Slow-Roll Inflation in Scalar-Tensor Models
SSlow-Roll Inflation in Scalar-Tensor Models
L.N. Granda ∗ , D. F. Jimenez † Departamento de Fisica, Universidad del ValleA.A. 25360, Cali, Colombia
Abstract
The linear and quadratic perturbations for a scalar-tensor model with non-minimal coupling to curvature, coupling to the Gauss-Bonnet invariant andnon-minimal kinetic coupling to the Einstein tensor are developed. The quadraticaction for the scalar and tensor perturbations is constructed and the power spec-tra for the primordial scalar and tensor fluctuations are given. A consistencyrelation that is useful to discriminate the model from the standard inflation withcanonical scalar field was found. For some power-law potentials it is shown thatthe Introduction of additional interactions, given by non-minimal, kinetic andGauss-Bonnet couplings, can lower the tensor-to-scalar ratio to values that areconsistent with latest observational constraints, and the problem of large fieldsin chaotic inflation can be avoided.
The improvement in the quality of the cosmological observations of the last years[1, 2, 3, 4] has reinforced the theory of cosmic inflation [5, 6, 7]. The inflationary the-ory gives by now the most likely scenario for the early universe, since it provides the ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] S e p xplanation to flatness, horizon and monopole problems, among others, for the stan-dard hot Bing Bang cosmology [8, 9, 10, 11, 12, 13, 14]. In other words, the inflationcan set the initial conditions for the subsequent hot Big Bang, by eliminating the fine-tuning condition needed for solving the horizon, flatness and other problems. Besidesthat, the quantum fluctuations during inflation could provide the seeds for the largescale structure and the observed CMB anisotropies [15, 16, 17, 18, 19, 20, 21, 22]. Inparticular, inflation allows as to understand how the scale-invariant power spectrumcan be generated, though it does not predict an exact scale invariant but nearly scaleinvariant power spectrum [22]. The deviation from scale invariance is connected withthe microphysics description of the inflationary theory which is still incomplete.The simplest and most studied model of inflation consists of minimally-coupled scalarfield with flat enough potential to provide the necessary conditions for slow-roll[6, 7]. But the inflation scenario can be realized in many other models like non-minimally coupled scalar field [23, 24, 25, 26, 27], kinetic inflation [28], vector infla-tion [29, 30, 31], inflaton potential in supergravity [32, 33, 34], string theory inspiredinflation [35, 36, 37, 38, 39, 40], Dirac-Born-Infeld inflation model [41, 42, 43, 44], α -attractor models originated in supergravity [45, 46, 47, 48, 49]. Apart from the DBImodels of inflation, another class of ghost-free models has been recently considered,named ”Galileon” models [50, 51]. In spite of the higher derivative nature of thesemodels, the gravitational and scalar field equations contain derivatives no higher thantwo. The effect of these Galileon terms is mostly reflected in the modification of thekinetic term compared to the standard canonical scalar field, which in turn can im-prove (or relax) the physical constraints on the potential. In the case of the Higgspotential, for instance, one of the effects of the higher derivative terms is the reduc-tion of the self coupling of the Higgs boson, so that the spectra of primordial densityperturbations are consistent with the present observational data [52, 53] (which isnot possible within the standard canonical scalar field inflation with Higgs potential).Different aspects of Galilean-inflation have been considered in [52, 53, 54, 55, 56, 57].A particular and important case belonging to the above class of models is the scalarfield with kinetic coupling to the Einstein tensor [58, 59, 60, 61] whose application inthe context of inflationary cosmology has been analyzed in [62, 63, 64, 65, 66, 67].2n the present paper we consider a scalar-tensor model with non-minimal coupling toscalar curvature, non-minimal kinetic coupling to the Einstein tensor and coupling ofthe scalar field to the Gauss-Bonnet 4-dimensional invariant, to study the slow-rollinflation and the observable magnitudes, the scalar espectral index and the tensor-to-scalar ratio, derived from it. These interaction terms have direct correspondence withterms presented in Galileon theories [57, 68]. This model is the simplest and moregeneral scalar-tensor theory (whose Lagrangian density contains up to first derivativesof the scalar field) leading to second-order field equations, avoiding the appearance ofOstrogradsky instabilities and leading to ghost-free theory. These couplings, includinglinear and second-order curvature corrections, arise in the low energy effective actionof string theory (in fact a remarkable peculiarity of the string effective action is theappearance of field-dependent couplings to curvature) [69, 70], where couplings suchas Gauss-Bonnet provide the possibility of avoiding the initial singularity [71, 72].Given that there exist non-singular cosmological solutions based on these couplings,it is pertinent to investigate the effect of these correction terms on the evolution ofprimordial fluctuations that leave the power-spectrum nearly scale-invariant. Also, inview of the accuracy of future observations, we expect that these corrections to thesimplest, canonical scalar field, inflation model become important in a high-curvatureregime typical of inflation. The effect of such corrections to the inflationary scenariocould provide a connection with fundamental theories like supergravity or string the-ory. For studies of inflation with GB coupling and modified gravity see, for instance[72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85].In the appendixes we develop in detail the linear and quadratical perturbations for allthe interaction terms of the model and deduce the second order action for the scalarand tensor perturbations. In appendix A we present the basic formulas for the firstorder perturbations, needed for the model, in the Newtonian gauge. In appendix Bwe deduce the gravitational and scalar field equations in a general background. Inappendix C and D we give the first order perturbations of the field equations in theNewtonian gauge. In Appendix E we give the details for constructing the second or-der action using the Xpand tool [86], and in appendix F we give a detailed descriptionof the slow-roll mechanism for the minimally coupled scalar field.3he expressions for the primordial density fluctuations in terms of the slow-roll param-eters and the corresponding power spectra were found. We have found a consistencyrelation that is useful to discriminate the model from the standard inflation withcanonical scalar field. The latest observational data disfavor monomial-type models V ∝ φ n with n ≥ We consider the scalar-tensor model with non-minimal coupling of the scalar field tocurvature, non-minimal kinetic coupling of the scalar field to the Einstein’s tensorand coupling of the scalar field to the Gauss-Bonnet (GB) 4-dimensional invariant S = (cid:90) d x √− g (cid:20) F ( φ ) R − ∂ µ φ∂ µ φ − V ( φ ) + F ( φ ) G µν ∂ µ φ∂ ν φ − F ( φ ) G (cid:21) (2.1)where G µν is the Einstein’s tensor, G is the GB 4-dimensional invariant given by G = R − R µν R µν + R µνλρ R µνλρ (2.2) F ( φ ) = 1 κ + f ( φ ) , (2.3)and κ = M − p = 8 πG . One remarkable characteristic of this model is that it yieldssecond-order field equations and can avoid Ostrogradski instabilities. Using the gen-4ral results of Appendix B, expanded on the flat FRW background ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) (2.4)one finds the following equations3 H F (cid:32) − F ˙ φ F − H ˙ F F (cid:33) = 12 ˙ φ + V − H ˙ F (2.5)2 ˙ HF (cid:32) − F ˙ φ F − H ˙ F F (cid:33) = − ˙ φ − ¨ F + H ˙ F + 8 H ¨ F − H ˙ F − H F ˙ φ + 4 HF ˙ φ ¨ φ + 2 H ˙ F ˙ φ (2.6)¨ φ + 3 H ˙ φ + V (cid:48) − F (cid:48) (cid:16) H + ˙ H (cid:17) + 24 H (cid:16) H + ˙ H (cid:17) F (cid:48) + 18 H F ˙ φ + 12 H ˙ HF ˙ φ + 6 H F ¨ φ + 3 H F (cid:48) ˙ φ = 0 (2.7)where ( (cid:48) ) denotes derivative with respect to the scalar field. Related to the differentterms in the action (2.1) we define the following slow-roll parameters (cid:15) = − ˙ HH , (cid:15) = ˙ (cid:15) H(cid:15) (2.8) (cid:96) = ˙ FHF , (cid:96) = ˙ (cid:96) H(cid:96) (2.9) k = 3 F ˙ φ F , k = ˙ k Hk (2.10)∆ = 8 H ˙ F F , ∆ = ˙∆ H ∆ (2.11)The slow-roll conditions in this model are satisfied if all these parameters are muchsmaller than one, and will be used in the next section. From the cosmological equa-tions (2.5) and (2.6) and using the parameters (2.8)-(2.11) we can write the followingexpressions for ˙ φ and VV = H F (cid:104) −
52 ∆ − k − (cid:15) + 52 (cid:96) + 12 (cid:96) ( (cid:96) − (cid:15) + (cid:96) ) −
12 ∆ (∆ − (cid:15) + (cid:96) ) − k ( k + (cid:96) − (cid:15) ) (cid:105) (2.12)5 φ = H F (cid:104) (cid:15) + (cid:96) − ∆ − k + ∆ (∆ − (cid:15) + (cid:96) ) − (cid:96) ( (cid:96) − (cid:15) + (cid:96) ) + 23 k ( k + (cid:96) − (cid:15) ) (cid:105) (2.13)where we used ¨ F = H F (cid:96) ( (cid:96) − (cid:15) + (cid:96) ) , ¨ F = F ∆ + (cid:15) + (cid:96) ) (2.14)It is also useful to define the variable Y from Eq. (2.13) as Y = ˙ φ H F (2.15)where it follows that Y = O ( ε ). Notice that for the simplest case of minimally coupledscalar field ( F = 1 /κ , F = F = 0), the Eqs. (2.12) and (2.13) give the standardequations H = 8 πG (cid:18)
12 ˙ φ + V ( φ ) (cid:19) , ˙ H = − πG ˙ φ Under the slow-roll conditions ¨ φ << H ˙ φ and (cid:96) i , k i , ∆ i <<
1, it follows from (2.5)-(2.7) 3 H F (cid:39) V, (2.16)2 ˙ HF (cid:39) − ˙ φ + H ˙ F − H F ˙ φ − H ˙ F , (2.17)3 H ˙ φ + V (cid:48) − H F (cid:48) + 18 H F ˙ φ + 24 H F (cid:48) (cid:39) V gives the dominant contribution to the Hubble param-eter, while Eqs. (2.17) and (2.18) determine the dynamics of the scalar field in theslow-roll approximation. The number of e -folds can be determined from N = (cid:90) φ E φ I H ˙ φ dφ = (cid:90) φ E φ I H + 6 H F H F (cid:48) − H F (cid:48) − V (cid:48) dφ (2.19)where φ I and φ E are the values of the scalar field at the beginning and end of inflationrespectively, and the expression for ˙ φ was taken from (2.18). The criteria for choosingthe initial values will be discussed below. 6 Quadratic action for the scalar and tensor per-turbations
Scalar Perturbations .After the computation of the second order perturbations we are able to write thesecond order action for the scalar perturbations as follows δS s = (cid:90) dtd xa (cid:20) G s ˙ ξ − F s a ( ∇ ξ ) (cid:21) (3.1)where G s = ΣΘ G T + 3 G T (3.2) F s = 1 a ddt (cid:16) a Θ G T (cid:17) − F T (3.3)with G T = F − F ˙ φ − H ˙ F . (3.4) F T = F + F ˙ φ − F (3.5)Θ = F H + 12 ˙ F − HF ˙ φ − H ˙ F (3.6)Σ = − F H − H ˙ F + 12 ˙ φ + 18 H F ˙ φ + 48 H ˙ F (3.7)And the sound speed of scalar perturbations is given by c S = F S G S (3.8)The conditions for avoidance of ghost and Laplacian instabilities as seen from theaction (3.1) are F > , G > G T , F T , Θ and Σ in terms of the slow-roll parameters (2.8)-(2.11) andusing Eqs. (2.13) and (2.14), as follows G T = F (cid:18) − k − ∆ (cid:19) (3.9)7 T = F (cid:18) k − ∆ (∆ + (cid:15) + (cid:96) ) (cid:19) (3.10)Θ = F H (cid:18) (cid:96) − k −
