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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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Gauge-invariant Boltzmann equation and the fluid limit
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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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Gauge-invariant Boltzmann equation and the fluid limit
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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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Gauge-invariant Boltzmann equation and the fluid limit
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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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Gauge-invariant Boltzmann equation and the fluid limit
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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
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Gauge-invariant Boltzmann equation and the fluid limit
6
fashion (see Ref. [32] or Ref. [31])
Φ⋆
X,λ(T) =
k=∞
?
k=0
λk
k!Lk
XT, (9)
for any tensor T.
A remark about coordinates changes is on order here. When the tensor T is a
coordinate xµ(once µ is fixed, it is a scalar field), the previous definition reduces to
the standard finite coordinates transformation.
λ,ξ(xµ) = xµ+ λξµ+λ2
x′µ≡ Φ⋆
2ξµ
,νξν+ ...(10)
This is the standard way of defining an active transformation on the manifold,
by transporting a point of coordinates xµto a point of coordinates x′µ.
transformation, when performed on the coordinate system - considering the
coordinates as a grid on the manifold that one would displace according to the active
transformation - induces a passive coordinates transformation, if we decide that the
new coordinates of a point q are the coordinates of the point p such that φλ(p) = q.
When considering a transformation induced by a field ξ, we will refer to the passive
coordinates transformation induced by the active transportation of the coordinates
system.
The expansion of Eq. (9) on P0(N) provides a way to compare a tensor field
on Pλ(N) to the corresponding one on the background space-time P0(N).
background value being T0≡ L0
perturbation
This
The
XT|P0(N), we obtain a natural definition for the tensor
∆XTλ≡
k=∞
?
k=1
λk
k!Lk
XT
???
P0(N)= Φ⋆
X,λ(T) − T0.(11)
The subscript X reminds the gauge dependence.
perturbation as
We can read the n-th order
δ(n)
XT ≡ Ln
XT
???
P0(N),(12)
which is consistent with the expansion of perturbation variables of the physical metric
in Eq. (4), since the physical space-time is labeled by λ = 1. However, the fact
that the intermediate space-time slices Pλ(N) are labeled by λ removes the absolute
meaning of order by order perturbations, as it can be seen from Eq. (11). The entire
structure embedded by N is more than just a convenient construction and this will
have important consequences in gauge changes as we will now detail.
2.4. Gauge transformations and gauge invariance
If we consider two gauge choices X and Y , a gauge transformation from X to Y is
defined as the diffeomorphism
φX→Y,λ= (φX,λ)−1(φY,λ),(13)
and it induces a pull-back which carries the tensor ∆XTλ, which is the perturbation
in the gauge X, to ∆YTλ, which is the perturbation in gauge Y since
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Gauge-invariant Boltzmann equation and the fluid limit
31
Appendix B.0.2. Second order
R(2)
00,X
R(2)
0ai,X
R(2)
ai0,X
R(2)
= Φ(2)− 3Φ2+ ∂iB∂iB
= − ∂aiB(2)+ (2Φ − 4Ψ)∂aiB + 4∂ajB?∂ai∂ajE + Eaiaj
= − S(2)
aiak,X= − S(2)
= Ψ(2)δaiak−
+ 3?∂ai∂alE + Eal
− 6Ψ(∂ai∂akE + Eaiak)
−S(2)
−S(2)
In these formulas, we have omitted the first order superscript as there is no possible
confusion. In the following, we will also omit the first order superscript.
transformations rules for the tetrads can be read, as we did for the first order case:
(B.3)
?
ai0,X= 0
aiak,X
?
∂ak∂aiE(2)+ E(2)
akai
?
+ 3Ψ2δaiak
ai
?(∂al∂akE + Ealak)
00,X= Φ(2)− Φ2+ ∂iB∂iB
0ai,X= − ∂aiB(2)− 2Ψ∂aiB + 2∂ajB?∂ai∂ajE + Eaiaj
?
The
T
?
δ(2)
Xeµ
0
?
= −
?
?
T (Φ(2)) − 3T (Φ)2+ ∂iT (B)∂iT (B)
− ∂aiT (B(2)) + [2T (Φ) − 4T (Ψ)]∂aiT (B)
+ 4∂ajT (B)
T (Ψ(2)) + 3T (Ψ)2?
+
?
− 6T (Ψ)?∂ak∂aiT (E) + Eak
?
¯ eµ
0
(B.4)
+
?
∂ai∂ajT (E) + Eai
¯ eµ
ai
aj
??
¯ eµ
ai
T
?
δ(2)
Xeµ
ai
?
=
?
− ∂ak∂aiT (E(2)) + 3?∂ai∂ajT (E) + Eaj
ai
??
∂ak∂ajT (E) + Eak
aj
?
ai
??
¯ eµ
ak.
Appendix C. Transformation of δ(2)f
T
??LTξ2+ L2
+
?
+
?
+ 2T
?
δ(2)
Xf
?
