Table Of ContentConsistency relations between the source terms in the
second-order Einstein equations for cosmological perturbations
Kouji NAKAMURA1,
Department of Astronomical Science, the Graduate University for Advanced Studies,
Mitaka, Tokyo 181-8588, Japan.
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0
0
Abstract
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Inadditiontothesecond-orderEinsteinequationsonfour-dimensionalhomogeneous
n
isotropic background universe filled with thesingle perfect fluid, we also derived the
a
J second-order perturbations of the continuity equation and the Euler equation for a
perfect fluid in gauge-invariant manner without ignoring any mode of perturbations.
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Theconsistency ofall equationsof thesecond-orderEinstein equation andtheequa-
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tionsofmotion formatterfieldsisconfirmed. Duetothisconsistency check,wemay
] say that the set of all equations of the second-order are self-consistent and they are
c
correct in this sense.
q
-
r 1 Introduction
g
[
The general relativistic second-order cosmological perturbation theory is one of topical subjects in the
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recent cosmology. By the recent observation[1], the first order approximation of the fluctuations of our
v
universe from a homogeneousisotropic one was revealed. The observationalresults also suggest that the
8
3 fluctuations of our universe are adiabatic and Gaussian at least in the first order approximation. We are
6 now on the stage to discuss the deviation from this first order approximation from the observational[2]
3 and the theoretical side[3] through the non-Gaussianity, the non-adiabaticity, and so on. To carry out
.
1 this, someanalysesbeyondlinearorderare required. The second-ordercosmologicalperturbationtheory
0 is one of such perturbation theories beyond linear order.
9 In this article, we confirm the consistency of all equations of the second-order Einstein equation and
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the equations of motion for matter fields, which are derived in Refs. [4, 5]. Since the Einstein equations
:
v includetheequationofmotionformatterfields,thesecond-orderperturbationsoftheequationsofmotion
i formatterfieldsarenotindependentequationsofthesecond-orderperturbationoftheEinsteinequations.
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Through this fact, we can check whether the derived equations of the second order are self-consistent or
r
a not. This confirmation implies that the all derived equations of the second order are self-consistent and
these equations are correct in this sense.
2 Metric perturbations
The background spacetime for the cosmological perturbations is a homogeneous isotropic background
spacetime. The background metric is given by
g =a2 −(dη) (dη) +γ (dxi) (dxj) , (1)
ab a b ij a b
where γ := γ (dxi) (dxj) is the(cid:8)metric on the maximally symm(cid:9)etric three-space and the indices
ab ij a b
i,j,k,... for the spatial components run from 1 to 3. On this background spacetime, we consider the
perturbative expansion of the metric as g¯ab =gab+λXhab+ λ22Xlab+O(λ3), where λ is the infinitesimal
parameter for perturbation and h and l are the first- and the second-order metric perturbations,
ab ab
respectively. As shown in Ref. [6], the metric perturbations h and l are decomposed as
ab ab
h =:H +£ g , l =:L +2£ h + £ −£2 g , (2)
ab ab X ab ab ab X ab Y X ab
1E-mail:[email protected] (cid:0) (cid:1)
1
where H and L are the gauge-invariant parts of h and l , respectively. The components of H
ab ab ab ab ab
and L can be chosen so that
ab
(1) (1) (1) (1)
Hab = a2 −2 Φ (dη)a(dη)b+2 νi (dη)(a(dxi)b)+ −2 Ψ γij+ χij (dxi)a(dxj)b , (3)
(cid:26) (cid:18) (cid:19) (cid:27)
(2) (2) (2) (2)
Lab = a2 −2 Φ (dη)a(dη)b+2 νi (dη)(a(dxi)b)+ −2 Ψ γij+ χij (dxi)a(dxj)b . (4)
(cid:26) (cid:18) (cid:19) (cid:27)
(p) (p)
In Eqs. (3) and (4), the vector-mode ν and the tensor-mode χ (p=1,2) satisfy the properties
i ij
(p) (p) (p) (p)
Di ν =γijD ν =0, χi =0, Di χ =0, (5)
i p j i ij
where γkj is the inverse of the metric γ .
ij
3 Background, First-, and Second-order Einstein equations
The Einstein equations of the background, the first order, and the second order on the above four-
dimensional homogeneous isotropic universe are summarized as follows.