32 ∆ (cid:19) (3.11)Σ = − F H (cid:104) − (cid:15) + 52 (cid:96) − k −
112 ∆ + 12 (cid:96) ( (cid:96) − (cid:15) + (cid:96) ) − k ( k − (cid:15) + (cid:96) ) −
12 ∆ (∆ − (cid:15) + (cid:96) ) (cid:105) (3.12)The expressions for G S and c S in terms of the slow roll parameters can be written as G S = F (cid:0) Y + k + W (1 − ∆ − k ) (cid:1)(cid:0) W (cid:1) (3.13) c S = 1 + W (cid:0) ∆ (∆ + ε + l − − k (cid:1) + W (cid:0) k (2 − k − l ) + 2∆ ε (cid:1) − k ε Y + 2 k + W (1 − ∆ − k ) (3.14)where W = (cid:96) − ∆ − k − ∆ − k (3.15)Notice that in general G S = F O ( ε ) and c S = 1 + O ( ε ). Also in absence of thekinetic coupling it follows that c S = 1 + O ( ε ). Keeping first order terms in slow-rollparameters, the expressions for G S y c S reduce to G S = F (cid:18) ε + 12 l −
12 ∆ (cid:19) (3.16) c S = 1 + k (cid:0) l − ∆ − k (cid:1) − k ε ε + l − ∆ (3.17)To normalize the scalar perturbations we perform the change of variables [57] (see(F.7)) dτ s = c S a dt, ˜ z = √ a ( F S G S ) / , ˜ U = ξ ˜ z (3.18)and the action (3.1) becomes δS s = 12 (cid:90) dτ s d x (cid:20)
12 ( ˜ U (cid:48) ) − D i ˜ U D i ˜ U + ˜ z (cid:48)(cid:48) ˜ z ˜ U (cid:21) (3.19)8here ”prima” indicates derivative with respect to τ s . Working in the Fourier repre-sentation, one can write ˜ U ( (cid:126)x, τ s ) = (cid:90) d k (2 π ) ˜ U (cid:126)k ( τ s ) e i(cid:126)k(cid:126)x (3.20)and the equation of motion for the action (3.19) takes the form˜ U (cid:48)(cid:48) (cid:126)k + (cid:18) k − ˜ z (cid:48)(cid:48) ˜ z (cid:19) ˜ U (cid:126)k = 0 (3.21)From (3.18), and keeping up to first-order terms in slow-roll variables using (3.16)and (3.17), we find the following expression for ˜ z (cid:48) ˜ z (cid:48) = 1 c S a z (cid:20) F df ( (cid:15) , (cid:96) , ∆ ) dt + 2 F ˙ F f ( (cid:15) , (cid:96) , ∆ ) (cid:21) + 1 c S aHz (3.22)where f ( (cid:15) , (cid:96) , ∆ ) = (cid:18) (cid:15) + 12 (cid:96) −
12 ∆ (cid:19) . Then, under the approximation of slowly varying c S and up to first-order in slow-rollvariables we find the following expression for ˜ z (cid:48)(cid:48) / ˜ z ˜ z (cid:48)(cid:48) ˜ z = a H c S (cid:104) − (cid:15) + 32 (cid:96) + 32 2 (cid:15) (cid:15) + (cid:96) (cid:96) − ∆ ∆ (cid:15) + (cid:96) − ∆ (cid:105) . (3.23)This expression reduces to the one of the canonical scalar field given in Appendix E,Eq. (F.24), in the case (cid:96) = ∆ = 0 where c S = 1 and (cid:15) = 2( (cid:15) − δ ), with δ definedin (F.22). In what follows the reasoning is similar to the simplest case, correspondingto minimally-coupled scalar field, which is analyzed in detail in Appendix E. So onsub-horizon scales when the k term dominates in Eq. (3.21) we choose the sameBunch-Davies vacuum solution defined for the scalar field, which leads to˜ U k = 1 √ k e − ikτ s (3.24)Note that from the expression a ddt (cid:18) aH (cid:19) = − (cid:15) = ⇒ c S ddτ s (cid:18) aH (cid:19) = − (cid:15) , (3.25)9n the approximation of slowly varying c S and (cid:15) one can integrate the last equationto obtain τ s = − aH c S − (cid:15) (3.26)Then in the limit (cid:15) → aH = − τ dS c S (3.27)In this last case and neglecting the slow-roll parameters (in this limit c S = 1) we canwrite from (3.23) ˜ z (cid:48)(cid:48) ˜ z (cid:39) a H c S = 2 τ dS (3.28)which allows the integration of Eq. (3.21), giving the known solution for the scalarperturbations in a de Sitter background. Taking into account the slow-roll parametersand using (3.26) we can rewrite the Eq. (3.21) in the form˜ U (cid:48)(cid:48) k + k ˜ U k + 1 τ s (cid:18) µ s − (cid:19) ˜ U k = 0 (3.29)where µ s = 94 (cid:20) (cid:15) + 23 (cid:96) + 23 2 (cid:15) (cid:15) + (cid:96) (cid:96) − ∆ ∆ (cid:15) + (cid:96) − ∆ (cid:21) (3.30)where we have expanded up to first order in slow-roll parameters. The general solutionof Eq. (3.30) for constant µ s (slowly varying slow-roll parameters) is˜ U k = √− τ s (cid:104) C k H (1) µ s ( − kτ s ) + C k H (2) µ s ( − kτ s ) (cid:105) (3.31)and after matching the boundary condition related with the choosing of the Bunch-Davies vacuum (3.24) we find the solution˜ U k = √ π e i π ( µ s + ) √− τ s H (1) µ s ( − kτ s ) (3.32)using the asymptotic behavior of H (1) µ s ( x ) at x >>
1, we find at super horizon scales( c S k << aH ) ˜ U k = 1 √ e i π ( µ s − ) µ s − Γ( µ s )Γ(3 / √− τ s ( − kτ s ) − µ s . (3.33)To evaluate the power spectra we use the relationship˜ z (cid:48) ˜ z = − − (cid:15) ) τ s (cid:104) (cid:96) + 12 2 (cid:15) (cid:15) + (cid:96) (cid:96) − ∆ ∆ (cid:15) + (cid:96) − ∆ (cid:105) = − τ s (cid:18) µ s − (cid:19) . (3.34)10here we used (3.26) for aH , and for the last equality we have expanded up to firstorder in slow-roll parameters, resulting in µ s = 32 + (cid:15) + 12 (cid:96) + 12 2 (cid:15) (cid:15) + (cid:96) (cid:96) − ∆ ∆ (cid:15) + (cid:96) − ∆ (3.35)Assuming again the approximation of slowly varying slow-roll parameters we canIntegrate this equation to find ˜ z ∝ τ − µ s s (3.36)which gives, in the super horizon regime, for the amplitude of the scalar perturbationsthe following expression ξ k = ˜ U k ˜ z ∝ k − µ s (3.37)where the τ s dependence disappears as expected from the solution (3.33) in superhorizon scales ( c s k << aH ). The power spectra for the scalar perturbations takesthe following k -dependence P ξ = k π | ξ k | ∝ k − µ s (3.38)and the scalar spectral index becomes n s − d ln P ξ d ln k = 3 − µ s = − (cid:15) − (cid:96) − (cid:15) (cid:15) + (cid:96) (cid:96) − ∆ ∆ (cid:15) + (cid:96) − ∆ (3.39)It is worth noticing that the slow-roll parameter k , related to the kinetic coupling,do not appear in the above expression for the scalar spectral index. This is because k appears only in second order terms (or higher) in the expressions for G S and F S (see (3.13) and (3.14)). Tensor perturbations .The second order action for the tensor perturbations takes the form δS = 18 (cid:90) d xdt G T a (cid:20)(cid:16) ˙ h ij (cid:17) − c T a ( ∇ h ij ) (cid:21) (3.40)where G T and F T are defined in (E.2) and (E.3) (in terms of the slow-roll variablesin (3.18) and (3.19)). The velocity of tensor perturbations is given by c T = F T G T = 3 + k − (∆ + (cid:15) + (cid:96) )3 − k − . (3.41)11s in the case of scalar perturbations, in order to canonically normalize the tensorperturbations the following variables are used [57] dτ T = c T a dt, z T = a F T G T ) / , v ij = z T h ij (3.42)leading to the quadratic action δS = 12 (cid:90) d xdτ T (cid:20)(cid:0) v (cid:48) ij (cid:1) − ( ∇ v ij ) + z (cid:48)(cid:48) T z T v ij (cid:21) (3.43)which gives the equation v (cid:48)(cid:48) ij − ∇ v ij − z (cid:48)(cid:48) T z T v ij = 0 . (3.44)Or for the corresponding Fourier modes v (cid:48)(cid:48) ( k ) ij + (cid:18) k − z (cid:48)(cid:48) T z T (cid:19) v ( k ) ij = 0 , (3.45)which is of the same nature as the equation for the scalar perturbations, and thereforethe perturbations h ij on super horizon scales behave exactly as the solutions (F.6).For the evaluation of the primordial power spectrum we follow the same steps as forthe scalar perturbations. To this end we write the expression for z (cid:48)(cid:48) T /z T , up to firstorder in slow-roll parameters, as follows z (cid:48)(cid:48) T z T = a H c T (cid:18) − (cid:15) + 32 (cid:96) (cid:19) (3.46)Then, the normalized solution of (3.45) in the approximation of slowly varying slow-roll parameters can be written in terms of the Hankel function of the first kind as v ( k ) ij = √ π √− τ T H (1) µ T ( − kτ T ) e ( k ) ij (3.47)where the tensor e ( k ) ij describe the polarization states of the tensor perturbations forthe k -mode, and µ T = 32 + (cid:15) + 12 (cid:96) . (3.48)At super horizon scales ( c T k << aH ) the tensor modes (3.47) have the same func-tional form for the asymptotic behavior as the scalar modes (3.33), and therefore wecan write power spectrum for tensor perturbations as P T = k π | h ( k ) ij | (3.49)12here h ( k ) ij = v ( k ) ij /z T , and the sum over the polarization states must be taken intoaccount. Then, the tensor spectral index will be given by n T = 3 − µ T = − (cid:15) − (cid:96) (3.50)An important quantity is the relative contribution to the power spectra of tensor andscalar perturbations, defined as the tensor/scalar ratio rr = P T ( k ) P ξ ( k ) . (3.51)For the scalar perturbations, using (3.38), we can write the power spectra as P ξ = A S H (2 π ) G / S F / S (3.52)where A S = 12 2 µ s − (cid:12)(cid:12)(cid:12) Γ( µ s )Γ(3 / (cid:12)(cid:12)(cid:12) and all magnitudes are evaluated at the moment of horizon exit when c s k = aH ( kτ s = − z we used (3.18) with a = c S k/H . In analogous way we can writethe power spectra for tensor perturbations as P T = 16 A T H (2 π ) G / T F / T (3.53)where A T = 12 2 µ T − (cid:12)(cid:12)(cid:12) Γ( µ T )Γ(3 / (cid:12)(cid:12)(cid:12) . Noticing that A T /A S (cid:39) (cid:15) , (cid:96) , ∆ , ... <<
1, as followsfrom (3.35) and (3.48), we can write the tensor/scalar ratio as follows r = 16 G / T F / S G / S F / T = 16 c S G S c T G T . (3.54)Taking into account the expressions for G T , F T , G S , F S up to first order obtained from(3.9), (3.10), (3.16) and (3.17), and using the condition (cid:15) , (cid:96) , k , ∆ <<
1, then wecan see that c T (cid:39) c S (cid:39) (cid:96) → c S = 1 independently of thevalues of (cid:15) and ∆ ) and we can make the approximation r = 8 (cid:18) (cid:15) + (cid:96) − ∆ − k − ∆ (cid:19) (cid:39) (cid:15) + (cid:96) − ∆ ) (3.55)13hich is a modified consistency relation due to the non-minimal and GB couplings.In the limit (cid:96) , ∆ → r = − n T , (3.56)with n T = − (cid:15) . Taking into account the non-minimal and GB couplings we find thedeviation from the standard consistency relation in the form r = − n T + δr, δr = − , (3.57)with n T given by (3.50). Thus, the consistency relation still valid in the case ofnon-minimal coupling, and if there is an observable appreciable deviation from thestandard consistency relation, it can reveal the effect of an interaction beyond thesimple canonical scalar field or even non-minimally coupled scalar field models ofinflation. It is worth noticing that in the first-order formalism the kinetic-couplingrelated slow-roll parameter k does not appear in the spectral index for the scalar andtensor perturbations and is also absent in the tensor-to-scalar ratio, appearing onlystarting form the second order expansion in slow-roll parameters. Nevertheless, allthe couplings are involved in the definition of the slow-roll parameters trough thefield equations. Of special interest are the cases of monomial potentials V ∝ φ n .These potentials are disfavored by the observational data for n ≥ Model I.