= (C.1)
Tξ1
??¯f(xν,apµ)?+ 2LTξ1
?
?
?
R(1)b
a,X
+ S(1)b
a,X
?
δ(1)
Xf(xν,apµ)
πa∂¯f
?
T
?
?
?
T
R(2)c
a,X
+ S(2)c
a,X+ 2S(1)d
a,XT
?
R(1)c
d,X
??
∂πc
T
R(1)b
a,X
T
πa∂
?
∂πbLTξ1
R(1)d
c,X
?
+ S(1)b
a,XS(1)d
c,X+ 2S(1)b
a,XT
?
?
R(1)d
c,X
??
πaπc
∂2¯f
∂πb∂πd
R(1)b
a,X
?¯f(xν,apµ)?+ 2LTξ1
πa∂
∂πbδ(1)
S(1)b
a,Xπa∂
∂πb¯f(xν,apµ)
?
+ 2
??
?
?
Xf(xν,apµ)
?
.
These individual terms are explicitly given by
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Gauge-invariant Boltzmann equation and the fluid limit
32
?
T
?
R(2)0
0,X
?
?
+ S(2)0
0,X+ 2S(1)0
0,XT
?
R(1)0
0,X
??
?
π0∂¯f
∂π0=
+ 4Φ(T′+ HT) + 3(T′+ HT)2
?
(C.2)
?
− 2∂iB∂i(−T + L′) − ∂i(−T + L′)∂i(−T + L′)
−
T(2)′+ HT(2)+ SΦ(T,L)
π0∂¯f
∂π0,
?
T
?
R(1)0
0,X
?
T
?
R(1)0
0,X
∂2¯f
∂ (π0)2(π0)2(T′+ HT)2,
?
+ S(1)0
0,XS(1)0
0,X+ 2S(1)0
0,XT
?
R(1)0
0,X
??
π0π0
∂2¯f
∂π0∂π0=
(C.3)
2
?
T
?
R(1)b
a,X
?
∂π0π0(T′+ HT) − 2∂δ(1)
− 2∂δ(1)
+ S(1)b
a,X
?
πa∂
∂πbδ(1)
Xf(xν,apµ) =(C.4)
− 2∂δ(1)
Xf
Xf
∂πiπ0(−∂iT + ∂iL′)
Xf
∂πi(πj∂i∂jL + HπiT),
2T
?
R(1)b
a,X
?
?
πa∂
∂πbLTξ1
∂2¯f
∂ (π0)2(π0)2+∂¯f
∂2¯f
∂ (π0)2(π0)2?ni∂iT?(Φ + T′+ HT) − 2T
− 2∂¯f
− 2∂¯f
?¯f(xν,apµ)?=
∂π0π0
(C.5)
− 2
?
(T′+ HT)(Φ + T′+ HT)
− 2
∂2¯f
∂η∂π0(Φ + T′+ HT)
∂π0π0(Bi− ∂iT + ∂iL′)∂iT
∂π0π0?nj∂i∂j(E + L) + njEi
j+ ni(−Ψ + HT)?∂iT,
2LTξ1
?
S(1)b
c,Xaπc∂
∂πb¯f(xν,apµ)
?
=(C.6)
2
∂2¯f
∂η∂π0π0ΦT + 2∂¯f
?
∂π0π0?Φ′T + ∂iΦ∂iL?
∂π0π0
+ 2
∂2¯f
∂ (π0)2(π0)2+∂¯f
?
Φ?T′+ HT + ni∂iT?,
?LTξ2+ L2
Tξ1
??¯f(xν,apµ)?=
∂π0π0(T(2)′+ HT(2)+ ni∂iT(2))
∂η2T2+∂¯f
(C.7)
T(2)∂¯f
∂η+∂¯f
+∂2¯f
∂η(TT′+ ∂iT∂iL) +
∂2¯f
∂π0∂ηπ02T(T′+ HT + ni∂iT)
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Gauge-invariant Boltzmann equation and the fluid limit
33
+
∂2¯f
∂ (π0)2(π0)2?2ni∂iT(HT + T′) + (ni∂iT)(nj∂jT) + H2T2+ 2HTT′+ (T′)2?
+∂¯f
∂π0π0?
+ ∂jT′∂jL + ∂jT∂jL′+ H∂jT∂jL + ∂jLni∂i∂jT + ∂jTni∂i∂jL + (T′)2?
2LTξ1
2?π0T′+ πj∂jT?∂δ(1)
+ 2∂iL∂δ(1)
∂xi
∂η
TT′′+ H′T2+ 3HTT′+ Tni∂iT′+ T′ni∂iT + H2T2+ 2HTni∂iT
,
?
δ(1)
Xf(xν,apµ)
?
=(C.8)
Xf
∂π0
?
+ 2?π0∂iL′+ πj∂i∂jL?∂δ(1)
∂δ(1)
Xf
+∂δ(1)
Xf
∂πi
Xf
+ 2T
Xf
∂π0Hπ0+∂δ(1)
Xf
∂πiHπi
?
