Propagation of Gravitational Waves in Chern-Simons Axion F(R) Gravity
Shin'ichi Nojiri, S.D. Odintsov, V.K. Oikonomou, Arkady A. Popov
aa r X i v : . [ g r- q c ] F e b Propagation of Gravitational Waves in Chern-Simons Axion F ( R ) Gravity
Shin’ichi Nojiri, , S.D. Odintsov, , , V.K. Oikonomou, , Arkady A. Popov, Department of Physics, Nagoya University, Nagoya 464-8602, Japan Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain Institute of Space Sciences (IEEC-CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain Tomsk State Pedagogical University, 634061 Tomsk, Russia Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece International Laboratory for Theoretical Cosmology,Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia N. I. Lobachevsky Institute of Mathematics and Mechanics,Kazan Federal University, 420008, Kremlevskaya street 18, Kazan, Russia
In this paper we shall study the evolution of cosmological gravitational waves in the context ofChern-Simons axion F ( R ) gravity. In the case of Chern-Simons axion F ( R ) gravity there existspin-0, spin-2 and spin-1 modes. As we demonstrate, from all the gravitational waves modes of theChern-Simons axion F ( R ) gravity, only the two tensor modes are affected, while the spin-0 andspin-1 modes are not affected at all. With regard to the two tensor modes, we show that thesemodes propagate in a non-equivalent way, so the resulting tensor modes are chiral. Notably, withregard to the propagation of the spin-2 graviton modes, the structure of the dispersion relationsbecomes more complicated in comparison with the Einstein gravity with the Chern-Simons axion,but the resulting qualitative features of the propagating modes are not changed. With regard to thespin-0 and spin-1 modes, the Chern-Simons axion F ( R ) gravity contains two spin-0 modes and novector spin-1 mode at all. We also find that for the very high energy mode, both the group velocityand the phase velocity are proportional to the inverse of the square root of the wave number, andtherefore the velocities become smaller for larger wave numbers or even vanish in the limit that thewave number goes to infinity. PACS numbers: 04.50.Kd, 95.36.+x, 98.80.-k, 98.80.Cq,11.25.-w
I. INTRODUCTION
The presence of a dark component of matter in the Universe was assumed early after the first galactic rotation curvesappeared. Since then, theoretical physics studies were focused on proposing massive particles weakly interacting withluminous matter, the so-called WIMPs (Weakly Interacting Massive Particles), and there exist examples coming fromvarious theoretical contexts, see for example [1–6]. To be honest, for the moment only indications exist that supportthe particle nature of dark matter, such as the observational data from the bullet cluster. However, nearly two decadesof searches did not result in finding any WIMP. A crucial point to stress is that all the dark matter searches focusedin mass ranges from a few MeV up to hundreds of GeV’s. However, only recently the experimentalists focused theirinterest in searching WIMPs having masses a few eV or even sub-eV.String theory is up to date the most prominent theory that may describe in a consistent way the quantum theoryof gravity. One of the interesting predictions of string theory is the existence of low-mass axion [7–10] particles, otherthan the QCD axions. The axions are particularly appealing as dark matter candidates, since the mass range thatthese may have, is not investigated yet, and only the last 5 years experimentalists turned their focus on the axions.There is a plethora of experimental [11–19] and theoretical proposals related to the axions [20–68]. Axions can induceparticularly interesting effects in the phenomenology of gravitational theories, since Chern-Simons terms of the form U ( φ ) ˜ RR are allowed in the theory [69–82]. The Chern-Simons terms produce non-equivalent polarizations in thetensor modes of the underlying gravitational theory [83, 84], and in the literature there exist timely studies on chiralgravitational waves [85–92].In this paper we shall study a general Chern-Simons Axion F ( R ) gravity [93–99], by mainly focusing on thepossibility of generating non-equivalent polarizations in the tensor modes of the gravitational waves. We shall beinterested in the primordial gravity waves propagation, assuming the conservation of the gravitational wave in thelarge scale [84]. Our approach will mainly be quantitative, since we will extract the gravitational wave equations,and we shall study the tensor, vector and scalar modes. As we demonstrate, the situation with the spin-0 scalarmode and spin-1 vector mode, is not changed from the standard F ( R ) gravity coupled with quintessence-like typescalar field and as we demonstrate, the Chern-Simons axion term does not affect the propagation of these modes atall. Particularly, there exist two propagating scalar modes, and no propagating vector mode. With regard to thetwo tensor modes, the resulting dispersion relations are more complicated compared with the Einstein gravity withthe Chern-Simons axion term, but the qualitative features of the propagating modes are not changed, and actuallynon-equivalent propagation occur in both Einstein Chern-Simons and F ( R ) gravity Chern-Simons theories.This paper is organized as follows: In section II we present the general features of Chern-Simons Axion F ( R )gravity models. In section III we extract the general gravitational wave equations, while in section IV the variousgravitational wave modes, and the corresponding polarizations, are studied. Finally, the conclusions follow in the endof the paper. II. THE CHERN-SIMONS CORRECTED AXION F ( R ) GRAVITY