First we consider the particular case of the non-minimal coupling ξφ with quadraticpotential and kinetic coupling with constant F . F ( φ ) = 1 κ − ξφ , V ( φ ) = 12 m φ , F ( φ ) = γ, F ( φ ) = 0 . (4.1)14sing the Eqs. (2.16) and (2.18) we can express the slow-roll parameters (2.8)-(2.11)in therms of the potential and the coupling functions, and once we specify the model,we can find the slow-roll parameters in terms of the scalar field and the couplingconstants. For the model (4.1) the slow-roll parameters take the form (cid:15) = 2 + 2 ξφ φ + ( m γ − ξ ) φ , (cid:15) = 4(1 − ξφ ) (( m γ − ξ )( ξφ + 2) φ + 1) φ (1 + ( m γ − ξ ) φ ) (cid:96) = 4 ξ ( ξφ + 1)( m γ − ξ ) φ + 1 , (cid:96) = − m γ − ξ )( ξφ − m γ − ξ ) φ + 1) . (4.2)where φ is dimensionless ( φ has been rescaled as κφ → φ to measure it in units of M p )and γ has dimension of mass − . Additionally, the scalar field at the end of inflationcan be evaluated under the condition (cid:15) ( φ E ) = 1. Sitting (cid:15) = 1 in (4.2) it follows φ E = (cid:112) m γ + 4 ξ − ξ + 1 + 2 ξ − m γ − ξ (4.3)From Eq. 2.19 it follows that the number of e -foldings can be evaluated as N = φ E (cid:90) φ I φ + ( m γ − ξ ) φ ξ φ − dφ = 18 ξ (cid:2) m γ ln (cid:0) − ξ φ (cid:1) − ξ ln (cid:0) − ξφ (cid:1)(cid:3) (cid:12)(cid:12)(cid:12) φ E φ I (4.4)This expression allows us to evaluate φ I for a given N . We can make some qualitativeanalysis by assuming that ξφ << m γ >> ξ . In this case from (4.3) it is foundthat φ E ≈ (cid:18) m γ (cid:19) / , (4.5)and from (4.4) we find for φ I φ I ≈ (cid:18) N + 2 m γ (cid:19) / (4.6)giving an approximate relation between the values of the scalar field at the beginningand end of inflation as φ I ≈ (4 N + 1) / φ E So, assuming N = 60 gives φ I ≈ . φ E . This will have sense only if the scalar spectralindex and the tensor-to-scalar ratio behave properly. In fact from (4.2) and replacing15n (3.39) and (3.55), we find (under the condition ξφ << m γ >> ξ ) n s ≈ N + 2) / ( m γ ) / − N + 2 − N + 2) / ( m γ ) / (4.7)and r ≈ N + 2 + 64 ξ (8 N + 2) / ( m γ ) / (4.8)where we have used (4.6) for φ I . Additional simplification can be made if we assumethat the scalar field at the beginning of inflation is of the order of M p ( φ (cid:39) m γ = 8 N + 2, as follows from (4.6), which gives n s ≈ − N + 2 − N + 2) , r ≈
32 + 64 ξ N + 2 (4.9)Thus, for 60 e -foldings we find n s ≈ .
98 and r ≈ .
067 ( ξ = 10 − ). In this casethe inflation begins with φ I = M p and ends with φ E ≈ . M p . For the numeri-cal analysis with the exact expressions, we assume N = 60, m = 10 − M p . In factfrom Eqs. (4.2) follows that the spectral index n s and the tensor-to-scalar ratio de-pend on the dimensionless combination m γ . Fig. 1 shows the behavior of n s and r where ξ takes two fixed values ξ = 1 / , ξ = 0 . m γ is running in the interval2 × ≤ m γ ≤ × , and in Fig. 2 we consider the two fixed values m γ = 1and m γ = 2 . ξ is varying in the interval − . ≤ ξ ≤ − . n s and r fall in a more acceptable range, according to the latest observationalbounds [3, 4]. Thus, keeping m γ ∼ ξ varying in the interval [ − . , − . Model II
The following example considers a model with kinetic and GB couplings F = 1 κ , V ( φ ) = 12 m φ , F ( φ ) = γ, F ( φ ) = ηφ (4.10)where the constant η has dimension of mass and φ is measured in units of M p . Theslow-roll parameters from (2.8)-(2.11), necessary to evaluate n s and r , take the form (cid:15) = 6 − m η φ (1 + m γφ ) , (cid:15) = 4(3 − m η )(1 + 2 m γφ )3 φ (1 + m γφ ) .9730 0.9735 0.9740 0.97450.0900.0950.1000.1050.1100.1150.120 n s r Figure 1: The behavior of the scalar spectral index n s and tensor-to-scalar ratio r for the two fixed values ξ = 1 / ξ = 0 . m γ in the interval2 × ≤ m γ ≤ × . m γ = m γ = n s r Figure 2: The variation of the scalar spectral index n s and tensor-to-scalar ratio r for the two cases m γ = 1 and m γ = 2 .
5, for ξ in the interval [ − . , − . n s and r are in the region bounded by the latest observations [3, 4].17 = 16 m η (3 − m η )9 φ (1 + m γφ ) , ∆ = 4(3 − m η )(1 + 2 m γφ )3 φ (1 + m γφ ) , (4.11)where theproduct m η is measured in units of M p and the product m γ is dimension-less. The scalar field at the end of inflation is obtained from the condition (cid:15) ( φ E ) = 1,which gives φ E = 16 m γ (cid:104)(cid:112) m γ − m γη + 9 − (cid:105) (4.12)And From Eq. (2.19), the number of e -foldings can be evaluated as N = 3 φ (2 + m γφ )8(4 m η − (cid:12)(cid:12)(cid:12) φ E φ I (4.13)which allows to find φ I for a given N and φ E from (4.12). From (3.39) and (4.11) wefind the scalar spectral index as n s = 3 m γφ ( γφ + 16 η ) + 6 m γφ ( φ −
6) + 3 φ + 32 m η − φ (1 + m γφ ) (cid:12)(cid:12)(cid:12) φ I (4.14)And from (3.55) and (4.11) we find the expression for the tensor-to-scalar ratio as r = 3 φ (1 + m γφ )8 m η − (cid:12)(cid:12)(cid:12) φ I (4.15)For N = 60 and taking m = 10 − M p we can find the behavior of n s and r in termsof the dimensionless parameter m γ . In Fig. 3 we show the behavior of the scalarfield at the beginning and end of inflation for 1 < m γ <
5. In Fig. 4 we show thecorresponding behavior for n s and r . Model III.
The following model considers the general power-law potential and non-minimalpower-law functions for the GB and kinetic couplings F = 1 κ , V ( φ ) = λn φ n , F ( φ ) = γφ n , F ( φ ) = ηφ n (4.16)The slow-roll parameters (2.8)-(2.11) for this model take the form ( κ = 1) (cid:15) = n (3 n − ηλ )6( n + 2 γλ ) φ , (cid:15) = 2 n (3 n − ηλ )3( n + 2 γλ ) φ , ∆ = 8 nηλ (3 n − ηλ )9( n + 2 γλ ) φ , η = m η = ϕ I ϕ E Figure 3: The scalar field at the beginning and at the end of inflation for 1 ≤ m γ ≤ m η = 0 , m η = 0 . M p ). At the end of inflation φ E < M p . m η = m η = n s r Figure 4: The behavior of the scalar spectral index n s and r for m γ , in the interval1 ≤ m γ ≤ m η = 0 ,
65 , m η = 0 . M p ).19 = 2 n (3 n − ηλ )3( n + 2 γλ ) φ , k = nγλ (3 n − ηλ ) n + 2 γλ ) φ , k = 2 n (3 n − ηλ )3( n + 2 γλ ) φ (4.17)The scalar field at the end of inflation ( (cid:15) = 1) takes the form φ E = n √ n − ηλ √ n + 12 γλ (4.18)The number of e-foldings from (2.19) is given by N = − n + 2 γλ )2 n (3 n − ηλ ) φ (cid:12)(cid:12)(cid:12) φ E φ I (4.19)which, using (4.18) allows to find the exact explicit form for the scalar field N e -foldsbefore the end of inflation as φ I = (cid:18) (4 N + n )(3 n − nηλ )6 n + 12 γλ (cid:19) / = (cid:114) (4 N + n ) n φ E (4.20)From (3.39) and (4.17) after replacing the value of the scalar field φ I from (4.20), wefind the scalar spectral index as n s = (6 γλ + 16 nηλ + 3 n ) φ − n − (6 − ηλ ) n n + 2 γλ ) φ (cid:12)(cid:12)(cid:12) φ I = 4 N − n − N + n (4.21)And from (3.55), (4.17) and (4.20) we find the expression for the tensor-to-scalar ratioas r = 8 n (3 n − ηλ ) n + 2 γλ ) φ (cid:12)(cid:12)(cid:12) φ I = 16(3 n − ηλ )3(4 N + n ) (4.22)The slow-roll parameters N e -folds before the end of inflation take the values (cid:15) = n N + n , (cid:15) = 44 N + n , ∆ = 16 ηλ N + n ) , ∆ = 44 N + n , k = 2 γλ (3 n − ηλ )3(4 N + n )( n + 2 γλ ) , k = 44 N + n (4.23)The Eq. (4.21) predicts the scalar spectral index n s in terms of the number of e -foldings N, and the power n , which is the same result as that obtained for the stan-dard chaotic inflation. However, the tensor-to-scalar ratio depends additionally onthe self coupling λ and the GB coupling constant η , but not on the kinetic couplingconstant. As can be seen from the expressions (4.18) and (4.20), the kinetic coupling20an lower the values of the scalar field at the end, and therefore at the beginning, ofinflation. Note also that the strong coupling regime of the GB coupling spoils theinflation (∆ and k break the slow-roll restrictions), while at the strong couplinglimit all slow-roll parameters and derived quantities are well defined. Note also thatall of the slow roll parameters (4.17), and therefore the quantities derived from them,depend on coupling constants through the products ηλ and γλ . The dimension of η is mass n , the dimension of λ is mass − n and the dimension of γ is mass n − , and there-fore independently of n , the product ηλ has constant dimension [ ηλ ] = mass andthe corresponding dimension of γλ is mass . This can be used to write ηλ = αM p where α is a dimensionless parameter that defines the behavior of r once n and N have been fixed. While the coupling λ is subject to different restrictions, dependingon the power n , one can vary the coupling η (and therefore α ) to find the appropriatevalue for the tensor-to-scalar ratio. On the other hand, the parameter β = γλ leadsto consistent inflation in the weak coupling, γ →
0, and strong coupling, γ → ∞ ,limits and can take any value between these limits. In table I we list some samplevalues for n s , r , for N = 60 and a range of α , for some power-law models includingmodels with fractional n that appear in string theory compactification [87, 88] andare favored by Planck 2018 data [3].Power n n s Parameter α range r in α range4 0.9508 1 ≤ α ≤ . . ≥ r ≥ . . ≤ α ≤ . . ≥ r ≥ . . ≤ α ≤ . . ≥ r ≥ . . ≤ α ≤ . . ≥ r ≥ . − ≤ α ≤ . . ≥ r ≥ . − ≤ α ≤ . . ≥ r ≥ . Table I . Some values of n s and r in an appropriate range for α in each case.It is noticeable the n = 2 case, which for minimally coupled scalar field is disfavoredby the latest observations [3, 4], but in the presence of GB coupling falls in the range21avored by the observational data. For all cases, the low tensor-to-scalar ratio is con-sistent with current observations. Since the parameter β = γλ is a free parameter,then one can use this freedom to set the values φ E , and therefore φ I , to any desiredvalue. Model IV.
