos θT cos θT 2 cosh χT ∂ T ∂ T ∇˜ T a2 = − 33 + 22 + 12 + 1 12 + 2 22 a sin3 θ sinh4 χ sin θ sinh4 χ sinh3 χ sinh2 χ sinh4 χ ∂ T + 3 23 = 0, sin2 θ sinh4 χ cos θT 2 cosh χT ∂ T ∂ T ∇˜ T a3 = 23 + 13 + 1 13 + 2 23 a sin3 θ sinh4 χ sin2 θ sinh3 χ sin2 θ sinh2 χ sin2 θ sinh4 χ ∂ T + 3 33 = 0. (4.156) sin4 θ sinh4 χ

˜ ˜ a ij Under these conditions the components of ∇a∇ T evaluate to

6 6 cosh χ∂ T cos θ∂ T ∇˜ ∇˜ aT 11 = T 6 + + 1 11 + ∂ ∂ T + 2 11 a 11 sinh2 χ sinh χ 1 1 11 sin θ sinh2 χ ∂ ∂ T ∂ ∂ T + 2 2 11 + 3 3 11 , sinh2 χ sin2 θ sinh2 χ 117

4T 4T 4T 2T 2T ∇˜ ∇˜ aT 22 = 22 − 22 + 11 − 22 − 11 a sinh6 χ sin2 θ sinh6 χ sinh4 χ sinh4 χ sin2 θ sinh4 χ 2T 2 cosh χ∂ T ∂ ∂ T 4 cosh χ∂ T + 11 − 1 22 + 1 1 22 + 2 12 sinh2 χ sinh5 χ sinh4 χ sinh5 χ cos θ∂ T ∂ ∂ T 4 cos θ∂ T ∂ ∂ T + 2 22 + 2 2 22 − 3 23 + 3 3 22 , sin θ sinh6 χ sinh6 χ sin3 θ sinh6 χ sin2 θ sinh6 χ 2T 2 2 ∇˜ ∇˜ aT 33 = 33 1 − sinh2 χ + T + a sin4 θ sinh6 χ 11 sin4 θ sinh4 χ sin2 θ sinh2 χ 4 cos θ cosh χT 4 cos θ∂ T 2 cosh χ∂ T ∂ ∂ T − 12 − 1 12 − 1 33 + 1 1 33 sin3 θ sinh5 χ sin3 θ sinh4 χ sin4 θ sinh5 χ sin4 θ sinh4 χ 4 cos θ∂ T cos θ∂ T ∂ ∂ T 4 cosh χ∂ T + 2 11 + 2 33 + 2 2 33 + 3 13 sin3 θ sinh4 χ sin5 θ sinh6 χ sin4 θ sinh6 χ sin4 θ sinh5 χ ∂ ∂ T + 3 3 33 , sin6 θ sinh6 χ 1 2 2 cosh χ∂ T ∂ ∂ T ∇˜ ∇˜ aT 12 = T − − + 1 12 + 1 1 12 a 12 sin2 θ sinh4 χ sinh2 χ sinh3 χ sinh2 χ 2 cosh χ∂ T cos θ∂ T ∂ ∂ T 2 cos θ∂ T + 2 11 + 2 12 + 2 2 12 − 3 13 sinh3 χ sin θ sinh4 χ sinh4 χ sin3 θ sinh4 χ ∂ ∂ T + 3 3 12 , sin2 θ sinh4 χ 2T 2 cosh χ∂ T ∂ ∂ T cos θ∂ T ∇˜ ∇˜ aT 13 = − 13 + 1 13 + 1 1 13 − 2 13 a sin2 θ sinh2 χ sin2 θ sinh3 χ sin2 θ sinh2 χ sin3 θ sinh4 χ ∂ ∂ T 2 cosh χ∂ T 2 cos θ∂ T ∂ ∂ T + 2 2 13 + 3 11 + 3 12 + 3 3 13 , sin2 θ sinh4 χ sin2 θ sinh3 χ sin3 θ sinh4 χ sin4 θ sinh4 χ 2(1 − sinh2 χ) 1 2 cos θ∂ T ∇˜ ∇˜ aT 23 = T − + 1 13 a 23 sin2 θ sinh6 χ sin4 θ sinh6 χ sin3 θ sinh4 χ 2 cosh χ∂ T ∂ ∂ T 2 cosh χ∂ T cos θ∂ T − 1 23 + 1 1 23 + 2 13 + 2 23 sin2 θ sinh5 χ sin2 θ sinh4 χ sin2 θ sinh5 χ sin3 θ sinh6 χ ∂ ∂ T 2 cosh χ∂ T 2 cos θ∂ T ∂ ∂ T + 2 2 23 + 3 12 + 3 22 + 3 3 23 . sin2 θ sinh6 χ sin2 θ sinh5 χ sin3 θ sinh6 χ sin4 θ sinh6 χ (4.157)

Following our analysis of the vector sector, in the k = −1 tensor sector we seek solutions to

˜ ˜ a (∇a∇ + AT )Tij = 0. (4.158) 118

(Here Tij is to denote the full combination of tensor components that appears in (4.112).) In (4.158) we have introduced a generic tensor sector constant AT , whose values in (4.112) are (2, 3, 6). Conveniently, we ﬁnd that the equation for

T11 involves no mixing with any other components of Tij, and can thus be solved

m directly. On setting T11(χ, θ, φ) = h11,`(χ)Y` (θ, φ), the equation for T11 reduces to

d2 cosh χ d 6 `(` + 1) + 6 + 6 + − + A h = 0. (4.159) dχ2 sinh χ dχ sinh2 χ sinh2 χ T 11,`

To determine the χ → ∞ and χ → 0 limits, we take the solutions to behave as eλχ (times an irrelevant polynomial in χ) and χn in these two limits. For (4.159) the limits give

2 1/2 λ + 6λ + 6 + AT , λ = −3 ± (3 − AT ) ,

λ(AT = 2) = (−4, − 2), √ λ(AT = 3) = (−3, − 3), λ(AT = 6) = −3 ± i 3,

n(n − 1) + 6n + 6 − `(` + 1) = 0, n = ` − 2, −` − 3. (4.160)

Thus for any allowed AT , every solution to (4.159) is bounded at χ = ∞, while for each AT one of the solutions will be well-behaved as χ → 0 for any ` ≥ 2.

Thus for ` = 2, 3, 4, .. there will always be one solution for any allowed AT that is bounded at χ = ∞ and well-behaved at χ = 0, with all solutions with ` = 0 or

` = 1 being excluded. 119

2 To solve (4.159) we set h11,` = γ`/ sinh χ to obtain:

d2 cosh χ d `(` + 1) + 2 − − 2 + A γ = 0. (4.161) dχ2 sinh χ dχ sinh2 χ T `

2 We recognize (4.161) as being (4.128), and can set ν = AT − 3 in (4.132), viz.

2 2 ν = (−1, 0, 3) for AT = 2, 3, 6. For AT = 2 we see that ν = −1. However in

2 2 2 the scalar case discussed above where ν = AS − 1, ν would also obey ν = −1

˜ ˜ a if AS = 0. Thus for AT = 2 we can obtain the solutions to (∇a∇ + 2)T11 = 0

2 (1) directly from (4.136), and after implementing h11,` = γ`/ sinh χ we obtain T` ,

(2) T` solutions to (4.159) of the form

cosh χ 1 Tˆ(1)(A = 2) = , Tˆ(2)(A = 2) = , 0 T sinh3 χ 0 T sinh2 χ 1 cosh χ χ Tˆ(1)(A = 2) = , Tˆ(2)(A = 2) = − , 1 T sinh4 χ 1 T sinh3 χ sinh4 χ cosh χ Tˆ(1)(A = 2) = , 2 T sinh5 χ 1 3 3χ cosh χ Tˆ(2)(A = 2) = + − , 2 T sinh2 χ sinh4 χ sinh5 χ 4 5 Tˆ(1)(A = 2) = + , 3 T sinh4 χ sinh6 χ 2 cosh χ 15 cosh χ 12χ 15χ Tˆ(2)(A = 2) = + − − . (4.162) 3 T sinh3 χ sinh5 χ sinh4 χ sinh6 χ

ˆ(2) All of these solutions are bounded at χ = ∞ and all T` (AT = 2) with ` ≥ 2 are

˜ ˜ a well-behaved at χ = 0. Thus if we implement (4.112) by (∇a∇ + 2)Tij = 0, we are not forced to Tij = 0, with the decomposition theorem not then following in the tensor sector.

2 For AT = 3 we see that ν = 0. However in the vector case discussed above 120

2 2 2 where ν = AV − 2, ν would also obey ν = 0 if AV = 2. Thus for AT = 3 we

˜ ˜ a can obtain the solutions to (∇a∇ + 3)T11 = 0 directly from (4.151), and after

11 implementing h` = α`/ sinh χ we obtain

1 χ Tˆ(1)(A = 3) = , Tˆ(2)(A = 3) = , 0 T sinh3 χ 0 T sinh3 χ cosh χ 1 χ cosh χ Tˆ(1)(A = 3) = , Tˆ(2)(A = 3) = − , 1 T sinh4 χ 1 T sinh3 χ sinh4 χ 2 3 Tˆ(1)(A = 3) = + , 2 T sinh3 χ sinh5 χ 3 cosh χ 2χ 3χ Tˆ(2)(A = 3) = − − , 2 T sinh4 χ sinh3 χ sinh5 χ 2 cosh χ 5 cosh χ Tˆ(1)(A = 3) = + , 3 T sinh4 χ sinh6 χ 11 15 6χ cosh χ 15χ cosh χ Tˆ(2)(A = 3) = + − − . (4.163) 3 T sinh3 χ sinh5 χ sinh4 χ sinh6 χ

ˆ(2) All of these solutions are bounded at χ = ∞ and all T` (AT = 3) with ` ≥ 2 are

˜ ˜ a well-behaved at χ = 0. Thus if implement (4.112) by (∇a∇ + 3)Tij = 0, we are

not forced to Tij = 0, with the decomposition theorem not then following.

A similar outcome occurs for AT = 6, and even though we do not evaluate

˜ ˜ a the AT = 6 solutions explicitly, according to (4.160) all solutions to (∇a∇ +

−3χ √ 6)T11 = 0 with AT = 6 are bounded at χ = ∞ (behaving as e cos( 3χ) and √ e−3χ sin( 3χ)), with one set of these solutions being well-behaved at χ = 0 for all

˜ ˜ a ` ≥ 2. Thus if we implement (4.112) by (∇a∇ + 6)Tij = 0, we are not forced to

Tij = 0, with the decomposition theorem again not following in the tensor sector. 121

Decomposition Theorem Analysis

In Sec. 4.1.4 we have seen that there are realizations of the evolution equations

in the scalar, vector, and tensor sectors that would not lead to a decomposition

theorem in those sectors. However, equally there are other realizations that given

the boundary conditions would lead to a decomposition theorem. Thus we need

to determine which realizations are the relevant ones. To this end we look not at

the individual higher-derivative equations obeyed by the separate scalar, vector,

and tensor sectors, but at how these various sectors interface with each other

in the original second-order ∆µν = 0 equations themselves. Any successful such interface would require that all the terms in ∆µν = 0 would have to have the same χ behavior. Noting that the scalar modes appear with two ∇˜ derivatives in

˜ ∆ij = 0, the vector sector appears with one ∇ derivative and the tensor appears with none, we need to compare derivatives of scalars with vectors and derivatives of vectors with tensors.

To see how to obtain such a needed common χ behavior we diﬀerentiate the scalar ﬁeld (4.128) with respect to χ, to obtain

d2 cosh χ d 2 `(` + 1) dS cosh χ + 4 + − + 4 + A ` + 2A S dχ2 sinh χ dχ sinh2 χ sinh2 χ S dχ S sinh χ ` = 0. (4.164)

Comparing with the vector (4.148) we see that up to an overall normalization we can identify dS`/dχ with the vector g1,` for modes that obey AS = 0 and AV = 2, 122

so that these particular scalar and vector modes can interface. As a check, with

ˆ(2) the vector sector needing ` ≥ 1 we diﬀerentiate S1 (AS = 0) to obtain

d d cosh χ χ Sˆ(2)(A = 0) = − dχ 1 S dχ sinh χ sinh2 χ 2 2χ cosh χ = − + = −2Vˆ (2)(A = 2). (4.165) sinh2 χ sinh3 χ 1 V

Similarly, if we diﬀerentiate the vector ﬁeld (4.148) with respect to χ we

obtain

d2 cosh χ d 6 `(` + 1) dg + 6 + 10 + A + − 1,` dχ2 sinh χ dχ V sinh2 χ sinh2 χ dχ cosh χ +2(2 + A ) g = 0. (4.166) V sinh χ 1,`

Comparing with the tensor (4.159) we see that up to an overall normalization

we can identify dg1,`/dχ with the tensor h11,` for modes that obey AV = −2 and

AT = 2, so that these particular vector and tensor modes can interface. As a

ˆ (1) check, with the tensor sector needing ` ≥ 2 we diﬀerentiate V2 (AV = −2) to

obtain

d d 2 cosh χ 3 cosh χ 3χ Vˆ (1)(A = −2) = − + dχ 2 V dχ sinh χ sinh3 χ sinh4 χ 4 12 12χ cosh χ = + − sinh2 χ sinh4 χ sinh5 χ ˆ(2) = 4T2 (AT = 2). (4.167)

Thus while we can interface AS = 0 and AV = 2, we cannot interface

AV = 2 with any of the tensor modes. Rather, we must interface the AV = −2

vector modes with the AT = 2 tensor modes. With none of the scalar sector 123 modes meeting the boundary conditions at both χ = ∞ and χ = 0 anyway, the scalar sector must satisfy ∆µν = 0 by itself, with the scalar term contribution to

∆µν = 0 then having to vanish, just as required of the decomposition theorem.

However, in the vector and tensor sectors we can achieve a common χ behavior if

˙ ˆ (1) ˆ(2) we set B1 − E1 = p1(τ)V2 (AV = −2), E11 = q11(τ)T2 (AT = 2), since then the

∆11 = 0 equation reduces to

1 3 3χ cosh χ ∆ = + − × 11 sinh2 χ sinh4 χ sinh5 χ ˙ −1 ˙ −1 8ΩΩ p1(τ) + 4p ˙1(τ) − q¨11(τ) + 2q11(τ) − 2ΩΩ q˙11(τ) − 2q11(τ) = 0.

(4.168)

This relation has a non-trivial solution of the form

1 4p (τ) − q˙ (τ) = , (4.169) 1 11 Ω2(τ) to thereby relate the τ dependencies of the vector and tensor sectors. With the other components of Vi and Tij being constructed in a similar manner, as such we have provided an exact interface solution in the vector and tensor sectors.

ˆ (1) ˆ(2) However, it only falls short in one regard. Both of V2 (AV = −2) and T2 (AT =

ˆ(2) 2) are well-behaved at χ = 0 and T2 (AT = 2) vanishes at χ = ∞. However,

ˆ (1) V2 (AV = −2) does not vanish at χ = ∞, as it limits to a constant value.

Imposing a boundary condition that the vector and tensor modes have to vanish at χ = ∞ then excludes this solution, with the decomposition theorem then being 124 recovered according to

1 ˙ ¨ ˙ −1 ˙ 2 (Bi − Ei) + ΩΩ (Bi − Ei) = 0,

.. ˙ ˙ −1 ˜ ˜ a −Eij + 2Eij − 2EijΩΩ + ∇a∇ Eij = 0, (4.170) with these being the equations that then serve to ﬁx the τ dependencies in the vector and tensor sectors. Consequently, we establish that the decomposition theorem does in fact hold for Robertson-Walker cosmologies with non-vanishing spatial 3-curvature after all.

4.1.5 δWµν Conformal to Flat

Since the SVT3 and SVT4 formulations are not contingent on the choice of evolution equations, we continue our study of cosmological ﬂuctuations by discussing how things work in an alternative to standard Einstein gravity, namely conformal gravity. For SVT3 ﬂuctuations around a Robertson-Walker background in the conformal gravity case we have found it more convenient not to use the metric given in (4.73), viz.

dr2 ds2 = a2(τ) dτ 2 − − r2dθ2 − r2 sin2 θdφ2 , (4.171) 1 − kr2 but to instead take advantage of the fact that via a general coordinate transformation a non-zero k Robertson-Walker metric can be brought into a form in which it is conformal to ﬂat. (With k = 0 the metric already is conformal to ﬂat.) The needed transformations for k < 0 and k > 0 may for instance be found in [42].

For the illustrative k < 0 case for instance, it is convenient to set k = −1/L2, and 125

introduce sinhχ = r/L and p = τ/L, with the metric given in (4.171) then taking

the form

ds2 = L2a2(p) dp2 − dχ2 − sinh2χdθ2 − sinh2χ sin2 θdφ2 . (4.172)

Next we introduce

p0 + r0 = tanh[(p + χ)/2], p0 − r0 = tanh[(p − χ)/2], sinh p sinh χ p0 = , r0 = , (4.173) cosh p + cosh χ cosh p + cosh χ so that

1 dp02 − dr02 = [dp2 − dχ2]sech2[(p + χ)/2]sech2[(p − χ)/2], 4 1 (cosh p + cosh χ)2 = cosh2[(p + χ)/2]cosh2[(p − χ)/2] 4 1 = . (4.174) [1 − (p0 + r0)2][1 − (p0 − r0)2]

With these transformations the line element takes the conformal to ﬂat form

4L2a2(p) ds2 = dp02 − dr02 − r02dθ2 − r02 sin2 θdφ2 . [1 − (p0 + r0)2][1 − (p0 − r0)2] (4.175)

The secpatial sector can then be written in Cartesian form

ds2 = L2a2(p)(cosh p + cosh χ)2 dp02 − dx02 − dy02 − dz02 , (4.176)

where r0 = (x02 + y02 + z02)1/2.

We note that while our interest in this section is in discussing ﬂuctuations in

conformal gravity, a theory that actually has an underlying conformal symmetry, 126 in transforming from (4.171) to (4.176) we have only made coordinate transformations and have not made any conformal transformation. However, since the Wµν gravitational Bach tensor introduced in (2.20) and (2.21) above is associated with

2 a conformal theory, under a conformal transformation of the form gµν → Ω (x)gµν,

−2 −2 Wµν and δWµν respectively transform into Ω (x)Wµν and Ω (x)δWµν. More- over, since this is the case for any background metric that is conformal to ﬂat we need not even restrict to Robertson-Walker or de Sitter, and can consider

ﬂuctuations around any background metric of the form

2 2 2 i j ds = Ω (x)[dt − δijdx dx ], (4.177)

where δij is the Kronecker delta function and Ω(x) is a completely arbitrary function of the four xµ coordinates.

In this background we take the SVT3 background plus ﬂuctuation line element to be of the form

2 2 2 ˜ i ˜ ˜ ds = Ω (x) (1 + 2φ)dt − 2(∇iB + Bi)dtdx − [(1 − 2ψ)δij + 2∇i∇jE ˜ ˜ i j +∇iEj + ∇jEi + 2Eij]dx dx , (4.178)

˜ i where Ω(x) is an arbitrary function of the coordinates, where ∇i = ∂/∂x (with

˜ i ij ˜ −2 ij ˜ Latin index) and ∇ = δ ∇j (i.e. not Ω δ ∇j) are deﬁned with respect to the background 3-space metric δij, and where the elements of (4.178) obey

ij ˜ ij ˜ jk ˜ ij δ ∇jBi = 0, δ ∇jEi = 0,Eij = Eji, δ ∇kEij = 0, δ Eij = 0. (4.179) 127

For these ﬂuctuations δWµν is readily calculated, and it is found to have the form

[42]

2 δW = − δmnδ`k∇˜ ∇˜ ∇˜ ∇˜ α, 00 3Ω2 m n ` k 2 1 h i δW = − δmn∇˜ ∇˜ ∇˜ ∂ α + δ`k∇˜ ∇˜ (δmn∇˜ ∇˜ − ∂2)(B − E˙ ) , 0i 3Ω2 i m n 0 2Ω2 ` k m n 0 i i 1 δW = δ δ`k∇˜ ∇˜ (∂2 − δmn∇˜ ∇˜ ) + (δ`k∇˜ ∇˜ − 3∂2)∇˜ ∇˜ α ij 3Ω2 ij ` k 0 m n ` k 0 i j 1 hh i h ii + δ`k∇˜ ∇˜ − ∂2 ∇˜ ∂ (B − E˙ ) + ∇˜ ∂ (B − E˙ ) 2Ω2 ` k 0 i 0 j j j 0 i i 1 h i2 + δmn∇˜ ∇˜ − ∂2 E . (4.180) Ω2 m n 0 ij where as before α = φ + ψ + B˙ − E¨. We note that the derivatives that appear in (4.180) are conveniently with respect to the ﬂat Minkowski metric and not

2 2 2 i j with respect to the full background ds = Ω (x)[dt − δijdx dx ] metric, with the Ω(x) dependence only appearing as an overall factor. This must be the case since the δWµν given in (4.180) is related to the δWµν given in (3.42) by an

−2 Ω (x) conformal transformation, and the δWµν given in (3.42) is associated with

ﬂuctuations around ﬂat spacetime.

Unlike the standard gravity case, in a background geometry that is conformal to ﬂat, namely in a background geometry in which the Weyl tensor vanishes, then according to the conformal transformation properties of Sec. 2.2.1 the background Wµν will vanish as well. The background Tµν thus vanishes also. Fluctua- tions are thus described by δTµν = 0, and thus by δWµν = 0, with δWµν as given

˙ in (4.180), and thus α, Bi − Ei and Eij, thus being gauge invariant. To check 128

whether a decomposition theorem might hold we thus need to solve the equation

δWµν = 0. To this end we note that since there are derivatives with respect to

`k ˜ ˜ the purely spatial δ ∇`∇k, on imposing spatial boundary conditions the relation

δW00 = 0 immediately sets α = 0. With α = 0, applying the spatial boundary

mn ˜ ˜ 2 ˙ conditions to the relation W0i = 0 immediately sets (δ ∇m∇n −∂0 )(Bi −Ei) = 0, h i2 mn ˜ ˜ 2 with δWij = 0 then realizing δ ∇m∇n − ∂0 Eij = 0. Thus with asymptotic boundary conditions the solution to δWµν = 0 is

h i2 mn ˜ ˜ 2 ˙ mn ˜ ˜ 2 α = 0, (δ ∇m∇n − ∂0 )(Bi − Ei) = 0, δ ∇m∇n − ∂0 Eij = 0. (4.181)

Since decomposition would require

mn `k ˜ ˜ ˜ ˜ δ δ ∇m∇n∇`∇kα = 0,

mn ˜ ˜ ˜ δ ∇i∇m∇n∂0α = 0,

`k ˜ ˜ mn ˜ ˜ 2 ˙ δ ∇`∇k(δ ∇m∇n − ∂0 )(Bi − Ei) = 0, `k ˜ ˜ 2 mn ˜ ˜ `k ˜ ˜ 2 ˜ ˜ δijδ ∇`∇k(∂0 − δ ∇m∇n) + (δ ∇`∇k − 3∂0 )∇i∇j α = 0,

h i h i `k ˜ ˜ 2 ˜ ˙ ˜ ˙ δ ∇`∇k − ∂0 ∇i∂0(Bj − Ej) + ∇j∂0(Bi − Ei) = 0, h i2 mn ˜ ˜ 2 δ ∇m∇n − ∂0 Eij = 0, (4.182) we see that the decomposition theorem is recovered.

To underscore this result we note that (4.180) can be inverted so as to write each gauge-invariant combination as a separate combination of components of

δWµν, viz.

2 2 ˜ ˜ b ˜ ˜ a Ω δW00 = − 3 ∇b∇ ∇a∇ α, (4.183) 129

.. ... ˜ ˜ a 2 ˜ ˜ a 2 1 ˜ ˜ b ˜ ˜ a ∇a∇ (Ω δW0i) − ∇i∇ (Ω δW0a) = − 2 ∇b∇ ∇a∇ (Bi − Ei) (4.184)

1 ˜ ˜ c ˜ ˜ b ˜ ˜ a ˙ + 2 ∇c∇ ∇b∇ ∇a∇ (Bi − Ei),

˜ ˜ a ˜ ˜ b 2 ˜ ˜ l 2 ˜ ˜ l 2 ∇a∇ ∇b∇ (Ω δWij) − ∇i∇ (Ω δWjl) − ∇j∇ (Ω δWil)

1 ˜ ˜ a ˜ k ˜ l 2 ˜ ˜ b 2 ab 1 ˜ ˜ ˜ k ˜ l 2 + 2 δij∇a∇ ∇ ∇ (Ω δWkl) − ∇b∇ (Ω δ δWab) + 2 ∇i∇j ∇ ∇ (Ω δWkl) h i2 ˜ ˜ a 2 ab ˜ ˜ a ˜ ˜ b ˜ ˜ c 2 +∇a∇ (Ω δ δWab) = ∇a∇ ∇b∇ ∇c∇ − ∂0 Eij. (4.185)

On setting δWµν = 0, each of these relations can now be integrated separately, with the decomposition theorem then following.

For completeness we also note that for SVT3 ﬂuctuations around the (4.74)

i j background k 6= 0 Robertson-Walker metric given in (4.73) withγ ˜ijdx dx = dr2/(1 − kr2) + r2dθ2 + r2 sin2 θdφ2 and with Ω actually now more generally being

i an arbitrary function of τ and x , the conformal gravity δWµν is given by

2 −2 ˜ ˜ a ˜ ˜ b δW00 = − 3 Ω (∇a∇ + 3k)∇b∇ α, 2 −2 ˜ ˜ ˜ a 1 −2 ˜ ˜ a ˜ ˜ b 2 ˙ δW0i = − 3 Ω ∇i(∇a∇ + 3k)α ˙ + 2 Ω ∇a∇ (∇b∇ − ∂0 )(Bi − Ei) 2 ˙ −2k(2k + ∂0 )(Bi − Ei) , h i 1 −2 ˜ ˜ a ˜ ˜ b 2 ˜ ˜ ˜ ˜ a 2 δWij = − 3 Ω γ˜ij∇a∇ (∇b∇ + 2k − ∂0 )α − ∇i∇j(∇a∇ − 3∂0 )α 1 −2 ˜ ˜ ˜ a 2 ˙ + 2 Ω ∇i(∇a∇ − 2k − ∂0 )∂0(Bj − Ej) ˜ ˜ ˜ a 2 ˙ +∇j(∇a∇ − 2k − ∂0 )∂0(Bi − Ei) h i −2 ˜ ˜ a 2 ˜ ˜ a 2 +Ω (∇a∇ − ∂0) Eij − 4k(∇a∇ − k − 2∂0 )Eij , (4.186) 130 where again α = φ + ψ + B˙ − E¨. These equations can be inverted, and yield

2 2 ˜ ˜ a ˜ ˜ b (Ω δW00) = − 3 (∇a∇ + 3k)∇b∇ α,

˜ ˜ a 2 ˜ ˜ a 2 1 ˜ ˜ a 2 (∇a∇ − 2k)(Ω δW0i) − ∇i∇ (Ω δW0a) = 2 (∇a∇ − 2k − ∂0 )×

˜ ˜ b ˜ ˜ c ˙ (∇b∇ + 2k)(∇c∇ − 2k)(Bi − Ei),

˜ ˜ a ˜ ˜ b 2 1 ˜ ˜ ˜ a ˜ b 2 (∇a∇ − 2k)(∇b∇ − 3k)(Ω δWij) + 2 ∇i∇j ∇ ∇ (Ω δWab)

˜ ˜ a bc 2 1 ˜ ˜ a ˜ b ˜ c 2 + (∇a∇ + 4k)(˜γ (Ω δWbc)) + 2 γ˜ij (∇a∇ − 4k)∇ ∇ (Ω δWbc)

˜ ˜ a ˜ ˜ b ˜ ˜ a 2 bc 2 − (∇a∇ ∇b∇ − 2k∇a∇ + 4k )(˜γ (Ω δWbc))

˜ ˜ a ˜ ˜ b 2 ˜ ˜ b 2 − (∇a∇ − 3k)(∇i∇ (Ω δWjb) + ∇j∇ (Ω δWib)) h i ˜ ˜ a ˜ ˜ b ˜ ˜ a 2 ˜ ˜ a 2 = (∇a∇ − 2k)(∇b∇ − 3k) (∇a∇ − ∂0) Eij − 4k(∇a∇ − k − 2∂0 )Eij .

(4.187)

With this separation of the gauge-invariant combinations we again have the decomposition theorem.

4.2 SVT4

We now proceed to carry through a similar analysis as performed in the SVT3

ﬂuctuations, but now as applied to the SVT4 formulation. We will again construct the evolution equations in de Sitter and Robertson Walker geometries, separate the scalar, vector, and tensor sectors by applying derivatives, and analyze the necessary conditions for the decomposition theorem to hold, with the discussion 131

beginning with Einstein gravity and afterward focusing on conformal gravity.

