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arXiv:0706.4383v3 [gr-qc] 1 May 2008

Gauge-invariant Boltzmann equation and the fluid

limit

Cyril Pitrou

Institut d’Astrophysique de Paris, Universit´ e Pierre & Marie Curie - Paris VI,

CNRS-UMR 7095, 98 bis, Bd Arago, 75014 Paris, France.

E-mail: pitrou@iap.fr

Abstract.

second order in the cosmological perturbations. It describes the gauge dependence

of the distribution function and the construction of a gauge-invariant distribution

function and brightness, and then derives the gauge-invariant fluid limit.

This article investigates the collisionless Boltzmann equation up to

PACS numbers: 98.80

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1. Introduction

The origin of the large-scale structure is nowadays understood from the gravitational

collapse of initial density perturbations which were produced by amplification of

the quantum fluctuations in the inflaton field [1]. The properties of the large-scale

structure depend both on the initial conditions at the end of inflation and on the

growth of perturbations in a universe filled with non relativistic matter and radiation.

The theory of cosmological perturbations is thus a cornerstone of our understanding

of the large-scale structure. The evolution of radiation (photons and neutrinos) needs

to be described by a Boltzmann equation [2, 3, 4]. Two types of perturbative schemes

have extensively been used in the literature in order to describe the evolution of

the cosmological perturbations. The first is a 1 + 3 covariant splitting of space-time

[5, 6, 7] and the second is a more pedestrian coordinate based approach. In the first

approach, exact equations on the physical space-time are derived and perturbative

solutions around a background solution are then calculated. In the second approach,

an averaging procedure is implicitly assumed and, starting from a background space-

time, perturbation variables satisfying the equations of motion order by order are

constructed. In the 1 + 3 approach, the variables used are readily covariant, but

the absence of background space-time can be a problem to simplify the resolution by

performing a mode expansion, since the Helmholtz function is in general not defined on

the physical space-time. In the coordinate based approach, all perturbation variables

live on the background space-time, and enjoy the advantages of its highly symmetric

properties. However, this extra mathematical structure is at the origin of the gauge

issue through the identification mapping that we needs to be defined between the

background space-time and the physical space-time. Thus, the gauge dependence

needs to be understood. An elegant solution is to construct gauge-invariant variables

` a la Bardeen both for the metric perturbation variables [8] and for the distribution

function [9, 10]. Since the Boltzmann and Einstein equations are covariant, they can

be expressed solely in terms of gauge invariant variables provided we have a full set

at hand. A full comparison of these two formalisms has been performed at first order

in Ref. [11], and for gravitational waves at second order in Ref. [12].

In the coordinate based approach, the true degrees of freedom identified from

the Lagrangian formalism, are quantized. They transfer to classical perturbations

which inherit a nearly scale invariant power spectrum and Gaussian statistics, when

their wavelength stretches outside the horizon, thus providing initial conditions for the

standard big-bang model. Conserved quantities [13, 14] enable to ignore the details of

the transition between inflation and the standard big-bang model (see however [15]),

and the evolution details need only to be known when the wavelength reenters the

horizon. A first step to extend this procedure in the 1 + 3 formalism has been taken

in Ref. [16] where conserved quantities were defined. As for the degrees of freedom

which need to be quantized, a first proposal was made in Ref. [17], in order to identify

them, but it has not yet been motivated by a Lagrangian formulation.

The properties of the observed cosmic microwave background (CMB) anisotropies

have confirmed the validity of the linear perturbation theory around a spatially

homogeneous and isotropic universe and have set strong constraints on the origin of

structures, as predicted by inflation. It now becomes necessary, with the forthcoming

increasing precision of data that may allow to detect deviation from Gaussianity

[18], to study the second-order approximation, in order to discuss the accuracy of

these first-order results. These non-Gaussian features are also of first importance,

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since they can help discriminating between different inflation theories. Indeed, one-

field driven inflation leads to very small levels of primordial non-Gaussianity [19],

whereas multifield inflation can present significant non-Gaussian features [20],[21].