The Einstein equations for this background spacetime filled with a perfect fluid are given by
(0) (0)
8πG
(p)E :=H2+K− a2ǫ=0, (p)E :=2∂ H+H2+K+8πGa2p=0, (6)
(1) 3 (2) η
where H = ∂ a/a, K is the curvature constant of the maximally symmetric three-space, ǫ and p are
η
energy density and pressure, respectively.
On the other hand, the second-order perturbations of the Einstein equation are summarized as
(2)
(2) (2) (2)
(p)E(1) := (−3H∂η+∆+3K) Ψ −3H2 Φ −4πGa2 E −Γ0 =0, (7)
(2)
1 (2) 1 (2)
(p)E(2) := ∂η2+2H∂η−K− 3∆ Ψ + H∂η+2∂ηH+H2+ 3∆ Φ
(cid:18) (cid:19) (cid:18) (cid:19)
(2) 1
−4πGa2 P − Γ k =0, (8)
6 k
(2)
(p)E(3) := (Ψ2)−(Φ2)−32(∆+3K)−1 ∆−1DiDjΓij − 31Γkk =0, (9)
(cid:18) (cid:19)
(2)
(p)E(4)i := ∂ηDi (Ψ2)+HDi (Φ2)−12Di∆−1DkΓk+4πGa2(ǫ+p)Di (v2)=0, (10)
(p)E(2) := (∆+2K)(ν2)+2 Γ −D ∆−1DkΓ −16πGa2(ǫ+p)(V2)=0, (11)
(5)i i i i k i
(p)E(2) := ∂ a2 (ν2) −2a2(cid:0)(∆+2K)−1 D ∆(cid:1)−1DkDlΓ −DkΓ =0, (12)
(6)i η i i kl ik
(cid:18) (cid:19)
(cid:8) (cid:9)
(2)
(p)E := ∂2+2H∂ +2K−∆ χ(2) −2Γ + 2γ Γ k
(7)ij η η ij ij 3 ij k
+(cid:0) 3 D D − 1γ ∆ ((cid:1)∆+3K)−1 ∆−1DkDlΓ − 1Γ k
i j 3 ij kl 3 k
(cid:18) (cid:19) (cid:18) (cid:19)
−4 D (∆+2K)−1D ∆−1DlDkΓ −D (∆+2K)−1DkΓ =0, (13)
(i j) lk (i j)k
(cid:16) (cid:17)
(2) (2)
where we denote Γ j = γjkΓ . In these equations, E and P are the second-order perturbations of the
i ik
(2) (2)
energy density and the pressure, respectively. Further, D v and V are the scalar-and the vector-parts
i i
2
of the spatial components of the covariant fluid four-velocity, in these equations. Γ , Γ , and Γ are the
0 i ij
collectionsofthequadratictermsofthelinear-orderperturbationsinthesecond-orderEinsteinequations
and these can be regardedas the source terms in the second-orderEinstein equations. The explicit form
of these source terms are given in Refs. [4, 7]. First-order perturbations of the Einstein equations are
given by the replacements (Φ2)→(Φ1), (Ψ2)→(Ψ1), (ν2i)→(ν1i), χ(2i)j→χ(1i)j, (E2)→(E1), (P2)→(P1), Di (v2)→ Di (v1), (V2i)→(V1i),
and Γ =Γ =Γ =0.
0 i ij
4 Consistency with the equations of motion for matter field
Now,weconsiderthesecond-orderperturbationoftheenergycontinuityequationandtheEulerequations.
In terms of gauge-invariant variables, the second-order perturbations of the energy continuity equation
and the Euler equation for a single perfect fluid are given by[5]
a(2)C0(p) := ∂η (E2)+3H (E2)+ (P2) +(ǫ+p) ∆(v2)−3∂η (Ψ2) −Ξ0 =0, (14)
(cid:18) (cid:19) (cid:18) (cid:19)
(2)Ci(p) := (ǫ+p) (∂η+H) Di (v2)+(V2i) +Di (Φ2) +Di (P2)+∂ηp Di (v2)+(V2i) −Ξ(ip) =0,(15)
( ! ) !
where Ξ and Ξ(p) are the collection of the quadratic terms of the linear order perturbations and its
0 i
explicit forms are given in Ref. [5, 7].
