Inclusion of the first-order vector- and tensor-modes in the second-order gauge-invariant cosmological perturbation theory

Inclusion of the first-order v ector- and tensor-mo des in the second-order gauge-in v arian t c osmo logical p erturbation theory Kouji Nak amura 1 Dep artment of Astr onomic al Scienc e, t h e Gr aduate University for A dvanc e d Studies, Mitaka 181-8588, Jap an Abstract Gauge-inv ariant treatments of the second- o rder cosmological pertu rbation in a four dimensional homogeneous isotropic universe are formula ted without any gauge fixing. W e hav e deriv ed the Einstein equations in the case of the single p erfect fluid without ignoring any mod es . These equations imply that any typ es of mode-coupling arise due to the second-order effects of the Einstein equations. The second-order general relativis tic cosmolo gical per turbation theory has v ery wide ph ysic a l moti- v ation. In particular , the first or der approximation of our univ erse from a homogeneous isotropic one is revealed by the recent obs e r v ations of Co s mic Microw av e Background (CMB) by Wilkinso n Microwa ve Anisotropy Pr obe[1], whic h sugg ests that the fluctuations of our universe are adiabatic a nd Gaussia n at least in the first o rder a ppro ximation. One of the next theor etical re searc hes is to clarify the accuracy o f this result through the non-Gaussia nit y , or non-adiabaticity , a nd s o on. T o ca rry out this, it is necessary to dis cuss the second- order cosmologica l p erturbations. How ever, general relativistic p erturbation theory req uires delicate treatments of “ gauges” and this situation becomes clea rer by the general a rgumen ts of per turbation theorie s . Therefore, it is worthwhile to formulate the higher -order gaug e-in v a rian t pe r turbation theory fr om gener al p oin t o f view. Accor ding to this motiv a tion, we prop osed the general framework o f the second-order g auge-in v ariant p erturbation theory on a g eneric background spacetime[2]. This ge neral framework was applied to cosmologica l p er- turbation theory[3] and all comp o nen ts of the second-or der per tur bation of the Einstein equation were derived in gauge inv a rian t manner. The der iv ed second-or der E instein equations are quite similar to the equations for the first-order o ne but there are s ource terms which consist of the quadratic terms o f the linear-or der p erturbations. In this ar ticle, we show the extension of the form ulation in Refs. [3] to include the firs t-order vector- and tensor-mo des in the so urce terms o f the second-or der Einstein equation, whic h were ig nored in Refs. [3]. As emphasized in Refs.