Consistency relations between the source terms in the second-order Einstein equations for cosmological perturbations

Consistency relations b et w een the source terms in the second-order Einstein equations for cosmological p erturbations Kouji NAKAMURA 1 , Dep artment of Astr onomic al Scienc e, the Gr aduate University for A dvanc e d Stu dies, Mitaka, T okyo 181-8588, J ap an. Abstract In addition to the second-order Einstein equations on four-d imensional h omogeneous isotropic bac k ground universe filled with the single p erfect fluid, we also d erived the second-order p erturb ations of the contin uity equ ation an d the Euler equation for a p erfect fluid in gauge-in vari ant manner without ignoring any mo de of p erturbations. The consistency of all equations of t h e second- order Einstein equation and the equa- tions of motion for matter fields is confi rmed. Due to this consistency c h ec k, we may sa y that the set of all equations of the second- order are self-consistent and they are correct in t his sense. 1 In tro duction The general relativistic second-or der cosmo lo gical p erturba tion theory is o ne o f topical sub jects in the recent cosmo logy . By the recent observ ation[1], the fir st order approximation o f the fluctuations of our universe fr o m a homog eneous isotropic one was re vealed. The obse rv ational r esults a lso sugg est that the fluctuations o f our universe are adiabatic and Gaussian at least in the first order appr oximation. W e are now on the s tage to dis cuss the devia tio n from this first o rder approximation from the obse r v ational[2] and the theoretical side[3 ] through the non-Ga us sianity , the no n-adiabaticity , and s o on. T o carr y out this, some ana ly ses be yond linear or der ar e required. The seco nd-order co smologica l p erturbatio n theor y is one of such p er tur bation theories beyond linear order. In this article, w e confirm the consistency of all e q uations of the second-orde r Einstein e q uation and the equatio ns of motion for matter fields , which are derived in Refs. [4, 5 ]. Since the E instein equations include the equation of mo tion for matter fields, the second-order p erturbatio ns o f the equa tions of motio n for matter fields are not indep endent equations of the second-or der perturba tion of the Einstein equations . Through this fact, we can c heck whether the derived e quations of the second order are self-co nsistent o r not. This co nfirmation implies tha t the a ll derived equations o f the seco nd order ar e self-cons istent and these equations are cor r ect in this sense. 2 Metric p erturbations The background spac e time for the cosmological p er turbations is a homogeneo us isotropic background spacetime. The background metric is given by g ab = a 2  − ( dη ) a ( dη ) b + γ ij ( dx i ) a ( dx j ) b  , (1) where γ ab := γ ij ( dx i ) a ( dx j ) b is the metric on the maxima lly symmetric three-spac e and the indices i, j, k , ... for the spatial comp onents r un from 1 to 3 . On this background spa cetime, we consider the per turbative expansion of the metric as ¯ g ab = g ab + λ X h ab + λ 2 2 X l ab + O ( λ 3 ), where λ is the infinitesimal parameter fo r pertur ba tion and h ab and l ab are the first- a nd the se cond-order metric per turbations, resp ectively . As shown in Ref. [6], the metric per turbations h ab and l ab are decomp osed as h ab =: H ab + £ X g ab , l ab =: L ab + 2 £ X h ab +  £ Y − £ 2 X  g ab , (2) 1 E-mail:kouc han@th.nao.ac.jp 1 where H ab and L ab are the gauge-inv ariant par ts o f h ab and l ab , resp ectively . The comp onents of H ab and L ab can be chosen so that H ab = a 2  − 2 (1) Φ ( dη ) a ( dη ) b + 2 (1) ν i ( dη ) ( a ( dx i ) b ) +  − 2 (1) Ψ γ ij + (1) χ ij  ( dx i ) a ( dx j ) b  , (3) L ab = a 2  − 2 (2) Φ ( dη ) a ( dη ) b + 2 (2) ν i ( dη ) ( a ( dx i ) b ) +  − 2 (2) Ψ γ ij + (2) χ ij  ( dx i ) a ( dx j ) b  . (4) In Eqs. (3) and (4), the vector-mo de ( p ) ν i and the tensor-mo de ( p ) χ ij ( p = 1 , 2) satisfy the prop erties D i ( p ) ν i = γ ij D p ( p ) ν j = 0 , ( p ) χ i i = 0 , D i ( p ) χ ij = 0 , (5) where γ kj is the in verse o f the metric γ ij . 