However, since non-Gaussian effects also appear through non-linear evolution, that

is from the second-order approximation and beyond of the evolution equations, the

study of second-order evolution equations is necessary in order to distinguish between

primordial and evolutionary non-Gaussianities (see Ref. [22] for a review on non-

Gaussianity). Second-order Einstein and Boltzmann equations have been written in

the 1+3 formalism [23, 24], but not integrated numerically, partly because the mode

expansion is not defined on the physical space-time, and this would then require a

four dimensional numerical integration. However, the promising formalism of Ref. [25],

which builds a bridge between the 1+3 formalism and the coordinate based approach,

might shed some light on these issues. Similarly, in the coordinate based approach,

the second-order Einstein equations have been written in terms of gauge-invariant

variables [26], and a first attempt has been made to write the Boltzmann equation in

a given gauge for the different species filling the universe, and to solve them analytically

[27, 28].

The goal of this paper is to provide the full mathematical framework for handling

distribution functions at second order in the coordinate based approach taking into

account the gauge issue. This will clarify the existing literature and point out some

existing mistakes. We first review briefly in section II the gauge transformations and

the procedure to build gauge invariant variables. We then present in section III the

transformation properties of the distribution function, and express them up to second

order. We define in section IV the gauge-invariant distribution function and the gauge

invariant brightness up to second order in the particular case of radiation (but this

is readily extendable to cold dark matter). We then deduce in section V, from the

Boltzmann equation, the evolution of the gauge invariant brightness in its simplest

collisionless form, at first and second orders. To finish, we express in section VI the

fluid limit as a consistency check of our results.

2. Overview on gauge transformations and gauge-invariant variables

2.1. First- and second-order perturbations

We assume that, at lowest order, the universe is well described by a Friedmann-

Lemaˆ ıtre space-time (FL) with flat spatial sections. The most general form of the

metric for an almost FL universe is

ds2= gµνdxµdxν

= a(η)2?− (1 + 2Φ)dη2+ 2ωidxidη + [(1 − 2Ψ)δij+ hij]dxidxj?,

where η is the conformal time and a the scale factor. We perform a scalar-vector-tensor

decomposition as

(1)

ωi= ∂iB + Bi, (2)

hij= 2Eij+ ∂iEj+ ∂jEi+ 2∂i∂jE,(3)

where Bi, Eiand Eij are transverse (∂iEi= ∂iBi= ∂iEij= 0), and Eij is traceless

(Eii= 0). There are four scalar degrees of freedom (Φ, Ψ, B, E), four vector degrees of

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freedom (Bi, Ei) and two tensor degrees of freedom (Eij). Each of these perturbation

variables can be split in first and second-order parts as

W = W(1)+1

2W(2).

This expansion scheme will refer, as we shall see, to the way gauge transformations

and gauge-invariant (GI) variables are defined. First-order variables are solutions of

first-order equations which have been extensively studied (see Ref. [29] for a review).

Second-order equations will involve purely second-order terms, e.g. W(2)and terms

quadratic in the first-order variables, e.g. [W(1)]2. There will thus never be any

ambiguity about the order of perturbation variables involved as long as the order of

the equation considered is known. Consequently, we will often omit to specify the

order superscript when there is no risk of confusion.

At first order, 4 of the 10 metric perturbations are gauge degrees of freedom and

the 6 remaining degrees of freedom reduce to 2 scalars, 2 vectors and 2 tensors. The

three types of perturbations decouple and can thus be treated separately. As long as

no vector source terms are present, which is generally the case when no magnetic field

or topological defect is taken into account, the vector modes decay as a−2. Thus, we

can safely discard them and set E(1)

i

= B(1)

shall not include vector modes for the sake of clarity. We checked that our arguments

and derivation can trivially (but at the expense of much lengthy expressions) take

them into account.

In the fluid description, we assume that the matter content of the universe can

be described by a mixture of fluids. The four-velocity of each fluid is decomposed as

uµ=1

a(δµ

The perturbation vµhas only three independent degrees of freedom since uµmust

satisfy uµuµ= −1. The spatial components can be decomposed as

vi= ∂iv + ¯ vi,

(4)

i

= 0. In the following of this work, we

0+ vµ). (5)

(6)

¯ vibeing the vector degree of freedom, and v the scalar degree of freedom. The stress-

energy tensor of this fluid is of the form

Tµν= ρuµuν+ P (gµν+ uµuν) , (7)

where the density and pressure are expanded as follows

ρ = ¯ ρ + δρ,P =¯P + δP. (8)

At the background level, the form of the stress-energy tensor is completely fixed by

the symmetry properties of the FL space-time. However, at the perturbation level,

one must consider an anisotropic stress component, πµν with πµ

pressure and density of the fluid are related by an equation of state, P = ρ/3, in the

case of radiation.