[2, 3], in any p erturbation theory , we alwa ys treat t w o spacetime manifolds. One is a physical spa cetime M λ and the other is the background spa cetime M 0 . In this article, the background spacetime M 0 is the F riedmann-Rob ertson-W alker univ erse filled with a p erfect fluid whose metric is g iv en by g ab = a 2 ( η )  − ( dη ) a ( dη ) b + γ ij ( dx i ) a ( dx j ) b  , (1) where γ ij is the metr ic o n max ima lly s ymmetric three space. The physical v a riable Q on the physical spacetime is pulled ba c k to X Q on the background spa cetime by a n a ppropriate gauge c hoice X whic h is an po in t-identification ma p from M 0 to M λ . The gauge transfor mation rules for the pulled-back v a riable X Q , which is expanded a s X Q λ = Q 0 + λ (1) X Q + 1 2 λ 2 (2) X Q , are g iv en b y (1) Y Q − (1) X Q = £ ξ (1) Q 0 , (2) Y Q − (2) X Q = 2 £ ξ (1) (1) X Q + n £ ξ (2) + £ 2 ξ (1) o Q 0 , (2) where X and Y represent tw o differen t gauge choices, ξ a (1) and ξ a (2) are generator s of the first- a nd the second-or der g a uge transformations, respec tively . The metric ¯ g ab on the physical spacetime M λ is a lso expanded as ¯ g ab = g ab + λh ab + λ 2 2 l ab under a g auge choice. Ins p ecting gauge transfor ma tion rules (2), the first-order metric pertur bation h ab is decompo s ed as h ab =: H ab + £ X g ab , where H ab and X a 1 E-mail:kouc han@th.nao.ac.jp 1 are transfo rmed as Y H ab − X H ab = 0, a nd Y X a − X X a = ξ (1) a under the g auge transformation (2), resp ectiv ely[3]. The gauge inv aria n t pa rt H ab of h ab is g iv en in the form H ab = − 2 a 2 (1) Φ ( dη ) a ( dη ) b + 2 a 2 (1) ν i ( dη ) ( a ( dx i ) b ) + a 2  − 2 (1) Ψ γ ij + (1) χ ij  ( dx i ) a ( dx j ) b , (3) where D i (1) ν i = (1) χ [ ij ] = (1) χ i i = D i (1) χ ij = 0 and D i := γ ij D j is the cov ariant deriv a tiv e asso ciate with the metric γ ij . In the cosmo logical p erturbations[5], { (1) Φ , (1) Ψ } , (1) ν i , and (1) χ ij are called the s calar-, vector- , and tensor -modes, resp ectiv ely . W e have to note tha t we us ed the exis tence of the Green functions ∆ − 1 =: ( D i D i ) − 1 , (∆ + 2 K ) − 1 , and (∆ + 3 K ) − 1 to accomplish the ab o ve decomp osition of h ab . As shown in Ref.[2], thro ug h the a bov e v ariables X a and h ab , the second order metric perturbatio n l ab is decomp osed as l ab =: L ab + 2 £ X h ab +  £ Y − £ 2 X  g ab The v ariables L ab and Y a are the gaug e inv ariant and v ar ian t parts of l ab , re s pectively . The vector field Y a is transfor med as Y Y a − X Y a = ξ a (2) + [ ξ (1) , X ] a under the ga uge transfor mations (2 ). The comp onen ts of L ab are g iv en by L ab = − 2 a 2 (2) Φ ( dη ) a ( dη ) b + 2 a 2 (2) ν i ( dη ) ( a ( dx i ) b ) + a 2  − 2 (2) Ψ γ ij + (2) χ ij  ( dx i ) a ( dx j ) b , (4) where D i (2) ν i = (2) χ [ ij ] = (2) χ i i = D i (2) χ ij = 0. As s ho wn in Ref.