In this paper we shall mainly consider a Axionic Chern-Simons corrected F ( R ) gravity model, whose action is givenby, S = 12 κ Z d x √− g (cid:20) F ( R ) − ω ( φ )2 ∂ µ φ∂ µ φ − V ( φ ) + U ( φ )˜ ǫ µνρσ R τλµν R λτρσ (cid:21) , (1)where we defined the totally antisymmetric Levi-Civita symbols ǫ µνρσ and ǫ µνρσ as follows, ǫ = − ǫ = 1 , (2)and in addition, ǫ µνρσ = η µµ ′ η νν ′ η ρρ ′ η σσ ′ ǫ µ ′ ν ′ ρ ′ σ ′ . (3)Then we obtain the following tensors,˜ ǫ µνρσ ≡ √− g ǫ µνρσ , ≡ g µµ ′ g νν ′ g ρρ ′ g σσ ′ ˜ ǫ µ ′ ν ′ ρ ′ σ ′ = √− gǫ µνρσ . (4)We should note that the following tensor identity holds true, ∇ σ ˜ ǫ ζηρξ = 0 . (5)Since, δ (cid:0) √− gU ( φ )˜ ǫ µνρσ R τλµν R λτρσ (cid:1) = 2 √− gU ( φ ) h ˜ ǫ ζηρµ R τνζη + ˜ ǫ ζηρν R τµζη i ∇ ρ ∇ τ δg µν , (6)upon varying the action (1) with respect to the metric, we obtain the equations as follows,0 = 12 g µν F ( R ) − R µν F ′ ( R ) + ∇ µ ∇ ν F ′ ( R ) − g µν (cid:3) F ′ ( R )+ 12 (cid:26) − ω ( φ )2 ∂ µ φ∂ µ φ − V ( φ ) (cid:27) g µν + ω ( φ )2 ∂ µ φ∂ ν φ + 2 ( g µξ g νσ + g µσ g νξ ) ∇ τ ∇ ρ (cid:0) U ( φ )˜ ǫ ζηρξ R τσζη (cid:1) . (7)The equation obtained from the variation of the action with respect to the scalar field φ is given by,0 = ∇ µ ( ω ( φ ) ∂ µ φ ) − V ′ ( φ ) + U ′ ( φ ) ǫ µνρσ R τλµν R λτρσ . (8)We now assume that the geometric background is a Friedmann-Robertson-Walker (FRW) spacetime with flat spatialpart, ds = − dt + a ( t ) X i =1 , , (cid:0) dx i (cid:1) , (9)and we also assume that the scalar field φ depends solely on the cosmological time t . In the FRW background, weobtain, Γ tij = a Hδ ij , Γ ijt = Γ itj = Hδ ij , Γ ijk = ˜Γ ijk , R itjt = − (cid:16) ˙ H + H (cid:17) a δ ij , R ijkl = a H ( δ ik δ lj − δ il δ kj ) ,R tt = − (cid:16) ˙ H + H (cid:17) , R ij = a (cid:16) ˙ H + 3 H (cid:17) δ ij , R = 6 ˙ H + 12 H , other components = 0 , (10)and from these we obtain the following equations,0 = − F ( R ) + 3 (cid:16) H + ˙ H (cid:17) F ′ ( R ) − (cid:16) H ˙ H + H ¨ H (cid:17) F ′′ ( R ) + ω ( φ )4 ˙ φ + V ( φ )2 , F ( R ) − (cid:16) ˙ H + 3 H (cid:17) F ′ ( R ) + 24 (cid:16) H ˙ H + ˙ H + 2 H ¨ H (cid:17) F ′′ ( R ) + 36 (cid:16) H ˙ H + ¨ H (cid:17) F ′′′ ( R )+ ω ( φ )4 ˙ φ − V ( φ )2 . (11)Here R = 12 H + 6 ˙ H and we have neglected the contributions from matter perfect fluids. We should note that theterm containing the scalar coupling function to the Chern-Simons term, namely, U ( φ ), does not contribute to theabove equations in (11). We may choose φ to be the cosmological time t , that is, φ = t . Then the equations in (11)can be rewritten as, ω ( φ ) = − HF ′ ( R ) − (cid:16) H ˙ H + 12 ˙ H + 15 H ¨ H (cid:17) F ′′ ( R ) − (cid:16) H ˙ H + ¨ H (cid:17) F ′′′ ( R ) ,V ( φ ) = F ( R ) − (cid:16) H + 3 H (cid:17) F ′ ( R ) + (cid:16) H ˙ H + 24 ˙ H + 66 H ¨ H (cid:17) F ′′ ( R ) + 36 (cid:16) H ˙ H + ¨ H (cid:17) F ′′′ ( R ) . (12)Then if we choose, ω ( φ ) = − f ′ ( φ ) HF ′ ( R f ) − (cid:0) f ( φ ) f ′ ( φ ) + 12 f ′ ( φ ) + 15 f ( φ ) f ′′ ( φ ) (cid:1) F ′′ ( R f ) −
72 (4 f ( φ ) f ′ ( φ ) + f ′′ ( φ )) F ′′′ ( R f ) ,V ( φ ) = F ( R f ) − (cid:0) f ′ ( φ ) + 3 f ( φ ) (cid:1) F ′ ( R f ) + (cid:0) f ( φ ) f ′ ( φ ) + 24 f ′ ( φ ) + 66 f ( φ ) f ′′ ( φ ) (cid:1) F ′′ ( R f )+ 36 (4 f ( φ ) f ′′ ( φ ) + f ′′ ( φ )) F ′′′ ( R f ) ,R f ≡ f ( φ ) + 6 f ′ ( φ ) . (13)a solution of the equations in (11) is given by H = f ( t ) and φ = t . III. GRAVITATIONAL WAVE EQUATIONS
We now investigate the propagation of gravitational waves in the Chern-Simons Axion F ( R ) gravity. In order tostudy the propagation of the gravitational waves, we consider the perturbation of Eq. (7), from the background whosemetric is g (0) µν , g µν = g (0) µν + h µν , φ = φ (0) + ϕ . (14)Then we can find the equations corresponding to the perturbed Einstein equation, which are presented in the Appendixin Eq. (44) due to the extended analytic form these have. On the other hand, Eq. (8) gives,0 = − g (0) µν h µν ∇ (0) ρ (cid:16) ω (cid:16) φ (0) (cid:17) ∂ ρ φ (0) (cid:17) − ∇ (0) ν (cid:16) h νµ ω (cid:16) φ (0) (cid:17) ∂ µ φ (0) (cid:17) + 12 ∇ (0) ρ (cid:16) g (0) µν h µν ω (cid:16) φ (0) (cid:17) ∂ ρ φ (0) (cid:17) + 2 U ′ (cid:16) φ (0) (cid:17) h ˜ ǫ (0) ζηρµ R (0) τνζη + ˜ ǫ (0) ζηρν R (0) τµζη i ∇ (0) ρ ∇ (0) τ h µν − g (0) ηζ h ηζ U ′ (cid:16) φ (0) (cid:17) ˜ ǫ (0) µνρσ R (0) τλµν R (0) λτρσ + ∇ (0) µ (cid:16) ω ′ (cid:16) φ (0) (cid:17) ϕ∂ µ φ (0) (cid:17) + ∇ (0) µ (cid:16) ω (cid:16) φ (0) (cid:17) ∂ µ ϕ (cid:17) − V ′′ (cid:16) φ (0) (cid:17) ϕ + U ′′ (cid:16) φ (0) (cid:17) ϕ ˜ ǫ (0) µνρσ R (0) τλµν R (0) λτρσ . (15)The explicit form of the ( t, t ), ( i, j ), and ( t, i ) components of the modified Einstein equations δG µν = 0 from Eq. (44)in the FRW background (9) are given as follows, δG tt ≡ g (0) tµ δG µt = F ′ (cid:26) − (cid:16) ˙ H + 3 H (cid:17) h tt + H (cid:0) ∂ t h ii + 2 ∂ i h ti (cid:1) + 12 ∂ i ∂ j h ji − ∂ i ∂ i h jj (cid:27) + F ′′ n − H h tt + ¨ H (cid:2) H (cid:0) − h tt − h ii (cid:1) − ∂ t h ii − ∂ i h ti (cid:3) −