4.2.1 Minkowski

In treating the ﬂuctuation equations there are two types of perturbation that one

needs to consider. If we start with the Einstein equations in the presence of some

general non-zero background Tµν, viz. Gµν + 8πGTµν = 0, the ﬁrst type is to consider perturbations δGµν and δTµν to the background and look for solutions to

δGµν + 8πGδTµν = 0 (4.188)

in a background that obeys Gµν + 8πGTµν = 0. If the background is not ﬂat, the

ﬂuctuation δGµν will not be gauge invariant but it will instead be the combination

δGµν + 8πGδTµν that will be expressible in the gauge-invariant SVT3 or SVT4 bases as appropriately generalized to a non-ﬂat background.

The second kind of perturbation is one in which we introduce some new

¯ ¯ perturbation δTµν to a background that obeys Gµν + 8πGTµν = 0. This δTµν will modify both the background Gµν and the background Tµν and will lead to a

ﬂuctuation equation of the form

¯ δGµν + 8πGδTµν = −8πGδTµν. (4.189)

In (4.189) the combination δGµν + 8πGδTµν will be gauge invariant since struc-

¯ turally it will be of the same form as it would be in the absence of δTµν, and would

¯ thus be gauge invariant since it already was in the absence of δTµν. In consequence 132

¯ of this any δTµν that we could introduce would have to be gauge invariant all on its own.

If there is no background Tµν so that the background metric is ﬂat, the only

¯ perturbation that one could consider is δTµν, with the ﬂuctuation equation then

being of the form

¯ δGµν = −8πGδTµν. (4.190)

¯ ν ¯ With δTµν obeying ∇ δTµν = 0, in analog to (3.40) in general in the SVT4 case

¯ α ¯ α δTµν must be of the form ∇α∇ Fµν + 2(gµν∇α∇ − ∇µ∇ν)¯χ, with (4.190) taking the form

α α α ¯ ∇α∇ Fµν + 2(gµν∇α∇ − ∇µ∇ν)χ = −8πG[∇α∇ Fµν (4.191)

α +2(gµν∇α∇ − ∇µ∇ν)¯χ].

With the idea behind the decomposition theorem being that the tensor and scalar

sectors of (4.192) satisfy (4.192) independently, so that one can set

α α ¯ ∇α∇ Fµν = −8πG∇α∇ Fµν,

α α (gµν∇α∇ − ∇µ∇ν)χ = −8πG(gµν∇α∇ − ∇µ∇ν)¯χ, (4.192)

to see whether this is the case we take the trace of (4.192), to obtain

α α ∇α∇ χ = −8πG∇α∇ χ.¯ (4.193)

α If we now apply ∇α∇ to (4.192), then given (4.193) we obtain

α β α β ¯ ∇α∇ ∇β∇ Fµν = −8πG∇α∇ ∇β∇ Fµν. (4.194) 133

Now while this does give us an equation that involves Fµν alone, this equation is

α α ¯ not the second-order derivative equation ∇α∇ Fµν = −8πG∇α∇ Fµν that one is looking for. Moreover, getting to (4.193) and (4.194) is initially as far as we can

α β α go, since according to (3.40) only ∇α∇ ∇β∇ Fµν and ∇α∇ χ are automatically gauge invariant.

Now initially (4.193) does not imply that χ is necessarily equal to −8πGχ¯,

α since they could diﬀer by any function f that obeys ∇α∇ f = 0, i.e., by any harmonic function of the form f(q·x−qt). However, it is the very introduction ofχ ¯ that is causing χ to be non-zero in the ﬁrst place, and thus χ must be proportional toχ ¯. Hence harmonic functions can be ignored. Then with χ = −8πGχ¯, it follows

α α ¯ from (4.192) that ∇α∇ Fµν = −8πG∇α∇ Fµν. And again, since it is the very

¯ introduction of Fµν that is causing δGµν to be non-zero in the ﬁrst place, it must

¯ be the case that Fµν = −8πGFµν. As we see, (4.192) does hold, and thus for

¯ an external δTµν perturbation to a ﬂat background we obtain the decomposition theorem.

¯ However, in the absence of any explicit external δTµν the discussion is dif-

ferent, and is only of relevance in those cases where there is a background Tµν, as otherwise δTµν would be zero. When the background Tµν is non-zero and ac- cordingly the background is not ﬂat, the ﬂuctuation quantity δGµν +8πGδTµν can still only depend on six gauge-invariant SVT4 combinations, viz. the curved space generalizations of the above Fµν and one combination of χ, F and Fµ. Thus in 134

the following we will explore the SVT4 formulation in some non-ﬂat backgrounds

that are of cosmological interest.

4.2.2 dS4

SVT4 dS4 Basis Without a Conformal Factor

Since a background de Sitter metric can be written as a comoving coordinate

system metric with no conformal prefactor [viz. ds2 = dt2 −e2Ht(dx2 +dy2 +dz2)], or written with a conformal prefactor as a conformal to ﬂat Minkowski metric

[ds2 = (1/τH)2(dτ 2 − dx2 − dy2 − dz2) where τ = e−Ht/H], in setting up the

SVT4 description of fluctuations around a de Sitter background there are then two options. One is to define the fluctuations in terms of χ, F , Fµ and Fµν with no multiplying conformal prefactor so that

hµν = −2gµνχ + 2∇µ∇νF + ∇µFν + ∇νFµ + 2Fµν, (4.195)

with the ∇µ derivatives being fully covariant with respect to the de Sitter back-

µ ν ground so that ∇ Fµ = 0, ∇ Fµν = 0. The second is to deﬁne the ﬂuctuations

with a conformal prefactor so that the ﬂuctuation metric is written as conformal

to a ﬂat Minkowski metric according to

1 h = [−2η χ + 2∇˜ ∇˜ F + ∇˜ F + ∇˜ F + 2F ], (4.196) µν (τH)2 µν µ ν µ ν ν µ µν

˜ ˜ µ with the ∇µ derivatives being with respect to ﬂat Minkowski so that ∇ Fµ = 0,

˜ ν ˙ ˜ j ˙ ˜ j ˙ ˜ j ∇ Fµν = 0, i.e. −F0 + ∇ Fj = 0, −F00 + ∇ F0j = 0, −F0i + ∇ Fij = 0. We shall discuss both options below starting with (4.195). 135

However, before doing so and in order to be as general as possible we shall initially work in D dimensions where the de Sitter space Riemann tensor takes the form

2 2 α 2 Rλµνκ = H (gµνgλκ − gλνgµκ),Rµκ = H (1 − D)gµκ,R α = H D(1 − D).

(4.197)

To construct ﬂuctuations we have found it convenient to generalize (3.29) to

2 − D h = 2F + ∇ W + ∇ W + ∇ ∇ + g H2 × µν µν ν µ µ ν D − 1 µ ν µν Z g dDy(−g)1/2D(E)(x, y)∇αW − µν (∇αW − h) α D − 1 α 1 Z − ∇ ∇ + g H2 dDy(−g)1/2D(E)(x, y)h, (4.198) D − 1 µ ν µν

where in the curved background the Green’s function obeys

ν 2 (E) −1/2 (D) ∇ν∇ + H D D (x, y) = (−g) δ (x − y). (4.199)

ν With this deﬁnition Fµν is automatically traceless. On applying ∇ to (4.198)

and recalling that for any vector or scalar in a de Sitter space we have

ν ν 2 (∇ ∇µ − ∇µ∇ )Wν = H (D − 1)Wµ,

ν ν 2 (∇ ∇µ∇ν − ∇µ∇ ∇ν)V = H (D − 1)∇µV, (4.200)

we obtain

ν ν 2 ∇ hµν = ∇ν∇ Wµ + H (D − 1)Wµ, (4.201) 136

with (4.201) serving to deﬁne Wµ. To decompose Wµ into transverse and longi-

µ µ µ tudinal components we set Wµ = Fµ + ∇µA where ∇ Fµ = 0, ∇ Wµ = ∇ ∇µA,

and thus set

Z D 1/2 (D) α Wµ = Fµ + ∇µ d y(−g) D (x, y)∇ Wα, (4.202)

where

ν (D) −1/2 (D) ∇ν∇ D (x, y) = (−g) δ (x − y). (4.203)

Finally, with

(2 − D)H2 Z ∇αW − h −2χ = dDy(−g)1/2D(E)(x, y)∇αW − α D − 1 α D − 1 H2 Z − dDy(−g)1/2D(E)(x, y)h, D − 1 2 − D Z 2F = dDy(−g)1/2D(E)(x, y)∇αW D − 1 α 1 Z Z − dDy(−g)1/2D(E)(x, y)h + 2 dDy(−g)1/2D(D)(x, y)∇αW , D − 1 α Z D 1/2 (D) α Fµ = Wµ − ∇µ d y(−g) D (x, y)∇ Wα, (4.204)

we can now write hµν as given (4.195), with Fµν being given by the transverse- traceless

2Fµν = hµν + 2gµνχ − 2∇µ∇νF − ∇µFν − ∇νFµ, (4.205)

where Wµ is determined from (4.201). In this way then we can decompose hµν into a covariant SVTD in the de Sitter background case.

As well as the above formulation, which involves the Green’s function D(E)(x, y),

we should note that there is also an alternate formulation that does not involve it 137

(D) at all, one that implements the tracelessness of Fµν using D (x, y) alone, though it does so at the expense of leading to a more complicated expression for Wµ. To this end we replace (4.198) by

2 − D Z h = 2F + ∇ W + ∇ W + ∇ ∇ dDy(−g)1/2D(D)(x, y)∇αW µν µν ν µ µ ν D − 1 µ ν α g 1 Z − µν (∇αW − h) − ∇ ∇ dDy(−g)1/2D(D)(x, y)h, (4.206) D − 1 α D − 1 µ ν

with Fµν automatically being traceless. To ﬁx Wµ we evaluate

ν ν 2 2 ∇ hµν = ∇ν∇ Wµ + H (D − 1)Wµ + H (2 − D) × Z D 1/2 (D) α ∇µ d y(−g) D (x, y)∇ Wα Z 2 D 1/2 (D) −H ∇µ d y(−g) D (x, y)h,

µ ν µ ν 2 ν ∇ ∇ hµν = ∇ ∇ν∇ Wµ + H (∇ Wν − h). (4.207)

In terms of (4.206) and (4.202) we can set

1 2χ = [∇αW − h] D − 1 α 1 Z 2F = dDy(−g)1/2D(D)(x, y)[D∇αW − h], D − 1 α Z D 1/2 (D) α Fµ = Wµ − ∇µ d y(−g) D (x, y)∇ Wα,

2Fµν = hµν + 2gµνχ − 2∇µ∇νF − ∇µFν − ∇νFµ, (4.208)

µ with ∇ Fµ = 0 as before, and with (4.195) following. Thus either way we are

led to (4.195) and we now apply it to ﬂuctuations around a background de Sitter

geometry. 138

Application of SVT4 to de Sitter Fluctuation Equations

We now restrict to four dimensions where in a de Sitter geometry the background

Einstein equations are given by

2 Gµν = −8πGTµν = 3H gµν. (4.209)

The ﬂuctuating Einstein tensor is given by

1 δG = [∇ ∇αh − ∇ ∇αh − ∇ ∇αh + ∇ ∇ h] µν 2 α µν ν αµ µ αν µ ν g H2 + µν ∇α∇βh − ∇ ∇αh + [4h − g h] , (4.210) 2 αβ α 2 µν µν

2 while the perturbation in the background Tµν is given by δTµν = −3H hµν (we conveniently set 8πG = 1). If we now reexpress these ﬂuctuations in the SVT4 basis given in (4.195) we obtain

α 2 2 2 δGµν = 2gµν∇α∇ χ − 2∇µ∇νχ + 6H ∇µ∇νF + 3H ∇µFν + 3H ∇νFµ +

α 2 (∇α∇ + 4H )Fµν, (4.211)

2 2 3H hµν = 3H [−2gµνχ + 2∇µ∇νF + ∇µFν + ∇νFµ + 2Fµν] , (4.212)

and thus

2 α 2 δGµν − 3H hµν = (∇α∇ − 2H )Fµν

α 2 +2(gµν∇α∇ − ∇µ∇ν + 3H gµν)χ. (4.213)

2 As we see, δGµν + 8πGδTµν = δGµν − 3H hµν only depends on Fµν and χ, with

it thus being these quantities that are gauge invariant, with the thus non-gauge-

invariant Fµ and F dropping out. 139

¯ In the event that there is an additional source term δTµν, it must be gauge

ν ¯ invariant on its own, and must obey ∇ δTµν = 0 in the de Sitter background, to thus be of the form

¯ ¯ α 2 δTµν = Fµν + 2(gµν∇α∇ − ∇µ∇ν + 3H gµν)¯χ. (4.214)

With this source the ﬂuctuation equations take the form

α 2 α 2 ¯ (∇α∇ − 2H )Fµν + 2(gµν∇α∇ − ∇µ∇ν + 3H gµν)χ = Fµν

α 2 +2(gµν∇α∇ − ∇µ∇ν + 3H gµν)¯χ, (4.215) with trace

α 2 α 2 6(∇α∇ + 4H )χ = 6(∇α∇ + 4H )¯χ. (4.216)

α While the trace condition would only set χ =χ ¯+f where f obeys (∇α∇ +

4H2)f = 0, when there is aχ ¯ source present then it is the cause of ﬂuctuations in the background in the ﬁrst place, and thus we can only have χ =χ ¯ with any possible f being zero. Then, from (4.215) we obtain

α 2 ¯ (∇α∇ − 2H )Fµν = Fµν, (4.217) and the decomposition theorem is achieved.

¯ However, if there is no δTµν source the ﬂuctuation equations take the form

α 2 α 2 (∇α∇ − 2H )Fµν + 2(gµν∇α∇ − ∇µ∇ν + 3H gµν)χ = 0, (4.218) 140 with the trace condition being given by

α 2 6(∇α∇ + 4H )χ = 0, (4.219) with spherical Bessel solution

X 2 2 ik·x χ = k τ [a2(k)j2(kτ) + b2(k)y2(kτ)]e . (4.220) k

(To obtain this solution for χ it is more straightforward to use ds2 = (1/τ 2H2)(dτ 2− dx2 − dy2 − dz2) as the background de Sitter metric, something we can do regardless of whether or not we include a conformal factor in the ﬂuctuations.) Given the trace condition we can rewrite the evolution equation given in (4.218) as

α 2 2 (∇α∇ − 2H )Fµν − 2(gµνH + ∇µ∇ν)χ = 0. (4.221)

2 Since it is not automatic that χ would obey (H gµν + ∇µ∇ν)χ = 0 even

µν 2 though it does obey g (H gµν + ∇µ∇ν)χ = 0, it is thus not automatic that

(4.218) and (4.221) could be replaced by

α 2 2 (∇α∇ − 2H )Fµν = 0, (gµνH + ∇µ∇ν)χ = 0, (4.222)

as would be required of a decomposition theorem. In fact, since gµνχ and ∇µ∇νχ behave totally diﬀerently (∇µ∇νχ is non-zero if µ 6= ν while gµνχ is not), the only way to get a decomposition theorem would be for χ, and thus a2(k) and b2(k), to be zero. As we now show, this can in fact be made to be the case, though it is only a particular solution to the full ﬂuctuation equations. 141

To explore this possibility we need to obtain an expression that only depends on Fµν, and we note that for any scalar in D = 4 de Sitter we have [11]

α α 2 α 2 ∇α∇ ∇µ∇νχ = ∇µ∇ν∇α∇ χ − 2H gµν∇α∇ χ + 8H ∇µ∇νχ. (4.223)

Given the trace condition shown in (4.219) we then ﬁnd that

α 2 2 2 α α 2 (∇α∇ − 4H )(gµνH + ∇µ∇ν)χ = (∇µ∇ν − H gµν)(∇ ∇ + 4H )χ = 0(4.224),

and from (4.221) we thus obtain the fourth-order derivative equation

α 2 α 2 (∇α∇ − 4H )(∇α∇ − 2H )Fµν = 0 (4.225)

for Fµν, with a decomposition for the components of the ﬂuctuations thus being

found, only in the higher-derivative form given in (4.224) and (4.225) rather than

α 2 in the second-derivative form given in (4.222). Now (∇α∇ − 2H )Fµν = 0 is a

particular solution to (4.225), and for this particular solution it would follow that

the only solution to (4.218) would then be χ = 0, with both the Fµν and χ sector

equations given in (4.222) then holding.

α 2 To determine the conditions under which (∇α∇ − 2H )Fµν = 0 might

actually hold we need to look for the general solution to (4.225), and since (4.225)

is a covariant equation we can evaluate it in any coordinate system, with conformal

to ﬂat Minkowski being the most convenient for the de Sitter background. To this

end we recall that in any metric that is conformal to ﬂat Minkowski (ds2 =

M N 2 µ ν −gMN dx dx = −Ω (x)ηµνdx dx ) one has the relation [11]

LR LR −2 −4 TQ g ∇L∇RAMN = η Ω ∂L∂RAMN − 2Ω ∂M Ω∂N Ωη ATQ 142

LR −3 LR −3 −2η Ω ∂L∂RΩAMN − 2η Ω ∂RΩ∂LAMN

−4 TX QY KQ −3 +2Ω ηMN η ∂X Ωη ∂Y ΩATQ + 2η Ω ∂QΩ∂N AKM

KQ −3 −1 L +2η Ω ∂QΩ∂M AKN − 2Ω ∂N Ω∇LA M

−1 L −2Ω ∂M Ω∇LA N , (4.226)

for any rank two tensor AMN , with the ∇L referring to covariant derivatives in the gMN geometry. For an AMN that is transverse and traceless, and for Ω = 1/τH

(4.226) reduces to (the dot denotes ∂/∂τ)

LR LR 2 2 2 2 ˙ 2 g ∇L∇RAMN = η τ H ∂L∂RAMN + 4H AMN − 2τH AMN + 2H ηMN A00

2 2 + 2τH ∂N A0M + 2τH ∂M A0N . (4.227)

While the general components of AMN are coupled in (4.227), this is not the case for A00, and so we look at A00 and obtain

L LR 2 2 2 2 ˙ ∇L∇ A00 = η τ H ∂L∂RA00 + 2H A00 + 2τH A00. (4.228)

Now in a de Sitter background the identity

K P K 2 P 2 P ∇P ∇K ∇ A M = [∇K ∇ + 5H ]∇P A M − 2H ∇M A P (4.229)

L holds [11]. Thus if any AMN is transverse and traceless then so is ∇L∇ AMN .

L 2 So let us deﬁne AMN = [∇L∇ − 2H ]FMN , with this AMN being traverse and traceless since FMN is. For this AMN (4.225) takes the form

α 2 (∇α∇ − 4H )Aµν = 0. (4.230) 143

Thus for A00 we have

LR 2 2 2 ˙ 2 η τ H ∂L∂RA00 + 2τH A00 − 2H A00 = 0. (4.231)

ik·x In a plane wave mode e the quantity A00 thus obeys

2 2 A¨ − A˙ + k2A + A = 0. (4.232) 00 τ 00 00 τ 2 00

The general solution to (4.231) is thus

L 2 X 4 2 ik·x A00 = [∇L∇ − 2H ]F00 = k τ [a00(k)j0(kτ) + b00(k)y0(kτ)]e k X 3 ik·x = k τ[a00(k) sin(kτ) + b00(k) cos(kτ)]e , (4.233) k where a00(k) and b00(k) are polarization tensors. (Here and throughout we leave

out the complex conjugate solution.)

To see if we can support this solution, or whether we are forced to (4.222),

we need to see whether (4.233) is compatible with (4.221), and thus require that

α 2 2 A00 = (∇α∇ − 2H )F00 = 2(g00H + ∇0∇0)χ 1 ∂2 = 2 − + − Γα ∂ χ, (4.234) τ 2 ∂τ 2 00 α

α when evaluated in the solution for χ as given in (4.220). On noting that Γ00 =

α −δ0 /τ in a background de Sitter geometry, we evaluate

1 ∂2 ∂2 1 ∂ 1 2 − + − Γα ∂ [k2τ 2j (kτ)] = 2 + − [k2τ 2j (kτ)] τ 2 ∂τ 2 00 α 2 ∂τ 2 τ ∂τ τ 2 2 = 2k3τ sin(kτ), (4.235) 144 where we have utilized properties of Bessel functions in the last step. With an analogous expression holding for the y2(kτ) term, we thus precisely do conﬁrm

(4.233), and on comparing (4.220) with (4.233) obtain

a00(k) = 2a2(k), b00(k) = 2b2(k). (4.236)

As we see, in the general solution we are not at all forced to χ = 0 as would be required by the decomposition theorem.

For completeness we note that once we have determined A00 we can use

L MN (4.227) and the ∇LA M = 0 and g AMN = 0 conditions to determine the other components of Aµν, and note only that they satisfy and behave as

2 2 ηµν∂ A + A = 0, ηµν∂ A + A = 0, µ 0ν τ 00 µ iν τ 0i ∂2 2 + k2 A = ∂ A , ∂τ 2 0i τ i 00 ∂2 2 ∂ 2 2 + + k2 A = δ A + (∂ A + ∂ A ) , ∂τ 2 τ ∂τ ij τ 2 ij 00 τ i 0j j 0i X A0i = ikik[−kτa00(k) cos(kτ) + kτb00(k) sin(kτ) + a00(k) sin(kτ) k ik·x +b00(k) cos(kτ)]e ,

X ik·x Aij = kikjkτ[a00(k) sin(kτ) + b00(k) cos(kτ)]e k X 2 + δijk − 3kikj − a00(k) cos(kτ) + b00(k) sin(kτ) k 1 + [a (k) sin(kτ) + b (k) cos(kτ) eik·x. (4.237) kτ 00 00

(In order to derive the solutions given in (4.237) we needed to include terms that would vanish identically in the left-hand sides of the second-order diﬀerential 145

L equations so that the ﬁrst-order ∇LA M = 0 conditions would then be satisﬁed.)

In this solution we then need to satisfy

α 2 (∇α∇ − 2H )Fµν = Aµν, (4.238)

which for the representative A00 and F00 is of the form

2 X k3 F¨ − F˙ + k2F = − [a (k) sin(kτ) + b (k) cos(kτ)]eik·x, (4.239) 00 τ 00 00 H2τ 00 00 k

with solution

X k2 F = [−a (k) cos(kτ) + b (k) sin(kτ)]eik·x. (4.240) 00 2H2 00 00 k

Now in order to get a decomposition theorem in the form given in (4.222) we would need χ to vanish, i.e. we would need a2(k) and b2(k) to vanish. And that would mean that a00(k) and b00(k) would have to vanish as well, and thus not only would A00 have to vanish but so would all the other components of Aµν

as well. A decomposition theorem would thus require that

α 2 (∇α∇ − 2H )Fµν = 0, (4.241)

for all components of Fµν. To look for a non-trivial solution to (4.241) in order to

show that the decomposition theorem does in fact have a solution, we note that

in a plane wave (4.241) reduces to

2 F¨ − F˙ + k2F = 0, (4.242) 00 τ 00 00 146

for the representative F00 component. The non-trivial solution to (4.242) is of the form

X 2 2 ik·x F00 = k τ [c00(k)j1(kτ) + d00(k)y1(kτ)]e . (4.243) k

The form for F00 given in (4.243) and its Fµν analogs together with χ = 0 thus constitute a non-trivial solution that corresponds to the decomposition theorem, so in this sense the decomposition theorem can be recovered, as it is a speciﬁc solution to the full evolution equations. However, there is no compelling reason to restrict the solutions to (4.230) to the trivial Aµν = 0, with it being (4.233),

(4.237), (4.240) and (4.243) that provide the most general solution in the F00 sector and its analogs according to

X k2 F = [−a (k) cos(kτ) + b (k) sin(kτ)]eik·x 00 2H2 00 00 k X 2 2 ik·x + k τ [c00(k)j1(kτ) + d00(k)y1(kτ)]e , (4.244) k while at the same time (4.220) is the most general solution in the χ sector as constrained by (4.236). Moreover, in this solution we can choose the coefficients in (4.236) so that Fµν and χ are localized in space. Thus no spatially asymptotic boundary coefficient could affect them. In fact suppose that we could have constrained the solutions by an asymptotic condition. We would need one that would

α 2 force Aµν to have to vanish in (∇α∇ − 4H )Aµν = 0 while not at the same time

α 2 forcing Fµν to have to vanish in (∇α∇ − 2H )Fµν = 0, something that would not obviously appear possible to achieve. Thus as we see, in this general solution the 147

decomposition theorem does not hold. And just as we noted in Sec. 3.4.2, in the

SVT4 case asymptotic boundary conditions do not force us to the decomposition

theorem, to thus provide a completely solvable cosmological model in which the

decomposition theorem does not hold. However, we should point out that while

we could not make χ vanish through spatial boundary conditions it would be possible to force χ to vanish at all times by judiciously choosing initial conditions at an initial time. However, there would not appear to be any compelling rationale for doing so, and to nonetheless do so would appear to be quite contrived. Thus absent any compelling rationale for such a judicious choice or for any other choice at all for that matter (i.e., no compelling rationale that would force χ to vanish)

the decomposition theorem would not hold for SVT4 ﬂuctuations around a de

Sitter background.

Deﬁning the SVT4 Fluctuations With a Conformal Factor

In a

1 ds2 = (dτ 2 − dx2 − dy2 − dz2) (4.245) (τH)2 de Sitter background with ﬂuctuations of the form

1 h = (−2g χ + 2∇˜ ∇˜ F + ∇˜ F + ∇˜ F + 2F ), (4.246) µν (τH)2 µν µ ν µ ν ν µ µν

˜ µ ˜ ν µν ˜ where ∇ Fµ = 0, ∇ Fµν = 0, g Fµν = 0, and where, as per (4.196), the ∇µ

µ ν denote derivatives with respect to the ﬂat Minkowski ηµνdx dx metric, we write 148

the ﬂuctuation Einstein tensor as

.. −1 −1 ˜ 2 ˙ ˜ 2 −1 ˜ 2 ˙ −1 ˜ 2 δG00 = −6χτ ˙ − 2τ ∇ F − 2∇ χ − 2τ ∇ F0 − F 00 − 2F00τ + ∇ F00,

.. −1 ˜ −2 ˜ ˙ ˜ −1 ˜ ˙ −2 δG0i = −2τ ∇iF + 6τ ∇iF − 2∇iχ˙ − 2τ ∇iχ + 3Fiτ

.. −1 ˜ ˙ −2 ˜ −2 ˜ 2 −1 ˜ −2τ ∇iF0 + 3τ ∇iF0 − F 0i + 6F0iτ + ∇ F0i − 2τ ∇iF00,

...... −2 −1 −1 −1 ˜ 2 ˙ δGij = −2χδij + 6F δijτ − 2F δijτ + 2χδ ˙ ijτ + 2δijτ ∇ F

˜ 2 −1 ˜ ˜ ˙ −2 ˜ ˜ ˜ ˜ ˙ −2 +2δij∇ χ − 2τ ∇j∇iF + 6τ ∇j∇iF − 2∇j∇iχ + 6F0δijτ

.. −1 −1 ˜ 2 −2 ˜ −2 ˜ −1 ˜ ˜ −2F 0δijτ + 2δijτ ∇ F0 + 3τ ∇iFj + 3τ ∇jFi − 2τ ∇j∇iF0

.. −2 −2 ˙ −1 ˜ 2 −1 ˜ −F ij + 6Fijτ + 6F00δijτ + 2Fijτ + ∇ Fij − 2τ ∇iF0j

−1 ˜ −2τ ∇jF0i,

.. ... µν 2 2 2 2 .. 2 2 ˜ 2 ˙ 2 ˜ 2 g δGµν = 18H F − 6H F τ + 12H χτ˙ − 6H χτ + 6H τ∇ F + 6H ∇ F

.. 2 2 ˜ 2 2 ˙ 2 2 ˜ 2 2 +6H τ ∇ χ + 24H F0 − 6H F 0τ + 6H τ∇kF0 + 24H F00. (4.247)

Here the dot denotes the derivative with respect to the conformal time τ and

˜ 2 ij ˜ ˜ 2 ∇ = δ ∇i∇j. With a 3H hµν perturbation the ﬂuctuation equations take the form

.. −2 −1 −2 −1 ˜ 2 ˙ ˜ 2 ˙ −2 ∆00 = −6F τ − 6χτ ˙ − 6τ χ − 2τ ∇ F − 2∇ χ − 6F0τ

.. −1 ˜ 2 −2 ˙ −1 ˜ 2 −2τ ∇ F0 − F 00 − 6F00τ − 2F00τ + ∇ F00 = 0,

.. .. −1 ˜ ˜ −1 ˜ −1 ˜ ˙ ˜ 2 ∆0i = −2τ ∇iF − 2∇iχ˙ − 2τ ∇iχ − 2τ ∇iF0 − F 0i + ∇ F0i

−1 ˜ −2τ ∇iF00 = 0,

...... −2 −1 −1 −2 −1 ˜ 2 ˙ ∆ij = −2χδij + 6F δijτ − 2F δijτ + 2χδ ˙ ijτ + 6δijτ χ + 2δijτ ∇ F 149

.. ˜ 2 −1 ˜ ˜ ˙ ˜ ˜ ˙ −2 −1 +2δij∇ χ − 2τ ∇j∇iF − 2∇j∇iχ + 6F0δijτ − 2F 0δijτ

.. −1 ˜ 2 −1 ˜ ˜ −2 ˙ −1 ˜ 2 +2δijτ ∇ F0 − 2τ ∇j∇iF0 − F ij + 6F00δijτ + 2Fijτ + ∇ Fij

−1 ˜ −1 ˜ −2τ ∇iF0j − 2τ ∇jF0i = 0,

.. ... µν 2 2 2 2 .. 2 2 2 ˜ 2 ˙ g ∆µν = 24H F − 6H F τ + 12H χτ˙ − 6H χτ + 24H χ + 6H τ∇ F

.. 2 2 ˜ 2 2 ˙ 2 2 ˜ 2 2 +6H τ ∇ χ + 24H F0 − 6H F 0τ + 6H τ∇ F0 + 24H F00 = 0,

(4.248)

˙ where ∆µν = δGµν + 8πGδTµν. On introducing α = F + τχ + F0 the perturbative equations simplify to

.. −2 −1 ˜ 2 −2 ˙ −1 ˜ 2 ∆00 = −6ατ ˙ − 2τ ∇ α − F 00 − 6F00τ − 2F00τ + ∇ F00 = 0,

.. −1 ˜ ˜ 2 −1 ˜ ∆0i = −2τ ∇iα˙ − F 0i + ∇ F0i − 2τ ∇iF00 = 0, h i −1 −2 −1 ˜ 2 −2 −1 ˜ ˜ ∆ij = δij −2¨ατ + 6ατ ˙ + 2τ ∇ α + 6F00τ − 2τ ∇i∇jα

.. ˙ −1 ˜ 2 −1 ˜ −1 ˜ −F ij + 2Fijτ + ∇ Fij − 2τ ∇iF0j − 2τ ∇jF0i = 0,

−2 µν .. ˜ 2 H g ∆µν = 24α ˙ − 6ατ + 6τ∇ α + 24F00 = 0. (4.249)

We thus see that α and Fµν are gauge invariant for a total of six (one plus ﬁve) gauge-invariant components, just as needed. (In passing we note that the gauge

˙ invariant α = F + τχ + F0 actually mixes scalars and vectors.)