[2], by using the a bov e v aria bles X a and Y a , we can find the gaug e inv aria n t v ariables for the p erturbations o f an arbitrar y field as (1) Q := (1) Q − £ X Q 0 , , (2) Q := (2) Q − 2 £ X (1) Q −  £ Y − £ 2 X  Q 0 . (5) As the matter conten ts, in this ar ticle , we consider a perfect fluid whose ener gy-momen tum tensor is given by ¯ T b a = (¯ ǫ + ¯ p ) ¯ u a ¯ u b + ¯ pδ b a . W e expand these fluid comp onent s ¯ ǫ , ¯ p , and ¯ u a as ¯ ǫ = ǫ + λ (1) ǫ + 1 2 λ 2 (2) ǫ , ¯ p = p + λ (1) p + 1 2 λ 2 (2) p , ¯ u a = u a + λ (1) u a + 1 2 λ 2 (2) u a p. (6) F ollowing the definitions (5), w e easily obtain the co rrespo nding gaug e inv ariant v a riables for these per turbations of the fluid co mponents: (1) E := (1) ǫ − £ X ǫ, (1) P := (1) p − £ X p, (1) U a := (1) ( u a ) − £ X u a , (2) E := (2) ǫ − 2 £ X (1) ǫ −  £ Y − £ 2 X  ǫ, (2) P := (2) p − 2 £ X (1) p −  £ Y − £ 2 X  p, (2) U a := (2) ( u a ) − 2 £ X (1) u a −  £ Y − £ 2 X  u a . Through ¯ g ab ¯ u a ¯ u b = g ab u a u b = − 1, the comp onen ts o f (1) U a and (2) U a are g iv en by (1) U a = − a (1) Φ ( dη ) a + a D i (1) v + (1) V i ! ( dx i ) a , (2) U a = (2) U η ( dη ) a + a D i (2) v + (2) V i ! ( dx i ) a , (7) (2) U η := a (  (1) Φ  2 − (2) Φ − D i (1) v + (1) V i − (1) ν i ! D i (1) v + (1) V i − (1) ν i !) (8) where D i (1) V i = D i (2) V i = 0. W e also expand the Einstein tensor as ¯ G b a = G b a + λ (1) G b a + 1 2 λ 2 (2) G b a . F r om the decomp osition of the first- and the second-order metric p erturbation in to g auge-in v ariant pa rts and g auge-v ariant parts, each order per tur bation of the Einstein tenso r is given by (1) G b a = (1) G b a [ H ] + £ X G b a , (2) G b a = (1) G b a [ L ] + (2) G b a [ H , H ] + 2 £ X (1) G b a +  £ Y − £ 2 X  G b a (9) as exp ected from Eqs . (5 ). Her e, (1) G b a [ H ] and (1) G b a [ L ] + (2) G b a [ H , H ] are gauge inv aria n t parts o f the first- and the se c o nd- order perturba tions of the Einstein tensor, r espectively . On the other hand, the 2 energy moment um tensor of the p erfect fluid is also expanded as ¯ T b a = T b a + λ (1) T b a + 1 2 λ 2 (2) T b a and (1) T b a and (2) T b a are a lso given in the form (1) T b a = (1) T b a + £ X T b a , (2) T b a = (2) T b a + 2 £ X (1) T b a +  £ Y − £ 2 X  T b a (10) through the definitions (7) o f the gauge inv ar ian t v ariables o f the fluid comp onen ts. Here, (1) T b a and (2) T b a are gaug e inv a r ian t par t o f the first- and the second-or der p erturbations of the energy momentum tensor, re s pectively . Then, the first- and the second-o rder p erturbations of the E instein equation a re necessarily given in term o f gaug e in v a rian t v ariables : (1) G b a [ H ] = 8 π G (1) T b a , (1) G b a [ L ] + (2) G b a [ H , H ] = 8 π G (2) T b a . (11) In the single p erfect fluid case, the traceles s sca la r part of the spatial comp onen t of the firs t equa tion in Eq.