Consistency 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. 9 0 0 Abstract 2 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. 3 Theconsistency ofall equationsof thesecond-orderEinstein equation andtheequa- 2 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 1 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 0 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. X 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. References [1] C.L. Bennett et al., Astrophys. J. Suppl. Ser. 148, (2003), 1. [2] E. Komatsu et al., arXiv:0803.0547[astro-ph], (2008). 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Theor. Phys. 117 (2007), 17. [7] K. Nakamura, preprint (arXiv:0812.4865[gr-qc]). [8] K. Nakamura, Prog. Theor. Phys. 110, (2003), 723; K. Nakamura, Prog. Theor. Phys. 113 (2005), 481. 4