While we have written ∆µν in the non-manifestly covariant form given

(4.249) as this will be convenient for actually solving ∆µν = 0 below, since the

SVT4 approach is covariant we are able to write the rank two tensor ∆µν in a manifestly covariant form. To do so we introduce a unit timelike four-vector U µ 150

whose only non-zero component is U 0. In terms of this U µ the gauge-invariant α is

µ µ now given by the manifestly general coordinate scalar α = U ∂µF +χ/HΩ+U Fµ,

µν µ ν while the F00 term in g ∆µν can be written as U U Fµν.

If there is to be a decomposition theorem then (4.249) would have to break up into

.. −2 −1 ˜ 2 −2 ˙ −1 ˜ 2 −6ατ ˙ − 2τ ∇ α = 0, −F 00 − 6F00τ − 2F00τ + ∇ F00 = 0,

.. −1 ˜ ˜ 2 −1 ˜ −2τ ∇iα˙ = 0, −F 0i + ∇ F0i − 2τ ∇iF00 = 0, h i −1 −2 −1 ˜ 2 −1 ˜ ˜ δij −2¨ατ + 6ατ ˙ + 2τ ∇ α − 2τ ∇i∇jα = 0,

.. −2 ˙ −1 ˜ 2 −1 ˜ −1 ˜ 6δijF00τ − F ij + 2Fijτ + ∇ Fij − 2τ ∇iF0j − 2τ ∇jF0i = 0,

.. ˜ 2 24α ˙ − 6ατ + 6τ∇ α = 0, 24F00 = 0,

(4.250) to then yield

˜ ˜ ˜ 2 ¨ ˜ 2 α˙ = 0, ∇i∇jα = 0, ∇ α = 0,F00 = 0, −F0i + ∇ F0i = 0,

.. ˙ −1 ˜ 2 −1 ˜ −1 ˜ −F ij + 2Fijτ + ∇ Fij − 2τ ∇iF0j − 2τ ∇jF0i = 0, (4.251)

ijk ˜ with the ∇j∆0k = 0 condition not being needed as it is satisﬁed identically.

The solution to (4.251) is the form

X ik·x−ikτ j α = 0,F00 = 0,F0i = f0i(k)e , ik f0j(k) = 0, k X ˆ ik·x−ikτ Fij = [fij(k) + τfij(k)]e , k 151

ˆ −ikfij(k) + fij(k) = ikjf0i(k) + ikif0j(k), (4.252)

ij ij ˆ j j ˆ δ fij(k) = 0, δ fij(k) = 0, ik fij(k) = −ikf0i(k), ik fij(k) = 0, and while the most general solution for α would be a constant, we have imposed an asymptotic spatial boundary condition, which sets the constant to zero.

We now solve the full (4.249) exactly to determine whether and under what

µν conditions (4.253) might hold. Eliminating F00 between the ∆00 = 0 and g ∆µν =

0 equations in (4.249) yields

τ ∂2 ∂2 − − ∇˜ 2 − ∇˜ 2 α = 0, (4.253) 4 ∂τ 2 ∂τ 2 with general solution

X ik·x−ikτ α = (ak + τbk) e , (4.254) k

where ak and bk are independent of x and τ. Given α, F00 then evaluates to

X ik·x−ikτ F00 = [a00(k) + τb00(k)] e , (4.255) k where

ik a (k) = ika − b , b (k) = b . (4.256) 00 k k 00 2 k

Inserting these solutions for α and F00 into ∆0i = 0 then yields

¨ ˜ 2 X ik·x−ikτ F0i − ∇ F0i = − kkibke , (4.257) k with solution

X ik·x−ikτ F0i = [a0i(k) + τb0i(k)] e , (4.258) k 152

where

ik b (k) = − i b . (4.259) 0i 2 k

i ˙ With F0i obeying the transverse condition ∂ F0i − F00 = 0, we obtain

3ik ikia (k) = k2a + b . (4.260) 0i k 2 k

Finally, from ∆ij = 0 we obtain

2 X 1 F¨ − F˙ − ∇˜ 2F = [δ ikb + 2k k a − 2ik a − 2ik a ] eik·x−ikτ . ij τ ij ij ij k i j k i 0j j 0i τ k (4.261)

We can thus set

X ik·x−ikτ Fij = [aij(k) + τbij(k)] e , (4.262) k where

2ikaij(k) − 2bij(k) = δijikbk + 2kikjak − 2ikia0j − 2ikja0i. (4.263)

j ˙ ij With Fij obeying the transverse and traceless conditions ∂ Fij = F0i, δ Fij −F00 =

0, we obtain

ik kk ikja (k) = −ika (k) − i b , ikjb (k) = − i b , ij 0i 2 k ij 2 k ik δija (k) = ika − b , δijb (k) = b . (4.264) ij k k ij 2 k

Equations (4.253) to (4.264) provide us with the most general solution to (4.249). 153

Having now obtained the exact solution, we see that we do not get the decomposition theorem solution given in (4.253). If we want to get the exact solution to reduce to the decomposition theorem solution we would need to set

α and F00 to zero, and this would be a particular solution to the ﬂuctuation equations. However, there is no reason to set them to zero, and certainly no spatial asymptotic condition that could do so. And even if there were to be one, then such an asymptotic condition would have to suppress α and F00 while at the same time not suppressing F0i and Fij, even though though all of the ﬂuctuation components have precisely the same asymptotic spatial behavior. We could possibly set α and

F00 to zero at all times via judiciously chosen initial conditions, but there would not appear to be any compelling reason for doing that either. As we had seen in our study of SVT4 without a conformal factor we would only be able to recover the decomposition theorem solution if we were to set χ to zero, and just as with wanting to set α and F00 to zero, for χ there is also no reason to do so. Thus in parallel with our analysis of SVT4 with no conformal factor, we ﬁnd that similarly for SVT4 with a conformal factor no decomposition theorem is obtained in the de

Sitter background case.

Some General Comments

While we have discussed SVT4 ﬂuctuations around a de Sitter background as this is a rich enough system to show that one does not in general get a decomposition theorem, this discussion is not the one that is relevant to the early universe 154

inﬂationary model since that model is not described by an explicit cosmological

constant but by a scalar field instead. Specifically, if we have a scalar field S(x)

with a Lagrangian density L(S) = K(S) − V (S), then at the S = S0 minimum of the V (S) potential with constant S0 the potential acts as a cosmological constant

V (S0) and one has a background de Sitter geometry. If we now perturb the back-

ground the potential will change to V (δS) even though V (S0) will not change at

all. With there also being a change K(δS) in the scalar ﬁeld kinetic energy, all of

the terms in the background Tµν = ∂µS∂νS − gµνL(S) will be perturbed and δTµν

will not be of the form δTµν = δgµνV (S0) that we studied above. Nonetheless, it

would not appear that there would obviously be an SVT4 decomposition theorem

in this more general scalar ﬁeld case.

To conclude this section we note that in the above study of Einstein gravity

SVT4 ﬂuctuations around a de Sitter background we found in the no conformal

prefactor case that the tensor ﬂuctuations obeyed (4.225), viz.

α 2 α 2 (∇α∇ − 4H )(∇α∇ − 2H )Fµν = 0. (4.265)

Even though (4.265) was obtained in Einstein gravity, this very same structure for

Fµν also appears in conformal gravity. In [11] the perturbative conformal gravity

Bach tensor δWµν was calculated for ﬂuctuations around a de Sitter background of

αβ the form hµν = Kµν + gµνg hαβ/4 (i.e. a traceless but not necessarily transverse 155

Kµν), and was found to take the form

1 δW = [∇ ∇α − 4H2][∇ ∇β − 2H2]K µν 2 α β µν 1 − [∇ ∇β − 4H2][∇ ∇ Kλ + ∇ ∇ Kλ ] 2 β µ λ ν ν λ µ 1 + [g ∇ ∇α + 2∇ ∇ − 6H2g ]∇ ∇ Kκλ. (4.266) 6 µν α µ ν µν κ λ

Evaluating δWµν for the ﬂuctuation hµν given in (4.195) in the same de Sitter background is found to yield

α 2 α 2 δWµν = (∇α∇ − 4H )(∇α∇ − 2H )Fµν, (4.267)

i.e. the same structure that we would have obtained from (4.266) had we made

Kµν transverse and replaced it by 2Fµν (even though the relation of Fµν to hµν is

not the same as that of Kµν to hµν). We recognize the structure of the conformal

gravity (4.267) as being none other than that of the standard gravity (4.265).

The transverse-traceless sector of standard gravity (viz. gravity waves) thus has

a conformal structure.

Now in a geometry that is conformal to ﬂat such as de Sitter, the back-

ground Wµν vanishes identically. Thus from the conformal gravity equation of

motion (2.20) for the Bach tensor it follows that the background Tµν also vanishes

¯ identically. In the absence of a new source δTµν, for conformal gravity ﬂuctuations around de Sitter we can thus set

4αgδWµν − δTµν = 0, 4αgδWµν = 0, (4.268) 156

since δTµν = 0. And since δTµν is zero, it follows that δWµν is gauge invariant all

on its own in a background that is conformal to ﬂat. And with it being traceless,

the ﬁve degree of freedom δWµν can only depend on the ﬁve degree of freedom

Fµν, just as we see in (4.267).

α 2 Now from (4.213) we can identify (∇α∇ − 2H )Fµν as the transverse-

2 traceless piece of δGµν − 3H hµν in a de Sitter background, and thus can set

α 2 T θ α 2 α 2 (∇α∇ − 4H )(δGµν + δTµν) = (∇α∇ − 4H )(∇α∇ − 2H )Fµν (4.269)

for the transverse (T ) traceless (θ) sector of δGµν + δTµν. Thus given (4.267) we

can set

α 2 T θ δWµν = (∇α∇ − 4H )(δGµν + δTµν) . (4.270)

α T θ We thus generalize the ﬂat space ﬂuctuation relation δWµν = ∇α∇ δGµν to the de ¯ Sitter case. Finally, we note that if the δTµν source is present, then its tracelessness

¯ ¯ in the conformal case restricts its form in (4.214) to δTµν = Fµν, with the conformal

gravity ﬂuctuation equation in a de Sitter background then taking the form

α 2 α 2 ¯ 4αg(∇α∇ − 4H )(∇α∇ − 2H )Fµν = Fµν. (4.271)

¯ Thus with or without δTµν, in the conformal gravity SVT4 de Sitter case δWµν

depends on Fµν alone, and with there being no dependence on χ the decomposition theorem is automatic. 157

4.2.3 General Robertson Walker The Background

Let us take the background metric and the 3-space Ricci tensor to be of the form

2 µ ν 2 2 i j ˜ ds = −gµνdx dx = Ω (τ) dτ − γ˜ijdx dx , Rij = −2kγ˜ij. (4.272)

Given the symmetry of the 4-geometry, the 4-space Ricci tensor and the 4-space

Einstein tensor can be written as

Rµν = (A + B)UµUν + gµνB,

1 1 Gµν = 2 Agµν − 2 Bgµν + AUµUν + BUµUν, (4.273) where A and B are functions of τ alone and U µ is a unit 4-vector that obeys

µ ν gµνU U = −1. With a background perfect ﬂuid radiation era or matter era

source of the form

Tµν = (ρ + p)UµUν + pgµν, (4.274)

where ρ and p are functions of τ, the background Einstein equations are of the

form

(0) 1 1 ∆µν = 2 Agµν − 2 Bgµν + gµνp + AUµUν + BUµUν + pUµUν

+UµUνρ = 0, (4.275) with solution

.. 1 ˙ 2 −4 −3 A = − 2 (3p + ρ) = −3Ω Ω + 3ΩΩ , 158

.. 1 ˙ 2 −4 −3 −2 B = 2 (p − ρ) = −Ω Ω − ΩΩ − 2kΩ ,

1 ˙ 2 −4 −2 ρ = 2 (−A − 3B) = 3Ω Ω + 3kΩ ,

.. 1 ˙ 2 −4 −3 −2 p = 2 (−A + B) = Ω Ω − 2ΩΩ − kΩ . (4.276)

The SVT4 Fluctuations

While we have incorporated a prefactor of Ω2(τ) in the background metric, we have found it more convenient to not include such a prefactor in the ﬂuctuations.

We thus take the background plus ﬂuctuation metric to be of the form

2 µ ν ds = −[gµν + hµν]dx dx ,

hµν = −2gµνχ + 2∇µ∇νF + ∇µFν + ∇νFµ + 2Fµν, (4.277)

where the ∇µ derivatives are with respect to the full background gµν, with respect

µ µ to which ∇ Fµ = 0, ∇ Fµν = 0. In analog to our discussion of SVT3 Robertson-

Walker ﬂuctuations given above, we set

α 1 α β δUµ = (Vµ + ∇µV ) + UµU (Vα + ∇αV ) − Uµ 2 U U hαβ ,Qµ = Fµ + ∇µF,

ˆ α V = V − U Qα,

α β α β δρˆ = δρ − (ρ + p)(Q Uα∇βU − Q U ∇αUβ),

1 α 1 α β δpˆ = δp − 3 Q ∇α(3p + ρ) + 3 (ρ + p)Q Uα∇βU . (4.278)

With these deﬁnitions and quite a bit of algebra we ﬁnd that we can write the

ﬂuctuation equation ∆µν = 0 as

∆µν = (gµν + UµUν)δpˆ + UµUνδρˆ + (A − B)gµν + 2(A + B)UµUν χ 159

α ˆ α −2(A + B)UµUνU ∇αV + 2gµν∇α∇ χ

ˆ ˆ α −(A + B)Uν∇µV − (A + B)Uµ∇νV − 2∇ν∇µχ − 2(A + B)UµUνU Vα

α β −(A + B)UνVµ − (A + B)UµVν + 2(A + B)UµUνU U Fαβ

α 1 α +2(A + B)UνU Fµα + ( 3 A + B)Fµν + 2(A + B)UµU Fνα

α +∇α∇ Fµν = 0,

µν α α β g ∆µν = 3δpˆ − δρˆ + 2(A − 3B)χ + 6∇α∇ χ + 2(A + B)U U Fαβ = 0,

(4.279) or as

∆µν = (gµν + UµUν)δpˆ + UµUνδρˆ + −2pgµν − 2(p + ρ)UµUν χ

α ˆ α ˆ +2(p + ρ)UµUνU ∇αV + 2gµν∇α∇ χ + (p + ρ)Uν∇µV

ˆ α +(p + ρ)Uµ∇νV − 2∇ν∇µχ + 2(p + ρ)UµUνU Vα

α β +(p + ρ)UνVµ + (p + ρ)UµVν − 2(p + ρ)UµUνU U Fαβ

α 2 α −2(p + ρ)UνU Fµα − 3 ρFµν − 2(p + ρ)UµU Fνα

α +∇α∇ Fµν = 0,

µν α g ∆µν = 3δpˆ − δρˆ + (−6p + 2ρ)χ + 6∇α∇ χ

α β −2(p + ρ)U U Fαβ = 0. (4.280) 160

As written, ∆µν only depends on the metric ﬂuctuations Fµν and χ and the source

ˆ ﬂuctuations δρˆ, δpˆ, V and Vi. Comparing with the SVT3 (4.85) to (4.89) where

˙ there are α, γ, Bi − Ei and Eij metric ﬂuctuations and the same set of source

ﬂuctuations, we ﬁnd, just as in the de Sitter background case, that in a general

Robertson-Walker background the SVT4 formalism is far more compact than the

SVT3 formalism.

As a check on our result we note that in a background de Sitter geometry with ρ = −p = 3H2, k = 0, Ω = 1/τH, δρˆ = δρ = 0, δpˆ = δp = 0, (4.280) reduces to

2 α 2 α ∆µν = 6H gµνχ + 2gµν∇α∇ χ − 2∇ν∇µχ − 2H Fµν + ∇α∇ Fµν = 0,

µν 2 α g ∆µν = 24H χ + 6∇α∇ χ = 0. (4.281)

We recognize (4.281) as (4.218) and (4.219), just as required.

α Finally, since the SVT4 ﬂuctuation equations involve the ∇α∇ operator with its curved space harmonic basis functions, as before there will again be no decomposition theorem unless we choose some judicious initial conditions.

4.2.4 δWµν Conformal to Flat

For conformal gravity SVT4 ﬂuctuations associated with the metric gµν + hµν

where the background metric gµν is of the conformal to ﬂat form given in (4.177),

we recall that for completely arbitrary conformal factor Ω(x) the ﬂuctuation δWµν

is given by the remarkably simple expression [42] 161

1 δW = Ω−2 ∂ ∂σ∂ ∂τ [Ω−2K ] − ∂ ∂σ∂ ∂α[Ω−2K ] − ∂ ∂σ∂ ∂α[Ω−2K ] µν 2 σ τ µν σ µ αν σ ν αµ 2 1 + ∂ ∂ ∂α∂β[Ω−2K ] + η ∂ ∂σ∂α∂β[Ω−2K ] , (4.282) 3 µ ν αβ 3 µν σ αβ

where all derivatives are four-dimensional derivatives with respect to a ﬂat Minkowski

αβ metric, and where Kµν is given by Kµν = hµν − (1/4)gµνg hαβ. If we now make

the SVT4 expansion

2 hµν = Ω (x)[−2ηµνχ + 2∂µ∂νF + ∂µFν + ∂νFµ + 2Fµν] , (4.283)

µ ν where the derivatives and the transverse and tracelessness ∂ Fµ = 0, ∂ Fµν = 0,

µν η Fµν = 0 conditions are with respect to a ﬂat Minkowski background, we ﬁnd that (4.282) reduces to

−2 σ τ δWµν = Ω ∂σ∂ ∂τ ∂ Fµν. (4.284)

This expression is remarkable not just in its simplicity but in the fact that all components of Fµν are completely decoupled from each other, with (4.284) being diagonal in the µ, ν indices. Since (4.284) only contains Fµν with none of χ, F or Fµ appearing in it, unlike in the Einstein gravity SVT4 case where one needs initial conditions to establish the decomposition theorem, in the conformal gravity

SVT4 case the decomposition theorem is automatic. 162

4.3 Gauge Invariants, SVT Mixing, and the Decomposition Theorem

In the SVT4 study of ﬂuctuations around a de Sitter background that we presented

in Sec. 4.2.2 we had found that one of the gauge-invariant combinations was given

˙ by α = F + τχ + F0 (see (4.249)). In this combination F and χ are scalars while

F0 is the fourth component of the vector Fµ. In solving the ﬂuctuation equations

in this case we actually solved for the gauge-invariant combinations and not for

the individual scalar, vector and tensor sectors. In the solution we found that

˙ α = 0 thus would entail only that F + τχ = −F0. However, decomposition with

respect to scalars, vectors and tensors would in addition entail that F˙ + τχ = 0,

and F0 = 0, something that would not be warranted as it is not required by the

ﬂuctuation equations, while moreover not being a gauge-invariant decomposition

of the gauge-invariant α. Thus given this example we in general see that one should only look for a decomposition theorem for gauge-invariant combinations and not look for one for the separate scalar, vector and tensor sectors as gauge invariance can in general intertwine them. Since it might perhaps be thought that this is an artifact of using SVT4 we now present two examples in which it occurs in SVT3. One is fluctuations around an anti de Sitter background, and the other is fluctuations around a completely general conformal to flat background. 163

4.3.1 SVT3 Fluctuations Around an Anti de Sitter Background

For an anti de Sitter background in four dimensions we have

1 ds2 = Ω2(z) dt2 − dx2 − dy2 − dz2 = −g dxµdxν, Ω(z) = , µν Hz

2 2 2 Rλµνκ = −H (gµνgλκ − gλνgµκ),Rµκ = 3H gµκ,R = 12H ,

2 2 Gµν = −3H gµν,Tµν = 3H gµν. (4.285)

We take the ﬂuctuations to have the standard SVT3 form of

2 2 µ ν ds = −Ω (z)(ηµν + fµν) dx dx ,

˜ ˜ i f00 = −2φ, f0i = ∇iB + Bi, ∇ Bi = 0,

˜ ˜ ˜ ˜ fij = −2ψδij + 2∇i∇jE + ∇iEj + ∇iEj + 2Eij,

˜ i ˜ i ij ∇ Ei = 0, ∇ Eij = 0, δ Eij = 0, (4.286)

˜ ˜ i ij ˜ where ∇i and ∇ = δ ∇j are deﬁned with respect to a ﬂat three-dimensional

i j i j background ηijdx dx = δijdx dx .

On deﬁning

2 α = φ + ψ + B˙ − E,¨ δ = φ − ψ + B˙ − E¨ + (∇˜ E + E ), (4.287) z 3 3 following some algebra we ﬁnd that the components of

2 2 ∆µν = δGµν + δTµν = δGµν + 3Ω H fµν (4.288) are given by

.. µν 2 2 2 .. 2 2 2 2 2 ˜ 2 2 2 ˜ 2 g ∆µν = −12H α − 3H z α + 3H z δ + 12H δ + H z ∇ α − 3H z ∇ δ 164

2 ˜ 2 ˙ ¨ 2 +6H z∇3δ + 6H z(B3 − E3) + 24H E33,

.. ij −2 .. −2 ˜ 2 −1 ˜ −1 ˜ δ ∆ij = −9z α − 3α + 3δ + 9z δ − 2∇ δ + z ∇3α + 5z ∇3δ

−1 ˙ ¨ −2 +6z (B3 − E3) + 18z E33,

−2 −2 ˜ 2 ˜ 2 −1 ˜ −1 ˜ −2 ∆00 = 3z α − 3z δ − ∇ α + ∇ δ + z ∇3α − z ∇3δ − 6z E33,

.. −2 .. −2 ˜ 2 ˜ ˜ −1 ˜ −1 ˜ ∆11 = −3z α − α + δ + 3z δ − ∇ δ + ∇1∇1δ + z ∇3α + z ∇3δ

.. −1 ˙ ¨ ˜ ˙ ¨ −2 ˜ 2 +2z (B3 − E3) + ∇1(B1 − E1) − E11 + 6z E33 + ∇ E11

−1 ˜ −1 ˜ +4z ∇1E13 − 2z ∇3E11,

.. −2 .. −2 ˜ 2 ˜ ˜ −1 ˜ −1 ˜ ∆22 = −3z α − α + δ + 3z δ − ∇ δ + ∇2∇2δ + z ∇3α + z ∇3δ

.. −1 ˙ ¨ ˜ ˙ ¨ −2 ˜ 2 +2z (B3 − E3) + ∇2(B2 − E2) − E22 + 6z E33 + ∇ E22

−1 ˜ −1 ˜ +4z ∇2E23 − 2z ∇3E22,

.. −2 .. −2 ˜ 2 −1 ˜ −1 ˜ ˜ ˜ ∆33 = −3z α − α + δ + 3z δ − ∇ δ − z ∇3α + 3z ∇3δ + ∇3∇3δ

−1 ˙ ¨ ˜ ˙ ¨ +2z (B3 − E3) + ∇3(B3 − E3)

.. −2 ˜ 2 −1 ˜ −E33 + 6z E33 + ∇ E33 + 2z ∇3E33,

˜ ˜ ˙ 1 ˜ 2 ˙ −1 ˜ ˙ −1 ˜ ˙ ∆01 = −∇1α˙ + ∇1δ + 2 ∇ (B1 − E1) + z ∇1(B3 − E3) − z ∇3(B1 − E1)

−1 ˙ +2z E13,

˜ ˜ ˙ 1 ˜ 2 ˙ −1 ˜ ˙ −1 ˜ ˙ ∆02 = −∇2α˙ + ∇2δ + 2 ∇ (B2 − E2) + z ∇2(B3 − E3) − z ∇3(B2 − E2)

−1 ˙ +2z E23,

−1 −1 ˙ ˜ ˜ ˙ 1 ˜ 2 ˙ −1 ˙ ∆03 = −z α˙ + z δ − ∇3α˙ + ∇3δ + 2 ∇ (B3 − E3) + 2z E33,

.. ˜ ˜ 1 ˜ ˙ ¨ 1 ˜ ˙ ¨ ˜ 2 ∆12 = ∇2∇1δ + 2 ∇1(B2 − E2) + 2 ∇2(B1 − E1) − E12 + ∇ E12 165

−1 ˜ −1 ˜ −1 ˜ +2z ∇1E23 + 2z ∇2E13 − 2z ∇3E12,

−1 ˜ −1 ˜ ˜ ˜ 1 ˜ ˙ ¨ 1 ˜ ˙ ¨ ∆13 = −z ∇1α + z ∇1δ + ∇3∇1δ + 2 ∇1(B3 − E3) + 2 ∇3(B1 − E1)

.. ˜ 2 −1 ˜ −E13 + ∇ E13 + 2z ∇1E33,

−1 ˜ −1 ˜ ˜ ˜ 1 ˜ ˙ ¨ 1 ˜ ˙ ¨ ∆23 = −z ∇2α + z ∇2δ + ∇3∇2δ + 2 ∇2(B3 − E3) + 2 ∇3(B2 − E2)

.. ˜ 2 −1 ˜ −E23 + ∇ E23 + 2z ∇2E33, (4.289)

˜ 2 ab ˜ ˜ ˙ where ∇ = δ ∇a∇b. With ∆µν being gauge invariant we recognize α, δ, Bi − Ei and Eij as being gauge invariant. We thus see that one of the gauge-invariant combinations, viz. δ, depends on both scalars and vectors. Since our only purpose here is in establishing that one of the gauge-invariant SVT3 combinations does depend on both scalars and vectors, we shall not seek to solve ∆µν = 0 in this particular case. Though if we were to we would only ﬁnd expressions for α, δ,

˙ Bi − Ei and Eij, and not for the separate scalar and vector components of δ.