(11) yields (1) Ψ = (1) Φ due to the absence of the anisotro pic stress in the first o rder p erturbation of the energy moment um tensor and the other comp onent s of Eq. (11) give well-known equations[5]. W e show the express ion of the se c o nd-order p erturbations of the Einstein equa tion after imp osing these first-or der per turbations of the Einstein equations. Though we have derived a ll compo ne nts o f the seco nd e q uation in Eq. (1 1 ), w e only show their scalar parts of it for simplicit y: 4 π Ga 2 (2) E =  − 3 H ∂ η + ∆ + 3 K − 3 H 2  (2) Φ − Γ 0 + 3 2  ∆ − 1 D i D j Γ j i − 1 3 Γ k k  − 9 2 H ∂ η (∆ + 3 K ) − 1  ∆ − 1 D i D j Γ j i − 1 3 Γ k k  , (12) 8 π Ga 2 ( ǫ + p ) D i (2) v = − 2 ∂ η D i (2) Φ − 2 H D i (2) Φ + D i ∆ − 1 D k Γ k − 3 ∂ η D i (∆ + 3 K ) − 1  ∆ − 1 D i D j Γ j i − 1 3 Γ k k  , (13) 4 π Ga 2 (2) P =  ∂ 2 η + 3 H ∂ η − K + 2 ∂ η H + H 2  (2) Φ − 1 2 ∆ − 1 D i D j Γ j i + 3 2  ∂ 2 η + 2 H ∂ η  (∆ + 3 K ) − 1  ∆ − 1 D i D j Γ j i − 1 3 Γ k k  , (14) (2) Ψ − (2) Φ = 3 2 (∆ + 3 K ) − 1  ∆ − 1 D i D j Γ j i − 1 3 Γ k k  . (15) where H := ∂ η a/a . Γ 0 , Γ i and Γ ij in Eqs. (12)-(15) ar e defined b y Γ 0 := +8 π Ga 2 ( ǫ + p ) D i (1) v D i (1) v − 3 D k (1) Φ D k (1) Φ − 8 (1) Φ ∆ (1) Φ − 3  ∂ η (1) Φ  2 − 12  K + H 2   (1) Φ  2 − 4  ∂ η D i (1) Φ + H D i (1) Φ  (1) V i − 2 H D k (1) Φ (1) ν k +8 π Ga 2 ( ǫ + p ) (1) V i (1) V i + 1 2 D k (1) ν l D ( k (1) ν l ) +3 H 2 (1) ν k (1) ν k + D l D k (1) Φ (1) χ lk − 2 H D k (1) ν l (1) χ kl − 1 2 D k (1) ν l ∂ η (1) χ lk + 1 8 ∂ η (1) χ lk ∂ η (1) χ kl + H (1) χ kl ∂ η (1) χ lk − 1 8 D k (1) χ lm D k (1) χ ml + 1 2 D k (1) χ lm D [ l (1) χ k ] m − 1 2 (1) χ lm (∆ − K ) (1) χ lm , Γ i := − 16 π Ga 2  (1) E + (1) P  D i (1) v +12 H (1) Φ D i (1) Φ − 4 (1) Φ ∂ η D i (1) Φ − 4 ∂ η (1) Φ D i (1) Φ − 16 π Ga 2  (1) E + (1) P  (1) V i − 2 D j (1) Φ D i (1) ν j +2 D i D j (1) Φ (1) ν j +2∆ (1) Φ (1) ν i + (1) Φ ∆ (1) ν i +2 K (1) Φ (1) ν i − 4 H (1) ν j D i (1) ν j +2 D j (1) Φ ∂ η (1) χ j i − 2 ∂ η D j (1) Ψ (1) χ ij +2 D k D [ i (1) ν m ] (1) χ km +2 D [ k (1) ν j ] D j (1) χ ki +2 K (1) ν j (1) χ ij − (1) ν j ∆ (1) χ j i − 1 2 ∂ η (1) χ j k D i (1) χ kj +2 (1) χ kj ∂ η D [ j (1) χ i ] k , (1 6) 3 Γ ij := 16 π Ga 2 ( ǫ + p ) D i (1) v D j (1) v − 4 D i (1) Φ D j (1) Φ − 8 (1) Φ D i D j (1) Φ + ( 6 D k (1) Φ D k (1) Φ +8 (1) Φ ∆ (1) Φ +2  ∂ η (1) Φ  2 + 16 H (1) Φ ∂ η (1) Φ +8  2 ∂ η H + K + H 2   (1) Φ  2 ) γ ij +32 π Ga 2 ( ǫ + p ) D ( i (1) v (1) V j ) − 4 ∂ η (1) Φ D ( i (1) ν j ) +4 ∂ η D ( i (1) Φ (1) ν j ) + 4 ∂ η D k (1) Φ (1) ν k +4 H D k (1) Φ (1) ν k ! γ ij +16 π Ga 2 ( ǫ + p ) (1) V i (1) V j − 2 (1) ν k D k D ( i (1) ν j ) +2 (1) ν k D i D j (1) ν k + D i (1) ν k D j (1) ν k + D k (1) ν i D k (1) ν j + − D k (1) ν l