18 ˙ H h tt + ˙ H (cid:2) H (cid:0) − h tt − h ii (cid:1) + H (cid:0) − ∂ t h tt − ∂ t h ii − ∂ i h ti (cid:1) − ∂ t h ii − ∂ i ∂ t h ti + 9 ∂ i ∂ i h tt − ∂ i ∂ j h ji + 3 ∂ i ∂ i h jj i + 36 H h tt + H (cid:0) − ∂ t h tt − ∂ t h ii − ∂ i h ti (cid:1) + H (cid:0) − ∂ t h tt + 9 ∂ t h ii + 21 ∂ i ∂ i h tt − ∂ i ∂ t h ti − ∂ i ∂ j h ji + 9 ∂ i ∂ i h jj (cid:17) + H (cid:16) ∂ t h ii + 6 ∂ i ∂ t h ti − ∂ i ∂ i ∂ t h jj + 3 ∂ i ∂ j ∂ t h ji − ∂ i ∂ j ∂ j h ti (cid:17) − ∂ i ∂ i ∂ t h jj + ∂ i ∂ i ∂ k ∂ k h tt − ∂ j ∂ j ∂ i ∂ t h ti + ∂ j ∂ j ∂ k ∂ k h ii − ∂ l ∂ l ∂ j ∂ i h ij o + F ′′′ n −
54 ¨ H h tt + ¨ H h −
540 ˙
HHh tt − H h tt + H (cid:0) − ∂ t h tt + 72 ∂ t h ii + 72 ∂ i h ti (cid:1) + H (cid:16) ∂ t h ii + 36 ∂ i ∂ t h ti − ∂ i ∂ i h tt + 18 ∂ i ∂ j h ji − ∂ i ∂ i h jj (cid:17)i − H H h tt + ˙ H (cid:2) − H h tt + H (cid:0) − ∂ t h tt + 288 ∂ t h ii + 288 ∂ i h ti (cid:1) + H (cid:0) ∂ t h ii − ∂ i ∂ i h tt +144 ∂ i ∂ t h ti + 72 ∂ i ∂ j h ji − ∂ i ∂ i h jj (cid:17)i + 12 (cid:26) ω (cid:16) φ (0) (cid:17) (cid:16) ˙ φ (0) (cid:17) − V (cid:16) φ (0) (cid:17)(cid:27) h tt − ( ω ′ (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) ϕ + ω (cid:16) φ (0) (cid:17) ˙ φ (0) ˙ ϕ + V ′ (cid:16) φ (0) (cid:17) ϕ ) =0 , (16)We define the Levi-Civita symbol, which is totally antisymmetric tensor, in three dimensions as follows, ǫ xyz ≡ a , ǫ ijk ≡ g (0) il g (0) jm g (0) kn ǫ lmn = a − ǫ ijk . (17)Then we can obtain the non-zero components of the perturbed Einstein tensor. Specifically, the components δG ij arepresented in the Appendix in Eq. (45) due to their extended form. The corresponding δG ti components are, δG ti ≡ g (0) tµ δG µi = 2 ˙ Ua ǫ ikl (cid:0) ∂ k ∂ m ∂ t h lm + ∂ m ∂ m ∂ k h tl (cid:1) + F ′ (cid:26) H∂ i h tt + 12 ∂ i ∂ t h kk − ∂ k ∂ t h ik + 12 ∂ k ∂ k h ti − ∂ k ∂ i h tk (cid:27) + F ′′ n ¨ H (cid:0) − Hh ti + 3 ∂ i h tt (cid:1) −
24 ˙ H h ti + ˙ H (cid:2) − H h ti + 6 H∂ i h tt + ∂ i ∂ t (cid:0) h tt − h kk (cid:1) − ∂ i ∂ k h tk (cid:3) − H ∂ i h tt + H (cid:2) ∂ i ∂ t (cid:0) h tt + 4 h kk (cid:1) + 12 ∂ i ∂ k h tx (cid:3) + H (cid:2) ∂ i ∂ t (cid:0) h tt − h kk (cid:1) − ∂ i ∂ k ∂ k (cid:0) h tt + h kk (cid:1) + 3 ∂ i ∂ k ∂ l h kl + 2 ∂ i ∂ k ∂ t h tk (cid:3) − ∂ i ∂ t h kk + ∂ i ∂ k ∂ k ∂ t (cid:0) h tt + h ll (cid:1) − ∂ i ∂ k ∂ l ∂ t h kl − ∂ i ∂ k ∂ t h tk (cid:9) + F ′′′ n −
36 ¨ H h ti + ¨ H h ˙ H (cid:0) − Hh ti + 36 ∂ i h tt (cid:1) + 72 H ∂ i h tt + H∂ i ∂ t (cid:0) h tt − h kk (cid:1) − H∂ i ∂ k h tk − ∂ i ∂ t h kk + 6 ∂ i ∂ k ∂ k (cid:0) h tt + h ll (cid:1) − ∂ i ∂ k ∂ l h kl − ∂ i ∂ k ∂ t h tk (cid:3) + ˙ H (cid:0) − H h ti + 144 H∂ i h tt (cid:1) + ˙ H (cid:2) H ∂ i h tt + H (cid:0) ∂ i ∂ t (cid:0) h tt − h kk (cid:1) − ∂ i ∂ k h tk − H∂ i ∂ t h kk + H (cid:0) ∂ i ∂ k ∂ k (cid:0) h tt + h ll (cid:1) − ∂ i ∂ k ∂ l h kl − ∂ i ∂ k h tk (cid:1)(cid:3)(cid:9) + 12 ( − ω (cid:0) φ (0) (cid:1) ∂ µ φ (0) ∂ µ φ (0) − V (cid:16) φ (0) (cid:17)) h ti − ω (cid:0) φ (0) (cid:1) φ (0) ∂ i ϕ =0 . (18)We should note that all the terms coming from the last term in Eq. (44) identically vanish. A more explicit form ofEq. (15) is given by,0 = 12 g (0) µν h µν ( ∂ t + 3 H ) (cid:16) ω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) − ∂ ν (cid:16) h νt ω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) − Hh tt ∂ t (cid:16) ω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) −
12 ( ∂ t + 3 H ) (cid:16) g (0) µν h µν ω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) − ( ∂ t + 3 H ) (cid:16) ω ′ (cid:16) φ (0) (cid:17) ϕ ˙ φ (0) (cid:17) − ( ∂ t + 3 H ) (cid:16) ω (cid:16) φ (0) (cid:17) ˙ ϕ (cid:17) + a ω (cid:16) φ (0) (cid:17) ∂ k ∂ k ϕ − V ′′ (cid:16) φ (0) (cid:17) ϕ . (19) IV. POLARIZATIONS OF GRAVITATIONAL WAVES
The most important study in the Chern-Simons Axion F ( R ) gravity is related to the polarization modes of thegravitational waves. We now consider the following modes, • Spin 2 tensor mode ˆ h ij , h it = h tt = 0 , h ii = 0 , ∂ j h ij = 0 , ( i = x, y, z ) , ϕ = 0 . (20) • Spin 1 vector mode A i ≡ ∂ j h ji , ∂ i ∂ j h ij = ∂ i A i = 0 , h it = h tt = 0 , h ii = 0 , ( i, j = x, y, z ) , ϕ = 0 . (21) • Spin 0 scalar mode h = h ii . h ij = ∂ i ∂ j B − g (0) ij ∂ k ∂ k B , ϕ . (22)Then the tensor h ij can be decomposed as follows, h ij = ˆ h ij + 12 (cid:0) ∂ k ∂ k (cid:1) − ( ∂ i A j + ∂ j A i ) + 13 g (0) ij h + ∂ i ∂ j B − g (0) ij ∂ k ∂ k B , (23)which gives, B = 32 (cid:0) ∂ k ∂ k (cid:1) − ∂ i ∂ j h ij − (cid:0) ∂ k ∂ k (cid:1) − h or ∂ i ∂ j h ij = 23 (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h . (24)For all the above modes we considered, we have implicitly chosen the gauge condition h tµ = 0. A. Spin 2 tensor mode
Let us study in some detail the spin 2 tensor mode. In the case of spin 2 mode, we find that δG tt in (16) and δG ti in Eq. (18) trivially vanish. In addition δG tt = δG ti = 0 and Eq. (19) is also trivially satisfied. On the other hand, δG ij in (45) has the following form, δG ij = − ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) n(cid:16) H ˙ U + 2 ¨ U (cid:17) ∂ l ∂ t ˆ h nm + 2 ˙ U ∂ l ∂ t ˆ h nm o + F ′ (cid:26) − H∂ t ˆ h ij − ∂ t ˆ h ij + 12 ∂ k ∂ k ˆ h ij (cid:27) + F ′′ n ¨ H h − H ˆ h ij + 3 ∂ t ˆ h ij i −
24 ˙ H ˆ h ij + ˙ H h − H ˆ h ij + 12 ∂ t ˆ h ij io + F ′′′ n −
36 ¨ H ˆ h ij −
288 ¨ H ˙ HH ˆ h ij −