4.3.2 SVT3 Fluctuations Around a General Conformal to Flat Background

In [42] it was shown that for the arbitrary conformal to ﬂat SVT3 ﬂuctuations of the form

2 2 2 i ds = Ω (x, t) (1 + 2φ)dt − 2(∂iB + Bi)dtdx − [(1 − 2ψ)δij + 2∂i∂jE i j +∂iEj + ∂jEi + 2Eij]dx dx (4.290) with general Ω(x, t), the metric sector gauge-invariant combinations are

˙ ¨ −1 ˙ ˙ −1 ˜ i ˜ α = φ + ψ + B − E, η = ψ − Ω Ω(B − E) + Ω ∇ Ω(Ei + ∇iE), 166

˙ Bi − Ei,Eij. (4.291)

Of these invariant combinations three are independent of Ω altogether and have been encountered frequently throughout this study, while only one, viz. η, actually

depends on Ω at all. (For speciﬁc choices of Ω the quantity −ΩΩ˙ −1η reduces to

the previously introduced β in the de Sitter (4.5) and to γ in the Robertson-

Walker (4.29) and (4.74), while α − 2η reduces to the anti de Sitter δ given in

(4.287).) The invariant α involves scalars alone, the invariant Bi − Ei involves

vectors alone, the invariant Eij involves tensors alone, and only the invariant η

actually involves more than just one of the scalar, vector and tensor sets of modes,

with it speciﬁcally involving both scalars and vectors. While η must always involve

scalars, if Ω has a spatial dependence η will also involve the vector Ei. A spatial

dependence for Ω is encountered in our study of anti de Sitter ﬂuctuations as shown

in (4.287), and is also encountered in SVT3 ﬂuctuations around a Robertson-

Walker background with k 6= 0, where the background Robertson-Walker metric

as shown in (4.176) for k < 0 (and in [42] for k > 0) is written in a conformal

to ﬂat form, with the conformal factor expressly being a function of both time

and space coordinates. Thus in such cases we must expect ∆µν to depend on η

itself and not be separable in separate scalar and vector sectors. While this issue

is met for k 6= 0 Robertson-Walker ﬂuctuations when the background metric is

written in the conformal to ﬂat form given in (4.176), we note that it is not in fact

met for ﬂuctuations around a background Robertson-Walker geometry with metric 167 ds2 = Ω2(τ)[dτ 2 −dr2/(1−kr2)−r2dθ2 −r2 sin2 θdφ2] as given in (4.73), since with

Ω only depending on τ in that case, the gauge-invariant γ = −Ω˙ −1Ωψ + B − E˙ as given in (4.84) does not involve the vector sector modes. While one can of course transform the background metric given in (4.176) into the background metric given in (4.73) by a coordinate transformation with fluctuations around the two metrics thus describing the same physics, the very structure of (4.291) shows that one cannot simply separate in scalar, vector tensor components at will. Rather one must first separate in gauge-invariant combinations, and only if these combinations turn out not to intertwine any of the scalar, vector and tensor sectors could one then separate in each of the scalar, vector and tensor sectors. Moreover, while one can find a form for the background metric in which there is no such intertwining in the k 6= 0 Robertson-Walker case (for k = 0

Ω only depends on τ and so there is no intertwining), this only occurs because of the speciﬁc purely τ-dependent form that the k 6= 0 Ω just happens to take.

For more complicated Ω there would be no coordinate transformation that would eliminate the intertwining, and so it is of interest to study the spatially dependent

Ω situation in and of itself.

While we of course do not need to explicitly solve for fluctuations around a k 6= 0 Robertson-Walker metric when written in a conformal to flat form since in Secs. 4.1.3, 4.1.4, and 4.1.4 we already have solved for fluctuations around the same geometry when written in the general coordinate equivalent form given in 168

(4.73), it is nonetheless of interest to explore the structure of ﬂuctuations around

the conformal to ﬂat form for a k 6= 0 Robertson-Walker geometry. In particular it is of interest to show that the η invariant given in (4.291) actually behaves

quite diﬀerently from all the other invariants. In (4.115) to (4.119) we had ob-

tained some kinematic relations (i.e., relations that do not involve the evolution

equations) that express the gauge-invariant combinations in terms of the fµν com-

ponents of the ﬂuctuation metric. Inspection of these relations and of (4.84) shows

there is only one, viz. that for the relevant η in that case, that depends on Ω. Now

the relations given in (4.115) to (4.119) were derived for ﬂuctuations around the

(4.73) metric. If we now set k = 0 in these relations so that the ∇˜ derivative now

refers to ﬂat spacetime, we would anticipate that for ﬂuctuations around (4.176)

the relations for the gauge-invariant combinations that do not involve Ω might be

replaced by

1 1 ∇˜ b∇˜ ∇˜ a∇˜ α = − ∇˜ b∇˜ ∇˜ i∇˜ f + ∇˜ a∇˜ −∇˜ b∇˜ f + ∇˜ m∇˜ nf b a 2 b i 00 4 a b mn 1 + ∂ ∇˜ b∇˜ ∇˜ if − ∂2 3∇˜ m∇˜ nf − ∇˜ a∇˜ f , (4.292) 0 b 0i 4 0 mn a

˜ a ˜ ˜ i ˜ ˙ ˜ i ˜ ˜ a ˜ ˜ ˜ a ˜ a ˜ ˜ i ∇ ∇a∇ ∇i(Bj − Ej) = ∇ ∇i(∇ ∇af0j − ∇j∇ f0a) − ∂0∇ ∇a∇ fij

˜ ˜ a ˜ b + ∂0∇j∇ ∇ fab, (4.293)

h i ˜ a ˜ ˜ b ˜ ˜ a ˜ ˜ b ˜ 1 ˜ ˜ ˜ a ˜ b ˜ a ˜ 2∇ ∇a∇ ∇bEij = ∇ ∇a∇ ∇bfij + 2 ∇i∇j ∇ ∇ fab + ∇ ∇af 169

˜ a ˜ ˜ ˜ b ˜ ˜ b − ∇ ∇a(∇i∇ fjb + ∇j∇ fib) h i 1 ˜ a ˜ ˜ b ˜ c ˜ ˜ a ˜ ˜ b + 2 γ˜ij ∇ ∇a∇ ∇ fbc − ∇a∇ ∇b∇ f , (4.294)

ab (where f = δ fab), with α still being given by (4.291). Explicit calculation shows

that this anticipation is actually borne out, with (4.292), (4.293) and (4.294)

being found to hold for the ﬂuctuations given in (4.290), no matter how arbitrary

Ω might be.

Now in our discussion of the ﬂuctuations associated with the conformal

gravity δWµν we had obtained the relations given in (4.180) and their inversion

as given in (4.183), (4.185) and (4.185). We now note that these relations involve

the same gauge-invariant combinations as the ones that appear in (4.292), (4.293)

˙ and (4.294), viz. α, Bi − Ei and Eij, with η not appearing. That η could not

appear in δWµν is because Wµν is traceless so that on allowing for four coordinate transformations δWµν can only involve ﬁve quantities (a one-component α, a two-

˙ component Bi −Ei, and a two-component Eij). In this sense then we should think

˙ of α, Bi − Ei and Eij as a unit, with η needing to be treated independently.

Since Wµν is zero in a conformal to ﬂat background, it is associated with a background Tµν that is zero, with δWµν then being gauge invariant on its own as

δT µν is then zero. Thus to determine a gauge-invariant relation that does involve

η we should look for a purely geometric gauge-invariant ﬂuctuation relation that does not involve δTµν. However, none is immediately available as we have already used up δWµν, and in general a ﬂuctuation such as δGµν would not be gauge 170

invariant on its own. However, there is one situation in which not δGµν but

µν δ(g Gµν) is gauge invariant on its own, namely in the radiation era where Tµν,

µν and thus Gµν, are traceless, with δ(g Tµν) then being zero, and with the quantity

µν δ(g Gµν) then being gauge invariant on its own.

Thus in a radiation era conformal to ﬂat k 6= 0 Robertson-Walker back-

ground case as given by (4.290) we evaluate

µν ¨ −3 −3 ˜ ˜ a g Gµν = 6ΩΩ − 6Ω ∇a∇ Ω = 0, (4.295)

¨ ˜ ˜ a and on setting Ω = ∇a∇ Ω obtain

.. µν ˙ −3 ˙ −3 −3 .. −2 −2 ˜ ˜ a δ(g Gµν) = −6α ˙ ΩΩ − 12η ˙ΩΩ − 12ΩαΩ − 6ηΩ − 2Ω ∇a∇ α

−2 ˜ ˜ a −3 ˜ ˜ a −3 ˜ ˜ a +6Ω ∇a∇ η − 6Ω ∇aΩ∇ α + 12Ω ∇aΩ∇ η (4.296)

a ˙ a −3 ˜ ˙ ˙ a ¨a −3 ˜ ab −3 ˜ ˜ −12(B − E )Ω ∇aΩ − 6(B − E )Ω ∇aΩ + 12E Ω ∇b∇aΩ,

˙ where α, Bi − Ei and Eij are as before, with η now being given by the form given in (4.291). We thus establish that in the conformal to ﬂat case the appropriate η is

−1 ˙ ˙ −1 ˜ i ˜ indeed given by the spatially-dependent η = ψ−Ω Ω(B−E)+Ω ∇ Ω(Ei+∇iE), just as required.

4.3.3 Imposition of SVT Gauge Conditions

In [42] inﬁnitesimal gauge transformations of the form

¯ hµν = hµν − ∇µν − ∇νµ (4.297) 171

acting on the conformal to ﬂat (4.290) with arbitrary Ω were considered. On

introducing gauge parameters

2 ˜ ij ˜ ˜ i µ = Ω (x)fµ, f0 = −T, fi = Li + ∇iL, δ ∇jLi = ∇ Li = 0, (4.298) the following transformation relations were found

¯ ˙ −1 ˜ ij ¯ ˙ φ = φ − T − Ω [T ∂0 + (Li + ∇iL)δ ∂j]Ω, B = B + T − L,

¯ −1 ˜ ij ψ = ψ + Ω [T ∂0 + (Li + ∇iL)δ ∂j]Ω,

¯ ¯ ˙ ¯ ¯ E = E − L, Bi = Bi − Li, Ei = Ei − Li, Eij = Eij, (4.299)

with the elimination of the gauge parameters leading directly to the gauge-invariant

combinations shown in (4.291). We now specialize to a particular gauge, and pick

the gauge parameters so that

˙ Li = Ei,L = E,B + T − L = 0. (4.300)

With this choice (4.291) simpliﬁes to

α = φ + ψ, η = ψ, Bi,Eij, (4.301)

and now η only depends on scalars. The combinations given in (4.301) constitute

the longitudinal or conformal-Newtonian gauge, a gauge that is often considered

in cosmological perturbation theory (see e.g. [4,6]). We thus see that using the

gauge freedom one can ﬁnd gauges in which there is no intertwining of scalars and

vectors, so that for them a decomposition theorem in the separate scalar, vector

and tensor sectors is achievable. Chapter 5

Construction and Imposition of Gauge Conditions

In the context of conformal gravity, we continue work done in obtaining solutions to the cosmological fluctuation equations [11] by constructing a gauge condition that is invariant under conformal transformations. Referred to as the conformal gauge, we find that in backgrounds that are conformal to flat, the conformal gravity cosmological fluctuation equations can be brought to an exceedingly simple form, comprised only of a single term. This calculation requires a series of many steps which are given in detail within the chapter, consisting of first motivating and constructing the conformal gauge, composing the fluctuation equations and using curvature identities and covariant derivatives to reduce its form, and finally imposing the conformal gauge itself, with expansion of covariant derivatives into

flat Minkowski partial derivatives to yield (5.32), one of the seminal results of this thesis. Using the conformal properties detailed in Sec. 2.2.1, we find the extra degree of symmetry afforded by conformal invariance provides significant simplifications regarding the trace and moreover allows a very straightforward treatment of the entire cosmological fluctuation equations.

172 173

Such a streamlined approach may be contrasted with the Einstein ﬂuctua- tions, which are also studied here in Sec. 5.2 within diﬀering choices of gauges.

Speciﬁcally, we form a generalized gauge constraint and vary the coeﬃcients in order to cast δGµν into as reduced and compact form possible, permitting readily

solvable integral solutions to be obtained in some simple choices of background

geometry (de Sitter and Minkowski). We ﬁnd that is the coupling of the trace of

the ﬂuctuations that prevents the Einstein ﬂuctuation equations from being able

to be completely decoupled, unlike the case of conformal gravity where the trace

is eﬃciently isolated as a consequence of conformal invariance.

To demonstrate a concrete application of the ﬂuctuation equations in con-

formal gravity, in Sec. 5.3 we evaluate and solve the ﬂuctuations in a k = −1

Robertson Walker radiation era cosmology. Here we ﬁnd that gravitational perturbations naturally have substantial growth, with a leading order time behavior

∝ t4 (with t the comoving time). Such may be contrasted with comparatively damped radiation era t1/2 leading order behavior one obtains in standard Einstein gravity [1]. 174

5.1 The Conformal Gauge and General Solutions in Conformal Gravity

5.1.1 The Conformal Gauge

As discussed in Sec. 2.1.1, we recall that under an inﬁnitesimal transformation of coordinates of the form

xµ → xµ + µ(x), (5.1)

the metric perturbation hµν will transform as

hµν − ∇νµ − ∇µν, (5.2)

with contravariant components transforming similarly as

hµν − ∇νµ − ∇µν. (5.3)

Here all covariant derivatives are taken with respect to the background metric

(0) gµν . For every solution hµν that satisﬁes the ﬂuctuation equations (i.e. δGµν =

0 δTµν or δWµν = δTµν), there exists a transformed hµν that will also serve as a

solution. Thus with the freedom of ﬁxing the four possible arbitrary space-time

µ functions (x), one may eliminate the four gauge degrees of freedom within hµν

by imposing a gauge condition satisfying four equations. Within Ch.2 we have

already explored the form of the Einstein ﬂuctuation equations in the harmonic

gauge as well as the conformal gravity ﬂuctuation equations in the transverse

gauge. 175

In continuing the discussion within conformal gravity, we also recall that in background that are conformal to ﬂat, the gravitational sector ﬂuctuation δWµν

(the Bach tensor, analogous to the Einstein tensor) depends only upon the trace

free contribution of hµν, to thus be able to be expressed entirely in terms of the

νσ (0) 9 component Kµν, obeying K gνσ = 0. Our focus is then to determine the most

appropriate gauge condition as applied to Kµν. We have seen in Sec. 2.2.2 that

µ imposition of the transverse gauge (i.e. ∇ Kµν = 0) was eﬀective in reducing the

ﬂuctuation equations into a readily solvable form within a ﬂat space background.

However, the analogous imposition of the transverse gauge in curved conformal

to ﬂat backgrounds does not provide the same degree of reduction. Rather, in the

more general background, the ﬂuctuations take a form where tensor components

are tightly coupled, and no simple solution is immediately available.

In an attempt to remedy the situation, we seek to ﬁnd a gauge condition

that makes contact with the desirable behavior within the ﬂat space background.

That is, we want a gauge condition that reduces to the transverse gauge if the

background geometry is ﬂat. As the content of this work concerns cosmology,

even more speciﬁcally we desire a gauge condition that is suitable for conformal

to ﬂat backgrounds. The space of all conformal ﬂat backgrounds is in fact quite

large, with cosmologically relevant geometries only comprising a small subspace

of all possible conformal ﬂat metrics. The above gauge constraint criteria will in

fact be satisﬁed if we can ﬁnd a gauge condition that is conformally covariant (i.e. 176 invariant under conformal transformation up to an overall scale factor).

To this end, we ﬁrst note that under the ﬁrst order coordinate transform of

(5.1), Kµν transforms as

1 Kµν → Kµν − ∇νµ − ∇µν + gµν ∇ α. (5.4) 2 (0) α

With the transverse gauge serving as a starting point, we ﬁnd its behavior under coordinate transformation, which takes the from

1 1 ∇ Kµν = ∂ Kµν + Kνσgµρ ∂ g(0) − Kνσgµρ ∂ g(0) + Kµσgνρ ∂ g(0), (5.5) ν ν (0) ν ρσ 2 (0) ρ νσ 2 (0) σ ρν

µν Under a conformal transformation ∇νK transforms as

µν −2 µν −3 µσ ∇νK → Ω ∇νK + 4Ω K ∂σΩ. (5.6)

µν Hence, as alluded to, the transverse gauge condition ∇νK = 0 is not conformal invariant. To determine such a coordinate gauge condition that is conformal invariant, we refer to [42] and note that under a conformal transformation the

µν αβ (0) quantity K g(0)∂νgαβ transforms as

µν αβ (0) −2 µν αβ (0) −3 µν K g(0)∂νgαβ → Ω K g(0)∂νgαβ + 8Ω K ∂νΩ. (5.7)

Hence, it then follows that

1 1 ∇ Kµν − Kµνgαβ∂ g(0) → Ω−2 ∇ Kµν − Kµνgαβ∂ g(0) ν 2 (0) ν αβ ν 2 (0) ν αβ 1 = ∇ Kµν − K¯ µνg¯αβ∂ g¯(0). (5.8) ν 2 (0) ν αβ 177

µν (0) Here, we clarify that ∇νK is to be evaluated in a geometry with metricg ¯µν

according to

1 1 ∇ Kµν = ∂ K¯ µν + K¯ νσg¯µρ ∂ g¯(0) − K¯ νσg¯µρ ∂ g¯(0) + K¯ µσg¯νρ ∂ g¯(0). (5.9) ν ν (0) ν ρσ 2 (0) ρ νσ 2 (0) σ ρν

µν We see that we have achieved the desired result, with the quantity ∇νK −

µν αβ (0) K g(0)∂νgαβ /2 conformal covariance. Expressed in three equivalent forms, the gauge condition

1 ∇ Kµν = Kµνgαβ∂ g(0), ν 2 (0) ν αβ

µν µ(0) σν ν(0) µσ µν α(0) ∂νK + Γνσ K + Γνσ K = K Γαν ,

µν µ(0) σν ∂νK + Γνσ K = 0, (5.10)

is aptly referred to as the conformal gauge.

(0) As a check, we note that when the background is ﬂat Minkowski (gαβ = ηαβ),

µν (5.10) indeed reduces to the transverse condition ∂νK = 0. Hence we may

construct ﬂuctuations around a conformal to ﬂat background in the conformal

gauge by conformally transforming ﬂuctuations around a ﬂat background in the

transverse gauge. Such a method is not shared within standard Einstein gravity,

but in conformal gravity it will be prove to be very beneﬁcial in simplifying the

ﬂuctuation equations. 178

5.1.2 δWµν in an Arbitrary Background Composing the Fluctuation Equations

With the conformal gauge in hand, we proceed in a sequence of steps in order to implement it with the ﬂuctuation equations. Prior to perturbing Wµν we present a useful identity

σ σ ∇β∇νTλµ = ∇ν∇βTλµ + RλσνβT µ − RσµνβTλ , (5.11)

which holds for any rank two tensor. We then express W µν as

1 1 W = − g ∇ ∇βRα + ∇ ∇βR − ∇ ∇ Rα − RβσR µν 6 µν β α β µν 3 µ ν α σµβν 1 2 1 − RβσR + g R Rαβ + Rα R − g (Rα )2. (5.12) σνβµ 2 µν αβ 3 α µν 6 µν α

We set the metric as the most general gµν + hµν, where gµν denotes an arbitrary background metric and δgµν = hµν an arbitrary and general ﬂuctuation. Upon perturbing W µν we then obtain

1 αβ αβ γ 2 αβ 1 αβ δWµν(hµν) = 2 hµνRαβR − gµνh Rα Rβγ − 3 h RαβRµν + 3 gµνh RαβR

1 2 αβ γ αβ γ 1 α αβ − 6 hµνR + h Rα Rµβνγ + h Rα Rµγνβ − 6 hµν∇α∇ R − h ∇β∇αRµν

1 αβ 1 αβ γ 1 αβ 1 α + 6 gµνh ∇β∇αR + 6 gµνh ∇γ∇ Rαβ + 3 h ∇µ∇νRαβ + 3 R∇α∇ hµν

γ αβ γ αβ 1 α 1 α +Rµβνγ∇α∇ h + Rµγνβ∇α∇ h − 3 R∇α∇µhν − 3 R∇α∇νhµ

1 α 1 α β αβ 2 αβ − 6 ∇αhµν∇ R + 6 gµν∇ R∇βhα − ∇αh ∇βRµν − 3 Rµν∇β∇αh

1 αβ 1 α β αβ 1 α β + 3 gµνR∇β∇αh + 2 Rν ∇β∇αhµ − R ∇β∇αhµν + 2 Rµ ∇β∇αhν

1 α β 1 α β 1 β α 1 β α − 2 Rν ∇β∇ hµα − 2 Rµ ∇β∇ hνα + 2 ∇β∇ ∇α∇ hµν − 2 ∇β∇ ∇α∇µhν 179

1 β α 1 α β αβ 1 α β − 2 ∇β∇ ∇α∇νhµ − 2 Rν ∇β∇µhα + R ∇β∇µhνα − 2 Rµ ∇β∇νhα

αβ β α β α β α +R ∇β∇νhµα + ∇αRνβ∇ hµ − ∇βRνα∇ hµ + ∇αRµβ∇ hν

β α αβ γ 2 αβ γ γ αβ −∇βRµα∇ hν − gµνR ∇γ∇βhα + 3 gµνR ∇γ∇ hαβ − Rµανβ∇γ∇ h

1 γ αβ 1 γ αβ αβ 1 α + 6 gµν∇γ∇ ∇β∇αh + 3 gµν∇γRαβ∇ h − ∇βRνα∇µh + 6 ∇ R∇µhνα

1 αβ αβ 1 αβ 1 α − 6 R ∇µ∇νhαβ − ∇βRµα∇νh + 3 ∇µRαβ∇νh + 6 ∇ R∇νhµα

1 αβ 1 αβ 1 αβ 2 α + 3 ∇µh ∇νRαβ − 2 R ∇ν∇µhαβ + 3 ∇ν∇µ∇β∇αh + 3 Rµν∇α∇ h

1 α 1 α 1 α 1 α − 3 gµνR∇α∇ h + 2 ∇α∇ ∇ν∇µh − 12 gµν∇αh∇ R + 2 ∇αRµν∇ h

1 αβ 1 β α β α 1 + 2 gµνR ∇β∇αh − 6 gµν∇β∇ ∇α∇ h − Rµανβ∇ ∇ h + 3 R∇ν∇µh

1 α − 3 ∇ν∇µ∇α∇ h. (5.13)

Here we clarify that all covariant derivatives are evaluated with respect to the background gµν (and use a more compact notation where the Ricci scalar R de-

α notes R α.) Inspecting (5.13), we observe a total of 62 diﬀerent terms, with ten

µν of these depending on the trace h = g hµν.

We now substitute hµν = Kµν + (1/4)gµνh in (5.13) and, as anticipated and discussed in Sec. 2.2.1, we ﬁnd that δWµν(hµν) breaks into two pieces; one that depends only on Kµν comprising 52 terms and one that depends only on

µν h = gµνh with 19 total terms. These two separate components have been obtained and take the form

1 αβ αβ γ 2 αβ δWµν(Kµν) = 2 KµνRαβR − gµνK Rα Rβγ − 3 K RαβRµν

1 αβ 1 2 αβ γ αβ γ 1 α + 3 gµνK RαβR − 6 KµνR + K Rα Rµβνγ + K Rα Rµγνβ − 6 Kµν∇α∇ R 180

αβ 1 αβ 1 αβ γ 1 αβ −K ∇β∇αRµν + 6 gµνK ∇β∇αR + 6 gµνK ∇γ∇ Rαβ + 3 K ∇µ∇νRαβ

1 α γ αβ γ αβ 1 α + 3 R∇α∇ Kµν + Rµβνγ∇α∇ K + Rµγνβ∇α∇ K − 3 R∇α∇µKν

1 α 1 α 1 α β αβ − 3 R∇α∇νKµ − 6 ∇αKµν∇ R + 6 gµν∇ R∇βKα − ∇αK ∇βRµν

2 αβ 1 αβ 1 α β αβ − 3 Rµν∇β∇αK + 3 gµνR∇β∇αK + 2 Rν ∇β∇αKµ − R ∇β∇αKµν

1 α β 1 α β 1 α β 1 β α + 2 Rµ ∇β∇αKν − 2 Rν ∇β∇ Kµα − 2 Rµ ∇β∇ Kνα + 2 ∇β∇ ∇α∇ Kµν

1 β α 1 β α 1 α β αβ − 2 ∇β∇ ∇α∇µKν − 2 ∇β∇ ∇α∇νKµ − 2 Rν ∇β∇µKα + R ∇β∇µKνα

1 α β αβ β α β α − 2 Rµ ∇β∇νKα + R ∇β∇νKµα + ∇αRνβ∇ Kµ − ∇βRνα∇ Kµ

β α β α αβ γ 2 αβ γ +∇αRµβ∇ Kν − ∇βRµα∇ Kν − gµνR ∇γ∇βKα + 3 gµνR ∇γ∇ Kαβ

γ αβ 1 γ αβ 1 γ αβ −Rµανβ∇γ∇ K + 6 gµν∇γ∇ ∇β∇αK + 3 gµν∇γRαβ∇ K

αβ 1 α 1 αβ αβ −∇βRνα∇µK + 6 ∇ R∇µKνα − 6 R ∇µ∇νKαβ − ∇βRµα∇νK

1 αβ 1 α 1 αβ 1 αβ + 3 ∇µRαβ∇νK + 6 ∇ R∇νKµα + 3 ∇µK ∇νRαβ − 2 R ∇ν∇µKαβ

1 αβ + 3 ∇ν∇µ∇β∇αK , (5.14)

1 αβ 1 1 2 1 αβ δWµν(h) = − 8 gµνRαβR h − 6 RµνRh + 24 gµνR h + 2 R Rµανβh

1 α 1 α 1 1 α − 4 h∇α∇ Rµν + 24 gµνh∇α∇ R + 12 h∇ν∇µR + 4 ∇α∇ ∇ν∇µh

1 α 1 β α 1 α 1 α − 4 ∇αRµν∇ h − 2 Rµανβ∇ ∇ h + 4 ∇µRνα∇ h − 4 ∇αRν ∇µh

1 α 1 α 1 1 α 1 + 4 Rν ∇µ∇αh + 4 ∇νRµα∇ h + 8 ∇νR∇µh − 4 ∇αRµ ∇νh + 8 ∇µR∇νh +

1 α 1 α 4 Rµ ∇ν∇αh − 4 ∇ν∇µ∇α∇ h. (5.15) 181

Trace Properties

Within Sec. 2.2.1, we illustrated an important property of the trace of the ﬂuc-

tation, here given within (5.15), where we have demonstrated its form being able

to represent as proportional to the background Wµν. To show and verify such a form, we ﬁrst note the identity

σ ∇κ∇νVλ − ∇ν∇κVλ = V Rλσνκ (5.16)

which holds for any vector Vλ. Upon setting Vλ = ∇λh and Tλµ = ∇λ∇µh in

(5.11) we obtain

α αβ σ ∇ν∇µ∇α∇ h = g ∇ν[∇α∇µ∇βh + Rβσαµ∇ h]

αβ σ = g ∇ν[∇α∇β∇µh + Rβσαµ∇ h]

αβ σ σ = g [∇α∇ν∇β∇µh + Rβσαν∇ ∇µh − Rσµαν∇β∇ h

σ σ +Rβσαµ∇ν∇ h + ∇νRβσαµ∇ h]

αβ σ σ = g [∇α[∇β∇ν∇µh + Rµσβν∇ h] + Rβσαν∇ ∇µh

σ σ σ −Rσµαν∇β∇ h + Rβσαµ∇ν∇ h + ∇νRβσαµ∇ h]. (5.17)

Recalling the curvature relation

ν ∇ Rνµκη = ∇κRµη − ∇ηRµκ, (5.18)

it then follows that

α α α σ σ σ ∇ν∇µ∇α∇ h − ∇α∇ ∇ν∇µh = Rµσαν∇ ∇ h + ∇µRνσ∇ h − ∇σRνµ∇ h 182

σ α σ σ σ +Rσν∇ ∇µh − Rσµαν∇ ∇ h + Rσµ∇ν∇ h + ∇νRσµ∇ h. (5.19)

α Finally, using the Bianchi identity ∇ Rµα = (1/2)∇µR we thus observe that all

twelve terms within (5.15) that contain the gradient of h must all cancel. For those

seven terms that remain, we can further observe that they are directly equivalent

to the deﬁnition of the background Wµν. Thus, the trace component δWµν(h)

reduces to the extremely compact form of

1 δW (h) = − W h. (5.20) µν 4 µν

To gain further insight into the form of (5.20), it is instructive to consider the

ﬂuctuation of the Weyl tensor itself. In evaluating the perturbed Weyl tensor around an arbitrary background, iwe obtain δCλµνκ = δCλµνκ(Kµν) + δCλµνκ(h), where

1 αβ 1 αβ 1 1 δCλµνκ(Kµν) = − 6 gκµgλνK Rαβ + 6 gκλgµνK Rαβ + 2 KµνRκλ − 2 KλνRκµ

1 1 1 1 1 1 − 2 KκµRλν + 2 KκλRµν − 6 gµνKκλR + 6 gλνKκµR + 6 gκµKλνR − 6 gκλKµνR

α 1 α 1 α 1 α +Kλ Rκνµα + 4 gµν∇α∇ Kκλ − 4 gλν∇α∇ Kκµ − 4 gκµ∇α∇ Kλν

1 α 1 α 1 α 1 α + 4 gκλ∇α∇ Kµν − 4 gµν∇α∇κKλ + 4 gλν∇α∇κKµ − 4 gµν∇α∇λKκ

1 α 1 α 1 α 1 α + 4 gκµ∇α∇λKν + 4 gλν∇α∇µKκ − 4 gκλ∇α∇µKν + 4 gκµ∇α∇νKλ

1 α 1 αβ 1 αβ 1 − 4 gκλ∇α∇νKµ − 6 gκµgλν∇β∇αK + 6 gκλgµν∇β∇αK − 2 ∇κ∇λKµν

1 1 1 1 1 + 2 ∇κ∇µKλν + 2 ∇κ∇νKλµ − 2 ∇ν∇κKλµ + 2 ∇ν∇λKκµ − 2 ∇ν∇µKκλ

(5.21) 183 and

1 1 1 1 δCλµνκ(h) = 8 gµνRκλ − 8 gλνRκµ − 8 gκµRλν + 8 gκλRµν 1 1 1 + 24 gκµgλνR − 24 gκλgµνR − 4 Rκνλµ h 1 = hC . (5.22) 4 λµνκ

Inspection of (5.22) reveals that if the background Weyl tensor Cλµνκ is zero then it follows that δCλµνκ has no dependence upon h. Given a vanishing background

Weyl tensor, we may also observe that from (2.20) δW µν can therebfore be expressed as

µν µλνκ µλνκ δW = 2∇κ∇λδC − RκλδC , (5.23) and thereby also be independent of h.