This model considers the general power-law potential and the non-minimal kinetic coupling of the form F = 1 κ , V ( φ ) = λn φ n , F ( φ ) = βφ n +2 , F ( φ ) = 0 (4.24)The slow-roll parameters (2.8)-(2.11) take the form ( κ = 1) (cid:15) = n nφ + 4 βλφ , (cid:15) = 2 n ( n + 4 βλφ ) φ ( n + 2 βλφ ) k = βλn ( n + 2 βλφ ) , k = 8 βλn ( n + 2 βλφ ) . (4.25)from above the first equation we fond the scalar field at the end of inflation as φ E = (cid:112) n (1 + 4 nβλ ) − n βλ . (4.26)The number of e-foldings from (2.19) is N = βλφ n − φ n (cid:12)(cid:12)(cid:12) φ E φ I (4.27)This equation allows to find the scalar field N e -foldings before the end of inflationas φ I = n βλ (cid:32) √ n (cid:114) n + 2 nβλ ( n + 8 N ) + n √ nβλn − (cid:33) (4.28)Using this result we find the expression for the scalar spectral index from (3.39) and(4.25) as n s = 1 − nβλ (cid:2) √ n ( n + 4) f ( n, N, β, λ ) − (cid:3) n f ( n, N, β, λ ) (cid:2) √ nf ( n, N, β, λ ) − (cid:3) , (4.29)where f ( n, N, β, λ ) = (cid:114) n + 2 nβλ ( n + 8 N ) + n √ nβλn .
22n for the tensor-to-scalar ratio it is found (from (3.55), (4.25) and (4.28)) r = 32 √ βλf ( n, N, β, λ ) (cid:2) √ nf ( n, N, β, λ ) − (cid:3) (4.30)As can be seen form above results, both the slow-roll parameters and all the observ-able quantities depend on the product βλ , which independently of the power n , hasdimensions of ( mass ) . The coupling λ takes different significance and undergoes dif-ferent restrictions depending on n , but we have some freedom in choosing the coupling β , so we can define the free parameter α = βλ (4.31)In table II we list some sample values for the power-law potentials considered in TableI. N = 60 is assumed and an appropriate range of α is chosen for each power n .Power n n s Parameter α range r in α range4 0 . ≤ n s ≤ . ≤ α ≤
20 0 . ≥ r ≥ . . ≤ n s ≤ .
971 10 ≤ α ≤ . ≥ r ≥ . . ≤ n s ≤ . . ≤ α ≤ . ≥ r ≥ . . ≤ n s ≤ . − ≤ α ≤ − . ≥ r ≥ . . ≤ n s ≤ .
977 10 − ≤ α ≤ .
05 0 . ≥ r ≥ . . ≥ n s ≥ . − ≤ α ≤ − . ≥ r ≥ . Table II . n s and r in an appropriate range for α in each case.Notice that n s varies in very narrow intervals, retaining almost the same value in eachcase. The quartic potential presents better values for n s compared to the previousmodel, but the tensor-to-scalar ratio becomes larger that in the previous model, mov-ing away from the values favored by the latest observations. The quadratic potentialmaintains its viability in the present model, although r increases a bit with respectto the model (4.16). From the expressions (4.29) and (4.30) we find the followingbehavior for n s and r in the strong coupling limit ( β → ∞ )lim β →∞ n s = 8 N − n − N + n , lim β →∞ r = 16 n N + n (4.32)23n the weak coupling limit, β →
0, it is foundlim β → n s = 4 N − n − N + n , lim β → r = 16 n N + n . (4.33)From the expressions for φ E and φ I we find that at the strong coupling limit φ E → (cid:18) n βλ (cid:19) / , φ I → (cid:18) n (8 N + n )4 βλ (cid:19) / (4.34)and at the weak coupling limit, from the slow-roll parameter (cid:15) and N from (4.27),the φ E and φ I fields tend to the constant values φ E → n √ , φ I → n √ N + n √ The slow-roll inflation driven by a single scalar field with non-minimal couplings ofdifferent nature, that lead to second order field equations, have been studied. Thedetailed analysis of the linear and quadratic perturbations for all the interaction termsin the model is given. The second oder action for scalar and tensor perturbationshave been constructed, and the expressions for the scalar and tensor power spectrain terms of the slow-roll parameters have been obtained. In Eq. (3.57) we givethe consistency relation that allows to discriminate the model from the standardinflation with minimally coupled scalar field. The results were applied to some modelswith power-law potential. For the scalar field with quadratic potential, non-minimalcoupling and kinetic coupling to the Einstein tensor (4.1), we have found that thescalar spectral index and the tensor-to-scalar ratio can take values in the regionfavored by the latest observational data [3, 4], as seen in Fig. 2. The quadraticpotential is also considered with kinetic and GB couplings (4.10). In this case n s takes values in the region 0 . (cid:46) n s (cid:46) . r isof the order ≈ − for 60 e -foldings, which falls in the region bounded by [3, 4]. A24eneral monomial potential V ∝ φ n with non-minimal kinetic coupling, F ∝ φ − n andnon-minimal GB coupling F ∝ φ − n , was considered (4.16). For this model it waspossible to find exact analytical expressions for the main quantities in the slow-rollapproximation, and some notable values of n were analyzed. While the predictionsfor n s correspond to the standard chaotic inflation, the results for r could be improveddue to the GB coupling, and particularly, for the quadratic potential it was foundthat the tensor-to-scalar ratio falls in an appropriate range according to the latestrestrictions, as can be seen in table I. Analyzing the behavior of the model (4.16)in the weak and strong coupling limits, it was shown that the inflation is not viablein the strong GB coupling limit, especially because ∆ and k break the slow-rollrestrictions (see (4.23)) and the tensor-to-scalar ratio (4.22) increases substantially,while the kinetic coupling remains consistent with inflation in the strong couplinglimit. The kinetic coupling constant, as a free parameter, can be used to lower thevalue of the scalar field to any desired value at the end, and therefore at the beginning,of inflation, avoiding in this way the problem of large fields in chaotic inflation.Another interesting situation is found when we consider the model (4.24) with apower-law potential V ∝ φ n and non-minimal kinetic coupling ( F = β/φ n +2 ). In thiscase both n s and r depend on the kinetic coupling constant and the model behavesappropriately for any value of the coupling between the weak and strong couplingregimes. In the weak coupling limit we recover the standard chaotic inflation results,and in the strong coupling limit we can see from (4.32) and (4.33) that n s incrementswith respect to its value in the weak limit, and r decreases with respect to its value inthe weak coupling limit. This effect is appreciable, in fact, in the intermediate regimeas seen in table II. Thus, for the quadratic potential the tensor-to-scalar ratio fallsin the region favored by the latest observations [3, 4], since n s can reach a maximumvalue of (4 N − / (4 N +1) and r can reach the minimum value of 16 / (4 N +1)). For thequartic potential V = λφ / n s can reach a maximum value of (2 N − / (2 N +1) and r reaches a minimum value of 16 / (2 N + 1) which, assuming N = 60 gives r = 0 . r < . V ∝ φ n with n ≥ λ , depending on the power n .It is clear that the inclusion of non-minimal kinetic and GB couplings in single scalarfield inflationary scenarios has important consequences for the observable magnitudes,as was shown in the case of monomial potentials (see also [53, 65], [72]-[80]). Furtheranalysis of different single scalar field cosmological scenarios will be considered in thepresence of these couplings. A Basic formulas for the first order perturbations