At first order, the formalism developed by the seminal work of Ref. [8] provides

a full set of gauge-invariant variables (GIV). Thanks to the general covariance of the

equations at hand (Einstein equations, conservation equations, Boltzmann equation),

it was shown that it was possible to get first-order equations involving only these

gauge-invariant variables. In addition, if these gauge invariant variables reduce, in a

particular gauge, to the perturbation variables that we use in this particular gauge,

then the computation of the equation can be simplified. Actually, we only need to

compute the equations in this particular gauge, as long as it is completely fixed, and

µ= uµπµν= 0. The

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then to promote by identification our perturbation variables to the gauge-invariant

variables. Thus, provided we know this full set of gauge invariant variables, the

apparent loss of generality by fixing the gauge in a calculation, is in fact just a way to

simplify computations. Eventually we will reinterpret the equations as being satisfied

by gauge invariant variables. The full set of first-order gauge-invariant variables is

well known and is reviewed in Ref. [29] and Ref. [30]. As gauge transformations up

to any order were developed, it remained uncertain [31], whether or not a full set of

gauge-invariant variables could be built for second and higher orders. This has been

recently clarified [26], and the autosimilarity of the transformation rules for different

orders can be used as a guide to build the gauge-invariant variables at any order. We

present a summary of the ideas presented in Ref. [31] about gauge transformations

and the construction of gauge-invariant variables [26].

2.2. Points identification on manifolds

When working with perturbations, we consider two manifolds: a background manifold,

M0, with associated metric ¯ g, which in our case is the FL space-time, and the physical

space-time M1with the metric g. Considering the variation of metric boils down to a

comparison between tensor fields on distinct manifolds. Thus, in order to give a sense

to “δg(P) = g(P) − ¯ g(¯P)”, we need to identify the points P and¯P between these

two manifolds and also to set up a procedure for comparing tensors. This will also be

necessary for the comparison of any tensor field.

One solution to this problem [31] is to consider an embedding 4 + 1 dimensional

manifold N = M × [0,1], endowed with the trivial differential structure induced,

and the projections Pλ on submanifolds with P0(N) = M × {0} = M0 and

P1(N) = M × {1} = M1. The collection of Mλ ≡ Pλ(N) is a foliation of N,

and each element is diffeomorphic to the physical space-time M1and the background

space-time M0. The gauge choice on this stack of space-times is defined as a vector

field X on N which satisfies X4= 1 (the component along the space-time slicing R).

A vector field defines integral curves that are always tangent to the vector field itself,

hence inducing a one parameter group of diffeomorphisms φ(λ,.), also noted φλ(.),

a flow, leading in our case from φ(0,p ∈ P0(N)) = p ∈ P0(N) along the integral

curves to φ(1,p ∈ P0(N)) = q ∈ P1(N). Due to the never vanishing last component

of X, the integral curves will always be transverse to the stack of space-times and

the points lying on the same integral curve, belonging to distinct space-times, will be

identified. Additionally the property X4= 1 ensures that φλ,X(P0(N)) = Pλ(N), i.e.

the flow carries a space-time slice to another. This points identification is necessary

when comparing tensors, but we already see that the arbitrariness in the choice of a

gauge vector field X should not have physical meaning, and this is the well known

gauge freedom.

2.3. Tensors comparison and perturbations

The induced transport, along the flow, of tensors living on the tangent bundle, is

determined by the push-forward φ⋆λ and the pull-back φ⋆

element φλ of the group of diffeomorphisms. These two functions encapsulate the

transformation properties of the tangent and co-tangent spaces at each point and its

image. Indeed, the pull-back can be linked to the local differential properties of the

vector field embedded by the Lie derivatives along the vector field in a Taylor-like

λ[32] associated with an