Diﬀerential Commutations

Before we can express (5.14) as a form ready for application of the conformal gauge condition, we must ﬁrst commute the diﬀerential operators as per (5.11) and (5.16). On performing the commutations for δWµν(Kµν) we obtain

1 αβ 1 α β 2 αβ αβ δWµν(Kµν) = 2 KµνRαβR − 2 Kν RαβRµ − 3 K RαβRµν + K RµαRνβ

1 α β 1 αβ 1 α 1 α 1 2 − 2 Kµ RαβRν + 3 gµνK RαβR + 3 Kν RµαR + 3 Kµ RναR − 6 KµνR

αβ γκ 2 αβ α βγ αβ γ −gµνK R Rαγβκ − 3 K RRµανβ − Kν R Rµβαγ + 2K Rα Rµγνβ

αβ γ κ α βγ 1 α 1 α +2K RαγβκRµ ν − Kµ R Rνβαγ + 3 R∇α∇ Kµν − 6 Kµν∇α∇ R

1 α β 1 α β 1 α 1 α β + 2 Rν ∇α∇βKµ + 2 Rµ ∇α∇βKν − 6 ∇αKµν∇ R + 6 gµν∇ R∇βKα 184

αβ 2 αβ 1 αβ αβ −∇αK ∇βRµν − 3 Rµν∇β∇αK + 3 gµνR∇β∇αK − R ∇β∇αKµν

αβ 1 αβ 1 α β 1 α β −K ∇β∇αRµν + 6 gµνK ∇β∇αR + 2 Kν ∇β∇ Rµα + 2 Kµ ∇β∇ Rνα

1 β α 1 β α 1 β α + 2 ∇β∇ ∇α∇ Kµν − 2 ∇β∇ ∇µ∇αKν − 2 ∇β∇ ∇ν∇αKµ

αβ γ β α β α 2 αβ γ −gµνR ∇β∇γKα + ∇αRνβ∇ Kµ + ∇αRµβ∇ Kν + 3 gµνR ∇γ∇ Kαβ

γ αβ 1 αβ γ αβ γ −2Rµανβ∇γ∇ K + 6 gµνK ∇γ∇ Rαβ − K ∇γ∇ Rµανβ

1 γ αβ 1 γ αβ γ αβ + 6 gµν∇γ∇ ∇β∇αK + 3 gµν∇γRαβ∇ K − 2∇γRµανβ∇ K

γ αβ γ αβ αβ 1 α +Rµβνγ∇ ∇αK + Rµγνβ∇ ∇αK − ∇βRνα∇µK + 6 ∇ R∇µKνα

1 α 1 α β αβ αβ − 3 R∇µ∇αKν − 2 Rν ∇µ∇βKα + R ∇µ∇βKνα − ∇βRµα∇νK

1 αβ 1 α 1 αβ 1 α + 3 ∇µRαβ∇νK + 6 ∇ R∇νKµα + 3 ∇µK ∇νRαβ − 3 R∇ν∇αKµ

1 α β αβ 2 αβ 1 αβ − 2 Rµ ∇ν∇βKα + R ∇ν∇βKµα − 3 R ∇ν∇µKαβ + 3 K ∇ν∇µRαβ

1 αβ + 3 ∇ν∇µ∇β∇αK . (5.24)

Comprising 59 terms, we note that when evaluated within a ﬂat background,

(5.24) reduces to

1 β α 1 β α 1 β α δWµν(Kµν) = 2 ∇β∇ ∇α∇ Kµν − 2 ∇β∇ ∇µ∇αKν − 2 ∇β∇ ∇ν∇αKµ

1 γ αβ 1 αβ + 6 gµν∇γ∇ ∇β∇αK + 3 ∇ν∇µ∇β∇αK . (5.25)

Hence, (5.24) reduces to the result of (2.42), namely the simplified flat fluctuation

equations without any imposition of a gauge condition - such a result provides an

aﬃrmative check on our calculation thus far. 185

5.1.3 δWµν in a Conformal to Flat Minkowski Background Implementing the Conformal Gauge Condition

With (5.24) now being ready for insertion of the conformal gauge, we evaluate

(5.24) in the conformal to ﬂat background given in (2.23) recalling that we take

Ω(x) to be a completely general and arbitrary spacetime function. In the gµν =

2 µν 1 µν αβ Ω (x)ηµν background the gauge condition ∇νK = 2 K g ∂νgαβ takes the form

1 ∇ Kµν − KµνΩ−2ηαβη ∂ Ω2 = ∇ Kµν − 4Ω−1Kµν∂ Ω ν 2 αβ ν ν ν

µν −1 µν −1 µν = ∂νK + 6Ω K ∂νΩ − 4Ω K ∂νΩ

µν −1 µν −2 2 µν = ∂νK + 2Ω K ∂νΩ = Ω ∂ν(Ω K )

= 0. (5.26)

We can factor out a contribution of Ω2(x) from the ﬂuctuation by setting Kµν =

−2 µν 2 µν Ω (x)k and Kµν = Ω (x)kµν. For clariﬁcation, here indices on k and kµν =

αβ ηµαηνβk are raised and lowered with ηµν. As a result, (5.26) can thus be ex-

µν pressed as the familiar transverse form ∂νk = 0, noting that the conformal gauge

condition is such that there is no longer a dependence upon the conformal factor.

1 Before we alas evaluate the (5.24) within a conformal Minkowski background in

the conformal gauge given in (5.26), we observe that the gauge condition

µν −1 µν ∇νK = 4Ω K ∂νΩ (5.27)

1 Within (5.26) we note that we have taken Ω(x) to be a general function of the coordinates so that we additionally encompass the special case of Robertson-Walker geometries with arbitrary spatial curvature k. As given in AppendixB, Ω( x) will only depend on the time coordinate t for k = 0, while for k 6= 0, Ω(x) will depend on both t and the radial coordinate r. 186

in fact possesses the same form of a covariant gauge condition for a background

2 metric Ω (x)gµν with any gµν. Hence our treatment is fully covariant. It follows that when (5.27) is imposed in a conformal to ﬂat but not necessarily

Minkowski background (e.g. polar coordinates of the form ds2 = dt2 − dr2 −

r2dθ2 − r2 sin2 θdφ2) then (5.24) will take the form

1 δW = Ω−4∇˜ ∇˜ β∇˜ ∇˜ αK − 4Ω−5∇˜ ∇˜ K ∇˜ β∇˜ αΩ µν 2 β α µν β α µν −5 ˜ ˜ α ˜ ˜ β −5 ˜ α ˜ ˜ β ˜ −2Ω ∇α∇ Ω∇β∇ Kµν − 4Ω ∇ Ω∇β∇ ∇αKµν

−5 ˜ ˜ β ˜ ˜ α −5 ˜ ˜ ˜ β ˜ α −6 ˜ ˜ α ˜ ˜ β −Ω Kµν∇β∇ ∇α∇ Ω − 4Ω ∇αKµν∇β∇ ∇ Ω + 6Ω ∇αΩ∇ Ω∇β∇ Kµν

−6 ˜ α ˜ ˜ ˜ β −6 ˜ ˜ α ˜ ˜ β +12Ω ∇ Ω∇β∇αKµν∇ Ω + 3Ω Kµν∇α∇ Ω∇β∇ Ω

−6 ˜ ˜ α ˜ ˜ β −6 ˜ α ˜ ˜ β ˜ +12Ω ∇αKµν∇ Ω∇β∇ Ω + 24Ω ∇ Ω∇βKµν∇ ∇αΩ

−6 ˜ ˜ ˜ β ˜ α −6 ˜ α ˜ ˜ β ˜ +6Ω Kµν∇β∇αΩ∇ ∇ Ω + 12Ω Kµν∇ Ω∇β∇ ∇αΩ

−7 ˜ ˜ α ˜ ˜ β −7 ˜ ˜ α ˜ ˜ β −24Ω Kµν∇αΩ∇ Ω∇β∇ Ω − 48Ω ∇αΩ∇ Ω∇βKµν∇ Ω

−7 ˜ α ˜ ˜ ˜ β −8 ˜ ˜ α ˜ ˜ β −48Ω Kµν∇ Ω∇β∇αΩ∇ Ω + 60Ω Kµν∇αΩ∇ Ω∇βΩ∇ Ω. (5.28)

˜ Here we have introduced ∇α to denote the covariant derivative with respect to

˜ α the ﬂat (but not necessarily Minkowski) background gµν such that ∇ is equal to

αβ ˜ g ∇α. Observing (5.28), we remarkably ﬁnd that the 17 terms can be factored

into one single compact expression

1 1 δW (K ) = Ω−2∇˜ ∇˜ α∇˜ ∇˜ β(Ω−2K ) = Ω−2∇˜ ∇˜ α∇˜ ∇˜ βk , (5.29) µν µν 2 α β µν 2 α β µν

−2 where we we again note that kµν = Ω (x)Kµν. Thus (5.29) embodies the rep-

resentation of ﬂuctuations around a geometry that is conformal to an arbitrary 187

µν −1 µν ﬂat background metric, within the ∇νK = 4Ω K ∂νΩ gauge. Upon setting

Ω(x) = 1, (5.29) reconciles our result within Sec. 2.2.2 as a representation of the

ﬂuctuation equations around a ﬂat background geometry in the transverse gauge

µν ∇νK = 0 as

1 δW (K ) = ∇˜ ∇˜ α∇˜ ∇˜ βK . (5.30) µν µν 2 α β µν

Further Fluctuation Reduction

Although the forms of (5.29) and (5.30) are exceedingly simple, from a practical perspective, these are not of straightforward use since they involve covariant derivatives that mix the various components of Kµν. To ﬁx this issue, we note that (5.29) and (5.30) also apply in the gauge given in (5.26). Hence within the conformal gauge, ﬂuctuations around a conformal to Minkowski geometry take the partial derivative form

1 δW (K ) = Ω−4∂ ∂β∂ ∂αK − 4Ω−5∂ ∂ K ∂β∂αΩ µν µν 2 β α µν β α µν

−5 α β −5 α β −5 β α −2Ω ∂α∂ Ω∂β∂ Kµν − 4Ω ∂ Ω∂β∂ ∂αKµν − Ω Kµν∂β∂ ∂α∂ Ω

−5 β α −6 α β −6 α β −4Ω ∂αKµν∂β∂ ∂ Ω + 6Ω ∂αΩ∂ Ω∂β∂ Kµν + 12Ω ∂ Ω∂β∂αKµν∂ Ω

−6 α β −6 α β −6 α β +3Ω Kµν∂α∂ Ω∂β∂ Ω + 12Ω ∂αKµν∂ Ω∂β∂ Ω + 24Ω ∂ Ω∂βKµν∂ ∂αΩ

−6 β α −6 α β −7 α β +6Ω Kµν∂β∂αΩ∂ ∂ Ω + 12Ω Kµν∂ Ω∂β∂ ∂αΩ − 24Ω Kµν∂αΩ∂ Ω∂β∂ Ω

−7 α β −7 α β −48Ω ∂αΩ∂ Ω∂βKµν∂ Ω − 48Ω Kµν∂ Ω∂β∂αΩ∂ Ω

−8 α β +60Ω Kµν∂αΩ∂ Ω∂βΩ∂ Ω. (5.31) 188

Inspection of (5.31) reveals that it remarkably simpliﬁes to

1 1 δW (K ) = Ω−2ησρηαβ∂ ∂ ∂ ∂ (Ω−2K ) = Ω−2ησρηαβ∂ ∂ ∂ ∂ k .(5.32) µν µν 2 σ ρ α β µν 2 σ ρ α β µν

As written, (5.32) may be recognized as being of the exact form given within Sec.

2.2.1 based on grounds on conformal invariance and transformation properties of

2 the Bach tensor. Regardless of the gµν = Ω (x)ηµν background not being ﬂat, in

(5.31) and (5.32) all derivatives are ﬂat Minkowski (i.e. associated with the metric

2 α β 2 2 2 2 ds = −ηαβdx dx = dt − dx − dy − dz ). In terms of these partial derivatives, we observe a signiﬁcant feature in that (5.31) and (5.32) are diagonal in the

(µ, ν) indices (i.e. there is no mixing of the components of kµν from the different operator). Consequently, the reductive significance of conformal symmetry is exemplified in our starting point with a the 62 term δWµν(hµν) given in (5.13) and arriving at the single term (5.32).

5.1.4 Calculation Summary

As we have covered numerous steps in the derivation of the ﬂuctuations within conformal gravity, we provide a summary overview of the procedure given. We began with a general Wµν in the form of in (5.12) and then perturbed Wµν to ﬁrst order around a general background with metric gµν with a perturbed metric of the form gµν + δgµν = gµν + hµν. With the identity

λ λ κ λ σ σ λ λ ν λ σ σ λ δR µνκ = ∂δΓµν/∂x + ΓκσδΓµν − ΓµκδΓνσ − ∂δΓµκ/∂x − ΓνσδΓµκ + ΓµνδΓκσ

λ λ = ∇κδΓµν − ∇νδΓµκ, (5.33) 189

λ λρ where δΓµν = (1/2)g [∇νδgρµ + ∇µδgρν − ∇ρδgµν], we then evaluate δWµν to lowest order in δgµν to obtain (5.13).

To address trace simpliﬁcations, in (5.13) we set hµν = Kµν + (1/4)gµνh

µν µν where h = g hµν and g Kµν = 0, from which it follows that δWµν may be expressed as two contributions viz. δWµν(Kµν) in (5.14) and δWµν(h) in (5.15).

Commuting covariant derivatives we observed and conﬁrmed that δWµν(h) =

−(1/4)Wµνh as shown in (5.20). Consequently, this has established that δWµν(h)

will vanish if the background Wµν vanishes (a result that applied to the back-

grounds of Robertson-Walker and de Sitter cosmologies).

For geometries in which the background Wµν vanishes, δWµν reduces to

δWµν(Kµν), with δWµν to then only be dependent on the traceless ﬂuctuation

Kµν as given in (5.14). With further commutation of covariant derivatives we

then expressed (5.14) in the form given in (5.24). Now in a form ready for con-

formal gauge implementation, we apply condition (5.10) within a conformal ﬂat

background to yield δWµν(Kµν) given as (5.32), the main result of this section.

5.2 Imposing Speciﬁc Gauges within δGµν

2 2 2 i j With background plus ﬂuctuation metric of the form ds = Ω (x)[dt −δijdx dx ]−

2 µ ν Ω (x)fµνdx dx , the ﬂuctuation in the Einstein tensor is given by

1 ˜ ˜ α 1 ˜ ˜ α αβ −1 ˜ ˜ αβ −1 ˜ ˜ δGµν = − 2 ∇α∇µfν − 2 ∇α∇νfµ − η ηµνΩ ∇αf∇βΩ + η Ω ∇αfµν∇βΩ

βα −2 ˜ ˜ 1 αβ ˜ ˜ 1 ˜ ˜ αβ 1 αβ ˜ ˜ +η fµνΩ ∇αΩ∇βΩ − 2 η ηµν∇β∇αf + 2 ηµν∇β∇αf + 2 η ∇β∇αfµν 190

αβ −1 ˜ ˜ αβ −1 ˜ ˜ αγ −1 ˜ β ˜ +2ηµνf Ω ∇β∇αΩ − 2η fµνΩ ∇β∇αΩ + 2η ηµνΩ ∇βfα ∇γΩ

αγ βκ −2 ˜ ˜ αβ −1 ˜ ˜ αβ −1 ˜ ˜ −η η ηµνfαβΩ ∇γΩ∇κΩ − η Ω ∇βΩ∇µfνα − η Ω ∇βΩ∇νfµα

1 ˜ ˜ + 2 ∇ν∇µf, (5.34)

where Ω(x) is an arbitrary function of xµ. In our exploration of various gauges

below, we make use of the trace free contribution of the conformally factored

αβ metric perturbation kµν = fµν − (1/4)η fαβ where the tracelessness is deﬁned

µν with respect to the background η kµν = 0. With the substitution of kµν, we may express (5.34) as

1 αβ −1 αβ −1 βα −2 δGµν = − 4 η ηµνΩ ∂αΩ∂βf + η Ω ∂αkµν∂βΩ + η kµνΩ ∂αΩ∂βΩ

1 αβ 1 αβ αβ −1 1 αβ + 2 η ∂β∂αkµν − 4 η ηµν∂β∂αf − 2η kµνΩ ∂β∂αΩ − 2 η ∂β∂µkνα

1 αβ αβ γκ −1 αγ βκ −2 − 2 η ∂β∂νkµα + 2η η ηµνΩ ∂βΩ∂κkαγ − η η ηµνkαβΩ ∂γΩ∂κΩ

1 αβ γκ αβ γκ −1 αβ −1 + 2 η η ηµν∂κ∂βkαγ + 2η η ηµνkαγΩ ∂κ∂βΩ − η Ω ∂βΩ∂µkνα

αβ −1 1 −1 1 −1 1 −η Ω ∂βΩ∂νkµα − 4 Ω ∂µΩ∂νf − 4 Ω ∂µf∂νΩ + 4 ∂ν∂µf, (5.35)

αβ with f denoting η fαβ. As our goal is to reduce the (5.35) into as compact

form as possible, we explore a most general set of possible gauges with variable

coeﬃcients in the form of the gauge condition

αβ −1 αβ −1 η ∂αkβν = Ω Jη kνα∂βΩ + P ∂νf + RΩ f∂νΩ. (5.36) 191

Here J, P , and R represent the constant coeﬃcients of which we will vary. On taking J = −2, P = 1/2, R = 0 the gauge condition becomes

αβ −1 αβ 1 η ∂αkβν = −2Ω η kνα∂βΩ + 2 ∂νf. (5.37)

If we elect to take an Ω that is strictly dependent upon conformal time τ, then

evaluation of δGµν leads to

−2 ˙ 2 −1 ¨ 1 µν −1 ˙ 1 −1 ˙ δG00 = (3Ω Ω − Ω Ω + 2 η ∂µ∂ν − Ω Ω∂0)k00 − 4 (Ω Ω∂0 + ∂0∂0)f,

−1 ¨ 1 µν −1 ˙ 1 −1 ˙ δG0i = (Ω Ω + 2 η ∂µ∂ν − Ω Ω∂0)k0i − 4 (Ω Ω∂i + ∂i∂0)f,

−2 ˙ 2 −1 ¨ −2 ˙ 2 −1 ¨ 1 µν δGij = δij(−2Ω Ω + Ω Ω)k00 + (−Ω Ω + 2Ω Ω + 2 η ∂µ∂ν

−1 ˙ 1 −1 ˙ −Ω Ω∂0)kij, − 4 (δijΩ Ω∂0 + ∂i∂j)f,

µν −2 ˙ 2 −1 ¨ 1 −1 ˙ µν η δGµν = (−10Ω Ω + 6Ω Ω)k00 − 4 (2Ω Ω∂0 + η ∂µ∂ν)f, (5.38)

where as usual a dot denotes a derivative with respect to τ, and where ∂0 denotes

∂τ . We see that δGµν is now expressed in a form that nearly decouples each tensor component, with it being the k00 and f dependence δGµν that prevents it from doing so. However, one can solve the equations exactly once δTµν is speciﬁed, as

µν one can use the δG00 and η δGµν equations to can determine k00 and f, and then from δGµν one can determine all remaining components of kµν.

To illustrate an example, if the background is the de Sitter geometry, i.e.

where we set Ω(τ) = 1/Hτ with H constant, then (5.38) takes the form

−2 1 µν −1 1 −1 δG00 = (τ + 2 η ∂µ∂ν + τ ∂0)k00 + 4 (τ ∂0 − ∂0∂0)f, 192

−2 1 µν −1 1 −1 δG0i = (2τ + 2 η ∂µ∂ν + τ ∂0)k0i + 4 (τ ∂i − ∂i∂0)f,

−2 1 µν −1 1 −1 δGij = (3τ + 2 η ∂µ∂ν + τ ∂0)kij + 4 (δijτ ∂0 − ∂i∂j)f,

µν −2 1 −1 µν η δGµν = 2τ k00 + 4 (2τ ∂0 − η ∂µ∂ν)f. (5.39)

These equations form a compact set of reduced terms, a form which was possible by the explicit incorporation of the conformal factor Ω(τ) into both the background and the ﬂuctuation (which also holds for (5.38). Hence, in this gauge the δGµν =

−8πGδTµν ﬂuctuation equations facilitate a readily integrable solution.

Another convenient decomposition of δGµν corresponds to the gauge choice

J = −4, R = 2P − 3/2, with P arbitrary. In the de Sitter background, the

ﬂuctuations take the form

−2 1 µν −1 3 −2 δG00 = (−2τ + 2 η ∂µ∂ν + 3τ ∂0)k00 + ( 4 − P )τ

1 µν −1 1 2 + 4 (1 − 2P )η ∂µ∂ν + P τ ∂0 + ( 4 − P )∂0 f,

−1 −2 1 µν −1 1 −1 δG0i = τ ∂ik00 + (τ + 2 η ∂µ∂ν + 2τ ∂0)k0i + (P − 2 )τ ∂i

1 +( 4 − P )∂i∂0 f,

−2 −1 −1 −2 1 µν −1 δGij = δijτ k00 + τ ∂jk0i + τ ∂ik0j + (3τ + 2 η ∂µ∂ν + τ ∂0)kij

3 −2 1 µν −1 + δij ( 4 − P )τ + 4 (2P − 1)η ∂µ∂ν + (P − 1)τ ∂0 f

1 +( 4 − P )∂i∂jf. (5.40)

αβ 3 αβ −1 −2 3 2 αβ −2 η δGαβ = (P − 4 )(η ∂α∂βf + 4τ ∂0f − 6τ f) = (P − 4 )τ η ∂α∂β(τ f).

µν 2 µν We see conveniently that η δGµν = Ω g δGµν depends only upon the trace f of

the ﬂuctuation. Hence we can foremost solve for f and then proceed to the other 193

components of the ﬂuctuation in turn. We may further observe the component

αβ η δGαβ takes the form of the ﬂat space free massless particle wave operator acting

−2 µν µν on τ f. Hence, we may solve g δGµν = −8πGg δTµν via integration by the

(4) αβ (4) 4 D (x − y) Green’s function obeying η ∂α∂βD (x − y) = δ (x − y). We thus can form the exact integral solution as

Z µν 8πG 2 4 (4) −2 µν f = η fµν = − 3 τ (x) d yD (x − y)τ (y)η δTµν(y), (5.41) (P − 4 )

where it is implied that τ(x) = x0, τ(y) = y0.

5.3 Robertson Walker Radiation Era Conformal Gravity Solution

Motivated by the excellent agreement between the ﬁts to the galactic rotation

curves of 138 spiral galaxies presented in [11, 15, 29] and to ﬁts of the accelerating

universe Hubble plot data presented in [10, 12], we thus consider conformal gravity

ﬂuctuations in Robertson-Walker cosmologies with negative k.

In the conformal gravity ﬁtting of the current era Hubble plot, we note

from [33] that the cosmological constant term is found to dominate over the per-

fect ﬂuid contribution. However, in the early universe, the radiation era perfect

ﬂuid is the dominant contribution since a(t) is small and A/a4(t) is large. More-

over, if the A/a4(t) radiation contribution is dominant, then since k is given

2 2 2 by k = −a˙ − 2A/a S0 when the α contribution in (B.19) is negligible, we

are phenomenologically steered towards negative k. Thus for studying ﬂuctu-

ation growth in the early universe the only relevant solution for a(t) is that 194

of a(t, α = 0, k < 0, A > 0). Using this solution, upon setting k = −1/L2,

2 4 2 2 2 2 2 d = 2AL /S0 , and L a (t) = (d + t ), we obtain

Z t dt t τ = L 2 2 1/2 = L arcsinh , t = d sinh p, (5.42) 0 (d + t ) d

where p = τ/L. We have seen that from (2.47), ﬂuctuations around a ﬂat back-

ground grow linearly in the relevant time variable, which according to the k < 0

(B.15) is p0. Upon setting Ω(p, χ) = La(p)(cosh p+cosh χ), we may express (B.15)

in the form

ds2 = Ω2(p, χ) dp02 − dx02 − dy02 − dz02 , (5.43)

i.e. the conformal to ﬂat form. From Sec. 5.1, we recall that the conformal gravity

ﬂuctuation (5.32) takes the form

−2 σρ αβ δWµν = (1/2)Ω η η ∂σ∂ρ∂α∂βkµν (5.44)

−2 where kµν = Ω (x)Kµν. Solving this fourth-order wave equation in terms of

momentum eiginstates as done in Sec. 2.2.2 (cf. (2.47)), we express the solution

to δWµν = 0 in the primed-variable form

0 ik0·x0 0 0 0 ik0·x0 0∗ −ik0·x0 0 0 0∗ −ik0·x0 kµν = Aµνe + (n · x )Bµνe + Aµνe + (n · x )Bµνe , (5.45)

µν 0 0 where η kµkν = 0. Within this coordinate basis, the conformal gauge condition

0 µν µν µα νβ takes the form ∂νk = 0, with k = η η kαβ. Substituting (5.45) into the

conformal gauge, it follows that

0ν h 0 ik0·x0 0 0 0 ik0·x0 0∗ −ik0·x0 0 0 0∗ −ik0·x0 i ik Aµνe + (n · x )Bµνe − Aµνe − (n · x )Bµνe 195

0ν h 0 ik0·x0 0∗ −ik0·x0 i +n Bµνe + Bµνe = 0. (5.46)

For these expression to hold for all x0, it then follows that we must have

0ν 0 0ν 0 0ν 0 ik Aµν + n Bµν = 0, ik Bµν = 0,

0ν 0∗ 0ν 0∗ 0ν 0∗ −ik Aµν + n Bµν = 0, −ik Bµν = 0. (5.47)

0 Inspection of (5.45) show that the Bµν term has leading order large time behavior via its n0·x0 prefactor. Consequently, continuing to work to leading order, we ignore

0 0 the non-leading Aµν modes and may take the Bµν modes to obey the transverse

0ν 0 0ν 0∗ momentum space conditions ik Bµν = 0 and −ik Bµν = 0. It follows then that

0 0 0 ik0·x0 components of the Bµν modes have the same (n · x )e leading behavior in coordinate space. More speciﬁcally, since n0µ = (1, 0, 0, 0) the ﬂuctuations grow linearly in the time variable p0 associated with (5.43).