576 ˙ H H ˆ h ij o + 12 ( ω (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) − V (cid:16) φ (0) (cid:17)) ˆ h ij . (25)We should note that the obtained equation (25) is a second order differential equation with respect to the cosmic time t , although the original equation (7) or (45) is a fourth order difference equation. This is not curious, because the F ( R ) gravity action can be rewritten in the form of a scalar-tensor theory, that is, the rewritten action is the sum ofthe Einstein-Hilbert action and the action of the scalar field with potential. So in effect, the spin-two mode originatesfrom the Einstein-Hilbert part, which gives the standard Einstein equation, that is, the second order differentialequation. The existence of U -terms in (25) give the mixing of +-mode and x -mode and the dispersion relation ofthe left-polarized mode is different from that of the right-polarized mode (see also [100], for example). We shouldnote that there appear terms including the first derivative with respect to the cosmic time, ∂ t ˆ h ij , which generate anenhancement or dissipation of the gravitational wave. By using (11), we may rewrite (25) as follows, δG ij = − ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) n(cid:16) H ˙ U + 2 ¨ U (cid:17) ∂ l ∂ t ˆ h nm + 2 ˙ U ∂ l ∂ t ˆ h nm o − F ˆ h ij + F ′ (cid:26) − H∂ t ˆ h ij − ∂ t ˆ h ij + 12 ∂ k ∂ k ˆ h ij + (cid:16) ˙ H + 3 H (cid:17) ˆ h ij (cid:27) + F ′′ n H∂ t ˆ h ij + 12 ˙ H ∂ t ˆ h ij − (cid:16) H ¨ H + 168 H ˙ H + 48 ˙ H + 48 H ¨ H (cid:17) ˆ h ij o + F ′′′ n −
72 ¨ H −
576 ¨ H ˙ HH − H H o ˆ h ij , (26)Just for simplicity, we consider the case that ˙ U and H , and therefore, F , F ′ , F ′′ , and F ′′′ can be regarded as constants.Then Eq. (26) can be reduced as follows, δG ij = − ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) ˙ U n H∂ l ∂ t ˆ h nm + 2 ∂ l ∂ t ˆ h nm o − F ˆ h ij + F ′ (cid:26) − H∂ t ˆ h ij − ∂ t ˆ h ij + 12 ∂ k ∂ k ˆ h ij + 3 H ˆ h ij (cid:27) , (27)We consider the plane wave propagating in the z -direction with the wave number k and frequency ω , ˆ h ij = h (0) ij e − iωt + ikz with constants h (0) ij . Then Eq. (20) tells h (0) zj = h (0) iz = 0 and in effect we find, δG xx =2 ωk ˙ U { H − iω } h (0) xy + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0) xx ,δG yy = − ωk ˙ U { H − iω } h (0) yx + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0) yy ,δG xy =2 ωk ˙ U { H − iω } h (0) yy + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0) xy ,δG yx = − ωk ˙ U { H − iω } h (0) xx + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0) yx ,δG zi = δG iz = 0 . (28)Usually we consider the following two modes, namely the + mode where h (0)+ ≡ h (0) xx = − h (0) yy and the × modewhere h (0) × ≡ h (0) xy = h (0) yx . In terms of this mode, the non-trivial equations in (28) take the following forms,0 =2 ωk ˙ U { H − iω } h (0) × + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0)+ , − ωk ˙ U { H − iω } h (0)+ + (cid:20) − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27)(cid:21) h (0) × . (29)The above equations indicate that there should always be a mixing between the + mode and × mode and they shouldappear in the forms of the left-handed or right handed mode, where h (0)+ = ± h (0) × . The equations in (29) give also thefollowing dispersion relation,0 = ± ωk ˙ U { H − iω } − F + F ′ (cid:26) iωH + 12 ω − k + 3 H (cid:27) . (30)The terms including iω in (30) come from the terms including ∂ t ˆ h ij in (25), which generate the enhancement ordissipation of the gravitational wave. In the case of the Chern-Simons axion Einstein gravity [22], we have F ′ = 1 and F = R ∼ H . Therefore the qualitative structure of the dispersion relation and the left- and right-handed modesare not so changed from those corresponding to the Chern-Simons axion Einstein gravity. In case of the Chern-Simonsaxion F ( R ) gravity, F ( R ) depends on the time coordinate t and therefore the full solution of the gravitational wave in(27) becomes rather complicated. We should note that the polarization of the gravitational wave in the early Universealso affects the polarization of CMB, and specifically the E-mode and B-modes, see for example [101].We now investigate the dispersion relation (29) in more detail. Eq. (29) can be solved as, ω = − (cid:16) iF ′ H + δ LR k ˙ U H (cid:17) ± r F ′ k − F ′ H + 64 k ˙ U H + F ′ F + iδ LR (cid:16) kF ′ ˙ U H − k ˙ U F − k ˙ U F ′ k (cid:17) (cid:16) F ′ − δ LR ik ˙ U (cid:17) . (31)Here δ LR = ± k ≫ H and by neglecting thecontribution from the Chern-Simons term, that is, k ≪ F ′ ˙ U , we obtain ω ∼ ± k . Therefore the propagating speedof the gravitational wave is not changed. When ω is real number, ω should be positive and we should choose theplus sign + of ± in (31). We should note, however, that for the very high energy mode when ˙ U does not vanish,˙ U = 0, that is, the mode k ≫ H and k ≫ F ′ ˙ U , Eq. (30) has the form 0 = − δ LR iω k ˙ U − F ′ k and therefore wefind ω ∼ − δ LR i F ′ U ′ k , which is rather strange dispersion relation. By assuming that the real part of ω is positive weobtain ω = e ± π q(cid:12)(cid:12) F ′ U ′ k (cid:12)(cid:12) . Therefore there always appear an amplified and a decaying mode. Furthermore the groupvelocity v g and the phase velocity v p are given by v g ≡ dωdk ∝ √ k and v p ≡ ωk ∝ √ k , which become smaller for larger k and much smaller than the velocity of light. B. Spin 1 Vector Mode