3 Bac kground, First-, and Second-ord er Einstein equations The Einstein equatio ns of the bac kground, the first order , and the se c ond order on the ab ov e four- dimensional homogeneo us iso tropic univ erse are summarized as follows. The Einstein equations for this background spacetime filled with a p erfect fluid a re g iven by (0) ( p ) E (1) := H 2 + K − 8 π G 3 a 2 ǫ = 0 , (0) ( p ) E (2) := 2 ∂ η H + H 2 + K + 8 π Ga 2 p = 0 , (6) where H = ∂ η a/a , K is the curv ature consta n t o f the ma ximally symmetric three-space, ǫ and p are energy density and press ure, resp ectively . On the other hand, the second-or der pertur bations of the Einstein equatio n are s umma r ized as (2) ( p ) E (1) := ( − 3 H ∂ η + ∆ + 3 K ) (2) Ψ − 3 H 2 (2) Φ − 4 π Ga 2 (2) E − Γ 0 = 0 , (7) (2) ( p ) E (2) :=  ∂ 2 η + 2 H ∂ η − K − 1 3 ∆  (2) Ψ +  H ∂ η + 2 ∂ η H + H 2 + 1 3 ∆  (2) Φ − 4 π Ga 2 (2) P − 1 6 Γ k k = 0 , (8) (2) ( p ) E (3) := (2) Ψ − (2) Φ − 3 2 (∆ + 3 K ) − 1  ∆ − 1 D i D j Γ ij − 1 3 Γ k k  = 0 , (9) (2) ( p ) E (4) i := ∂ η D i (2) Ψ + H D i (2) Φ − 1 2 D i ∆ − 1 D k Γ k + 4 π Ga 2 ( ǫ + p ) D i (2) v = 0 , (10) (2) ( p ) E (5) i := (∆ + 2 K ) (2) ν i +2  Γ i − D i ∆ − 1 D k Γ k  − 16 π Ga 2 ( ǫ + p ) (2) V i = 0 , (11) (2) ( p ) E (6) i := ∂ η  a 2 (2) ν i  − 2 a 2 (∆ + 2 K ) − 1  D i ∆ − 1 D k D l Γ kl − D k Γ ik  = 0 , (12) (2) ( p ) E (7) ij :=  ∂ 2 η + 2 H ∂ η + 2 K − ∆  (2) χ ij − 2Γ ij + 2 3 γ ij Γ k k +3  D i D j − 1 3 γ ij ∆  (∆ + 3 K ) − 1  ∆ − 1 D k D l Γ kl − 1 3 Γ k k  − 4  D ( i (∆ + 2 K ) − 1 D j ) ∆ − 1 D l D k Γ lk − D ( i (∆ + 2 K ) − 1 D k Γ j ) k  = 0 , (13) where we denote Γ j i = γ j k Γ ik . In these equations, (2) E and (2) P are the second-or der pertur bations of the energy density and the pressure, resp ectively . F urther, D i (2) v and (2) V i are the scalar - and the vector-parts 2 of the spatial comp onents of the c ov a riant fluid four-velo city , in these equations. Γ 0 , Γ i , and Γ ij are the collections of the quadratic terms o f the linear - order p erturbations in the second-o rder Einstein equations and these can b e r e garded as the s ource terms in the s econd-orde r E instein equa tions. The ex plicit form of these so urce terms are given in Refs. [4, 7]. First-order p erturbations o f the Einstein equa tions are given by the r eplacements (2) Φ → (1) Φ , (2) Ψ → (1) Ψ , (2) ν i → (1) ν i , (2) χ ij → (1) χ ij , (2) E → (1) E , (2) P → (1) P , D i (2) v → D i (1) v , (2) V i → (1) V i , and Γ 0 = Γ i = Γ ij = 0. 4 Consistency with the equations of motion for matter fi eld Now, we consider the second-o rder p erturbatio n of the energ y contin uity equation a nd the E uler equatio ns. In terms of ga uge-inv ariant v ar iables, the second-order p ertur ba tions o f the energy contin uity equatio n and the Euler equation for a single per fect fluid are given by[5] a (2) C ( p ) 0 := ∂ η (2) E +3 H  (2) E + (2) P  + ( ǫ + p )  ∆ (2) v − 3 ∂ η (2) Ψ  − Ξ 0 = 0 , (14) (2) C ( p ) i := ( ǫ + p ) ( ( ∂ η + H ) D i (2) v + (2) V i ! + D i (2) Φ ) + D i (2) P + ∂ η p D i (2) v + (2) V i ! − Ξ ( p ) i = 0 , (15 ) where Ξ 0 and Ξ ( p ) i are the collection of the quadratic terms of the linear order p erturbations and its explicit forms are given in Ref. [5 , 7]. T o confirm the consistency of the background and the p erturbations of the Einstein equation a nd the energy contin uity equatio n (14), w e first substitute the seco nd-order E instein equations (6)–(8), a nd (10) int o E q. (14). F or simplicity , we first imp ose the first-order version of E q. (9) on all equations. Then, w e obtain 4 π Ga 3 (2) C ( p ) 0 = − ∂ η (2) ( p ) E (1) −H (2) ( p ) E (1) − 3 H (2) ( p ) E (2) + D i (2) ( p ) E (4) i + 3 2 3 (0) ( p ) E (1) − (0) ( p ) E (2) ! ∂ η (2) Ψ − ∂ η Γ 0 − H Γ 0 − 1 2 H Γ k k + 1 2 D k Γ k − 4 π Ga 2 Ξ 0 . (16) This equa tion shows that the second-order p ertur bation (1 4) of the ener g y co ntin uity equation is consis- ten t with the second-o rder a nd the background Einstein eq uations if the equation 4 π Ga 2 Ξ 0 + ( ∂ η + H ) Γ 0 + 1 2 H Γ k k − 1 2 D k Γ k = 0 (17) is