At present, we have expressed the solutions in terms of the conformal to ﬂat coordinate system; however, to make contact with their leading order behavior in terms of the canonical comoving coordinates t and r, we need to both reexpress

Ω(p, χ) in terms of the comoving coordinates (t, r) and transform the components of the ﬂuctuation Kµν to the comoving coordinates. Starting ﬁrst with Ω(p, χ),

2 we note that from Kµν = Ω kµν and (5.32) that the Kµν ﬂuctuations grow as

Ω2(p, χ)p0. Making use of (B.15) and (B.12) from AppendixB, and coordinate transformation (5.42) one calculates Ω2(p, χ)p0 to obtain

Ω2(p, χ)p0 = L2a2(p)(cosh p + cosh χ)2p0 = L2a2(p) sinh p(cosh p + cosh χ) 196 " # t t2 1/2 r2 1/2 = (d2 + t2) 1 + + 1 + . (5.48) d d2 L2

Analysis of (5.48) reveals that Ω2(p, χ)p0 for t d grows linearly in t, for later times and grows as t4 when t d. Consequently, in the given conformal to ﬂat

Minkowski coordinate system of (B.15), we see that the leading time behavior

4 (t d speciﬁcally) of all the components of Kµν is proportional to t .

The second task that remains is to transform the fluctuation Kµν itself. To this end, one can note that in transforming from (B.15) to (B.14), there is no alteration of the behavior of p0, with all dependence relegated to the conformal factor. However, one must additional be careful to note that the final transformation between spatial Cartesian coordinates (B.15) and polar coordinates (B.14) does in fact introduce a dependence on r0. One can observe that, any angular component induces a dependence on the comoving t. For instance, in terms of the Kx0x0 type fluctuations in the (B.15) coordinate system, we have the following transformation relations in the (B.14) coordinate system

0 2 0 2 0 2 Kθθ = (r cos θ cos φ) Kx0x0 + (r cos θ sin φ) Ky0y0 + (r sin θ) Kz0z0

2 sinh χ 2 2 2 2 2 = [cos θ cos φK 0 0 + cos θ sin φK 0 0 + sin θK 0 0 ] (cosh p + cosh χ)2 x x y y z z r2d2 = × [L(d2 + t2)1/2 + d(L2 + r2)1/2]2

2 2 2 2 2 [cos θ cos φKx0x0 + cos θ sin φKy0y0 + sin θKz0z0 ],

0 2 0 2 0 Kr0θ = r cos θ sin θ cos φKx0x0 + r cos θ sin θ sin φKy0y0 − r sin θ cos θKz0z0 sinh χ = × (cosh p + cosh χ) 197

2 2 [cos θ sin θ cos φKx0x0 + cos θ sin θ sin φKy0y0 − sin θ cos θKz0z0 ],

0 0 0 Kp0θ = r cos θ cos φKp0x0 + r cos θ sin φKp0y0 − r sin θKp0z0 sinh χ = [cos θ cos φK 0 0 + cos θ sin φK 0 0 − sin θK 0 0 ]. (cosh p + cosh χ) p x p y p z

(5.49)

We may continue to compute the analogous expressions for Kθφ, Kφφ, Kr0φ and

0 Kp0φ. Within (5.49), we see that the r = sinh χ/(cosh p + cosh χ) prefactor has leading comoving time behavior as t0 if p = χ, (i.e. t = r with both t and r large) or as t−1 if p χ (i.e. t r). These correspond to the lightlike and timelike modes respectively.

In transforming from from (B.14) to (B.4), we note that the angular sector is unaﬀected by the transformation. Thus the angular sector ﬂuctuations Kθθ, Kθφ,

Kφφ associated with the comoving Robertson-Walker geometry given in (B.4) will have leading order growth as t4 when including the contribution of the prefactor in (5.49) for lightlike coordinates and as t2 for the timelike case. Moreover, since the lightlike ds2 = 0 is both general coordinate invariant and conformal invariant, lightlike modes associated with the (B.15) metric will transform into lightlike modes associated with the metric (B.4). Finally, we note that a t4 growth for the lightlike modes is a rather signiﬁcant growth rate, one that cannot be obtained from standard Einstein gravity when using the same radiation matter source.

To further address the non-angular modes, we recall that the full coordinate 198

transformation of the ﬂuctuation is given by

∂x0α ∂x0β K = K0 , (5.50) µν ∂xµ ∂xν αβ

and transformations between the (p0, r0), (p, χ) and (t, r) take the form

∂p0 ∂r0 1 + cosh p cosh χ ∂p0 ∂r0 sinh p sinh χ = = , = = − , ∂p ∂χ [cosh p + cosh χ]2 ∂χ ∂p [cosh p + cosh χ]2 ∂p 1 ∂χ 1 = , = . (5.51) ∂t La(t) ∂r L cosh χ

To interpret (5.51), we must determine the relation between p and χ, namely whether p = χ or p χ. Inspecting (5.42), we may observe that t = d sinh p

when La(t) = (d2 + t2)1/2. Hence for large t with p = χ also large, we obtain

∂p0 ∂r0 ∂p0 ∂r0 ∂p 1 ∂χ d = = 1, = = 1, = , = . (5.52) ∂p ∂χ ∂χ ∂p ∂t t ∂r Lt

Consequently, as we go from the (p0, r0) coordinates to (p, χ) coordinates, the lead-

ing time behavior is unchanged. In further transforming (B.11) to the comoving

2 (B.4) ((p, χ) to (t, r)) we obtain 1/t suppression in the Ktt, Ktr and Krr sectors,

a 1/t suppression for Ktθ, Ktφ, Krθ and Krφ, and no suppression for Kθθ, Kθφ and

Kφφ.

Lastly, we must include the contribution of the Ω2(p, χ)p0 ∝ t4 prefactor

2 3 and obtain Ktt, Ktr and Krr growing as t , Ktθ, Ktφ, Krθ and Krφ growing as t ,

4 and Kθθ, Kθφ and Kφφ growing as t . For this large t and large p = χ scenario,

the Ktt, Ktr, Krr, Ktθ, Ktφ, Krθ and Krφ are suppressed with respect to Kθθ, Kθφ

4 and Kφφ, so the leading growth will be the t growth associated with the angular

Kθθ, Kθφ and Kφφ. 199

Now investigating the leading order behavior for large p χ, the transformations take the form

∂p0 ∂r0 d cosh χ ∂p0 ∂r0 d sinh χ = = , = = − , ∂p ∂χ t ∂χ ∂p t ∂p 1 ∂χ 1 = , = . (5.53) ∂t t ∂r L cosh χ

As we go from the (p0, r0) coordinates to (p, χ) coordinates, incorporating the t4 dependence of Ω2(p, χ)p0 and including the prefactor in (5.49), we proceed

0 1 1 analogous to before and determine growth of Ktt ∼ t , Ktr ∼ t , Ktθ ∼ t ,

1 2 2 2 2 2 2 2 Ktφ ∼ t , while Krr ∼ t , Krθ ∼ t , Krφ ∼ t , Kθθ ∼ t , Kθφ ∼ t and Kφφ ∼ t .

1 Hence, for large p χ, Ktr, Ktθ and Ktφ all have t leading order time behavior,

2 2 and Krr, Krθ, Krφ, Kθθ, Kθφ and Kφφ all have t leading order behavior, with t being the overall leading growth.

2 Such suppression in this case can be viewed as the following: if χ is negligible then so is r, and thus the spatial part of the metric in (B.4) eﬀectively becomes ﬂat. Chapter 6

Conclusions

In the analysis of cosmological perturbations, we have delved into the two methods of constructing and solving the equations of motion: the SVT decomposition of the fluctuations and the imposition of specific gauge constraints. As regards the former, we have seen that the SVT basis provides a very convenient formalism in expressing the 10 degrees of freedom within the metric perturbation hµν. With the SVT3 quantities being defined as spatial integrals of hµν, the SVT construction itself is thus intrinsically non-local, with the existence of the integrals themselves requiring proper asymptotic convergence. Upon implementing the SVT basis within the cosmological fluctuation equations as applied to de Sitter and

Roberston Walker backgrounds, one can readily form a set of six gauge invariant combinations comprised of two scalars, a two-component transverse vector, and a traceless rank two tensor.

Expressing the ﬂuctuation equations solely in terms of these gauge invariants, one can solve them exactly and achieve a decoupling of the gauge invariants by application of higher derivatives which serve to project out the longitudinal and

200 201

transverse components. By solving these higher derivative equations of motion ex-

actly, we are able to assess the necessary conditions required for the decomposition

theorem - a theorem asserting that scalars, vectors, and tensors decouple in the

equations of motion themselves - to hold. We ﬁnd that solely by imposing the

same asymptotic boundary conditions at r = ∞ that are needed to set up the

scalar, vector, tensor basis in the ﬁrst place, one then indeed does get the decom-

position theorem for ﬂuctuations around a ﬂat background, around a de Sitter

background or around a spatially ﬂat Robertson-Walker background with k = 0.

However, for ﬂuctuations around a Robertson-Walker background with k 6= 0 one

additionally has to require that ﬂuctuations be well-behaved at r = 0 in order to

get the decomposition theorem to hold. The distinction between these two classes

of solution is that in the ﬁrst three (ﬂat, de Sitter, k = 0 Robertson-Walker) the

background metric can be written as a conformal to ﬂat metric with a conformal

factor is solely time dependent, i.e. Ω(τ). In this class, the SVT gauge invariant

˙ ¨ ˙ −1 ˙ ˙ combinations are φ + ψ + B − E, Bi − Ei, Eij, and −ψ + Ω Ω(B − E) (up to a minor variation given the selected linear combination of scalars). In the second class (k 6= 0 Robertson-Walker) the background metric can be written as a conformal to ﬂat metric with the conformal factor depending on both r and τ,

˙ ¨ ˙ i.e. Ω(r, τ). Here the gauge invariant combinations are φ + ψ + B − E, Bi − Ei,

−1 ˙ ˙ −1 ij Eij, and ψ − Ω Ω(B − E) − Ω δ ∂jΩ(Ei + ∂iE) (up to a minor variation given the selected linear combination of scalars), where the evident intertwining of SVT 202

scalars and vectors thus prevents establishing a decomposition theorem. Hence,

in conformal to ﬂat backgrounds asymptotic boundary conditions alone will in

general only secure the decomposition theorem if the conformal factor depends

solely on the conformal time. However using the underlying gauge freedom in the

theory one can ﬁnd gauges in which the scalars and vectors are not intertwined

to thus recover the decomposition theorem.

Given that the SVT3 basis is not manifestly covariant with, for instance,

scalars transforming into vectors under the full four dimensional transformations,

we have introduced an alternative SVT4 formalism (and more generally devel-

oped the D dimensional SVTD basis) that appropriately matches the underlying transformation group. When contrasted with SVT3, implementation of the SVT4 formalism provides significant simplification, with the fluctuation equations being able to be expressed in more compact and readily solvable forms. Similar to

SVT3, one can apply higher derivatives in order to separate the scalar, vector, and tensor sectors at the level of the equations of motion. However, to obtain a decomposition theorem, one need not only impose good asymptotic spatial behavior but also must require initial conditions as well.

As regards the second method of solution where one directly imposes gauge conditions (without enacting a decomposition), we have constructed a gauge condition appropriate to conformal gravity that remains invariant under conformal transformations. Referred to as the conformal gauge, upon implementing it within 203

a generical conformal to ﬂat background, one can express the entire perturbative

Bach tensor succinctly as

1 δW (K ) = Ω−2ησρηαβ∂ ∂ ∂ ∂ (Ω−2K ) µν µν 2 σ ρ α β µν 1 = Ω−2ησρηαβ∂ ∂ ∂ ∂ k . 2 σ ρ α β µν

with Kµν being the traceless component of the general metric perturbation hµν.

With the k = −1 Roberston Walker geometry being the relevant cosmology within conformal gravity, we solve the radiation era ﬂuctuation equations exactly. In terms of momentum eigenstates of the fourth order equation of motion, we ﬁnd that the solutions grow as t4, with t the comoving time. Such can be contrasted with the t1/2 growth one obtains in standard Einstein radiation era cosmology.

Finally, we evaluate a large set of possible gauges within Einstein gravity with a

de Sitter background, ﬁnding compact forms that facilitate exact integral solu-

tions, with coupling only occurring between the trace and the remaining metric

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SVT by Projection

A.1 The 3 + 1 Decomposition

For geometries with constant spatial curvature, (e.g. Roberston Walker, Minkowski) it is especially convenient to utilize projections that decouple the time and spatial components of symmetric rank two tensors. Thus, we introduce the standard covariant 3+1 decomposition of a symmetric rank two tensor Tµν in a 4-dimensional geometry with metric gµν. To facilitate decomposition, we need only make use of µ µ ν a 4-vector U obeying gµνU U = −1 and a projector

Pµν = gµν + UµUν (A.1) that obeys µν µν µν σ UµP = 0,PµνP = gµνP = 3,PµσP ν = Pµν. (A.2) In terms of the projector we can write σ τ σ τ σ τ Tµν = gµ gν Tστ = Pµ Pν Tστ − UµU Pν Tστ σ τ σ τ −Pµ UνU Tστ + UµUνU U Tστ . (A.3) On introducing 1 ρ = U σU τ T , p = P στ T , q = −P σU τ T , στ 3 στ µ µ στ 1 1 1 π = P σP τ + P σP τ − P P στ T , (A.4) µν 2 µ ν 2 ν µ 3 µν στ which obey µ ν µν µν U qµ = 0,U πµν = 0, πµν = πνµ, g πµν = P πµν = 0, (A.5) we can rewrite Tµν as

Tµν = (ρ + p)UµUν + pgµν + Uµqν + Uνqµ + πµν, (A.6) a familiar form that may for instance be found in [5]. As constructed, the ten-component Tµν has been covariantly decomposed into two one-component 4-scalars, one three-component 4-vector that is orthogonal to Uµ and one ﬁve- component traceless, rank two tensor that is also orthogonal to Uµ.

209 210

A.2 Vector Fields

We recall from Sec. 2.1 that the general hµν has ten components and with the freedom to impose four coordinate transformations, these may be reduced to six physical components. In terms of the SVT decompositions, the SVT3 and SVT4 formalisms yield ﬂuctuation equations that only depend on six SVT combinations in the SVT3 or SVT4 expansions of hµν. We recall that the SVT3 and SVT4 components are related to the components of hµν via integral relations such as that given in (3.16), for example Z Z 3 (3) ˜ i ˜ 3 (3) ˜ i B = d yD (x − y)∇yh0i,Bi = h0i − ∇i d yD (x − y)∇yh0i. (A.7)

From (A.7) one can observe that components are intrinsically non-local, with their very existence requiring that the associated integrals exist. Consequently, we see that asymptotic boundary conditions are a necessary and irreducible component in their construction. As discussed in [43] and [42], we introduce another method to implement gauge invariance using non-local operators, namely the projection operator approach, with such an approach being equivalent to the SVT formalism used in Ch.3. To discuss the application of the projection operator approach to rank two tensors such as hµν we ﬁrst apply it to a four-dimensional gauge ﬁeld Aµ. Thus in analog to (A.7) we set Z T 4 0 0 α T L Aµ = Aµ + ∂µ d x D(x − x )∂ Aα = Aµ + Aµ , (A.8) Z Z T 4 0 0 α L 4 0 0 α Aµ = Aµ − ∂µ d x D(x − x )∂ Aα,Aµ = ∂µ d x D(x − x )∂ Aα,

µ 0 4 0 T where ∂µ∂ D(x − x ) = δ (x − x ), where Aµ obeys the transverse condition µ T L ∂ Aµ = 0, and where Aµ is longitudinal. The utility of this expansion is that T under Aµ → Aµ + ∂µχ the transverse Aµ transforms as Z Z T 4 0 0 α 4 0 0 α Aµ → Aµ + ∂µχ − ∂µ d x D(x − x )∂ Aα − ∂µ d x D(x − x )∂ ∂αχ

T = Aµ , (A.9) where we perform an integration by parts. Thus with integration by parts the T T transverse Aµ is automatically gauge invariant. In addition we note that Aµ obeys

ν T ν ν ν ∂ν∂ Aµ = ∂ν∂ Aµ − ∂µ∂ Aν = ∂ Fνµ. (A.10) Thus just as with the use of the non-local SVT formalism for gravity, the use of T the non-local Aµ enables us to write the Maxwell equations entirely in terms of 211

L gauge-invariant quantities. With Aµ being the derivative of a scalar function it is pure gauge, and thus cannot appear in the gauge-invariant Maxwell equations. T Moreover, while there may be an integration by parts issue for Aµ , there is none ν T ν α for ∂ν∂ Aµ as it is equal to the gauge-invariant quantity ∂ν∂ Aµ −∂µ∂ Aα, just as it must be since the Maxwell equations are gauge invariant. In the SVT language, with (A.7) and (A.9) only involving scalars and vectors, we can think of (A.7) as an SV3 decomposition of the 3-component h0i, and (A.9) as an SV4 decomposition of the 4-component Aµ. An alternate way of understanding these results is to introduce a projection operator ∂ Z ∂ Π = η − d4x0D(x − x0) , (A.11) µν µν ∂xµ ∂x0ν

T as we can then rewrite Aµ as

T ν Aµ = ΠµνA . (A.12)

As introduced, Πµν obeys the projector algebra relations

ν ΠµνΠ σ = Πµσ, Z T ν T 4 0 0 T ν 0 T ΠµνA = Aµ − ∂µ d x D(x − x )∂νA (x ) = Aµ , Z Z Lν 4 0 0 ν 0 4 0 0 ΠµνA = ∂µ d x D(x − x )∂νA (x ) − ∂µ d x D(x − x )× Z ν 4 00 0 00 σ 00 ∂ν∂ d x D(x − x )∂σA (x ) = 0. (A.13)

T In the SVT4 language we set Aµ = Aµ + ∂µA, and can thus identify

T ν L ν Aµ = ΠµνA ,Aµ = ∂µA = (ηµν − Πµν)A . (A.14) For vector fields the SVT formalism is thus equivalent to the projector formalism. Having now established this equivalence for vector fields, we turn now to tensor fields.

A.3 Transverse and Longitudinal Projection Operators for Flat Spacetime Tensor Fields

For tensor ﬁelds we introduce 4-dimensional ﬂat spacetime transverse and longitudinal projection operators [42, 43]: Z Z 4 0 0 4 0 0 Tµνστ = ηµσηντ − ∂µ d x D(x − x )ηντ ∂σ − ∂ν d x D(x − x )ηµσ∂τ 212

Z Z 4 0 0 4 00 0 00 + ∂µ∂ν d x D(x − x )∂σ d x D(x − x )∂τ , Z Z 4 0 0 4 0 0 Lµνστ = ∂µ d x D(x − x )ηντ ∂σ + ∂ν d x D(x − x )ηµσ∂τ Z Z 4 0 0 4 00 0 00 − ∂µ∂ν d x D(x − x )∂σ d x D(x − x )∂τ . (A.15)

As constructed, these projectors obey a standard projector algebra

στ στ Tµνστ T αβ = Tµναβ,Lµνστ L αβ = Lµναβ, στ στ Tµνστ L αβ = 0,Lµνστ T αβ = 0,Lµνστ + Tµνστ = ηµσηντ . (A.16)

T In terms of these projectors we deﬁne transverse and longitudinal components hµν L and hµν of hµν according to Z στ T 4 0 0 σ 0 Tµνστ h = hµν = hµν − ∂µ d x D(x − x )∂σh ν(x ) Z 4 0 0 κ 0 −∂ν d x D(x − x )∂κh µ(x ) Z Z 4 0 0 4 00 0 00 σκ 00 +∂µ∂ν d x D(x − x )∂σ d x D(x − x )∂κh (x ), Z Z στ L 4 0 0 σ 0 4 0 0 κ 0 Lµνστ h = hµν = ∂µ d x D(x − x )∂σh ν(x ) + ∂ν d x D(x − x )∂κh µ(x ) Z Z 4 0 0 4 00 0 00 σκ 00 −∂µ∂ν d x D(x − x )∂σ d x D(x − x )∂κh (x ). (A.17)

Assuming integration by parts these components obey

T µν Lµν µν ∂νh = 0, ∂νh = ∂νh . (A.18)

T T T With hµν transforming as hµν → hµν under hµν → hµν − ∂µν − ∂νµ as long as T we can integrate by parts, we see that, as introduced, hµν is both transverse and gauge invariant. On evaluation we obtain 1 1 1 [∂ ∂ hT + ∂ ∂αhT ] − η ∂ ∂σhT = [∂ ∂ h − ∂ ∂ hλ − ∂ ∂ hλ 2 µ ν α µν 2 µν σ 2 µ ν µ λ ν ν λ µ 1 +∂ ∂αh ] − η [∂ ∂αh − ∂ ∂ hσλ], α µν 2 µν α σ λ (A.19) where hT is given by Z T αβ T 4 0 0 σν 0 h = η hαβ = h − ∂ν d x D(x − x )∂σh (x ), (A.20) 213

αβ 1 with h = η hαβ. On recognizing the right-hand side of (A.19) as δRµν − 2 ηµνδR = δGµν, we obtain 1 δG = 1 [∂ ∂ hT + ∂ ∂αhT ] − η ∂ ∂σhT . (A.21) µν 2 µ ν α µν 2 µν σ We thus write the perturbed Einstein tensor entirely in terms of the non-local, T gauge invariant, six degree of freedom hµν. To make contact with the SVT4 expansion we insert

hµν = −2ηµνχ + 2∂µ∂νF + ∂µFν + ∂νFµ + 2Fµν (A.22)

T into hµν, to obtain Z T 4 0 0 T hµν = −2ηµνχ + 2Fµν + 2∂µ∂ν d D(x − x )χ(x ), h = −6χ. (A.23)

T With δGµν being written in terms of the projected hµν, we see that it is written in T terms of the SVT4 Fµν and χ. However as written, hµν contains an integral term in (A.23). To eliminate it we extend transverse projection to transverse-traceless projection.

A.4 Transverse-Traceless Projection Operators for Flat Spacetime Tensor Fields

In [43] and [42] two further projectors were introduced

1 Z Z Q = η − ∂ ∂ d4x0D(x − x0) η − ∂0 d4x00D(x0 − x00)∂00 , µνστ 3 µν µ ν στ σ τ

Pµνστ = Tµνστ − Qµνστ . (A.24)

They obey the projector algebra

στ στ στ Tµνστ Q αβ = Qµναβ,Qµνστ T αβ = Qµναβ,Qµνστ Q αβ = Qµναβ, σταβ σταβ στ Pµνστ Q = 0,Qµνστ P = 0,Pµνστ P αβ = Pµναβ. (A.25)

T The projector Pµνστ projects out the traceless piece of hµν, while Qµνστ projects out its complement, and they implement

στ T T θ στ T T T θ Pµν hστ = hµν ,Qµν hστ = hµν − hµν , (A.26)

T θ στ στ L with hµν being both traceless and transverse. With Qµν implementing Qµν hστ = στ στ L 0, Pµν implements Pµν hστ = 0 as well, to thus implement

στ T θ Pµν hστ = hµν . (A.27) 214

T Pµνστ is thus a traceless projector not just for the transverse hµν but for the full hµν as well. We can thus introduce its complementary projection operator Uµνστ = ηµσηντ − Pµνστ , as it obeys

σταβ σταβ στ Pµνστ U = 0,Uµνστ P = 0,Uµνστ U αβ = Uµναβ, 1 U στ h = h − hT θ = hLθ + η ηστ h µν στ µν µν µν 3 µν στ 1 Z − ∂ ∂ d4yD(x − y)ηστ h , (A.28) 3 µ ν στ

Given (A.26) and (A.24) we obtain

1 1 Z hT θ = hT − η ησκhT + ∂ ∂ d4yD(x − y)ησκhT , (A.29) µν µν 3 µν σκ 3 µ ν σκ

Inserting (A.23) into (A.29) yields

T θ hµν = 2Fµν, (A.30)

T θ with χ dropping out. Finally, in terms of hµν we can rewrite (A.21) as

1 α T θ 1 σ T 1 T δGµν = 2 ∂α∂ hµν − 3 ηµν∂σ∂ h + 3 ∂µ∂νh . (A.31) Then with

1 T θ 1 T Fµν = 2 hµν , χ = − 6 h , (A.32) we can rewrite (A.31) as

α α δGµν = ∂α∂ Fµν + 2ηµν∂α∂ χ − 2∂µ∂νχ. (A.33)

We recognize (A.33) as the expression for δGµν as given in (3.40) when D = 4, T T θ T and with hµν and thus hµν and h being gauge invariant, we confirm that given integration by parts Fµν and χ are gauge invariant, just as noted in Sec. 3.2. Thus with (A.32) we establish the equivalence of the SVT4 decomposition and the projection operator technique. As a further example of this equivalence we note that for conformal gravity fluctuations around a flat spacetime background (4.282) takes the form

1 δW = ∂ ∂σ∂ ∂τ K − ∂ ∂σ∂ ∂αK − ∂ ∂σ∂ ∂αK µν 2 σ τ µν σ µ αν σ ν αµ 2 1 + ∂ ∂ ∂α∂βK + η ∂ ∂σ∂α∂βK , (A.34) 3 µ ν αβ 3 µν σ αβ 215

where all derivatives are four-dimensional derivatives with respect to a ﬂat Minkowski αβ metric, and where Kµν is given by Kµν = hµν − (1/4)ηµνη hαβ. Inserting (A.17) and (A.29) into (A.34) yields 1 δW = ∂ ∂σ∂ ∂τ hT θ. (A.35) µν 2 σ τ µν With the insertion of (A.22) into (A.34) yielding

σ τ δWµν = ∂σ∂ ∂τ ∂ Fµν, (A.36)

(cf. (4.284) with Ω = 1), we recover (A.30), and again conﬁrm the equivalence of the SVT4 decomposition and the projection operator technique.

A.5 Transverse and Longitudinal Projection Operators for Curved Spacetime Tensor Fields

For curved spacetime with background metric gµν it is convenient to deﬁne a 2-index propagator

ν τ ν β 0 ν −1/2 4 0 [g β∇τ ∇ + ∇β∇ ]D σ(x, x ) = g σ(−g) δ (x − x ). (A.37) In terms of it we introduce [43] Z 4 0 1/2 0 Tµνστ = gµσgντ − ∇µ d x (−g) Dνσ(x, x )∇τ Z 4 0 1/2 0 −∇ν d x (−g) Dµσ(x, x )∇τ , Z Z 4 0 1/2 0 4 0 1/2 0 Lµνστ = ∇µ d x (−g) Dνσ(x, x )∇τ + ∇ν d x (−g) Dµσ(x, x )∇τ . (A.38)

These projection operators close on the projector algebra given in (A.16). As στ T στ L such, they eﬀect Tµνστ h = hµν and Lµνστ h = hµν, where Z Z T 4 0 1/2 ν 0 στ 4 0 1/2 µ 0 στ hµν = hµν−∇µ d x (−g) D σ(x, x )∇τ h −∇ν d x (−g) D σ(x, x )∇τ h , (A.39)

Z Z L 4 0 1/2 ν 0 στ 4 0 1/2 µ 0 στ hµν = ∇µ d x (−g) D σ(x, x )∇τ h + ∇ν d x (−g) D σ(x, x )∇τ h . (A.40) 216

The utility of constructing these projected states is that under a gauge transformation hµν transforms into hµν − ∇µν − ∇νµ. However, we see that this L is precisely the structure of hµν. The longitudinal component of hµν can thus be removed by a gauge transformation, and the fluctuation Einstein equations can T only depend on the 6-component hµν. However, unlike the flat background case T where one can write δGµν itself entirely in terms of hµν, in the curved background case there must be a background Tµν, and thus it is only in the full δGµν +8πGδTµν T that the metric fluctuations can be described entirely by hµν. If we introduce a T quantity δTµν in which the dependence on µ has been excluded (i.e. under a T gauge transformation δTµν → δTµν plus a function of µ, and this function of µ cancels against an identical function of µ in δGµν), then following the commuting of some derivatives, the fluctuation equations take the form [43] 1 δG + 8πGδT = [∇ ∇ hT + Rσ hT + Rσ hT − 2R hT λσ + ∇ ∇αhT ] µν µν 2 µ ν µ σν ν σµ µλνσ α µν 1 1 1 − Rσ hT + g R hT αβ − g ∇ ∇αhT + 8πGδT T = 0. 2 σ µν 2 µν αβ 2 µν α µν (A.41)

The SVT4 ﬂuctuations around a de Sitter background as given in (4.210) to (4.213) and around a general Robertson-Walker background as given in (4.280) are special cases of (A.41), with the only metric ﬂuctuations that appear in (4.213) and (4.280) being Fµν and χ, viz. just the six degrees of freedom associated with T hµν.