In the case of the spin 1 mode, we find δG tt in (16) vanishes and Eq. (19) is also trivially satisfied, again, δG tt = 0.On the other hand, δG ti in (18) has the following form, δG ti = 2 ˙ Ua ǫ ikl ∂ k ∂ t A l − F ′ ∂ t A i = 0 , (32)and δG ij in Eq. (45) has the following form, δG ij = ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) h − n(cid:16) H ˙ U + 2 ¨ U (cid:17) ∂ l ∂ t h ( A ) nm + 2 ˙ U ∂ l ∂ t h ( A ) nm o − U ∂ l ∂ n A m i + F ′ (cid:26) − H∂ t h ( A ) ij − ∂ t h ( A ) ij + 12 ∂ k ∂ k h ( A ) ij − (cid:16) ∂ i A j + g (0) im g (0) jl ∂ l A m (cid:17)(cid:27) + F ′′ n ¨ H h − Hh ( A ) ij + 3 ∂ t h ( A ) ij i −
24 ˙ H h ( A ) ij + ˙ H h − H h ( A ) ij + 12 ∂ t h ( A ) ij io + F ′′′ n −
36 ¨ H h ( A ) ij −
288 ¨ H ˙ HHh ( A ) ij −
576 ˙ H H h ( A ) ij o + 12 ( ω (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) − V (cid:16) φ (0) (cid:17)) h ( A ) ij =0 , (33)where h ( A ) ij is, h ( A ) ij ≡ (cid:0) ∂ k ∂ k (cid:1) − ( ∂ i A j + ∂ j A i ) , (34)We now discuss the qualitative implications of Eq. (32). When U = 0 as in the standard F ( R ) gravity, we obtain˙ A i = 0. In effect, there is no time evolution of A i , or no propagating mode and therefore A i is determined by the initialconditions consistent with (33). When U = 0, since there is a rotational symmetry, we may consider the plane wavepropagating in the z -direction with the wave number k , A i = α i ( t )e ikz . Eq. (21) also indicates that A z = α z ( t ) = 0.Then the i = z component in Eq. (32) is trivially satisfied and we obtain the following non-trivial equations,0 = − ik ˙ Ua ∂ t (cid:0) a − α y ( t ) (cid:1) + 12 F ′ ˙ α x ( t ) , ik ˙ Ua ∂ t (cid:0) a − α x ( t ) (cid:1) + 12 F ′ ˙ α y ( t ) , (35)which can be rewritten in a matrix form as follows, ik ˙ Ua F ′ F ′ − ik ˙ Ua ! (cid:18) ˙ α x ˙ α y (cid:19) = 2 ik ˙ U Ha (cid:18) − α x α y (cid:19) . (36)As an example, we consider the case that H , ˙ Ua and F ′ are constant. Then by assuming α x = α (0) x e − iωt and α y = α (0) y e − iωt with constants ω , α (0) x and α (0) y , we obtain, ik ˙ Ua ( − iω + H ) F ′ F ′ − ik ˙ Ua ( − iω − H ) ! α (0) x α (0) y ! = 0 . (37)In order that Eq. (37) has non-trivial solutions for α (0) x and α (0) y , the determinant of the matrix should vanish, andthis constraint gives the following dispersion relation,0 = 4 k ˙ U a (cid:0) ω + H (cid:1) + 14 F ′ , (38)which indicates that ω must be purely imaginary and therefore there is no propagating mode. The above result istrue for the high frequency mode, even if H , ˙ Ua and F ′ are not constant. Therefore, there is no propagating mode ofSpin 1, which is consistent with the standard requirement coming from the general covariance. C. Spin 0 scalar mode
Let us now consider the spin-0 mode, which is also present in the pure F ( R ) gravity. For the spin-0 mode, we find, δG tt = 13 F ′ n(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h o + F ′′ n ¨ H [ − Hh − ∂ t h ] + ˙ H h − H h − H∂ t h − ∂ t h − (cid:0) ∂ k ∂ k (cid:1) B + 2 ∂ k ∂ k h i − H ∂ t h + H (cid:16) ∂ t h − (cid:0) ∂ k ∂ k (cid:1) B + 6 ∂ k ∂ k h (cid:17) + H (cid:16) ∂ t h + ( ∂ t + 2 H ) (cid:16) (cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)(cid:17) − ∂ i ∂ i ∂ t h − ∂ l ∂ l (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)(cid:27) + F ′′′ n ¨ H h H ∂ t h + 18 H (cid:16) ∂ t h + 12 (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)(cid:17)i + ˙ H (cid:20) H ∂ t h + 72 H (cid:18) ∂ t h + (cid:18) (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19) − ∂ i ∂ i h (cid:19)(cid:21) − ( ω ′ (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) ϕ + ω (cid:16) φ (0) (cid:17) ˙ φ (0) ˙ ϕ + V ′ (cid:16) φ (0) (cid:17) ϕ ) =0 , (39) δG ij = ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) h − n(cid:16) H ˙ U + 2 ¨ U (cid:17) ∂ l ∂ t h ( S ) nm + 2 ˙ U ∂ l ∂ t h ( S ) nm o +2 ˙ U (cid:16) ∂ k ∂ k ∂ l h ( S ) nm − ∂ l ∂ k ∂ n h ( S ) km (cid:17)i + F ′ (cid:26) H (cid:20) ∂ t hδ ij − ∂ t h ( S ) ij (cid:21) − ∂ t h ( S ) ij + (cid:18) ∂ t h − ∂ k ∂ k h + 23 (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19) δ ij + 12 ∂ k ∂ k h ( S ) ij + 12 ∂ i ∂ j h − (cid:16) ∂ i ∂ k h ( S ) kj + g (0) im g (0) jl ∂ l ∂ k h ( S ) km (cid:17)(cid:27) + F ′′ n ¨ H h − Hh ( S ) ij + ( − Hh + ∂ t h ) δ ij + 3 ∂ t h ( S ) ij i −
24 ˙ H h ( S ) ij + ˙ H h − H h ( S ) ij + 12 ∂ t h ( S ) ij + (cid:0) − H h − H∂ t h (cid:1) δ ij + (cid:16) ∂ t h + 2 ∂ k ∂ k h − (cid:0) ∂ k ∂ k (cid:1) B (cid:17) δ ij i − H hδ ij + H h ∂ t h + 2 ∂ k ∂ k h − (cid:0) ∂ k ∂ k (cid:1) B i δ ij + H (cid:20) ∂ i ∂ j ∂ t h + (cid:18) h − ∂ k ∂ k ∂ t h − ∂ t + 4 H ) (cid:18) (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19)(cid:19) δ ij (cid:21) + ∂ i ∂ j (cid:0) ∂ t h − ∂ k ∂ k h (cid:1) + (cid:0) ∂ t h − ∂ k ∂ k ∂ t h + ( ∂ t + 4 H ) (cid:18) (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19) + ∂ k ∂ k ∂ l ∂ l h − ∂ l ∂ l (cid:18) (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19)(cid:19) δ ij (cid:27) + F ′′′ n ... H h H∂ t h + 6 ∂ t h + 4 (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)i δ ij −
36 ¨ H h ( S ) ij + ¨ H h ˙ H (cid:16) − Hh ( S ) ij + 48 ∂ t h (cid:17) δ ij + 144 H ∂ t hδ ij + H (cid:16) ∂ t h + 8 (cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17) δ ij + (cid:18) ∂ t h −