To analyze the physical phenomena during the period of inflation and make contactwith the observables that originated at that period, we start with the perturbationsaround the homogeneous FRW background of the scalar field and the metric (includ-ing the geometrical quantities derived from it) involved in the inflation. The metricwith its first order perturbation is written as g µν = ¯ g µν ( t ) + h µν ( (cid:126)x, t ) (A.1)where ¯ g µν is the background FRW metric with components¯ g = − , ¯ g i = ¯ g i = 0 , ¯ g ij = a ( t ) δ ij (A.2)and h µν = h νµ is the small perturbation of the metric which satisfies the followingfirst order relation h µν = − ¯ g µρ ¯ g νλ h ρλ , (A.3)that follows from the metric property g µν g νρ = δ ρµ . Writing in components we find h ij = − a ( t ) − hij, h i = a ( t ) − h i , h = − h . (A.4)26he background Christoffel symbols are given by¯Γ ij = ¯Γ i j = ˙ aa δ ij , ¯Γ ij = a ˙ aδij, ¯Γ ijk = 0 (A.5)performing the first order perturbation in the Christoffel symbols for the metric (A.1)we find the following components δ Γ ijk = 12 a ( − a ˙ aδ jk h i + ∂ k h ij + ∂ j h ik − ∂ i h jk ) (A.6) δ Γ ij = 12 a (cid:18) − aa h ij + ˙ h ij + ∂ j h i − ∂ i h j (cid:19) (A.7) δ Γ ij = 12 (cid:16) a ˙ ah δ ij − ∂ i h j − ∂ j h i + ˙ h ij (cid:17) (A.8) δ Γ i = 12 a (cid:16) h i − ∂ i h (cid:17) (A.9) δ Γ i = ˙ aa h i − ∂ i h (A.10) δ Γ = −
12 ˙ h (A.11)and there is a useful formula for the trace of δ Γ δ Γ λλµ = ∂ µ (cid:18) a h ii − h (cid:19) (A.12)In what follows all the calculations will be performed in the Newtonian gauge. Thefirst-order perturbation formalism will be applied to all terms in the general scalar-tensor model described bellow, and here we describe the result for the basic geo-metrical quantities. In the Newtonian gauge, after the standard scalar-vector-tensordecomposition of the metric perturbations (see [89]), it is obtained B = F = 0 , E = 2Φ , A = − , And the metric perturbations take the form h = − , h i = h i = 0 , h ij = − a Ψ δ ij h = 2Φ , h i = h i = 0 , h ij = 2 a − Ψ δ ij (A.13)27eplacing these expressions into the results for the perturbations of the Christoffelsymbols given in Eqs. (A.6)-(A.12) we find δ Γ = ˙Φ , δ Γ i = − ∂ i Φ , δ Γ i = 1 a ∂ i Φ , δ Γ ij = − a ˙ a Φ δ ij − a ˙ a Ψ δ ij − a ˙Ψ δijδ Γ ij = − ˙Ψ δ ij , δ Γ ijk = − ∂ k Ψ δ ij − ∂ j Ψ δ ik + ∂ i Ψ δ jk (A.14)For the curvature tensor R ρσµν = ∂ µ Γ ρσν − ∂ ν Γ ρσµ + Γ ρµλ Γ λσν − Γ ρνλ Γ λσµ , (A.15)The background components are given by¯ R i j = − ¯ R i j = − (cid:16) H + ˙ H (cid:17) , ¯ R i j = − ¯ R ij = a ¨ aδ ij , ¯ R ijk = 0¯ R ijkl = ˙ a ( δ ik δ lj − δ il δ jk ) , ¯ R = ¯ R i = ¯ R i = ¯ R i = ¯ R i = 0 . (A.16)The first order perturbations are given by δR ρσµν = ∂ µ δ Γ ρσν − ∂ ν δ Γ ρσµ + ¯Γ ρµλ δ Γ λσν + δ Γ ρµλ ¯Γ λσν − ¯Γ ρνλ δ Γ λσµ − − δ Γ ρνλ ¯Γ λσµ (A.17)Using (A.5) and (A.14) in (A.17) we find the first-order perturbations for the com-ponents of the curvature tensor δR i j = − δR i j = 1 a ∂ i ∂ j Φ + (cid:16) ¨Ψ + H ˙Φ + 2 H ˙Ψ (cid:17) δ ij (A.18) δR ijk = − δR ij k = ∂ j ˙Ψ δ ik − ∂ i ˙Ψ δ jk + H∂ j Φ δ ik − H∂ i Φ δ jk (A.19) δR i j = − δR ij = − (cid:16) a ¨ a Φ + a ˙ a ˙Φ + 2 a ¨ a Ψ + a ¨Ψ (cid:17) δ ij − ∂ i ∂ j Φ (A.20) δR ijkl = − ∂ k ∂ j Ψ δ il + ∂ k ∂ i Ψ δ jl + ∂ l ∂ j Ψ δ ik − ∂ l ∂ i Ψ δ jk − a ˙ a ˙Ψ δ ik δ lj − a Φ δ ik δ lj − a Ψ δ ik δ lj + 2 a ˙ a ˙Ψ δ il δ kj + 2 ˙ a Φ δ il δ kj + 2 ˙ a Ψ δ il δ kj (A.21) δR = δR i = δR i = δR i = δR i = 0 (A.22)Contracting (A.17) we find the different components of the perturbation of Riccitensor as δR = 1 a ∇ Φ + 3 ¨Ψ + 3 H ˙Φ + 6 ˙ H ˙Ψ , (A.23) δR i = 2 ∂ i ˙Ψ + 2 H∂ i Φ , (A.24)28 R ij = − (cid:16) a ¨ a Φ + 4 ˙ a Φ + a ˙ a ˙Φ + 4 ˙ a Ψ + 2 a ¨ a Ψ + 6 a ˙ a ˙Ψ + a ¨Ψ (cid:17) δ ij − ∂ i ∂ j Φ + ∂ i ∂ j Ψ + ∇ Ψ δ ij . (A.25)For the mixed components it is found δR = − H Φ − H Φ − a ∇ Φ − H ˙Φ − H ˙Ψ − δR i = − ∂ i (cid:16) H Φ + ˙Ψ (cid:17) (A.27) δR ij = − (cid:16) H Φ + 2 ˙ H Φ + H ˙Φ + 6 H ˙Ψ + ¨Ψ (cid:17) δ ij + 1 a ∇ Ψ δ ij − a ∂ i ∂ j (Φ − Ψ) (A.28)And the perturbation for the scalar curvature is given by δR = − (cid:16) H + ˙ H (cid:17) Φ − H ˙Φ − a ∇ Φ − H ˙Ψ − a ∇ Ψ (A.29)For the scalar-tensor models that involve non-minimal couplings of the scalar field tocurvatures, given by general functions f ( φ ), the energy momentum tensor containscovariant derivatives of these functions of the scalar field. Here we give the perturba-tions for expressions that involve two covariant derivatives of functions of the scalarfield. Let’s consider the following derivatives ∇ µ ∇ ν f ( φ ) = ∂ µ ∂ ν f ( φ ) − Γ λµν ∂ λ f ( φ ) , ∇ µ ∇ ν f ( φ ) = g µλ ∇ λ ∇ ν f ( φ ) . (A.30)Then ∇ ∇ f ( φ ) = ∂ ∂ f ( φ ) = ˙ φ f (cid:48)(cid:48) ( φ ) + ¨ φf (cid:48) ( φ ) ∇ ∇ i f ( φ ) = ∇ i ∇ f ( φ ) = 0 ∇ i ∇ j f ( φ ) = − a ˙ a ˙ φf (cid:48) ( φ ) δ ij ∇ ∇ f ( φ ) = − ˙ φ f (cid:48)(cid:48) ( φ ) − ¨ φf (cid:48) ( φ ) ∇ ∇ i f ( φ ) = −∇ ∇ i f ( φ ) = 0 ∇ i ∇ j f ( φ ) = − H ˙ φf (cid:48) ( φ ) δ ij ∇ µ ∇ µ f ( φ ) = − H ˙ φf (cid:48) ( φ ) − ˙ φ f (cid:48)(cid:48) ( φ ) − ¨ φf (cid:48) ( φ ) ∇ ∇ f ( φ ) = ˙ φ f (cid:48)(cid:48) ( φ ) + ¨ φf (cid:48) ( φ ) ∇ ∇ i f ( φ ) = ∇ i ∇ f ( φ ) = 0 ∇ i ∇ j f ( φ ) = − ˙ aa ˙ φf (cid:48) ( φ ) δ ij . (A.31)29ere ’ represents derivative w.r.t. the scalar field φ . Let us consider the perturbationsof the above derivative terms δ (cid:104) ∇ µ ∇ ν f ( φ ) (cid:105) = ∂ µ ∂ ν (cid:104) f (cid:48) ( φ ) δφ (cid:105) − δ Γ λµν ∂ λ f ( φ ) − ¯Γ λµν ∂ λ (cid:104) f (cid:48) ( φ ) δφ (cid:105) (A.32)For the different components we find δ [ ∇ ∇ f ( φ )] = ˙ φ f (cid:48)(cid:48)(cid:48) ( φ ) δφ + ¨ φf (cid:48)(cid:48) ( φ ) δφ + 2 ˙ φf (cid:48)(cid:48) ( φ ) ˙ δφ − ˙Φ f (cid:48) ( φ ) ˙ φ (A.33) δ (cid:104) ∇ ∇ i f ( φ ) (cid:105) = f (cid:48)(cid:48) ( φ ) ˙ φ∂ i δφ + f (cid:48) (0) ∂ ∂ i δφ − ˙ aa f (cid:48) ( φ ) ∂ i δφ − f (cid:48) ( φ ) ˙ φ∂ i Φ (A.34) δ (cid:104) ∇ i ∇ j f ( φ ) (cid:105) = f (cid:48) ( φ ) ∂ i ∂ j δφ + (cid:16) a ˙ a Φ + 2 a ˙ a Ψ + a ˙Ψ (cid:17) ˙ φf (cid:48) ( φ ) δ ij − a ˙ a (cid:16) f (cid:48)(cid:48) ( φ ) ˙ φδφ + f (cid:48) ( φ ) ˙ δφ (cid:17) δij (A.35) δ (cid:104) ∇ ∇ f ( φ ) (cid:105) = − (cid:16) f (cid:48)(cid:48)(cid:48) ( φ ) ˙ φ δφ + f (cid:48)(cid:48) ( φ ) ¨ φδφ + 2 f (cid:48)(cid:48) ( φ ) ˙ φ ˙ δφ + f (cid:48) ( φ ) ¨ δφ − f (cid:48) ( φ ) ˙ φ ˙Φ (cid:17) + 2 (cid:16) f (cid:48)(cid:48) ( φ ) ˙ φ + f (cid:48) ( φ ) ¨ φ (cid:17) Φ (A.36) δ (cid:104) ∇ ∇ i f ( φ ) (cid:105) = − f (cid:48)(cid:48) ( φ ) ˙ φ∂ i δφ − f (cid:48) ( φ ) ∂ i ˙ δφ + Hf (cid:48) ( φ ) ∂ i δφ + f (cid:48) ( φ ) ˙ φ∂ i Φ (A.37) δ (cid:104) ∇ i ∇ j f ( φ ) (cid:105) = 1 a f (cid:48) ( φ ) ∂ i ∂ j δφ + (cid:16) Hf (cid:48) ( φ ) ˙ φ Φ + f (cid:48) ( φ ) ˙ φ ˙Ψ − Hf (cid:48)(cid:48) ( φ ) ˙ φδφ − Hf (cid:48) ( φ ) ˙ δφ (cid:17) δ ij (A.38) δ (cid:104) ∇ µ ∇ µ f ( φ ) (cid:105) = − f (cid:48)(cid:48)(cid:48) ( φ ) ˙ φ δφ − f (cid:48)(cid:48) ( φ ) ¨ φδφ − f (cid:48)(cid:48) ( φ ) ˙ φ ˙ δφ − f (cid:48) ( φ ) ¨ δφ + f (cid:48) ( φ ) ˙ φ ˙Φ+2 f (cid:48)(cid:48) ( φ ) ˙ φ Φ + 2 f (cid:48) ( φ ) ¨ φ Φ + f (cid:48) ( φ ) a ∇ δφ + 6 Hf (cid:48) ( φ ) ˙ φ Φ + 3 f (cid:48) ( φ ) ˙ φ ˙Ψ − Hf (cid:48)(cid:48) ( φ ) ˙ φδφ − Hf (cid:48) ( φ ) ˙ δφ (A.39) δ (cid:104) ∇ ∇ f ( φ ) (cid:105) = − (cid:16) f (cid:48)(cid:48) ( φ ) ˙ φ + f (cid:48) ( φ ) ¨ φ (cid:17) + f (cid:48)(cid:48)(cid:48) ( φ ) ˙ φ δφ + f (cid:48)(cid:48) ( φ ) ¨ φδφ + 2 f (cid:48)(cid:48) ( φ ) ˙ φ ˙ δφ + f (cid:48) ( φ ) ¨ δφ − f (cid:48) ( φ ) ˙ φ ˙Φ (A.40) δ (cid:104) ∇ ∇ i f ( φ ) (cid:105) = 1 a (cid:16) − f (cid:48)(cid:48) ( φ ) ˙ φ∂ i δφ − f (cid:48) ( φ ) ∂ i ˙ δφ + Hf (cid:48) ( φ ) ∂ i δφ + f (cid:48) ( φ ) ˙ φ∂ i Φ (cid:17) (A.41) δ (cid:104) ∇ i ∇ j f ( φ ) (cid:105) = − a Hf (cid:48) ( φ ) ˙ φ Ψ δ ij + 1 a (cid:16) f (cid:48) ( φ ) a ∂ i ∂ j δφ + 2 Hf (cid:48) ( φ ) ˙ φ Φ δ ij + f (cid:48) ( φ ) ˙ φ ˙Ψ δ ij − Hf (cid:48)(cid:48) ( φ ) ˙ φδφδ ij − Hf (cid:48) ( φ ) ˙ δφδ ij (cid:17) (A.42)30 The scalar-tensor model and the equations ofmotion