A.6 D-dimensional SVTD Transverse-Traceless Projection Operators for Curved Spacetime Tensor Fields

Rather than generalize the general curved spacetime transverse and longitudinal projection technique to the general transverse-traceless case, we have instead found it more convenient to generalize the SVTD discussion given in Secs. 3.2 and 4.2.2 to general curved spacetime background ﬂuctuations. To this end we take hµν to be of the form:

hµν = 2Fµν + Wµν + Sµν, (A.42)

where 2 W = ∇ W + ∇ W − g ∇αW , µν µ ν ν µ D µν α 1 Z S = (g ∇ ∇α − ∇ ∇ ) dDx0[−g(x0)]1/2D(D)(x, x0)h(x0), µν D − 1 µν α µ ν (A.43) 217

with D(x, x0) obeying

α (D) 0 −1/2 (D) 0 ∇α∇ D (x, x ) = [−g(x)] δ (x − x ). (A.44)

From (A.43) we obtain

µν µν g Wµν = 0, g Sµν = h, (A.45)

ν ν ν ∇ hµν = ∇ Wµν + ∇ Sµν (A.46) as the conditions that Fµν be transverse and traceless. From (A.46) we obtain

2 1 g ∇ ∇β + ∇ ∇ − ∇ ∇ W α = ∇αh − (∇ ∇ ∇α να β α ν D ν α αν D − 1 ν α α − ∇α∇ ∇ν)× Z dDx0[−g(x0)]1/2D(D)(x, x0)h(x0), (A.47)

and by commuting derivatives can rewrite (A.47) as

D − 2 1 g ∇ ∇β + ∇ ∇ − R W α = ∇αh − R ∇α× να β D ν α να αν D − 1 να Z dDx0[−g(x0)]1/2D(D)(x, x0)h(x0). (A.48)

To solve for Wµ it is convenient to use the bitensor formalism in which we (D)β 0 a β 0 a define Gα (x, x ) = eα(x)ea (x ) where the D-dimensional eα(x) vierbeins obey a b gµν(x) = ηabeµ(x)eν(x), with a and b referring to a fixed D-dimensional basis. a β 0 With this bitensor definition eα(x) and ea (x ) are acting in separate spaces, but 0 (D)β β at x = x we obtain Gα (x, x) = gα (x). On the introducing the propagator that satisfies D − 2 g ∇ ∇β + ∇ ∇ − R Dαγ (x, x0) = G(D)γ(x, x0)[−g(x0)]−1/2 × να β D ν α να (D) ν δ(D)(x − x0), (A.49)

we can solve for Wµ as Z D 0 0 1/2 (D)σ 0 ρ 0 1 0 ρ W (x) = d x [−g(x )] D (x, x ) ∇ 0 h (x ) − R (x )∇ 0 × µ µ x σρ D − 1 σρ x 218

Z dDx00[−g(x00)]1/2D(D)(x0, x00)h(x00) . (A.50)

Next we decompose Wµ into transverse and longitudinal components viz.

T L µ Wµ = Wµ + Wµ = Fµ + ∇µH, ∇ Fµ = 0, Z D 0 0 1/2 (D) 0 σ 0 H = d x [−g(x )] D (x, x )∇ Wσ(x ), (A.51) with hµν then taking the form 2 h = 2F + ∇ F + ∇ F + 2∇ ∇ H − g ∇ ∇αH µν µν µ ν ν µ µ ν D µν α 1 Z + (g ∇ ∇α − ∇ ∇ ) dDx0[−g(x0)]1/2D(D)(x, x0)h(x0). (A.52) D − 1 µν α µ ν Upon further deﬁning 1 Z F = H − dDx0[−g(x0)]1/2D(D)(x, x0)h(x0), 2(D − 1) 1 1 Z χ = ∇ ∇αH − ∇ ∇α dDx0[−g(x0)]1/2D(D)(x, x0)h(x0), D α 2(D − 1) α (A.53) we may express hµν in the SVTD form:

hµν = −2gµνχ + 2∇µ∇νF + ∇µFν + ∇νFµ + 2Fµν, (A.54) where 1 1 χ = ∇σW − h, D σ 2(D − 1) Z T D 0 0 1/2 (D) 0 σ 0 Fµ = Wµ = Wµ − ∇µ d x [−g(x )] D (x, x )∇ Wσ(x ), Z 1 F = dDx0[−g(x0)]1/2D(D)(x, x0) ∇σW (x0) − h(x0) , σ 2(D − 1)

2Fµν = hµν + 2gµνχ − 2∇µ∇νF − ∇µFν − ∇νFµ. (A.55)

We thus generalize the SVTD approach to the arbitrary D-dimensional curved spacetime background. Appendix B

Conformal to Flat Cosmological Geometries

B.1 Robertson-Walker k = 0

In order to apply (5.32) to cosmology we need to write the Robertson-Walker and de Sitter background geometries in a conformal to ﬂat Minkowski form. For a k = 0 Robertson-Walker background the comoving coordinate system metric takes the form

ds2(comoving) = dt2 − a2(t)[dx2 + dy2 + dz2]. (B.1)

The straightforward introduction of the conformal time Z dt dτ = (B.2) a(t)

then allows us to write the conformal time metric as

ds2(conformal time) = a2(τ)[dτ 2 − dx2 − dy2 − dz2]. (B.3)

B.2 Robertson-Walker k > 0

For a k > 0 or a k < 0 Robertson-Walker background the comoving and conformal time coordinate system metrics take the form

dr2 ds2(comoving) = dt2 − a2(t) + r2dθ2 + r2 sin2 θdφ2 , 1 − kr2 dr2 ds2(conformal time) = a2(τ) dτ 2 − − r2dθ2 − r2 sin2 θdφ2 .(B.4) 1 − kr2

To bring the RW geometries with non-zero k to a conformal to ﬂat form requires coordinate transformations that involve both τ and r. For the k > 0 case ﬁrst, it is convenient to set k = 1/L2, and introduce sin χ = r/L, with the conformal time metric given in (B.4) then taking the form

ds2 = L2a2(p) dp2 − dχ2 − sin2 χdθ2 − sin2 χ sin2 θdφ2 , (B.5)

219 220 where p = τ/L. Following e.g. [11] we introduce

p0 + r0 = tan[(p + χ)/2], p0 − r0 = tan[(p − χ)/2], sin p sin χ p0 = , r0 = , (B.6) cos p + cos χ cos p + cos χ so that 1 dp02 − dr02 = [dp2 − dχ2] sec2[(p + χ)/2] sec2[(p − χ)/2], (B.7) 4 1 (cos p + cos χ)2 = cos2[(p + χ)/2] cos2[(p − χ)/2] 4 1 = . (B.8) [1 + (p0 + r0)2][1 + (p0 − r0)2]

With these transformations the k > 0 line element then takes the conformal to ﬂat form 4L2a2(p) ds2 = dp02 − dr02 − r02dθ2 − r02 sin2 θdφ2 . (B.9) [1 + (p0 + r0)2][1 + (p0 − r0)2]

To bring the spatial sector of (B.9) to Cartesian coordinates we set x0 = r0 sin θ cos φ, y0 = r0 sin θ sin φ, z0 = r0 cos θ and thus bring the line element to the form

ds2 = L2a2(p)(cos p + cos χ)2 dp02 − dx02 − dy02 − dz02 , (B.10) where now r0 = (x02 + y02 + z02)1/2. With these transformations (B.10) is now in the form given in (2.23).

B.3 Robertson-Walker k < 0

For the k < 0 case, it is convenient to set k = −1/L2, and introduce sinhχ = r/L, with the conformal time metric given in (B.4) then taking the form

ds2 = L2a2(p) dp2 − dχ2 − sinh2χdθ2 − sinh2χ sin2 θdφ2 , (B.11) where p = τ/L. Next we introduce

p0 + r0 = tanh[(p + χ)/2], p0 − r0 = tanh[(p − χ)/2], sinh p sinh χ p0 = , r0 = , (B.12) cosh p + cosh χ cosh p + cosh χ so that 1 dp02 − dr02 = [dp2 − dχ2]sech2[(p + χ)/2]sech2[(p − χ)/2], 4 221

1 (cosh p + cosh χ)2 = cosh2[(p + χ)/2]cosh2[(p − χ)/2] 4 1 = . (B.13) [1 − (p0 + r0)2][1 − (p0 − r0)2] With these transformations the line element takes the conformal to ﬂat form 4L2a2(p) ds2 = dp02 − dr02 − r02dθ2 − r02 sin2 θdφ2 .(B.14) [1 − (p0 + r0)2][1 − (p0 − r0)2] The spatial sector can then be written in Cartesian form ds2 = L2a2(p)(cosh p + cosh χ)2 dp02 − dx02 − dy02 − dz02 , (B.15) where again r0 = (x02 + y02 + z02)1/2. We note that in transforming from (B.4) to (B.10) or to (B.15) we have only made coordinate transformations and not made any conformal transformation.

B.4 dS4 and AdS4 Background Solutions

While the conformal to flat Minkowski structures given in (B.3), (B.10) and (B.15) are purely kinematical, the explicit form of a(t) can be determined once a dynamics has been specified. Thus in regard to a de Sitter or anti-de Sitter cosmology, a de Sitter or an anti-de Sitter geometry is just a particular case of a Robertson-Walker geometry in which a(t) has a specific assigned value for each possible choice of spatial 3-curvature k. On writing the maximally 4-symmetric geometry condition Rµν = −3αgµν in Robertson-Walker form one obtains a˙ 2(t) + k = αa2(t). (B.16) (In terms of the scalar field model described in (2.48)–(2.52) we have K = α = 2 −2λSS0 .) Here α is positive for de Sitter and negative for anti-de Sitter. Allowable solutions to (B.16) depend on the values of α and k, and are of the form (see e.g. [10]) k 1/2 a(t, α > 0, k < 0) = − sinh(α1/2t), α a(t, α > 0, k = 0) = a(t = 0) exp(α1/2t), k 1/2 a(t, α > 0, k > 0) = cosh(α1/2t), α a(t, α = 0, k < 0) = (−k)1/2t, k 1/2 a(t, α < 0, k < 0) = sin((−α)1/2t). (B.17) α In these solutions (B.3), (B.10), and (B.15) all apply to a de Sitter or an anti-de Sitter cosmology. 222

B.5 dS4 and AdS4 Background Solutions - Radiation Era

For Robertson-Walker cosmologies we note that with slight modification we can extend the scalar field model given above to include a perfect fluid, with the energy-momentum tensor then being given by [35]

1 1 T µν = (ρ + p)U U + pg − S2 Rµν − gµνRα − gµνλ S4, (B.18) S µ ν µν 6 0 2 α S 0

µν with the background conformal cosmology still obeying TS = 0 since the background Robertson-Walker geometry continues to obey Wµν = 0. On taking the perfect ﬂuid energy-momentum tensor to be traceless radiation (viz. ρ = 3p, 4 2 ρ = A/a (t), A > 0) as needed in the early universe, and with α = −2λSS0 as before, the evolution equation takes the form

2 2 2A a˙ + k = αa − 2 2 , (B.19) S0 a with allowed solutions to the cosmology being given by [35]

k(β − 1) kβ 1/2 a(t, α > 0, k < 0, A > 0) = − − sinh2(α1/2t) , 2α α 1/4 A 1/2 1/2 a(t, α > 0, k = 0, A > 0) = − 4 cosh (2α t), λSS0 −k(β − 1) kβ 1/2 a(t, α > 0, k > 0, A > 0) = + cosh2(α1/2t) , 2α α 1/2 2A 2 a(t, α = 0, k < 0, A > 0) = − 2 − kt , kS0 k(β − 1) kβ 1/2 a(t, α < 0, k < 0, A > 0) = − + sin2((−α)1/2t) ,(B.20) 2α α

2 2 1/2 where β = (1 + 8Aα/k S0 ) . Appendix C

Computational Methods

C.1 GitHub Repository

All Mathematica notebooks, programs, and algorithms used within this work are freely available for use or modiﬁcation and can be found at: https://github.com/phelpsmatthew/Cosmological-Fluctuations

C.2 Mathematica and xAct

In composing the fluctuations equations in cosmology, there are numerous steps that pose enormous difficulty to being able to do calculations non-computationally, largely attributed to the sheer number of terms and manipulations involved. Such factors include dummy index refactoring, commutation of covariant derivatives, expressing curvature tensors as Christoffel symbols (functions of the metric), performing a 3+1 splitting of the fluctuations, substituting the SVT decomposition of the perturbed metric, collecting combinations of SVT components to form gauge invariants, testing covariant conservation, performing conformal transformations, and evaluating the fluctuations within an explicit choice of coordinate basis. As the focus of this thesis revolves around the discussion of the underlying physics and not the means of computation, the equations given in the main body have been presented in a significantly reduced form, with all auxiliary computations hidden. However, it is of practical interest to demonstrate how such calculations are performed. Therefore, to serve as a representative example, in Sec. C.3 we present a mostly-complete A to Z calculation of a SVT3 Roberston Walker k 6= 0 gauge invariant formulation of the Einstein fluctuation equations. Before illustrating an example calculation, we describe the computational tools underlying the results in this thesis. In the field of cosmological perturbations, one needs software capable of doing abstract symbolic tensor calculus on curved manifolds. In regard to symbolic computation alone, the Wolfram Mathe- matica language serves as one of the most effective frameworks for doing so, with especially large popularity and support within the fields theoretical physics and mathematics. Natively, Mathematica does not provide support for GR specific applications and therefore one must turn to user-created 3rd-party packages that

223 224

have been developed for use within Mathematica. To fulfill this need, the package xAct [45], developed originally by JoséM. Mart´ın-Garc´ıa,provides a powerful set of methods and capabilities for performing abstract tensor calculus within GR. It is composed of a suite of free packages that implement state-of-the-art algorithms for fast manipulation of indices in a highly programmable and configurable form. The set of packages used within this work, along with their purpose is described below (for more information, cf. [45]).

• xTensor: Provides core functionality and capabilities in abstract tensor calculus within General Relativity. Works with tensors with arbitrary symmetries under permutations of indices, defined on specified manifolds (or even several different manifolds). It computes covariant derivatives, Lie derivatives and parametric derivatives. Additionally it allows the presence of a metric in each manifold and defines all the associated tensors (Riemann, Ricci, Einstein, Weyl, etc.)

• xCoba: Provides several tools for working with bases and components. It allows one to define bases on one or more manifolds and to handle basis vectors, using basis indices notation. It performs component calculations such as expanding a tensor in a specified basis, changing the basis of an expression or tracing the contraction of basis dummies. The package knows how to express derivatives and brackets of basis vectors in terms of Christoffel and torsion tensors and how to assign values to the components of a tensor.

• xPert: Used for fast construction and manipulation of the equations of metric perturbation theory. Based on the use of explicit formulas for the construction of the n-th order perturbation of the most relevant curvature tensors.

• TexAct: Produces nice TeX code for tensor expressions. For any given tensor expression, the command TexPrint produces a single string with tensors, indices, derivatives, etc. formatted as one would express within in an article. A second command TexBreak allows breaking this string in various lines in a highly ﬂexible way; this function can also make line-breaks according to the lengths of the actual typeset expressions. Lists of equations can be typeset with automatic line breaking and alignment.

• xTras [46]: Provides functions and methods frequently needed when doing (classical) ﬁeld theory, e.g. Killing vectors, symmetrizing covariant derivatives, converting contractions of Riemann tensors to Ricci tensors. 225

C.3 Detailed Calculation Example

To convey the methods and procedural steps underyling computations within perturbative cosmology, here we calculate the perturbed Einstein equations of motion within a Roberton Walker background with unspeciﬁed 3-curvature k. All steps, including simpliﬁcations, are done entirely programmatically.

C.3.1 Conformal Transformation and Perturbation of Gµν

As we seek to describe the perturbed δGµν within the Roberston Walker geometry

1 ds2 = Ω2(τ)˜g dxµdxν, g˜ = diag −1, , r2, r2 sin2 θ , (C.1) µν µν 1 − kr2

we need to determine the form of δGµν under conformal transformation. To this 1 end, we begin with the full Gµν = Rµν − 2 gµνR and express the curvature tensors in terms of the Christoﬀel symbols,

α β α β Gµν = −Γ µβΓ αν + Γ αβΓ µν + Rµν 1 αβ γ ζ γ ζ − 2 gµν g (Γ γζ Γ αβ − Γ αζ Γ γβ + Rαβ γ γ α α −∂αΓ γβ + ∂γΓ αβ) + ∂αΓ µν − ∂µΓ αν. (C.2)

2 Under conformal transformation gµν → Ω (x)˜gµν, the Christoﬀel symbol transforms as

λ ˜λ −1 λ ˜ λ ˜ λσ ˜ Γµν → Γµν + Ω (δν ∇µ + δµ∇ν − g˜µνg˜ ∇σ)Ω(x), (C.3)

˜λ where Γµν is evaluated within the geometryg ˜µν. Now, transforming gµν → 2 Ω (x)˜gµν and (C.3) within (C.2), we ﬁnd that under conformal transformation Gµν transforms into

˜ 1 ˜α ˜ −1 ˜ ˜ α −2 ˜ ˜ α Gµν → Gµν − 2 g˜µνR α + Rµν − 2˜gµνΩ ∇α∇ Ω +g ˜µνΩ ∇αΩ∇ Ω. −2 ˜ ˜ −1 ˜ ˜ −4Ω ∇µΩ∇νΩ + 2Ω ∇ν∇µΩ. (C.4)

To determine the ﬂuctuation of Gµν, we need to perturb the metric according to (0) 2 gµν = gµν + hµν + O(h ) + ..., keeping terms only up to ﬁrst order. Denoting 2 hµν ≡ Ω (x)fµν, we obtain

1 ˜ 1 αβ ˜ 1 ˜ ˜ α 1 ˜ ˜ α −1 ˜ ˜ α δGµν = − 2 fµνR + 2 g˜µνf Rαβ + 2 ∇α∇ fµν − 2 g˜µν∇α∇ f − 2fµνΩ ∇α∇ Ω 1 ˜ ˜ α 1 ˜ ˜ α −1 ˜ ˜ α −1 ˜ ˜ α − 2 ∇α∇µfν − 2 ∇α∇νfµ − g˜µνΩ ∇αΩ∇ f + Ω ∇αfµν∇ Ω −2 ˜ ˜ α −1 ˜ α ˜ β 1 ˜ ˜ αβ +fµνΩ ∇αΩ∇ Ω + 2˜gµνΩ ∇ Ω∇βfα + 2 g˜µν∇β∇αf αβ −1 ˜ ˜ −2 ˜ α ˜ β +2˜gµνf Ω ∇β∇αΩ − g˜µνfαβΩ ∇ Ω∇ Ω 226

−1 ˜ α ˜ −1 ˜ α ˜ 1 ˜ ˜ −Ω ∇ Ω∇µfνα − Ω ∇ Ω∇νfµα + 2 ∇ν∇µf, (C.5) representing the perturbed Einstein tensor in a generic conformal geometry. In order to express (C.5) in the SVT3 basis, it must be prepared according to the 3+1 decomposition. Here we perform a splitting via a a projection based on the four α vector U as outlined in Sec. A.1. We evaluateg ˜00 = −1 and express all 3-space curvature tensors in terms of k. Finally, we decompose δGµν itself according to the 3+1 decomposition to obtain ba ˙ ˙ −1 ˙ ˙ −1 ˙ ba −2 ˜ δG00 = kg˜ fab + 3kf00 − f00ΩΩ − fΩΩ − 2Ω˜g f0bΩ ∇aΩ ba −1 ˜ ˙ ba −1 ˜ ˜ ˙ ba −1 ˜ +4˜g f0aΩ ∇bΩ +g ˜ Ω ∇aΩ∇bf00 + 2Ω˜g Ω ∇bf0a ba −1 ˜ ˜ ca db −2 ˜ ˜ ba −2 ˜ ˜ +˜g Ω ∇af∇bΩ +g ˜ g˜ fcdΩ ∇aΩ∇bΩ +g ˜ f00Ω ∇aΩ∇bΩ 1 ba ˜ ˜ 1 ba ˜ ˜ ba −1 ˜ ˜ + 2 g˜ ∇b∇af00 + 2 g˜ ∇b∇af − 2˜g f00Ω ∇b∇aΩ ca db −1 ˜ ˜ 1 ca db ˜ ˜ ca db −1 ˜ ˜ −2˜g g˜ Ω ∇aΩ∇dfcb − 2 g˜ g˜ ∇d∇cfab − 2˜g g˜ fabΩ ∇d∇cΩ (C.6) .. ˙ 2 −2 −1 ˙ ba −1 ˜ 1 ba ˜ ˙ δG0i = 3kf0i − Ω f0iΩ + 2Ωf0iΩ − fibg˜ Ω ∇aΩ − 2 g˜ ∇bfia ba −1 ˜ ˜ ba −2 ˜ ˜ 1 ba ˜ ˜ +˜g Ω ∇aΩ∇bf0i +g ˜ f0iΩ ∇aΩ∇bΩ + 2 g˜ ∇b∇af0i ba −1 ˜ ˜ 1 ba ˜ ˜ 1 ˜ ˙ 1 ˜ ˙ −2˜g f0iΩ ∇b∇aΩ − 2 g˜ ∇b∇if0a + 2 ∇if00 + 2 ∇if ˙ −1 ˜ ba −1 ˜ ˜ +ΩΩ ∇if00 − g˜ Ω ∇aΩ∇if0b (C.7)

...... 1 1 1 ba ˙ 2 −2 ˙ 2 −2 δGij = − 2 f ij + 2 f 00g˜ij + 2 fg˜ij − kg˜ g˜ijfab + 3kfij − Ω fijΩ − Ω g˜ijf00Ω .. .. ˙ ˙ −1 ˙ ˙ −1 ˙ ˙ −1 −1 −1 −fijΩΩ + 2f00Ω˜gijΩ + fΩ˜gijΩ + 2ΩfijΩ + 2Ω˜gijf00Ω ˙ ba −2 ˜ ˙ ba −1 ˜ ba ˜ ˙ +2Ω˜g g˜ijf0bΩ ∇aΩ − 2f0bg˜ g˜ijΩ ∇aΩ − g˜ g˜ij∇bf0a ba −1 ˜ ˙ ba −1 ˜ ˜ ˙ ba −1 ˜ −4˜g g˜ijf0aΩ ∇bΩ +g ˜ Ω ∇aΩ∇bfij − 2Ω˜g g˜ijΩ ∇bf0a ba −1 ˜ ˜ ca db −2 ˜ ˜ −g˜ g˜ijΩ ∇af∇bΩ − g˜ g˜ g˜ijfcdΩ ∇aΩ∇bΩ ba −2 ˜ ˜ 1 ba ˜ ˜ 1 ba ˜ ˜ ba −1 ˜ ˜ +˜g fijΩ ∇aΩ∇bΩ + 2 g˜ ∇b∇afij − 2 g˜ g˜ij∇b∇af − 2˜g fijΩ ∇b∇aΩ 1 ba ˜ ˜ 1 ba ˜ ˜ ca db −1 ˜ ˜ − 2 g˜ ∇b∇ifja − 2 g˜ ∇b∇jfia + 2˜g g˜ g˜ijΩ ∇aΩ∇dfcb 1 ca db ˜ ˜ ca db −1 ˜ ˜ 1 ˜ ˙ + 2 g˜ g˜ g˜ij∇d∇cfab + 2˜g g˜ g˜ijfabΩ ∇d∇cΩ + 2 ∇if0j ba −1 ˜ ˜ ˙ −1 ˜ 1 ˜ ˙ ba −1 ˜ ˜ −g˜ Ω ∇aΩ∇ifjb + ΩΩ ∇if0j + 2 ∇jf0i − g˜ Ω ∇aΩ∇jfib ˙ −1 ˜ 1 ˜ ˜ +ΩΩ ∇jf0i + 2 ∇j∇if, (C.8) with dots denoting time derivatives.

C.3.2 Background Equations of Motion Before we can use the form of (C.8), we note that in geometries with a non- vanishing background Tµν, we must first solve the background equations of motion and substitute the solutions for ρ and p in terms of Ω(τ) into the fluctuations 227 themselves. Recovering the background solution does not pose the same difficulties as the fluctuations, and thus potentially can all be done non-programmatically. We express here the core results in solving ∆µν = Gµν + Tµν = 0, including covariant conservation: 1 ds2 = Ω2(τ)˜g dxµdxν, g˜ = diag −1, , r2, r2 sin2 θ (C.9) µν µν 1 − kr2

˙ 2 −2 ˙ 2 −2 ¨ −1 G00 = −3k − 3Ω Ω Gij = kg˜ij − Ω Ω g˜ij + 2ΩΩ g˜ij (C.10)

2 0 Tµν = (ρ + p)UµUν + pΩ g˜µν,Uµ = −Ωδµ (C.11)

(0) ˙ 2 −2 2 ∆00 = −3k − 3Ω Ω + Ω ρ (C.12)

→ ρ = 3kΩ−2 + 3Ω˙ 2Ω−4 (C.13)

(0) ˙ 2 −2 ¨ −1 2 ∆ij = kg˜ij − Ω Ω g˜ij + 2ΩΩ g˜ij + Ω pg˜ij (C.14)

→ p = −kΩ−2 + Ω˙ 2Ω−4 − 2ΩΩ¨ −3 (C.15)

µ0 −5 ab ˙ ˙ ˙ ˜ a ∇µT = Ω g˜ TabΩ + T00Ω + T00Ω − Ω∇aT0 = 3ΩΩ˙ −3p + 3ΩΩ˙ −3ρ + Ω−2ρ˙ (C.16)

µi −5 i ˙ ˙ i ˜ ia ∇µT = Ω −2T0 Ω − T0 Ω + Ω∇aT = 0. (C.17)

C.3.3 δGµν and δTµν Fluctuations We now perturb the perfect ﬂuid energy momentum tensor and subsequently substitute the background quantities ρ and p which have been given in terms of Ω(x) above. We also utilize the conformal (C.8) and initiate the SVT3 decomposition of the fµν, to obtain the entire SVT3 ﬂuctuations for δGµν and δTµν: 1 ds2 = Ω2(τ)[˜g + f ]dxµdxν g˜ = diag −1, , r2, r2 sin2 θ µν µν µν 1 − kr2 (C.18) ˜ f00 = −2φ, f0i = ∇iB + Bi, ˜ ˜ ˜ ˜ fij = −2ψg˜ij + 2∇i∇jE + ∇iEj + ∇jEi + 2Eij (C.19) 228

2 ˙ 2 −2 δT00 = Ω δρ + (6k + 6Ω Ω )φ (C.20)

.. ˙ 2 −2 −1 ˜ δT0i = (−k + Ω Ω − 2ΩΩ )∇iB .. .. ˙ 2 −3 −2 −1 ˜ ˙ 2 −2 −1 +(−4Ω Ω + 2ΩΩ − 2kΩ )∇iV + (−k + Ω Ω − 2ΩΩ )Bi .. ˙ 2 −3 −2 −1 +(−4Ω Ω + 2ΩΩ − 2kΩ )Vi (C.21)

.. 2 ˙ 2 −2 −1 δTij = Ω δpg˜ij + (2kg˜ij − 2Ω Ω g˜ij + 4ΩΩ g˜ij)ψ (C.22) .. .. ˙ 2 −2 −1 ˜ ˜ ˙ 2 −2 −1 ˜ +(−2k + 2Ω Ω − 4ΩΩ )∇i∇jE + (−k + Ω Ω − 2ΩΩ )∇iEj .. .. ˙ 2 −2 −1 ˜ ˙ 2 −2 −1 +(−k + Ω Ω − 2ΩΩ )∇jEi + (−2k + 2Ω Ω − 4ΩΩ )Eij

.. µν ˙ 2 −4 −2 ˙ 2 −4 −3 −2 g δTµν = 3δp − δρ + (−6Ω Ω − 6kΩ )φ + (−6Ω Ω + 12ΩΩ + 6kΩ )ψ .. ˙ 2 −4 −3 −2 ˜ ˜ a +(2Ω Ω − 4ΩΩ − 2kΩ )∇a∇ E (C.23)

˙ −1 ˙ ˙ −1 ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ˜ ˜ a δG00 = 6ΩΩ ψ − 6kφ − 6kψ + 2ΩΩ ∇a∇ B − 2ΩΩ ∇a∇ E − 2∇a∇ ψ (C.24) .. ˙ 2 −2 −1 ˜ ˜ ˙ ˜ ˙ ˙ −1 ˜ δG0i = (3k − Ω Ω + 2ΩΩ )∇iB − 2k∇iE − 2∇iψ − 2ΩΩ ∇iφ .. ˙ 2 −2 −1 ˙ 1 ˜ ˜ a 1 ˜ ˜ a ˙ +(2k − Ω Ω + 2ΩΩ )Bi − kEi + 2 ∇a∇ Bi − 2 ∇a∇ Ei (C.25)