D [ k (1) ν l ] − D k (1) ν l D k (1) ν l − 2 (1) ν k ∆ (1) ν k − 4 ∂ η H (1) ν k (1) ν k +6 H 2 (1) ν k (1) ν k ! γ ij − 4 H ∂ η (1) Φ (1) χ ij − 2 ∂ 2 η (1) Φ (1) χ ij − 4 D k (1) Φ D ( i (1) χ j ) k +4 D k (1) Φ D k (1) χ ij − 8 K (1) Φ (1) χ ij +4 (1) Φ ∆ (1) χ ij − 4 D k D ( i (1) Φ (1) χ j ) k +2∆ (1) Φ (1) χ ij +2 D l D k (1) Φ (1) χ lk γ ij − 2 D k (1) ν ( i ∂ η (1) χ j ) k − 2 (1) ν k ∂ η D ( i (1) χ j ) k +2 (1) ν k ∂ η D k (1) χ ij + D k (1) ν l ∂ η (1) χ lk γ ij + ∂ η (1) χ ik ∂ η (1) χ k j +2 D [ l (1) χ k ] i D k (1) χ l j − 1 2 D j (1) χ lk D i (1) χ lk − (1) χ lm D i D j (1) χ ml +2 (1) χ lm D l D ( i (1) χ j ) m − (1) χ lm D m D l (1) χ ij + − 3 4 ∂ η (1) χ lk ∂ η (1) χ kl + 3 4 D k (1) χ lm D k (1) χ ml − 1 2 D k (1) χ lm D l (1) χ mk + K (1) χ lm (1) χ lm ! γ ij , and Γ j i := γ j k Γ ik . These e q uations (12)-(15) coincide with the equations derived in Refs.[3] ex cept fo r the definition of the so urce ter ms Γ 0 , Γ i , and Γ ij . F urther, as sho wn in Refs.[3], the equations (12 ) a nd (15) a re reduced to the single equation for (2) Φ . W e also derived the s imilar equations in the case where the matter co n tent o f the univ erse is a single scalar fie ld[4 ]. In summary , we ha ve extended our formulation witho ut ignoring the first-order v ector- and tensor- mo des. As the result, these equations imply that any types of mo de-coupling arise due to the seco nd-order effects o f the E instein equations, in principle. In some inflationary s c enario, the tensor mo de ar e also generated by the quantum fluctuations. This extension will be useful to clarify the evolution o f the seco nd order pe r turbation in the ex istence of the first- o rder tensor -mode. F urther, to apply this formulation to clarify the non-linea r effects in CMB ph ysics[6], we have to extend our formulation to m ulti-field system and to the Einstein-B oltzmann system. These extensions will b e one o f our future w orks. References [1] C.L. Bennett et al., Astroph ys. J. Suppl. Ser. 148 , (2003), 1 . [2] K. Nak amura, Pr o g. Theor. Phys. 110 (2003), 723; K. Nak am ura, Prog. Theor . P h ys. 113 (2005), 481. [3] K. Nak amura, P h ys . Rev. D 74 (2006 ), 101301 (R) ; K. Nak amura, Pro g. Theor. Phys. 117 (2007), 17. [4] K. Nak amura, in prepara tion. [5] J. M. Bardeen, Phys. Rev. D 22 (1 980), 1882; H. Ko dama a nd M. Sa s aki, Pro g. Theor. Phys. Suppl. No.78, 1 (1984); V. F. Mukhanov, H. A. F e ldman a nd R. H. Brandenberg er, Phys. Rep. 215 , 2 03 (1992). [6] N. Ba rtolo, S. Matarres e a nd A. Riotto, JCAP 0401 , 003 (20 04); N. Ba rtolo, S. Matarres e and A. Riotto, P h ys. Rev. Lett. 93 (2004), 23 1301; N. Bartolo, E. K omatsu, S. Matarrese and A. Riotto, Phys. Rept. 402 , 10 3 (2004 ); N. Bar tolo, S. Matarrese, and A. Riotto, arXiv:astro- ph/ 0512481 . 4