Using the above basic results for the fundamental geometrical quantities, we canproceed to evaluate the fist order perturbations for the following scalar-tensor modelwith non-minimal coupling to scalar curvature R , non-minimal kinetic coupling to theRicci and scalar curvature through the Einstein tensor G µν and non-minimal couplingto the 4-dimensional Gauss-Bonnet invariant G S = (cid:90) d x √− g (cid:20) F ( φ ) R − g µρ ∂ µ φ∂ ρ φ − V ( φ ) + F ( φ ) G µν ∂ µ φ∂ ν φ − F ( φ ) G (cid:21) (B.1)where G = R − R µν R µν + R µνλρ R µνλρ ,G µν = R µν − g µν R and F ( φ ) = 1 κ + f ( φ ). To obtain the field equations we use the following basic variations δg µν = − g µρ g νσ δg ρσ , (B.2) δ √− g = − √− gg µν δg µν , (B.3) δR = R µν δg µν + g µν ∇ σ ∇ σ δg µν − ∇ µ ∇ ν δg µν , (B.4) δR µν = 12 (cid:0) g µα g νβ ∇ λ ∇ λ δg αβ + g αβ ∇ ν ∇ µ δg αβ − g µβ ∇ α ∇ ν δg αβ − g να ∇ β ∇ µ δg αβ (cid:1) , (B.5) δR αβκλ = 12 (cid:16) ∇ κ ∇ β δg λα + ∇ λ ∇ α δg κβ − ∇ κ ∇ α δg λβ − ∇ λ ∇ β δg κα + R γβκλ δg γα − R γακλ δg βγ (cid:17) . The variation of the GB term requires, additionally, the use of the following Bianchi-related identities ∇ ρ R ρσµν = ∇ µ R σν − ∇ ν R σµ (B.6)31 ρ R ρµ = 12 ∇ µ R (B.7) ∇ ρ ∇ σ R σρ = 12 (cid:3) R (B.8) ∇ ρ ∇ σ R µρνσ = ∇ ρ ∇ ρ R µν − ∇ µ ∇ ν R + R γµλν R λγ − R γµ R γν (B.9) ∇ ρ ∇ µ R ρν + ∇ ρ ∇ ν R ρµ = 12 ( ∇ µ ∇ ν R + ∇ ν ∇ µ R ) − R λµγν R γλ + 2 R λν R λµ , (B.10)which can be obtained directly from the Bianchi identity.Variation with respect to metric gives the field equations R µν − g µν R = κ T µν = κ (cid:0) T φµν + T NMµν + T Kµν + T GBµν (cid:1) , (B.11)where T φµν = − √− g δS φ δg µν , T NMµν = − √− g δS NM δg µν T Kµν = − √− g δS K δg µν , T GBµν = − √− g δS GB δg µν , (B.12)with S φ = (cid:90) d x √− g (cid:20) − g µρ ∂ µ φ∂ ρ φ − V ( φ ) (cid:21) , (B.13) S NM = 12 (cid:90) d x √− gf ( φ ) R, (B.14) S K = (cid:90) d x √− gF ( φ ) G µν ∂ µ φ∂ ν φ, (B.15) S GB = − (cid:90) d x √− gF ( φ ) G , (B.16)where T φµν = ∂ µ φ∂ ν φ − g µν ∂ ρ φ∂ ρ φ − g µν V ( φ ) , (B.17) T NMµν = − f ( φ ) (cid:18) R µν − g µν R (cid:19) − g µν ∇ σ ∇ σ f ( φ ) + ∇ µ ∇ ν f ( φ ) , (B.18) T Kµν = F ∂ ρ φ∂ ρ φ (cid:18) R µν − g µν R (cid:19) + g µν ∇ σ ∇ σ (cid:16) F ∂ ρ φ∂ ρ φ (cid:17) − ∇ ν ∇ µ (cid:16) F ∂ ρ φ∂ ρ φ (cid:17) + F R∂ µ φ∂ ν φ − F (cid:16) R µρ ∂ ν φ∂ ρ φ + R νρ ∂ µ φ∂ ρ φ (cid:17) + F g µν R ρσ ∂ ρ φ∂ σ φ + ∇ ρ ∇ µ (cid:16) F ∂ ν φ∂ ρ φ (cid:17) + ∇ ρ ∇ ν (cid:16) F ∂ µ φ∂ ρ φ (cid:17) − ∇ σ ∇ σ (cid:16) F ∂ µ φ∂ ν φ (cid:17) − g µν ∇ ρ ∇ σ (cid:16) F ∂ ρ φ∂ σ φ (cid:17) , (B.19)32nd for the variation of the GB we find the expression, valid in four dimensions T GBµν = − (cid:16) [ ∇ ν ∇ µ F ] R − g µν [ ∇ σ ∇ σ F ] R − ∇ φ ∇ µ F ] R φν − ∇ φ ∇ ν F ] R φµ + 2[ ∇ λ ∇ λ F ] R µν + 2 g µν [ ∇ φ ∇ γ F ] R φγ − ∇ σ ∇ φ F ] R µφνσ (cid:17) . Taking into account the variations of all the terms in the action (B.1) we can writethe generalized Einstein equations in an arbitrary background as F ( φ ) G µν = ∂ µ φ∂ ν φ − g µν ∂ ρ φ∂ ρ φ − g µν V ( φ ) − g µν ∇ σ ∇ σ f ( φ ) + ∇ µ ∇ ν f ( φ ) F ∂ ρ φ∂ ρ φ (cid:18) R µν − g µν R (cid:19) + g µν ∇ σ ∇ σ (cid:16) F ∂ ρ φ∂ ρ φ (cid:17) − ∇ ν ∇ µ (cid:16) F ∂ ρ φ∂ ρ φ (cid:17) + F R∂ µ φ∂ ν φ − F (cid:16) R µρ ∂ ν φ∂ ρ φ + R νρ ∂ µ φ∂ ρ φ (cid:17) + F g µν R ρσ ∂ ρ φ∂ σ φ + ∇ ρ ∇ µ (cid:16) F ∂ ν φ∂ ρ φ (cid:17) + ∇ ρ ∇ ν (cid:16) F ∂ µ φ∂ ρ φ (cid:17) − ∇ σ ∇ σ (cid:16) F ∂ µ φ∂ ν φ (cid:17) − g µν ∇ ρ ∇ σ (cid:16) F ∂ ρ φ∂ σ φ (cid:17) − (cid:16) [ ∇ ν ∇ µ F ] R − g µν [ ∇ σ ∇ σ F ] R − ∇ φ ∇ µ F ] R φν − ∇ φ ∇ ν F ] R φµ + 2[ ∇ λ ∇ λ F ] R µν + 2 g µν [ ∇ φ ∇ γ F ] R φγ − ∇ σ ∇ φ F ] R µφνσ (cid:17) . (B.20) C First order perturbations of the field equationsin the Newtonian gauge
Notice that in compact notation and using the non-minimal coupling F ( φ ) (instead of f ( φ )) as it appears in the action (B.1) we can write the field equations, after variationof (B.1) with respect to the metric, as T NMCµν + T φµν + T Kµν + T GBµν = 0 (C.1)where T NMCµν is now defined as the energy momentum tensor for the action S NMC = (cid:90) √− gF ( φ ) R. (C.2)Expanding the equation (C.1) on the perturbed metric (A.1), up to first order wefind ˜ T NMCµν + ˜ T φµν + ˜ T Kµν + ˜ T GBµν + δT NMCµν + δT φµν + δT Kµν + δT GBµν = 0 (C.3)33here ”tilde”corresponds to the expressions evaluated on the background metric.Then the first order perturbations of the field equations satisfy the following equation δT µ ( φ ) ν + δT µ ( NMC ) ν + δT µ ( GB ) ν + δT µ ( K ) ν = 0 . (C.4)And now we use the Newtonian gauge to write the perturbations for the energy-momentum tensors. For δT µ ( φ ) ν we find δT φ )0 = ˙ φ Φ − ˙ φδ ˙ φ − V (cid:48) δφδT φ ) i = ∂ i (cid:16) − ˙ φδφ (cid:17) δT i ( φ ) j − δ ij δT k ( φ ) k = 0 δT k ( φ ) k − δT φ )0 = −
4Φ ˙ φ + 4 ˙ φδ ˙ φ − V (cid:48) δφ. (C.5)For δ T µ ( NM ) ν δT NM )0 = − F (cid:18) H (3 H Φ + 3 ˙Ψ) − a ∇ Ψ (cid:19) − ˙ F (3 ˙Ψ+6 H Φ)+3 H δF +3 Hδ ˙ F − a ∇ δF, (C.6) δT NM ) i = ∂ i (cid:16) F ( H Φ + ˙Ψ) + ˙ F Φ − δ ˙ F + HδF (cid:17) , (C.7) δT i ( NM ) j − δ ij δT k ( NM ) k = 1 a (cid:18) ∂ i ∂ j − δ ij ∇ (cid:19) ( F ( − Ψ + Φ) + δF ) . (C.8) δT k ( NM ) k − δT NM )0 = − F (cid:18) (3 H ˙Φ + 3 ¨Ψ) + 2 H (3 H Φ + 3 ˙Ψ) + 6 ˙ H Φ + 1 a ∇ Φ (cid:19) − ˙ F (3 ˙Ψ + 6 H Φ) − F ˙Φ − F Φ + 6( ˙ H + H ) δF + 3 δ ¨ F + 3 Hδ ˙ F − a ∇ δF. (C.9)For δ T µ ( K ) ν δT K )0 = − φ (cid:18) − F ˙ φ (cid:18) − a ∇ Ψ + 18Φ H + 9 H ˙Ψ) (cid:19) − a F H ∇ δφ + 9 H F δ ˙ φ + 92 H ˙ φδF (cid:19) , (C.10) δT K ) i = ∂ i (cid:104) − φ (cid:16) − HF δ ˙ φ + 3 H F δφ − H ˙ φδF + F ˙ φ (cid:16) ˙Ψ + 3 H Φ (cid:17)(cid:17)(cid:105) , (C.11) δT i ( K ) j − δ ij δT k ( K ) k = 1 a (cid:18) ∂ i ∂ j − δ ij ∇ (cid:19) [ − ˙ φ δF − F ¨ φ + HF ˙ φ ) δφ + F ˙ φ ( − Ψ − Φ)] , (C.12)34 T k ( K ) k − δT K )0 = − H ˙ F ˙ φδ ˙ φ −
12 ˙ HF ˙ φδ ˙ φ − HF ˙ φδ ¨ φ + 2 a F ˙ φ ∇ Φ + 2 a ˙ φ ∇ δF + 4 a F ˙ φ ∇ Ψ − H ˙ φ δF − H ˙ φ δ ˙ F + 2 ˙ F ˙ φ (12 H Φ + 3 ˙Ψ)+ 2 F ˙ φ (12 ˙ H Φ + 9 H ˙Φ + 3 ¨Ψ) + 4 a F ¨ φ ∇ δφ − F H ¨ φδ ˙ φ − H ˙ φ ¨ φδF + 4 F ˙ φ ¨ φ (12 H Φ + 3 ˙Ψ) (C.13)For δ T µ ( GB ) ν The perturbations of the GB energy momentum tensor from (B.20) are given by δT µGBν = 4 (cid:16) δ [ ∇ µ ∇ ν f ( φ )] R + [ ∇ µ ∇ ν f ( φ )] δR − δ [ ∇ ρ ∇ ρ f ( φ )] δ µν R − [ ∇ ρ ∇ ρ f ( φ )] δ µν δR − δ [ ∇ µ ∇ ρ f ( φ )] R νρ − ∇ µ ∇ ρ f ( φ )] δR νρ − δ [ ∇ ρ ∇ ν f ( φ )] R µρ − ∇ ρ ∇ ν f ( φ )] δR µρ + 2 δ [ ∇ ρ ∇ ρ f ( φ )] R µν + 2 [ ∇ ρ ∇ ρ f ( φ )] δR µν + 2 δ [ ∇ ρ ∇ σ f ( φ )] δ µν R ρσ +2 [ ∇ ρ ∇ σ f ( φ )] δ µν δR ρσ − δ [ ∇ ρ ∇ σ f ( φ )] R µρνσ − ∇ ρ ∇ σ f ( φ )] δR µρνσ (cid:17) . (C.14)Then using (A.32) and the components (A.33)-(A.42) we find after the correspondingsimplifications δT GB = 24 H ˙ δf ( φ ) − H ˙ f ( φ )Φ − H ˙ f ( φ ) ˙Ψ − a H ∇ δf ( φ ) + 16 a H ˙ f ( φ ) ∇ Ψ , (C.15) δT iGBj = 8 a ∂ i ∂ j (cid:104) − (cid:16) f (cid:48)(cid:48) ( φ ) ˙ φ + f (cid:48) ( φ ) ¨ φ (cid:17) Ψ + Hf (cid:48) ( φ ) ˙ φ Φ + (cid:16) H + ˙ H (cid:17) f (cid:48) ( φ ) δφ (cid:105) + 8 (cid:104) H ¨ δf ( φ ) − a H f (cid:48) ( φ ) ∇ δφ − a ˙ Hf (cid:48) ( φ ) ∇ δφ + 2 H ˙ δf ( φ ) + 2 H ˙ H ˙ δf ( φ ) − a H ˙ f ( φ ) ∇ Φ − H ˙ f ( φ )Φ − H ˙ H ˙ f ( φ )Φ + 1 a ¨ f ( φ ) ∇ Ψ − H ¨ f ( φ )Φ − H ˙ f ( φ ) ˙Φ − H ˙ f ( φ ) ˙Ψ − H ˙ f ( φ ) ˙Ψ − H ¨ f ( φ ) ˙Ψ − H ˙ f ( φ ) ¨Ψ (cid:105) δ ij , (C.16) δT kGBk = 16 a ¨ f ( φ ) ∇ Ψ − a H ˙ f ( φ ) ∇ Φ − a (cid:16) H + ˙ H (cid:17) ∇ δf ( φ ) + 24 H ¨ δf ( φ )+48 H ˙ δf ( φ ) + 48 H ˙ H ˙ δf ( φ ) − H ˙ f ( φ )Φ − H ˙ H ˙ f ( φ )Φ − H ¨ f ( φ )Φ − H ˙ f ( φ ) ˙Φ − H ˙ f ( φ ) ˙Ψ −
48 ˙ H ˙ f ( φ ) ˙Ψ − H ¨ f ( φ ) ˙Ψ − H ˙ f ( φ ) ¨Ψ , (C.17) δT GBi = 8 ∂ i (cid:104) H δf ( φ ) − H ˙ δf ( φ ) + 2 H ˙ f ( φ ) ˙Ψ + 3 H ˙ f ( φ )Φ (cid:105) , (C.18)35 T iGB = 8 a ∂ i (cid:104) H ˙ δf ( φ ) − H δf ( φ ) − H ˙ f ( φ ) ˙Ψ − H ˙ f ( φ )Φ (cid:105) . (C.19) D First order perturbations for the scalar fieldequation of motion.