satisfied under the ba ckground, the first-o rder E instein equations. Actually , through the ba ckground Einstein equations (6) a nd the first-order version of the Einstein equations (7)–(13), we ca n e a sily see that Eq. (17) is satis fie d under the Eins tein equations of the background and o f the first order [7]. Next, we consider the s econd-order pertur ba tions o f the Euler equations. F or simplicit y , we first impo se the fir s t-order version of Eq. (9) on all equations, aga in. Through the background Einstein equations (6) and the Einstein equatio ns of the second order (8)–(10), we ca n obtain 8 π Ga 2 (2) C ( p ) i = − 8 π Ga 3 (0) C ( p ) 0 D i (2) v + (2) V i ! − D i (2) Φ 3 (0) ( p ) E (1) − (0) ( p ) E (2) ! − 2 D i (2) ( p ) E (2) − 2 3 D i (∆ + 3 K ) (2) ( p ) E (3) + 1 2 ( ∂ η + 2 H ) +4 (2) ( p ) E (4) i − (2) ( p ) E (5) i ! + 1 2 a 2 (∆ + 2 K ) (2) ( p ) E (6) i − 8 π Ga 2 Ξ ( p ) j + ( ∂ η + 2 H ) Γ j − D l Γ j l . (18) This equation shows that the second-or der p er turbations of the Euler equa tions is consistent with the Einstein equations of the background a nd the second orde r if the equa tio n ( ∂ η + 2 H ) Γ j − D l Γ j l − 8 π Ga 2 Ξ ( p ) j = 0 (19) 3 is satisfie d under the E instein equa tio ns of the background a nd the firs t order . Actually , we can easily confirm E q . (19) due to the background Einstein equations a nd the first-or der per turbations o f the Einstein equations[7], and implies that the s econd-or de r per turbation of the Euler equation is cons istent with the set of the background, the first-order , and the seco nd-order Einstein equations. The co ns istency of equations for per turbations s hown her e is just a well-kno wn result, i.e., the E instein equation includes the equa tio ns of motion for matter field due to the Bianchi iden tit y . How ever, the ab ov e verification of the identities (17) and (19) implies that our derived second-or der p e rturbations of the E ins tein equa tion, the equation of contin uity , and the Euler equation are co nsistent. In this sense, we may say that the derived se c ond-order E ins tein equations, esp e cially , the derived formulae for the source terms Γ 0 , Γ i , Γ ij , Ξ 0 , and Ξ i in Ref. [7] are cor r ect. 5 Summary In summary , we show the all comp onents o f the second-or de r p erturbation of the Einstein equation without igno r ing any mo des of p er turbation in the ca se of a pe r fect fluid. The deriv ation is ba sed o n the general framework of the second-o rder ga ug e-inv arian t p erturba tion theory develop ed in Refs. [8]. In this formulation, any ga uge fixing is not necessar y and we can obtain any equation in the g auge-inv ariant form which is equiv alent to the c o mplete ga uge fixing. In other words, our formulation gives complete gauge fixed equations without any gauge fixing. Therefore , equations which ar e obtained in g auge-inv ariant manner cannot b e reduced without physical re s trictions any more. In this sens e, these equations are irreducible. This is o ne of the adv antages of the ga uge-inv ariant p erturbation theor y . W e hav e also chec ked the consistency o f the set of equations of the second-o rder p erturbatio n of the Einstein equations and the evolution equation of the matter field in the cases of a p erfect fluid. Therefore, in the case of the sing le matter field, we may say that we hav e b een re a dy to clarify the physical b ehaviors of the second-order cosmolo gical p erturbations . The physical behavior of the second-o rder p er turbations in the universe filled with a sing le matter field will b e instructive to c la rify those of the second-order per turbations in mor e realistic cos mological situations. W e leav e these issues as future works. References [1] C.L. Be nnett et al., Astrophys. J . Suppl. Ser. 148 , (2003), 1. [2] E. Komatsu et al., arXiv:08 03.05 47 [a stro-ph], (2008). [3] V. Acquaviv a, N. Bartolo, S. Mata rrese, and A. Riotto, Nucl. Phys. B 667 (2003 ), 119; J. Maldacena , JHEP , 0305 (2 003), 0 13; K . A. Malik and D. W ands, Class . 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