12 ( ∂ t + 2 H ) (cid:18) ∂ k ∂ k h + (cid:18) (cid:0) ∂ k ∂ k (cid:1) B + 13 ∂ k ∂ k h (cid:19)(cid:19)(cid:19) δ ij (cid:21) + ˙ H h − H h ( S ) ij + (cid:16) H∂ t h + 24 ∂ t h + 16 (cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17) δ ij i + ˙ H h H ∂ t h + 240 H ∂ t h + H (cid:16) − (cid:0) ∂ k ∂ k (cid:1) B + 32 ∂ k ∂ k h (cid:17) +48 H∂ t h + 32 H∂ t (cid:16) − (cid:0) ∂ k ∂ k (cid:1) B + ∂ k ∂ k h (cid:17)i δ ij o + F ′′′′ n ¨ H h Hh + 36 ∂ t h + 24 (cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h i δ ij + ¨ H ˙ H h H h + 288 H∂ t h + 192 H (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)i δ ij + ˙ H h H h + 576 H ∂ t h + 384 H (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)i δ ij o + 12 ( ω (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) − V (cid:16) φ (0) (cid:17)) h ( S ) ij + 12 ( ω ′ (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) ϕ + ω (cid:16) φ (0) (cid:17) ˙ φ (0) ˙ ϕ − V ′ (cid:16) φ (0) (cid:17) ϕ ) δ ij =0 , (40) δG ti = 2 ˙ Ua ǫ ikl ∂ k ∂ m ∂ t h ( S ) lm + F ′ (cid:26) ∂ i ∂ t h − ∂ k ∂ t h ( S ) ik (cid:27) + F ′′ n − H∂ i ∂ t h + 4 H ∂ i ∂ t h + H h − ∂ i ∂ t h + 2 (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)i − ∂ i ∂ t h + 23 ∂ i ( ∂ t + 2 H ) ∂ k ∂ k ∂ t (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)(cid:27) + F ′′′ n ¨ H h − H∂ i ∂ t h kk − ∂ i ∂ t h − ∂ i ∂ k (cid:16)(cid:0) ∂ k ∂ k (cid:1) B + ∂ k ∂ k h (cid:17)i + ˙ H h − H h − H∂ i ∂ t h + 16 H (cid:16)(cid:0) ∂ k ∂ k (cid:1) B − ∂ k ∂ k h (cid:17)io − ω (cid:0) φ (0) (cid:1) φ (0) ∂ i ϕ =0 , (41)where h ( S ) ij is equal to, h ( S ) ij = 13 g (0) ij h + ∂ i ∂ j B − g (0) ij ∂ k ∂ k B . (42)Eq. (19) takes the following form,0 = 12 h ( ∂ t + 3 H ) (cid:16) ω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) −
12 ( ∂ t + 3 H ) (cid:16) hω (cid:16) φ (0) (cid:17) ˙ φ (0) (cid:17) − ( ∂ t + 3 H ) (cid:16) ω ′ (cid:16) φ (0) (cid:17) ϕ ˙ φ (0) (cid:17) − ( ∂ t + 3 H ) (cid:16) ω (cid:16) φ (0) (cid:17) ˙ ϕ (cid:17) + a ω (cid:16) φ (0) (cid:17) ∂ k ∂ k ϕ − V ′′ (cid:16) φ (0) (cid:17) ϕ . (43)Since the Chern-Simons term does not contribute to the above equations, as in the F ( R ) gravity with a scalar field,there appear two propagating scalar modes. It is notable, and expected though, that the Chern-Simons term affectssolely the tensor gravitational wave modes, and not the spin-0 mode.0 V. SUMMARY
In summary, we have investigated the gravitational wave in the context of Chern-Simons axion F ( R ) gravity. Forthe spin-0 scalar mode and the spin-1 vector mode, we demonstrated that for these modes, the situation is not changedfrom the standard F ( R ) gravity coupled with quintessence type scalar field, and in addition, the Chern-Simons axionterm does not affect the propagation of these modes. This result was also known for the case of Chern-Simonsaxion Einstein gravity, as it was shown in Ref. [83]. Actually, the Chern-Simons term does not affect the scalarperturbations at all, and it affects solely the tensor perturbations. As a result, we have two propagating scalar modesand no propagating vector mode. With regard to the propagation of the spin-2 graviton mode, the structure ofthe dispersion relations become more complicated in comparison with the Einstein gravity with the Chern-Simonsaxion, but the qualitative features of the propagating modes are not changed, and actually non-equivalent polarizationmodes occur in both Einstein Chern-Simons and F ( R ) gravity Chern-Simons theory. Our study is focused mainly onprimordial gravitational modes, so in a future work we shall address several related issues, such as the conservationof the amplitude of gravitational waves at large scales, and the effect of the function U ( φ ) and of the F ( R ) gravityitself on the polarization asymmetry of the primordial gravitational waves.Although the dispersion relation (30) tells that the qualitative structure of the tensor modes is not extensivelychanged in comparison to that corresponding to the Chern-Simons axion Einstein gravity, there appear rather strangebehaviors in the very high energy mode where k ≫ H and k ≫ F ′ ˙ U . In the mode the frequency ω is always complexand proportional to the square root of the wave number k . Therefore an amplified and a decaying gravitational wavemode always occur, and also both the group velocity and the phase velocity are proportional to √ k . Then both thegroup velocity and the propagating velocity become smaller for larger k , and even vanish in the limit of k → ∞ .Finally let us note that in the present work we have found two propagating scalar modes, with the one being thescalar mode which appears commonly in the context of higher derivative gravity [102], and with the other being thepseudo-scalar mode corresponding to the axion scalar. Usually the scalar mode does not mix with the pseudo-scalarmode, but if the parity symmetry is broken by the non-trivial value of the Chern-Simons term, a mixing can occur ingeneral. Acknowledgments
This work is supported by MINECO (Spain), FIS2016-76363-P, and by project 2017 SGR247 (AGAUR, Catalonia)(S.D.O). This work is also supported by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas“Cosmic Acceleration” No. 15H05890 (S.N.) and the JSPS Grant-in-Aid for Scientific Research (C) No. 18K03615(S.N.). The work of A.P. is performed according to the Russian Government Program of Competitive Growth ofKazan Federal University. The work of A.P. was also supported by the Russian Foundation for Basic Research GrantNo 19-02-00496.