Toconfirmthe consistencyof the backgroundandthe perturbationsofthe Einsteinequationandthe
energycontinuityequation(14), we firstsubstitute the second-orderEinsteinequations(6)–(8), and(10)
into Eq.(14). For simplicity, we firstimpose the first-orderversionof Eq.(9) onall equations. Then, we
obtain
(2) (2) (2) (2) (0) (0)
3 (2)
4πGa3(2)C0(p) = −∂η (p)E(1) −H(p)E(1) −3H(p)E(2) +Di (p)E(4)i +2 3(p)E(1) −(p)E(2) ∂η Ψ
!
1 1
−∂ Γ −HΓ − HΓ k+ DkΓ −4πGa2Ξ . (16)
η 0 0 2 k 2 k 0
This equation shows that the second-orderperturbation (14) of the energy continuity equationis consis-
tent with the second-order and the backgroundEinstein equations if the equation
1 1
4πGa2Ξ +(∂ +H)Γ + HΓ k− DkΓ =0 (17)
0 η 0 2 k 2 k
is satisfied under the background, the first-order Einstein equations. Actually, through the background
Einstein equations (6) and the first-order version of the Einstein equations (7)–(13), we can easily see
that Eq. (17) is satisfied under the Einstein equations of the background and of the first order[7].
Next, we consider the second-order perturbations of the Euler equations. For simplicity, we first
impose the first-order version of Eq. (9) on all equations, again. Through the background Einstein
equations (6) and the Einstein equations of the second order (8)–(10), we can obtain
8πGa2(2)Ci(p) = −8πGa3 C(00(p)) Di (v2)+(V2i) −Di (Φ2) 3(p)(E0)(1) −(p)(E0)(2) −2Di (p)(E2)(2)
! !
(2) (2) (2)
2 1
− D (∆+3K)(p)E + (∂ +2H) +4(p)E −(p)E
3 i (3) 2 η (4)i (5)i
!
(2)
1
+ (∆+2K)(p)E −8πGa2Ξ(p)+(∂ +2H)Γ −DlΓ . (18)
2a2 (6)i j η j jl
This equation shows that the second-order perturbations of the Euler equations is consistent with the
Einstein equations of the backgroundand the second order if the equation
(∂ +2H)Γ −DlΓ −8πGa2Ξ(p) = 0 (19)
η j jl j
3
is satisfied under the Einstein equations of the background and the first order. Actually, we can easily
confirm Eq. (19) due to the background Einstein equations and the first-order perturbations of the
Einstein equations[7], and implies that the second-order perturbation of the Euler equation is consistent
with the set of the background, the first-order, and the second-order Einstein equations.
Theconsistencyofequationsforperturbationsshownhereisjustawell-knownresult,i.e.,theEinstein
equation includes the equations of motion for matter field due to the Bianchi identity. However, the
above verification of the identities (17) and (19) implies that our derived second-order perturbations of
theEinsteinequation,the equationofcontinuity,andtheEulerequationareconsistent. Inthissense,we
may say that the derivedsecond-orderEinsteinequations, especially,the derivedformulaefor the source
terms Γ , Γ , Γ , Ξ , and Ξ in Ref. [7] are correct.
0 i ij 0 i
5 Summary
In summary, we show the all components of the second-order perturbation of the Einstein equation
without ignoring any modes of perturbationin the case ofa perfect fluid. The derivationis based onthe
generalframeworkofthesecond-ordergauge-invariantperturbationtheorydevelopedinRefs.[8]. Inthis
formulation,anygaugefixingisnotnecessaryandwecanobtainanyequationinthegauge-invariantform
which is equivalent to the complete gauge fixing. In other words, our formulation gives complete gauge
fixed equations without any gauge fixing. Therefore, equations which are obtained in gauge-invariant
manner cannot be reduced without physical restrictions any more. In this sense, these equations are
irreducible. This is one of the advantages of the gauge-invariantperturbation theory.
We have also checked the consistency of the set of equations of the second-order perturbation of the
Einsteinequationsandtheevolutionequationofthematterfieldinthecasesofaperfectfluid. Therefore,
inthecaseofthesinglematterfield,wemaysaythatwehavebeenreadytoclarifythephysicalbehaviors
of the second-ordercosmologicalperturbations. The physicalbehaviorof the second-orderperturbations
in the universe filled with a single matter field will be instructive to clarify those of the second-order
perturbations in more realistic cosmologicalsituations. We leave these issues as future works.
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