.. .. ˙ −1 ˙ ˙ −1 ˙ ˙ 2 −2 −1 δGij = −2ψg˜ij − 2ΩΩ φg˜ij − 4ΩΩ ψg˜ij + (2Ω Ω g˜ij − 4ΩΩ g˜ij)φ .. ˙ 2 −2 −1 ˙ −1 ˜ ˜ a ˜ ˜ a ˙ +(2Ω Ω g˜ij − 4ΩΩ g˜ij)ψ − 2ΩΩ g˜ij∇a∇ B − g˜ij∇a∇ B .. ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ˜ ˜ a ˜ ˜ a +˜gij∇a∇ E + 2ΩΩ g˜ij∇a∇ E − g˜ij∇a∇ φ +g ˜ij∇a∇ ψ .. ˙ −1 ˜ ˜ ˜ ˜ ˙ ˜ ˜ ˙ −1 ˜ ˜ ˙ +2ΩΩ ∇j∇iB + ∇j∇iB − ∇j∇iE − 2ΩΩ ∇j∇iE .. ˙ 2 −2 −1 ˜ ˜ ˜ ˜ ˜ ˜ ˙ −1 ˜ +(2k − 2Ω Ω + 4ΩΩ )∇j∇iE + ∇j∇iφ − ∇j∇iψ + ΩΩ ∇iBj .. .. 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ ˙ 2 −2 −1 ˜ + 2 ∇iBj − 2 ∇iEj − ΩΩ ∇iEj + (k − Ω Ω + 2ΩΩ )∇iEj .. ˙ −1 ˜ 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ +ΩΩ ∇jBi + 2 ∇jBi − 2 ∇jEi − ΩΩ ∇jEi .. .. ˙ 2 −2 −1 ˜ ˙ −1 ˙ +(k − Ω Ω + 2ΩΩ )∇jEi − Eij − 2ΩΩ Eij .. ˙ 2 −2 −1 ˜ ˜ a +(−2Ω Ω + 4ΩΩ )Eij + ∇a∇ Eij (C.26)

.. .. µν −2 ˙ −3 ˙ ˙ −3 ˙ ˙ 2 −4 −3 −2 g δGµν = −6Ω ψ − 6ΩΩ φ − 18ΩΩ ψ + (6Ω Ω − 12ΩΩ + 6kΩ )φ .. ˙ 2 −4 −3 −2 ˙ −3 ˜ ˜ a −2 ˜ ˜ a ˙ +(6Ω Ω − 12ΩΩ + 6kΩ )ψ − 6ΩΩ ∇a∇ B − 2Ω ∇a∇ B .. −2 ˜ ˜ a ˙ −3 ˜ ˜ a ˙ −2 ˜ ˜ a −2 ˜ ˜ a +2Ω ∇a∇ E + 6ΩΩ ∇a∇ E − 2Ω ∇a∇ φ + 4Ω ∇a∇ ψ .. ˙ 2 −4 −3 −2 ˜ ˜ a +(−2Ω Ω + 4ΩΩ + 2kΩ )∇a∇ E (C.27) 229

C.3.4 Field Equations

Next we add δGµν to δTµν while simplifying common terms to obtain the ﬁeld equations ∆µν = 0:

∆µν ≡ δGµν + δTµν (C.28)

2 ˙ −1 ˙ ˙ 2 −2 ˙ −1 ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ∆00 = Ω δρ + 6ΩΩ ψ + 6Ω Ω φ − 6kψ + 2ΩΩ ∇a∇ B − 2ΩΩ ∇a∇ E ˜ ˜ a −2∇a∇ ψ (C.29)

.. ˜ ˜ ˙ ˜ ˙ ˙ 2 −3 −2 −1 ˜ ∆0i = 2k∇iB − 2k∇iE − 2∇iψ + (−4Ω Ω + 2ΩΩ − 2kΩ )∇iV .. ˙ −1 ˜ ˙ ˙ 2 −3 −2 −1 −2ΩΩ ∇iφ + kBi − kEi + (−4Ω Ω + 2ΩΩ − 2kΩ )Vi 1 ˜ ˜ a 1 ˜ ˜ a ˙ + 2 ∇a∇ Bi − 2 ∇a∇ Ei (C.30)

.. 2 ˙ −1 ˙ ˙ −1 ˙ ∆ij = −2ψg˜ij + Ω δpg˜ij − 2ΩΩ φg˜ij − 4ΩΩ ψg˜ij + 2kg˜ijψ .. ˙ 2 −2 −1 ˙ −1 ˜ ˜ a ˜ ˜ a ˙ +(2Ω Ω g˜ij − 4ΩΩ g˜ij)φ − 2ΩΩ g˜ij∇a∇ B − g˜ij∇a∇ B .. ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ˜ ˜ a ˜ ˜ a +˜gij∇a∇ E + 2ΩΩ g˜ij∇a∇ E − g˜ij∇a∇ φ +g ˜ij∇a∇ ψ .. ˙ 2 −2 −1 ˜ ˜ ˙ −1 ˜ ˜ ˜ ˜ ˙ +(−2k + 2Ω Ω − 4ΩΩ )∇i∇jE + 2ΩΩ ∇j∇iB + ∇j∇iB .. .. ˜ ˜ ˙ −1 ˜ ˜ ˙ ˙ 2 −2 −1 ˜ ˜ −∇j∇iE − 2ΩΩ ∇j∇iE + (2k − 2Ω Ω + 4ΩΩ )∇j∇iE .. ˜ ˜ ˜ ˜ ˙ −1 ˜ 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ +∇j∇iφ − ∇j∇iψ + ΩΩ ∇iBj + 2 ∇iBj − 2 ∇iEj − ΩΩ ∇iEj .. .. ˙ −1 ˜ 1 ˜ ˙ 1 ˜ ˙ −1 ˜ ˙ ˙ −1 ˙ +ΩΩ ∇jBi + 2 ∇jBi − 2 ∇jEi − ΩΩ ∇jEi − Eij − 2ΩΩ Eij ˜ ˜ a −2kEij + ∇a∇ Eij (C.31)

.. .. µν −2 ˙ −3 ˙ ˙ −3 ˙ −3 −2 g ∆µν = −6Ω ψ + 3δp − δρ − 6ΩΩ φ − 18ΩΩ ψ − 12ΩΩ φ + 12kΩ ψ .. ˙ −3 ˜ ˜ a −2 ˜ ˜ a ˙ −2 ˜ ˜ a −6ΩΩ ∇a∇ B − 2Ω ∇a∇ B + 2Ω ∇a∇ E ˙ −3 ˜ ˜ a ˙ −2 ˜ ˜ a −2 ˜ ˜ a +6ΩΩ ∇a∇ E − 2Ω ∇a∇ φ + 4Ω ∇a∇ ψ. (C.32)

C.3.5 Field Equations (Gauge Invariant Form) Finally, we form the set of 10 gauge invariants and express the field equations entirely terms of them. While straightforward in theory, in practice one must take special care in the order in which one substitutes the gauge invariants; otherwise, the remaining fluctuations will be expressed in a different basis of gauge invariants than desired.

˙ ¨ ˙ −1 ˙ ˙ α = φ + ψ + B − E, γ = −Ω Ωψ + B − E,Bi − Ei,Eij,Vi (C.33) 230

V GI = V − Ω2Ω˙ −1ψ (C.34)

.. δρGI = δρ − 12Ω˙ 2ψΩ−4 + 6ΩψΩ−3 − 6kψΩ−2 (C.35)

.. ... δpGI = δp − 4Ω˙ 2ψΩ−4 + 8ΩψΩ−3 + 2kψΩ−2 − 2ΩΩ˙ −1ψΩ−2 (C.36)

2 GI ˙ 2 −2 ˙ 2 −2 ˙ −1 ˜ ˜ a ∆00 = Ω δρ − 6Ω Ω γ˙ + 6Ω Ω α + 2ΩΩ ∇a∇ γ (C.37)

˙ −1 ˜ ˙ −1 ˜ ˜ ∆0i = 2ΩΩ ∇iγ˙ − 2ΩΩ ∇iα + 2k∇iγ + kQi .. ˙ 2 −3 −2 −1 1 ˜ ˜ a +(−4Ω Ω + 2ΩΩ − 2kΩ )Vi + 2 ∇a∇ Qi .. ˙ 2 −3 −2 −1 ˜ GI +(−4Ω Ω + 2ΩΩ − 2kΩ )∇iV (C.38)

.. ˙ −1 .. 2 GI ˙ −1 ˙ 2 −2 −1 ∆ij = 2ΩΩ γg˜ij + Ω δp g˜ij − 2ΩΩ α˙ g˜ij +γ ˙ (−2Ω Ω g˜ij + 4ΩΩ g˜ij) .. ˙ 2 −2 −1 ˜ ˜ a ˙ −1 ˜ ˜ a ˜ ˜ +(2Ω Ω g˜ij − 4ΩΩ g˜ij)α − g˜ij∇a∇ α − 2ΩΩ g˜ij∇a∇ γ + ∇j∇iα .. ˙ −1 ˜ ˜ 1 ˜ ˙ ˙ −1 ˜ 1 ˜ ˙ ˙ −1 ˜ +2ΩΩ ∇j∇iγ + 2 ∇iQj + ΩΩ ∇iQj + 2 ∇jQi + ΩΩ ∇jQi − Eij ˙ −1 ˙ ˜ ˜ a −2ΩΩ Eij − 2kEij + ∇a∇ Eij (C.39)

.. .. µν ˙ −3 .. GI GI ˙ −3 −3 −3 g ∆µν = 6ΩΩ γ + 3δp − δρ − 6ΩΩ α˙ + 12ΩΩ γ˙ − 12ΩΩ α −2 ˜ ˜ a ˙ −3 ˜ ˜ a −2Ω ∇a∇ α − 6ΩΩ ∇a∇ γ (C.40)

C.3.6 Conservation While the ability to express the entire ﬁeld equations solely in terms of gauge invariants quantities serves as a good check on the accuracy of the calculation, one should also check the conditions of covariant conservation. Here, we note that ˜ µν one cannot simply apply the covariant derivative to the perturbation viz. ∇µδT since the background geometry is not ﬂat. Rather, one must perturb the entire µν δ(∇µT ). Calculating the conservation conditions, we obtain:

µ0 a ˙ −5 ˙ −5 ab ˙ −5 a ˙ −5 δ(∇µT ) = δT aΩΩ + δT00ΩΩ − T ΩfabΩ + T aΩf00Ω ˙ −5 a ˙ −5 ˙ −4 1 ab ˙ −4 +2T00Ωf00Ω − 2T0 Ωf0aΩ + δT 00Ω + 2 T fabΩ 3 ˙ −4 a ˙ −4 1 ˙ −4 ˙ −4 + 2 T00f00Ω − 2T0 f0aΩ + 2 T00fΩ + 2T00f00Ω ˙ a −4 −4 ˜ a a −4 ˜ −4 ˜ a −2T0 f0aΩ − Ω ∇aδT0 − f0 Ω ∇aT00 − f00Ω ∇aT0 −4 ˜ a 1 a −4 ˜ a −4 ˜ b a −4 ˜ b −T00Ω ∇af0 − 2 T0 Ω ∇af + f0 Ω ∇bTa + T0 Ω ∇bfa −4 ˜ b a +fabΩ ∇ T0 (C.41) 231

= Ω−2δρ˙ + 3ΩΩ˙ −3δp + 3ΩΩ˙ −3δρ .. +(−12Ω˙ 2Ω−6 + 6ΩΩ−5 − 6kΩ−4)ψ˙ .. ˙ 2 −6 −5 −4 ˜ ˜ a +(−4Ω Ω + 2ΩΩ − 2kΩ )∇a∇ B .. ˙ 2 −6 −5 −4 ˜ ˜ a ˙ +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ E .. ˙ 2 −7 −6 −5 ˜ ˜ a +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ V (C.42)

GI = Ω−2δρ˙ + 3ΩΩ˙ −3δpGI + 3ΩΩ˙ −3δρGI .. ˙ 2 −6 −5 −4 ˜ ˜ a +(−4Ω Ω + 2ΩΩ − 2kΩ )∇a∇ γ .. ˙ 2 −7 −6 −5 ˜ ˜ a GI +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ V (C.43)

µi i ˙ −5 a ˙ i −5 i ˙ −5 i ˙ a −5 δ(∇µT ) = −2δT0 ΩΩ + 2T0 Ωf aΩ − 2T0 Ωf00Ω + 2T aΩf0 Ω a ˙ i −5 ˙ i −5 ˙ i −4 i ˙ −4 −T aΩf0 Ω − T00Ωf0 Ω − δT 0 Ω − T0 f00Ω i ˙ a −4 1 i ˙ −4 ˙ a i −4 ˙ i −4 ˙ i a −4 +T af0 Ω − 2 T0 fΩ + T0 f aΩ − T0 f00Ω + T af0 Ω ˙ i −4 −4 ˜ ia i −4 ˜ a a −4 ˜ i −T00f0 Ω + Ω ∇aδT + f0 Ω ∇aT0 + f0 Ω ∇aT0 i −4 ˜ a 1 i −4 ˜ a ia −4 ˜ b ab −4 ˜ i +T0 Ω ∇af0 + 2 T aΩ ∇ f − f Ω ∇bTa − f Ω ∇bT a ia −4 ˜ b 1 ab −4 ˜ i 1 −4 ˜ i −T Ω ∇bfa − 2 T Ω ∇ fab − 2 T00Ω ∇ f00 a −4 ˜ i +T0 Ω ∇ f0a (C.44)

.. = Ω−2∇˜ iδp + (4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)∇˜ iV˙ .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)∇˜ iV .. +(4Ω˙ 2Ω−6 − 2ΩΩ−5 + 2kΩ−4)∇˜ iφ .. +(4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)V˙ i .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)V i (C.45)

.. = Ω−2∇˜ iδpGI + (−4Ω˙ 2Ω−6 + 2ΩΩ−5 − 2kΩ−4)∇˜ iγ˙ .. +(4Ω˙ 2Ω−6 − 2ΩΩ−5 + 2kΩ−4)∇˜ iα .. +(4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)V˙ i .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)V i .. +(4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)∇˜ iV˙ GI .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)∇˜ iV GI (C.46)

µ0 a ˙ −5 ˙ −5 a ˙ −5 ab ˙ −5 δ(∇µG ) = δG aΩΩ + δG00ΩΩ − 2B G0aΩΩ − 2G ΩEabΩ 232

a ˙ −5 ˙ −5 a ˙ −5 ˙ −4 −2G aΩφΩ − 4G00ΩφΩ + 2G aΩψΩ + δG00Ω a ˙ −4 ˙ a −4 ab ˙ −4 ˙ −4 −2B G0aΩ − 2B G0aΩ + G EabΩ − 2G00φΩ a ˙ −4 ˙ −4 a −4 ˙ −4 −G aψΩ − 3G00ψΩ + 2kG0 EaΩ − 4G00φΩ a ˙ −5 ˜ ˙ a −4 ˜ a −4 ˜ ˙ −2G0 ΩΩ ∇aB − 2G0 Ω ∇aB − 2G0 Ω ∇aB −4 ˜ a a −4 ˜ −4 ˜ a −4 ˜ a −Ω ∇aδG0 − B Ω ∇aG00 + 2φΩ ∇aG0 − 2ψΩ ∇aG0 a −4 ˜ a −4 ˜ a −4 ˜ +2kG0 Ω ∇aE − G0 Ω ∇aφ + G0 Ω ∇aψ −4 ˜ ˜ a −4 ˜ ˜ a ˙ −4 ˜ ˜ a −G00Ω ∇a∇ B + G00Ω ∇a∇ E − Ω ∇aG00∇ B a −4 ˜ b −4 ˜ a ˜ b a −4 ˜ ˜ b +B Ω ∇bGa + Ω ∇ B∇bGa + G0 Ω ∇b∇ Ea a −4 ˜ ˜ b ˜ −4 ˜ b a −4 ˜ ˜ b a +G0 Ω ∇b∇ ∇aE + 2EabΩ ∇ G0 + Ω ∇aEb∇ G0 −4 ˜ ˜ b a −4 ˜ ˜ ˜ b a −4 ˜ b ˙ a +Ω ∇bEa∇ G0 + 2Ω ∇b∇aE∇ G0 + GabΩ ∇ E ˙ −5 ˜ b a −4 ˜ b ˜ a ˙ ˙ −5 ˜ b ˜ a −2GabΩΩ ∇ E + GabΩ ∇ ∇ E − 2GabΩΩ ∇ ∇ E

= 0 (C.47)

µi i ˙ −5 i a ˙ −5 a i ˙ −5 i ˙ −5 δ(∇µG ) = −2δG0 ΩΩ − B G aΩΩ + 2B G aΩΩ − B G00ΩΩ a ˙ i −5 i ˙ −5 i ˙ −5 ˙ i −4 +4G0 ΩE aΩ + 4G0 ΩφΩ − 4G0 ΩψΩ − δG0 Ω a ˙ i −4 i ˙ −4 ˙ a i −4 i ˙ −4 i ˙ −4 +B G aΩ − B G00Ω + B G aΩ + G0 φΩ + 3G0 ψΩ ˙ a i −4 i a −4 ˙ i −4 ˙ i −4 +2G0 E aΩ − 2kG aE Ω + 2G0 φΩ − 2G0 ψΩ −4 ˜ ia −4 ˜ ia i −4 ˜ a a −4 ˜ i +Ω ∇aδG + 4ψΩ ∇aG + B Ω ∇aG0 + B Ω ∇aG0 a ˙ −5 ˜ i ˙ a −4 ˜ i i −4 ˜ ˜ a +2G0 ΩΩ ∇aE + G0 Ω ∇aE + G0 Ω ∇a∇ B i −4 ˜ ˜ a ˙ i ˙ −5 ˜ a ˙ i −4 ˜ a −G0 Ω ∇a∇ E + 2G aΩΩ ∇ B + G aΩ ∇ B −4 ˜ i ˜ a i −4 ˜ a ˙ i −4 ˜ a +Ω ∇aG0 ∇ B + G aΩ ∇ B − 2kG aΩ ∇ E i −4 ˜ a i −4 ˜ a ia −4 ˜ b +G aΩ ∇ φ − G aΩ ∇ ψ − 2E Ω ∇bGa −4 ˜ a i ˜ b ab −4 ˜ i i −4 ˜ ˜ b a −Ω ∇ E ∇bGa − 2E Ω ∇bG a − G aΩ ∇b∇ E i −4 ˜ ˜ b ˜ a −4 ˜ i ˜ b a −4 ˜ i ˜ b a −G aΩ ∇b∇ ∇ E − Ω ∇aG b∇ E − Ω ∇bG a∇ E −4 ˜ i ˜ b ˜ a a ˙ −5 ˜ i ˙ −5 ˜ i −2Ω ∇bG a∇ ∇ E − G aΩΩ ∇ B − G00ΩΩ ∇ B ˙ −4 ˜ i −4 ˜ a ˜ i a −4 ˜ i −G00Ω ∇ B + Ω ∇aG0 ∇ B + G0 Ω ∇ Ba ab −4 ˜ i a ˙ −5 ˜ i ˙ a −4 ˜ i −G Ω ∇ Eab + 2G0 ΩΩ ∇ Ea + G0 Ω ∇ Ea −4 ˜ b ˜ i a −4 ˜ i a −4 ˜ i −Ω ∇bGa ∇ E + G00Ω ∇ φ + G aΩ ∇ ψ a −4 ˜ i ˜ a ˙ −5 ˜ i ˜ ˙ a −4 ˜ i ˜ +G0 Ω ∇ ∇aB + 4G0 ΩΩ ∇ ∇aE + 2G0 Ω ∇ ∇aE −4 ˜ b ˜ i ˜ a −4 ˜ i ˜ b a −4 ˜ i ˜ b ˜ a −2Ω ∇bGa ∇ ∇ E − GabΩ ∇ ∇ E − GabΩ ∇ ∇ ∇ E

= 0 (C.48)

µ0 −2 ˙ GI ˙ −3 GI ˙ −3 GI δ(∇µ∆ ) = Ω δρ + 3ΩΩ δp + 3ΩΩ δρ 233

.. ˙ 2 −6 −5 −4 ˜ ˜ a +(−4Ω Ω + 2ΩΩ − 2kΩ )∇a∇ γ .. ˙ 2 −7 −6 −5 ˜ ˜ a GI +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ V (C.49)

.. µi −2 ˜ i GI ˙ 2 −6 −5 −4 ˜ i δ(∇µ∆ ) = Ω ∇ δp + (−4Ω Ω + 2ΩΩ − 2kΩ )∇ γ˙ .. +(4Ω˙ 2Ω−6 − 2ΩΩ−5 + 2kΩ−4)∇˜ iα .. +(4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)V˙ i .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)V i .. +(4Ω˙ 2Ω−7 − 2ΩΩ−6 + 2kΩ−5)∇˜ iV˙ GI .. ... +(−4Ω˙ 3Ω−8 + 8ΩΩΩ˙ −7 − 2ΩΩ−6 + 2Ω˙ kΩ−6)∇˜ iV GI (C.50)

.. µi −2 ˜ ˜ a GI ˙ 2 −6 −5 −4 ˜ ˜ a ∇iδ(∇µ∆ ) = Ω ∇a∇ δp + (−4Ω Ω + 2ΩΩ − 2kΩ )∇a∇ γ˙ .. ˙ 2 −6 −5 −4 ˜ ˜ a +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ α .. ˙ 2 −7 −6 −5 ˜ ˜ a ˙ GI +(4Ω Ω − 2ΩΩ + 2kΩ )∇a∇ V .. ... ˙ 3 −8 ˙ −7 −6 ˙ −6 ˜ ˜ a GI +(−4Ω Ω + 8ΩΩΩ − 2ΩΩ + 2ΩkΩ )∇a∇ V (C.51)

µν Confirming our calculations, we find that δ(∇µG ) evaluates to zero as expected µν µν from the Bianchi identity and δ(∇µ∆ ) = δ(∇µT ). The above thus serves to define the perturbed covariant conservation condition for a RW perfect fluid.

C.3.7 Higher Derivative Relations In order to assess the status of the decomposition theorem, one lastly needs to apply higher derivatives to the ﬁeld equations. With the background 3-space geometry having constant spatial curvature, we programatically perform a series of covariant derivative commutations to bring the higher-derivative equations in as simpliﬁed form as possible.

˜ i ˙ −1 ˜ ˜ a ˙ −1 ˜ ˜ a ˜ ˜ a ∇ ∆0i = 2ΩΩ ∇a∇ γ˙ − 2ΩΩ ∇a∇ α + 2k∇a∇ γ .. ˙ 2 −3 −2 −1 ˜ ˜ a GI +(−4Ω Ω + 2ΩΩ − 2kΩ )∇a∇ V (C.52)

ij ˙ −1 .. 2 GI ˙ −1 g˜ ∆ij = 6ΩΩ γ + 3Ω δp − 6ΩΩ α˙ .. .. +(−6Ω˙ 2Ω−2 + 12ΩΩ−1)γ ˙ + (6Ω˙ 2Ω−2 − 12ΩΩ−1)α ˜ ˜ a ˙ −1 ˜ ˜ a −2∇a∇ α − 4ΩΩ ∇a∇ γ (C.53)

˜ i ˙ −1 ˜ .. 2 ˜ GI ˙ −1 ˜ ∇ ∆ij = 2ΩΩ ∇jγ + Ω ∇jδp − 2ΩΩ ∇jα˙ .. .. ˙ 2 −2 −1 ˜ ˙ 2 −2 −1 ˜ +(−2Ω Ω + 4ΩΩ )∇jγ˙ + (2k + 2Ω Ω − 4ΩΩ )∇jα 234

˙ −1 ˜ ˙ ˙ −1 1 ˜ ˜ a ˙ +4ΩkΩ ∇jγ + kQj + 2ΩkΩ Qj + 2 ∇a∇ Qj ˙ −1 ˜ ˜ a +ΩΩ ∇a∇ Qj (C.54)

˜ i ˜ j ˙ −1 ˜ ˜ a .. 2 ˜ ˜ a GI ˙ −1 ˜ ˜ a ∇ ∇ ∆ij = 2ΩΩ ∇a∇ γ + Ω ∇a∇ δp − 2ΩΩ ∇a∇ α˙ .. ˙ 2 −2 −1 ˜ ˜ a +(−2Ω Ω + 4ΩΩ )∇a∇ γ˙ .. ˙ 2 −2 −1 ˜ ˜ a ˙ −1 ˜ ˜ a +(2k + 2Ω Ω − 4ΩΩ )∇a∇ α + 4ΩkΩ ∇a∇ γ (C.55)

˜ ˜ a ˙ −1 ˜ ˜ a .. 2 ˜ ˜ a GI ˙ −1 ˜ ˜ a ∇a∇ ∆ij = 2ΩΩ g˜ij∇a∇ γ + Ω g˜ij∇a∇ δp − 2ΩΩ g˜ij∇a∇ α˙ .. ˙ 2 −2 −1 ˜ ˜ a +(−2Ω Ω g˜ij + 4ΩΩ g˜ij)∇a∇ γ˙ .. ˙ 2 −2 −1 ˜ ˜ a ˜ ˜ a ˜ ˜ +(2Ω Ω g˜ij − 4ΩΩ g˜ij)∇a∇ α + ∇a∇ ∇j∇iα ˙ −1 ˜ ˜ a ˜ ˜ ˜ ˜ b ˜ ˜ a +2ΩΩ ∇a∇ ∇j∇iγ − g˜ij∇b∇ ∇a∇ α ˙ −1 ˜ ˜ b ˜ ˜ a 1 ˜ ˜ a ˜ ˙ −2ΩΩ g˜ij∇b∇ ∇a∇ γ + 2 ∇a∇ ∇iQj ˙ −1 ˜ ˜ a ˜ 1 ˜ ˜ a ˜ ˙ ˙ −1 ˜ ˜ a ˜ +ΩΩ ∇a∇ ∇iQj + 2 ∇a∇ ∇jQi + ΩΩ ∇a∇ ∇jQi .. ˜ ˜ a ˙ −1 ˜ ˜ a ˙ ˜ ˜ a ˜ ˜ b ˜ ˜ a −∇a∇ Eij − 2ΩΩ ∇a∇ Eij − 2k∇a∇ Eij + ∇b∇ ∇a∇ Eij (C.56) ˜ ˜ a ˜ ˜ b ˙ −1 ˜ ˜ b ˜ ˜ a .. 2 ˜ ˜ b ˜ ˜ a GI ∇a∇ ∇b∇ ∆ij = 2ΩΩ g˜ij∇b∇ ∇a∇ γ + Ω g˜ij∇b∇ ∇a∇ δp ˙ −1 ˜ ˜ b ˜ ˜ a ˙ 2 −2 ˜ ˜ b ˜ ˜ a −2ΩΩ g˜ij∇b∇ ∇a∇ α˙ − 2Ω Ω g˜ij∇b∇ ∇a∇ γ˙ .. −1 ˜ ˜ b ˜ ˜ a ˙ 2 −2 ˜ ˜ b ˜ ˜ a +4ΩΩ g˜ij∇b∇ ∇a∇ γ˙ + 2Ω Ω g˜ij∇b∇ ∇a∇ α .. −1 ˜ ˜ b ˜ ˜ a ˜ ˜ b ˜ ˜ a ˜ ˜ −4ΩΩ g˜ij∇b∇ ∇a∇ α + ∇b∇ ∇a∇ ∇j∇iα ˙ −1 ˜ ˜ b ˜ ˜ a ˜ ˜ ˜ ˜ c ˜ ˜ b ˜ ˜ a +2ΩΩ ∇b∇ ∇a∇ ∇j∇iγ − g˜ij∇c∇ ∇b∇ ∇a∇ α ˙ −1 ˜ ˜ c ˜ ˜ b ˜ ˜ a 1 ˜ ˜ b ˜ ˜ a ˜ ˙ −2ΩΩ g˜ij∇c∇ ∇b∇ ∇a∇ γ + 2 ∇b∇ ∇a∇ ∇iQj ˙ −1 ˜ ˜ b ˜ ˜ a ˜ 1 ˜ ˜ b ˜ ˜ a ˜ ˙ +ΩΩ ∇b∇ ∇a∇ ∇iQj + 2 ∇b∇ ∇a∇ ∇jQi .. ˙ −1 ˜ ˜ b ˜ ˜ a ˜ ˜ ˜ b ˜ ˜ a +ΩΩ ∇b∇ ∇a∇ ∇jQi − ∇b∇ ∇a∇ Eij ˙ −1 ˜ ˜ b ˜ ˜ a ˙ ˜ ˜ b ˜ ˜ a −2ΩΩ ∇b∇ ∇a∇ Eij − 2k∇b∇ ∇a∇ Eij ˜ ˜ c ˜ ˜ b ˜ ˜ a +∇c∇ ∇b∇ ∇a∇ Eij. (C.57)

Having obtained the perturbed ﬂuctuation equations expressed entirely in terms of gauge invariants, checking conservation conditions, and generating the higher- derivative equations of motion for testing the decomposition theorem, we conclude our calculation for the standard Einstein ﬂuctuations in the Robertson Walker background geometry.