Appendix: Detailed Form of Perturbed Einstein Tensor Components
In this Appendix we present the detailed form of the tensor expressions needed in the text, but have quite extendedform. The full expression of the perturbed Einstein tensor is, δG µν = 12 F (cid:16) R (0) (cid:17) h µν − (cid:16) ∇ (0) µ ∇ (0) ρ h νρ + ∇ (0) ν ∇ (0) ρ h µρ − (cid:3) (0) h µν − ∇ (0) µ ∇ (0) ν (cid:16) g (0) ρλ h ρλ (cid:17) − R (0) λ ρν µ h λρ + R (0) ρµ h ρν + R (0) ρν h ρµ (cid:17) F ′ (cid:16) R (0) (cid:17) + 12 g (0) µν F ′ (cid:16) R (0) (cid:17) (cid:16) − h ρσ R (0) ρσ + ∇ (0) ρ ∇ (0) σ h ρσ − (cid:3) (0) (cid:16) g (0) ρσ h ρσ (cid:17)(cid:17) + (cid:16) − R (0) µν + ∇ (0) µ ∇ (0) ν − g (0) µν (cid:3) (0) (cid:17) (cid:16) F ′′ (cid:16) R (0) (cid:17) (cid:16) − h ρσ R (0) ρσ + ∇ (0) ρ ∇ (0) σ h ρσ − (cid:3) (0) (cid:16) g (0) ρσ h ρσ (cid:17)(cid:17)(cid:17) + 12 g (0) κλ (cid:16) ∇ (0) µ h νλ + ∇ (0) ν h µλ − ∇ λ h µν (cid:17) ∂ κ F ′ (cid:16) R (0) (cid:17) + g (0) µν g (0) ρτ g (0) ση h ρσ ∇ (0) τ ∇ (0) ηF ′ (cid:16) R (0) (cid:17) − g (0) µν g (0) ρσ g (0) κλ (cid:16) ∇ (0) ρ h σλ + ∇ (0) σ h ρλ − ∇ λ h ρσ (cid:17) ∂ κ F ′ (cid:16) R (0) (cid:17) + 12 − ω (cid:0) φ (0) (cid:1) g (0) ρσ ∂ ρ φ (0) ∂ σ φ (0) − V (cid:16) φ (0) (cid:17)! h µν + ω (cid:0) φ (0) (cid:1) g (0) µν h ρσ ∂ ρ φ (0) ∂ σ φ (0)
1+ 2 (cid:16) h µξ g (0) νσ + h µσ g (0) νξ + g (0) µξ h νσ + g (0) µσ h νξ (cid:17) ˜ ǫ (0) ζηρξ ∇ (0) τ ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) τσζη (cid:17) + 2 (cid:16) g (0) µξ g (0) νσ + g (0) µσ g (0) νξ (cid:17) (cid:26) − g (0) αβ h αβ ˜ ǫ (0) ζηρξ ∇ (0) τ ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) τσζη (cid:17) + ˜ ǫ (0) ζηρξ h αβ ∇ (0) α ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) σβζη (cid:17) −
12 ˜ ǫ (0) ζηρξ g (0) τα ∇ (0) τ ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) g (0) σβ (cid:16) ∇ (0) ζ (cid:16) ∇ (0) η h αβ + ∇ (0) α h ηβ − ∇ (0) β h ηα (cid:17)(cid:17)(cid:17) −
12 ˜ ǫ (0) ζηρξ g (0) τα ∇ (0) τ (cid:16) U (cid:16) φ (0) (cid:17) (cid:16) g (0) σβ (cid:16) ∇ (0) ρ h βγ + ∇ (0) γ h ρβ − ∇ (0) β h ργ (cid:17) R (0) γαζη − g (0) γβ (cid:16) ∇ (0) ρ h βα + ∇ (0) α h ρβ − ∇ (0) β h ρα (cid:17) R (0) σγζη − g (0) γβ (cid:16) ∇ (0) ρ h βζ + ∇ (0) ζ h ρβ − ∇ (0) β h ρζ (cid:17) R (0) σαγη − g (0) γβ (cid:16) ∇ (0) ρ h βη + ∇ (0) η h ρβ − ∇ (0) β hg ρη (cid:17) R (0) σαζγ (cid:17)(cid:17) −
12 ˜ ǫ (0) ζηρξ g (0) τα (cid:16) − g (0) βγ (cid:16) ∇ (0) τ h γρ + ∇ (0) ρ h τγ − ∇ (0) γ h τρ (cid:17) ∇ (0) β (cid:16) U (cid:16) φ (0) (cid:17) R (0) σαζη (cid:17) + g (0) σγ (cid:16) ∇ (0) τ h γβ + ∇ (0) β h τγ − ∇ (0) γ h τβ (cid:17) ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) βαζη (cid:17) − g (0) βγ (cid:16) ∇ (0) τ h γα + ∇ (0) α h τγ − ∇ (0) γ h τα (cid:17) ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) σβζη (cid:17) − g (0) βγ (cid:16) ∇ (0) τ h γζ + ∇ (0) ζ h τγ − ∇ (0) γ h τζ (cid:17) ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) σαβη (cid:17) − g (0) βγ (cid:16) ∇ (0) τ h γη + ∇ (0) η h τγ − ∇ (0) γ h τη (cid:17) ∇ (0) ρ (cid:16) U (cid:16) φ (0) (cid:17) R (0) σαζβ (cid:17)(cid:17)o + ω (cid:0) φ (0) (cid:1) ∂ ρ φ (0) ∂ σ φ (0) h ρσ g (0) µν + 12 ( − ω (cid:0) φ (0) (cid:1) ∂ µ φ (0) ∂ µ φ (0) − V (cid:16) φ (0) (cid:17)) h µν + 12 ( − ω ′ (cid:0) φ (0) (cid:1) ∂ µ φ (0) ∂ µ φ (0) ϕ − ω (cid:16) φ (0) (cid:17) ∂ µ φ (0) ∂ µ ϕ − V ′ (cid:16) φ (0) (cid:17) ϕ ) g (0) µν + ω ′ (cid:0) φ (0) (cid:1) ∂ µ φ (0) ∂ ν φ (0) ϕ + ω (cid:0) φ (0) (cid:1) (cid:16) ∂ µ ϕ∂ ν φ (0) + ∂ µ φ (0) ∂ ν ϕ (cid:17) + 2 (cid:16) g (0) µξ g (0) νσ + g (0) µσ g (0) νξ (cid:17) ∇ (0) τ ∇ (0) ρ (cid:16) U ′ (cid:16) φ (0) (cid:17) ϕ ˜ ǫ (0) ζηρξ R (0) τσζη (cid:17) =0 . (44)Moreover, the non-zero components of the perturbed Einstein tensor are δG ij , δG ij ≡ g (0) iµ δG µj = ǫ klm (cid:16) δ ik g (0) jn + δ in g (0) jk (cid:17) h − n(cid:16) H ˙ U + 2 ¨ U (cid:17) ∂ l ∂ t h nm + 2 ˙ U ∂ l ∂ t h nm o + 2 ¨ U ∂ l ∂ n h tm +2 ˙ U (cid:0) − ∂ n ∂ l ∂ t h tm + ∂ k ∂ k ∂ l h nm − ∂ l ∂ k ∂ n h km (cid:1)i + F ′ (cid:26) − (cid:16) ˙ H + 3 H (cid:17) h tt δ ij + H (cid:20)(cid:18) ∂ t (cid:18) h kk − h tt (cid:19) + ∂ k h tk (cid:19) δ ij − ∂ t h ij − ∂ i h tj − ∂ j h ti (cid:21) − ∂ t h ij + (cid:18) ∂ t h kk − ∂ k ∂ k (cid:0) h tt + h ll (cid:1) + ∂ k ∂ l h kl + 2 ∂ t ∂ k h tk (cid:19) δ ij + 12 ∂ k ∂ k h ij + 12 ∂ i ∂ j (cid:0) h tt + h kk (cid:1) − (cid:16) ∂ i ∂ k h kj + ∂ t ∂ i h tj + g (0) im g (0) jl (cid:0) ∂ l ∂ k h km + ∂ t ∂ l h tm (cid:1)(cid:17)(cid:27) + F ′′ n −
12 ...