.
In the above formulas, we have omitted to write the fact that the derivatives with
respect to η or xiare taken at fixed πa.
Appendix D. Integral relations necessary to derive the fluid limit
The integrations on angular directions can be handled with the general formulas (see
Ref. [43])
?
?
ni1...nikd2Ω
4π
= 0 ifk = 2p + 1(D.1)
ni1...nikd2Ω
4π
=
1
k + 1
?
δ(i1i2...δi(k−1)ik)?
ifk = 2p. (D.2)
By successive integration by parts, we also obtain the following useful results
?
?
?
¯f(xµ,π0)(π0)3dπ0d2Ω = ¯ ρ(xµ),
∂¯f(xµ,π0)
∂π0
∂2¯f(xµ,π0)
∂2π0
(π0)4dπ0d2Ω = − 4¯ ρ(xµ),
(π0)5dπ0d2Ω = 20¯ ρ(xµ). (D.3)
Appendix E. The fluid limit for radiation
As explained in section 2.1, second order quantities involve either purely second order
perturbation variables or terms quadratic in first order perturbation variables. As
long as the order of the quantity is known we can omit the order superscript in order
to simplify notations.
Appendix E.1. Geometric quantities
In the Newtonian gauge, ignoring vector perturbations for simplicity, the non-
vanishing Christoffel symbols associated with the metric (1) are for the background
(0)Γ0
00= H,
(0)Γ0
jk= Hδjk,
(0)Γi
0j= Hδi
j.(E.1)
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Gauge-invariant Boltzmann equation and the fluid limit
34
At first order, we get
(1)Γ0
00= Φ′,
(1)Γ0
0j= ∂jΦ,
(1)Γi
00= ∂iΦ,(E.2)
(1)Γ0
jk= 2HEjk+ E′
(1)Γi
jk− (2HΦ + Ψ′+ 2HΨ)δjk,
j,
(E.3)
0j= E′ij− Ψ′δi
(1)Γi
(E.4)
jk= 2∂(k[Ei
j)− Ψδi
j)] − ∂i(Ejk− Ψδjk), (E.5)
where A(ij)≡ (Aij+ Aji)/2. At second order, we obtain
(2)Γ0
00= Φ′− 4ΦΦ′,
(2)Γi
(2)Γ0
0j= ∂jΦ − 4Φ∂jΦ,(E.6)
00= ∂iΦ − 4Eij∂jΦ + 4Ψ∂iΦ,
(2)Γ0
+ 2HEjk− 8ΦHEjk+ E′
(2)Γi
(E.7)
jk= [−2HΨ − Ψ′+ 4ΦΨ′− 2HΦ + 8HΦ(Φ + Ψ)]δjk
jk− 4ΦE′
jk,(E.8)
0j= Ei′
j+ 4Ψ′Ei
− 4EikE′
jk= 2∂(k[Ei
+ 4?Eil− Ψδil??
j− Ψ′δi
kj+ 4ΨEi
j− 4ΨΨ′δi
′,
j
j
(E.9)
(2)Γi
j)− Ψδi
j)] − ∂i(Ejk− Ψδjk)
∂l(Ejk− Ψδjk)
− ∂k(Elj− Ψδlj) − ∂j(Ekl− Ψδkl)
(E.10)
?
.
Appendix E.2. The radiation fluid equations
The conservation equation ∇µTµνfor the stress energy tensor (7) with a radiation
equation of state P = ρ/3 and where we assume πµν= 0, are the conservation
equation
?′
?
and the Euler equation
v(1)′+ Φ(1)+δ(1)ρ
4¯ ρ
v(2)′+ Φ(2)+δ(2)ρ
4¯ ρ
where the source terms in the second order equations, which are quadratic in first
order perturbation variables, are given by
=8
3
+ ∂iv?−∂iδρ − 2¯ ρ∂iv′− 2¯ ρ∂iΦ + 3¯ ρ∂iΨ??
∂iSe= − 2
+ 2Φ∂iv′− 2δρ
?
δ(1)ρ
+ 4Hδ(1)ρ +4
+ 4Hδ(2)ρ +4
3¯ ρ
?
?
∆v(1)− 3Ψ(1)′?
∆v(2)− 3Ψ(2)′?
= 0,(E.11)
δ(2)ρ
?′
3¯ ρ
= Sc,
= 0,(E.12)
= Se,
Sc
?
δρΨ′+ 6¯ ρΨΨ′− (Φ + δ)¯ ρ∆v
, (E.13)
?δρ
¯ ρ∂iv
?′
+ 10Ψ′∂iv + 4Ψ∂iv′− 2∂j
?∂jv∂iv?
ρ∂iΦ + 4Φ∂iΦ. (E.14)
Page 35
Gauge-invariant Boltzmann equation and the fluid limit
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