H h tt δ ij + ¨ H (cid:2) − Hh ij + (cid:0) H (cid:0) − h tt − h kk (cid:1) − ∂ t h tt + ∂ t h kk − ∂ k h tk (cid:1) δ ij + 3 ∂ t h ij +3 ∂ i h tj + 3 ∂ j h ti (cid:3) + ˙ H (cid:0) − h tt δ ij − h ij (cid:1) + ˙ H (cid:2) − H h ij + 12 ∂ t h ij + 12 ∂ i h tj + 12 ∂ j h ti + (cid:0) H (cid:0) h tt − h kk (cid:1) + H (cid:0) − ∂ t h tt − ∂ t h kk − ∂ k h tk (cid:1)(cid:1) δ ij − ∂ i ∂ j h tt + (cid:0) ∂ t (cid:0) − h tt + 7 h kk (cid:1) + 9 ∂ k ∂ k h tt + 2 ∂ t ∂ k h tk + 3 ∂ k ∂ k h ll − ∂ k ∂ l h kl (cid:1) δ ij (cid:3) + 36 H h tt δ ij + H (cid:2) ∂ t (cid:0) − h tt − h kk (cid:1) − ∂ k h tk (cid:3) δ ij + H (cid:2) ∂ i ∂ j h tt + (cid:0) ∂ t (cid:0) − h tt + 5 h kk (cid:1) + ∂ k ∂ k (cid:0) h tt + 3 h kk (cid:1) − ∂ k ∂ l h kl − ∂ t ∂ k h tk (cid:1) δ ij (cid:3) + H (cid:2) ∂ i ∂ j (cid:0) − ∂ t h tt + 4 ∂ t h kk + 4 h tl (cid:1) (cid:0) ∂ t (cid:0) − h tt + 6 h kk (cid:1) + 5 ∂ k ∂ k ∂ t h tt − ∂ k ∂ k ∂ t h ll − ∂ k ∂ l ∂ t h kl − ∂ k ∂ k ∂ l h tl (cid:1) δ ij (cid:3) + ∂ i ∂ j (cid:0) ∂ t h kk − ∂ k ∂ k h tt − ∂ k ∂ k h mm + 2 ∂ l ∂ t h tl (cid:1) + (cid:0) ∂ t h kk + 2 ∂ k ∂ t h tk − ∂ k ∂ k ∂ t h tt − ∂ k ∂ k ∂ t h ll + ∂ k ∂ l ∂ t h kl + ∂ k ∂ k ∂ l ∂ l h tt + ∂ k ∂ k ∂ l ∂ l h mm − ∂ k ∂ k ∂ l ∂ m h lm − ∂ k ∂ k ∂ l ∂ t h tl (cid:1) δ ij (cid:9) + F ′′′ n ... H h −
36 ˙ Hh tt − H h tt + H (cid:0) ∂ t (cid:0) − h tt + 24 h kk (cid:1) + 24 ∂ k h tk (cid:1) +6 ∂ t h kk − ∂ k ∂ k (cid:0) h tt + h ll (cid:1) + 6 ∂ k ∂ l h kl + 12 ∂ k ∂ t h tk (cid:3) δ ij + ¨ H (cid:2) − h tt δ ij − h ij (cid:3) + ¨ H h ˙ H (cid:0) − Hh ij + (cid:0) − Hh tt + ∂ t (cid:0) − h tt + 48 h kk (cid:1) + 48 ∂ k h tk (cid:1) δ ij (cid:1) − H h tt δ ij + H (cid:0) ∂ t (cid:0) − h tt + 144 h kk (cid:1) + 48 ∂ k h tk (cid:1) δ ij + H (cid:0) ∂ t (cid:0) − h tt + 84 h kk (cid:1) − ∂ k ∂ k ∂ t (cid:0) h tt + h kk (cid:1) + 12 ∂ k ∂ l h kl + 72 ∂ k ∂ t h tk (cid:1) δ ij + (cid:0) ∂ t h kk + 24 ∂ k ∂ t h tk − ∂ k ∂ k (cid:0) h tt + h kk (cid:1) + 12 ∂ k ∂ l ∂ t h kl (cid:1) δ ij (cid:3) + ˙ H (cid:2) − H h ij + (cid:8) − H h tt + H (cid:0) ∂ t (cid:0) − h tt + 288 h kk (cid:1) +288 ∂ k h tk (cid:1) + 24 ∂ t h kk − ∂ k ∂ k (cid:0) h tt + h kk (cid:1) + 24 ∂ k ∂ l h kl + 48 ∂ k ∂ t h tk (cid:1) δ ij (cid:3) + ˙ H (cid:2) − H h tt + H (cid:0) ∂ t (cid:0) − h tt + 192 h kk (cid:1) − ∂ k h tk (cid:1) + H ∂ t (cid:0) − h tt + 240 h kk (cid:1) + H (cid:0) ∂ k ∂ k (cid:0) h tt + h kk (cid:1) − ∂ k ∂ l h kl + 96 ∂ k ∂ t h tk (cid:1) +48 H∂ t h kk + H (cid:0) ∂ k ∂ t h tk − ∂ k ∂ k ∂ t (cid:0) h tt + h kk (cid:1) + 48 ∂ k ∂ l ∂ t h kl (cid:1)(cid:3) δ ij (cid:9) + F ′′′′ n ¨ H h −
216 ˙ Hh tt − H h tt + H (cid:0) ∂ t ( − h tt + 144 h kk (cid:1) +144 ∂ k h tk (cid:1) + 36 ∂ t h kk − ∂ k ∂ k (cid:0) h tt + h kk (cid:1) + 36 ∂ k ∂ l h kl + 72 ∂ k ∂ t h tk (cid:3) δ ij − H ˙ H h tt δ ij + ¨ H ˙ H (cid:2) − H h tt + H (cid:0) ∂ t (cid:0) − h tt + 1152 h kk (cid:1) + 1152 ∂ k h tk (cid:1) +288 H∂ t h kk + 288 H (cid:0) − ∂ k ∂ k (cid:0) h tt + h ll (cid:1) + ∂ k ∂ l h kl + 2 ∂ k ∂ t h tk (cid:1)(cid:3) δ ij − H H h tt δ ij + ˙ H (cid:2) − H h tt + H (cid:0) ∂ t (cid:0) − h tt + 2304 h kk (cid:1) + 2304 ∂ k h tk (cid:1) +576 H ∂ t h kk + 576 H (cid:0) − ∂ k ∂ k (cid:0) h tt + h ll (cid:1) + ∂ k ∂ l h kl + 2 ∂ k ∂ t h tk (cid:1)(cid:3) δ ij (cid:9) + ω (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) h tt δ ij + 12 ( ω (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) − V (cid:16) φ (0) (cid:17)) h ij + 12 ( ω ′ (cid:0) φ (0) (cid:1) (cid:16) ˙ φ (0) (cid:17) ϕ + ω (cid:16) φ (0) (cid:17) ˙ φ (0) ˙ ϕ − V ′ (cid:16) φ (0) (cid:17) ϕ ) δ ij =0 . (45) [1] G. Bertone, D. 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