From the action (B.1) we find the equation of motion for the scalar field as12 F (cid:48) ( φ ) R + ∇ µ ∇ µ φ − V (cid:48) ( φ ) − F (cid:48) ( φ ) G µν ∇ µ φ ∇ ν φ − F ( φ ) G µν ∇ µ ∇ ν φ − F (cid:48) ( φ ) G = 0(D.1)In order to calculate the perturbation of this equation we need the perturbation ofthe GB invariant, which can be evaluated as follows δ G =2 RδR − δg µρ g νσ R µν R ρσ − g µρ g νσ δR µν R ρσ − δg µα g σδ R ναγδ R γσµν + 2 g νβ g ργ g σδ δg αη R ηβγδ R ανρσ − g ργ g σδ δR µβγδ R βµρσ . (D.2)Using the expressions for the perturbation of the metric (A.13) and of the curvatures(A.18)-(A.22) and (A.23)-(A.29) in the Newtonian gauge, and after some algebra wefind δ G = − a H ∇ Φ + 16 a H ∇ Ψ + 16 a ˙ H ∇ Ψ − H Φ − H ˙ H Φ − H ˙Φ − H ˙Ψ − H ˙ H ˙Ψ − H ¨Ψ (D.3)The perturbations of the Einstein tensor, using (A.26)-(A.29), are given by δG = 6 H ˙Ψ + 6 H Φ − a ∇ Ψ (D.4) δG i = − ∂ i (cid:16) ˙Ψ + H Φ (cid:17) (D.5) δG ij = (cid:18) H Φ + 2 H ˙Φ + 6 H ˙Ψ + 6 H Φ + 1 a ∇ (Φ − Ψ) (cid:19) − a ∂ i ∂ j (Φ − Ψ)(D.6)Using the above results, the first-order perturbation for the equation of motion of thescalar field (D.1), in the Newtonian gauge takes the form36 Hδ ˙ φ + 18 F H δ ˙ φ + 12 F H ˙ Hδ ˙ φ + δ ¨ φ + 6 F H δ ¨ φ − H ˙ φ Φ − F H ˙ φ Φ − F H ˙ H ˙ φ Φ − φ Φ − F H ¨ φ Φ − ˙ φ ˙Φ − F H ˙ φ ˙Φ − φ ˙Ψ − F H ˙ φ ˙Ψ − F ˙ H ˙ φ ˙Ψ − F H ¨ φ ˙Ψ − F H ˙ φ ¨Ψ − a ∇ δφ − a F H ∇ δφ − a F ˙ H ∇ δφ − a F H ˙ φ ∇ Φ + 4 a F H ˙ φ ∇ Ψ + 4 a F ¨ φ ∇ Ψ + 12 H Φ F (cid:48) + 6 ˙ H Φ F (cid:48) + 3 H ˙Φ F (cid:48) + 12 H ˙Ψ F (cid:48) + 3 ¨Ψ F (cid:48) + 1 a ∇ Φ F (cid:48) − a ∇ Ψ F (cid:48) + 18 H ˙ φδφF (cid:48) + 12 H ˙ H ˙ φδφF (cid:48) + 6 H ¨ φδφF (cid:48) + 6 H ˙ φδ ˙ φF (cid:48) − H ˙ φ Φ F (cid:48) − H ˙ φ ˙Ψ F (cid:48) + 2 a ˙ φ ∇ Ψ F (cid:48) − H Φ F (cid:48) − H ˙ H Φ F (cid:48) − H ˙Φ F (cid:48) − H ˙Ψ F (cid:48) − H ˙ H ˙Ψ F (cid:48) − H ¨Ψ F (cid:48) − a H ∇ Φ F (cid:48) + 16 a H ∇ Ψ F (cid:48) + 16 a ˙ H ∇ Ψ F (cid:48) − H δφF (cid:48)(cid:48) − HδφF (cid:48)(cid:48) + 3 H ˙ φ δφF (cid:48)(cid:48) + 24 H δφF (cid:48)(cid:48) + 24 H ˙ HδφF (cid:48)(cid:48) + δφV (cid:48)(cid:48) = 0 (D.7) E Second order action for the cosmological per-turbations.
In this section we briefly show the use of the tool
Xpand [86, 90, 91] to verify theresults of the second-order action as presented in [57] and apply this tool to find thesecond order action for the model (2.1) . We use the gauge of the uniform field andthe expression for the perturbed metric ds = − N dt + γ ij ( dx i + N i dt )( dx j + N j dt )where N = 1 + A, N i = ∂ i B, γ ij = a ( t ) e ξ (cid:18) δ ij + h ij + 12 h ik h kj (cid:19) with A , B and ξ scalar perturbations and h ij the tensor perturbation satisfying h ii = 0, h ij = h ji y ∂ i h ij = 0. Let us first focus in the scalar case ( h ij = 0).The above metric can be implemented in Xpand [86] as follows: <
Sqrt[-Detg[]] (cid:0) (1/2)F[ ϕ []]RicciScalarCD[] - (1/2)CD[ − µ ][ ϕ []]CD[ µ ][ ϕ []]- V[ ϕ []] + F1[ ϕ []]EinsteinCD[ − µ , − ν ]CD[ µ ][ ϕ []]CD[ ν ][ ϕ []] - F2[ ϕ []](RicciScalarCD[]^2 - 4 RicciCD[- α ,- β ] RicciCD[ α , β ]+ RiemannCD[- α ,- β , - γ ,- λ ] RiemannCD[ α , β , γ , λ ]) (cid:1)(cid:3) , 2]]];ExtractOrder[ExtractComponents[SplitPerturbations[Lag, MyMetricRules, h], h], 2]//Expand; canceling a large number of the boundary terms and using Eqs. (2.5) y (2.6), the38esult obtained with Xpand can be reduced to: δS s = (cid:90) dtd xa (cid:104) − G T ˙ ξ + F T a ∂ i ξ∂ i ξ + Σ A − A ∂ i ∂ i Ba + 2 G T ˙ ξ ∂ i ∂ i Ba + 6Θ A ˙ ξ − G T A ∂ i ∂ i ξa (cid:105) (E.1)where G T = F − F ˙ φ − H ˙ F . (E.2) F T = F + F ˙ φ − F (E.3)Θ = F H + 12 ˙ F − HF ˙ φ − H ˙ F (E.4)Σ = − F H − H ˙ F + 12 ˙ φ + 18 H F ˙ φ + 48 H ˙ F (E.5)From (E.1) it is easy to obtain the equations of motion for A and B , which are givenby Σ A + 3Θ ˙ ξ − Θ ∂ i ∂ i Ba − G T ∂ i ∂ i ξa = 0 (E.6) A = G T Θ ˙ ξ (E.7)By replacing the equation (E.7) in (E.6) it is obtained ∂ i ∂ i Ba = ΣΘ G T ˙ ξ + 3 ˙ ξ − G T Θ ∂ i ∂ i ξa (E.8)Replacing Eqs. (E.7) and (E.8) in (E.1) after simplifying it is obtained: δS s = (cid:90) dtd xa (cid:34)(cid:32) G T + Σ (cid:18) G T Θ (cid:19) (cid:33) ˙ ξ + F T a ∂ i ξ∂ i ξ − G T Θ ˙ ξ ∂ i ∂ i ξa (cid:35) Omitting total spatial derivatives in the last term, the previous expression can berewritten as δS s = (cid:90) dtd xa (cid:34)(cid:32) G T + Σ (cid:18) G T Θ (cid:19) (cid:33) ˙ ξ + F T a ∂ i ξ∂ i ξ + 2 G T Θ ∂ i ˙ ξ∂ i ξa (cid:35) (E.9)From ddt (cid:20) a G T Θ ∂ i ξ∂ i ξ (cid:21) = ddt (cid:20) a G T Θ (cid:21) ∂ i ξ∂ i ξ + 2 a G T Θ ∂ i ˙ ξ∂ i ξ,
39t follows that the last tern of (E.9) can be rewritten by using the previous expression(omitting total derivative). In this manner one obtains: δS s = (cid:90) dtd x (cid:34) a (cid:32)(cid:32) G T + Σ (cid:18) G T Θ (cid:19) (cid:33) ˙ ξ + F T a ∂ i ξ∂ i ξ (cid:33) − ddt (cid:18) a G T Θ (cid:19) ∂ i ξ∂ i ξ (cid:35) Organizing terms this expression takes the form δS s = (cid:90) dtd xa (cid:34)(cid:32) G T + Σ (cid:18) G T Θ (cid:19) (cid:33) ˙ ξ − a (cid:18) a ddt (cid:18) a G T Θ (cid:19) − F T (cid:19) ∂ i ξ∂ i ξ (cid:35) Defining the quantities F s = 1 a ddt (cid:18) a G T Θ (cid:19) − F T G s = 3 G T + Σ (cid:18) G T Θ (cid:19) , The expression for the second order action takes the form δS s = (cid:90) dtd xa (cid:20) G s ˙ ξ − F s a ∂ i ξ∂ i ξ (cid:21) = (cid:90) dtd xa G s (cid:20) ˙ ξ − c s a ∂ i ξ∂ i ξ (cid:21) where c s = F s G s In order to implement the tensor perturbations h ij in the Xpand algorithm, thefunction
MyMetricRules must be modified as follows:
MyMetricRules = { dg[LI[1], − µ , − ν ] :> Eth[LI[1], − µ , − ν ],dg[LI[2], − µ , − ν ] :> Module[ { α } , Eth[LI[1], − µ , α ]Eth[LI[1], − ν , − α ]] } ; where the fluctuations h ij are implemented with Eth . The explicit expression obtainedfrom the algorithm for the second-order action is: δS T = (cid:90) dtd x (cid:34) a ˙ h ij ˙ h ij F a ˙ h ij ¨ h ij F H + 3 a ˙ h ij ˙ h ij F H + a ˙ h ij ˙ h ij F ˙ H − a ˙ h ij ˙ h ij F ˙ φ − a ¨ h ij F ∂ k ∂ k h ij − a ˙ h ij F H∂ k ∂ k h ij − a∂ k h ij ∂ k h ij F aF H ∂ k h ij ∂ k h ij + aF ˙ H∂ k h ij ∂ k h ij − a∂ k h ij ∂ k h ij F ˙ φ − aF ∂ j ˙ h ik ∂ k ˙ h ij + 2 aF ∂ k ˙ h ij ∂ k ˙ h ij + F ∂ k ∂ k h ij ∂ l ∂ l h ij a − F ∂ j ∂ i h kl ∂ l ∂ k h ij a + 2 F ∂ l ∂ j h ik ∂ l ∂ k h ij a − F ∂ l ∂ k h ij ∂ l ∂ k h ij a (cid:35) ◦ , 15 ◦ and 16 ◦ are zero since ∂ i h ij = 0 (omitting surfaceterms). In addition, the terms 14 ◦ y 17 ◦ cancel each other (omitting surface terms).In this manner it is obtained δS T = (cid:90) dtd x (cid:34) a ˙ h ij ˙ h ij F a ˙ h ij ¨ h ij F H + 3 a ˙ h ij ˙ h ij F H + a ˙ h ij ˙ h ij F ˙ H − a ˙ h ij ˙ h ij F ˙ φ − a ¨ h ij F ∂ k ∂ k h ij − a ˙ h ij F H∂ k ∂ k h ij − a∂ k h ij ∂ k h ij F aF H ∂ k h ij ∂ k h ij + aF ˙ H∂ k h ij ∂ k h ij − a∂ k h ij ∂ k h ij F ˙ φ aF ∂ k ˙ h ij ∂ k ˙ h ij (cid:35) (E.10)Since ddt (cid:16) a ˙ h ij ˙ h ij F H (cid:17) = 3 a H ˙ h ij ˙ h ij F + 2 a ˙ h ij ¨ h ij F H + a ˙ h ij ˙ h ij ˙ F H + a ˙ h ij ˙ h ij F ˙ H then, the fourth term of (E.10) can be rewritten by using the previous expression (upto total derivatives). In this way it is found: δS T = (cid:90) dtd x (cid:34) a (cid:32) F − ˙ F H − F ˙ φ (cid:33) ˙ h ij ˙ h ij − a ¨ h ij F ∂ k ∂ k h ij − a ˙ h ij F H∂ k ∂ k h ij − a∂ k h ij ∂ k h ij F aF H ∂ k h ij ∂ k h ij + aF ˙ H∂ k h ij ∂ k h ij − a∂ k h ij ∂ k h ij F ˙ φ
8+ 2 aF ∂ k ˙ h ij ∂ k ˙ h ij (cid:35) (E.11)The third and sixth terms can be expressed, taking into account the following expres-sions ∂ k (cid:16) a ˙ h ij F H∂ k h ij (cid:17) = 4 aF H∂ k ˙ h ij ∂ k h ij + 4 a ˙ h ij F H∂ k ∂ k h ij (E.12) ddt ( aF H∂ k h ij ∂ k h ij ) = aF H ∂ k h ij ∂ k h ij + a ˙ F H∂ k h ij ∂ k h ij + aF ˙ H∂ k h ij ∂ k h ij + 2 aF H∂ k ˙ h ij ∂ k h ij (E.13)41mitting surface terms and total derivatives, the Eq. (E.11) takes the form δS T = (cid:90) dtd x (cid:34) a (cid:32) F − ˙ F H − F ˙ φ (cid:33) ˙ h ij ˙ h ij − a ¨ h ij F ∂ k ∂ k h ij + 2 aF H∂ k ˙ h ij ∂ k h ij − a∂ k h ij ∂ k h ij F − a ˙ F H∂ k h ij ∂ k h ij − a∂ k h ij ∂ k h ij F ˙ φ aF ∂ k ˙ h ij ∂ k ˙ h ij (cid:35) (E.14)The second term can be rewritten if taking into account the following expressions ∂ k (cid:16) a ¨ h ij F ∂ k h ij (cid:17) = 2 a∂ k ¨ h ij F ∂ k h ij + 2 a ¨ h ij F ∂ k ∂ k h ij ddt (cid:16) a∂ k ˙ h ij F ∂ k h ij (cid:17) = aH∂ k ˙ h ij F ∂ k h ij + a∂ k ¨ h ij F ∂ k h ij + a∂ k ˙ h ij ˙ F ∂ k h ij + a∂ k ˙ h ij F ∂ k ˙ h ij After which, the action (E.14) becomes δS T = (cid:90) dtd x (cid:34) a (cid:32) F − ˙ F H − F ˙ φ (cid:33) ˙ h ij ˙ h ij − a∂ k ˙ h ij ˙ F ∂ k h ij − a∂ k h ij ∂ k h ij F − a ˙ F H∂ k h ij ∂ k h ij − a∂ k h ij ∂ k h ij F ˙ φ (cid:35) (E.15)The fourth term can be rewritten if taking into account that ddt (cid:16) a ˙ F ∂ k h ij ∂ k h ij (cid:17) = a ˙ F H∂ k h ij ∂ k h ij + a ¨ F ∂ k h ij ∂ k h ij + 2 a ˙ F ∂ k ˙ h ij ∂ k h ij using this expression and simplifying, it follows that δS T = 18 (cid:90) dtd xa (cid:34) (cid:16) F − F H − F ˙ φ (cid:17) ˙ h ij ˙ h ij − a (cid:16) F − F + F ˙ φ (cid:17) ∂ k h ij ∂ k h ij (cid:35) , and using the definitions (E.2) y (E.3), the final expression for the second-order actiontakes the form δS T = 18 (cid:90) dtd xa (cid:34) G T ˙ h ij ˙ h ij − F T a ∂ k h ij ∂ k h ij (cid:35) (E.16)which gives the velocity of the tensor perturbations as c T = F T G T (E.17)42 The slow-roll inflation for the minimally coupledscalar field
In general for a second order action S (2) = (cid:90) dtd xa G s (cid:20) ˙ ξ − c s a ( ∇ ξ ) (cid:21) (F.1)one finds the equation of motion of the scalar perturbation as ddt (cid:0) a G s (cid:1) ˙ ξ + a G s ¨ ξ − ac s G s ∇ ξ = 0 (F.2)which can be written as ¨ ξ + 1 a G s ddt (cid:0) a G s (cid:1) ˙ ξ − c s a ∇ ξ = 0 (F.3)or in Fourier modes ( ∇ → − k )¨ ξ k + 1 a G s ddt (cid:0) a G s (cid:1) ˙ ξ k + c s a k ξ k = 0 (F.4)where k is the wave number k = 2 π/λ . For small k beyond the horizon, i.e. c s k < For the canonical scalar field the Friedman and field equations can be reduced to H = 13 M p (cid:18) 12 ˙ φ + V ( φ ) (cid:19) (F.12)˙ H = − M p ˙ φ (F.13)44o analyze the second order action in this case we set F = 1 /κ = M p , F = F = 0in (3.1)-(E.5), which givesΣ = − F H + 12 ˙ φ , Θ = M p H, F T = M p , G T = M p . (F.14)Therefore G s = ˙ φ H (F.15)and F s = M p a ddt (cid:16) aH (cid:17) − M p = − M p ˙ HH , (F.16)giving c s = 1. The second order action for this simplified case takes the form S (2) = (cid:90) dtd xa ˙ φ H (cid:20) ˙ ξ − a ( ∇ ξ ) (cid:21) (F.17)which in the Mukhanov variables becomes S (2) = (cid:90) dτ d x (cid:20) v (cid:48) + z (cid:48)(cid:48) z v − ( ∇ v ) (cid:21) (F.18)The Mukhanov equation (F.9) for the case of canonical scalar field simplifies takinginto account (F.15) and z = a ˙ φH . (F.19)Then, z (cid:48) = dzdτ = a dzdt = a (cid:32) ˙ a ˙ φH + a ¨ φH − a ˙ φ ˙ HH (cid:33) . On the other hand, the standard slow roll parameters for this case are defined as (cid:15) = − ˙ HH = ˙ φ M p H (F.20)and η = ˙ (cid:15)H(cid:15) = 2 ¨ φ ˙ φH − ˙ HH = 2 ( (cid:15) − δ ) (F.21)where δ = − ¨ φ ˙ φH (F.22)45o, if (cid:15), δ << η << z (cid:48) /z may be written as z (cid:48) z = aH (cid:32) φH ˙ φ − ˙ HH (cid:33) = aH (1 − δ + (cid:15) ) (F.23)Taking the derivative with respect to τ one finds ddτ (cid:18) z (cid:48) z (cid:19) = z (cid:48)(cid:48) z − (cid:18) z (cid:48) z (cid:19) = a ddt [ aH (1 − δ + (cid:15) )]= a H (cid:34) − δ + (cid:15) − (cid:15) (1 − δ + (cid:15) ) + ˙ (cid:15)H − ˙ δH (cid:35) (cid:39) a H (cid:20) − δ + (cid:15) − (cid:15) (1 − δ + (cid:15) ) + ˙ (cid:15)H (cid:21) where we have used ˙ a = aH and in the last equality we have neglected ˙ δ . Then, using(F.23), up to first order in slow roll parameters one can write z (cid:48)(cid:48) z (cid:39) a H (2 + 2 (cid:15) − δ ) . (F.24)Note that form the equality ddτ ( aH ) − = (cid:15) − , if we consider that (cid:15) varies very slowly with time, i.e. is quasi constant, then onefinds ( aH ) − = ( (cid:15) − τ ⇒ τ = 1 aH (cid:15) − 1) (F.25)which is the conformal time. Note that in de Sitter (cid:15) = 0 and one has τ dS = − aH so the comovil horizon is equal to the conformal time, and then neglecting the slowroll parameters, the following approximation z (cid:48)(cid:48) z (cid:39) a H (cid:18) (cid:15) − δ (cid:19) (cid:39) a H = 2 τ dS takes place in de Sitter. But taking into account the slow roll parameters and using(F.25) we find z (cid:48)(cid:48) z (cid:39) τ (cid:15) − δ (1 − (cid:15) ) = 1 τ (cid:18) µ − (cid:19) (F.26)46here µ = 14 + 2 + 2 (cid:15) − δ (1 − (cid:15) ) (cid:39) 94 + 6 (cid:15) − δ. Then µ (cid:39) 32 + 2 (cid:15) − δ Using (F.26) in the Mukhanov equation (F.11) with c s = 1 we find in the Fouriermodes v (cid:48)(cid:48) k + k v k − τ (cid:18) µ − (cid:19) v k = 0 . (F.27)First note that deep inside the horizon, when the condition k >> aH or τ → −∞ isfulfilled, the mode equation becomes v (cid:48)(cid:48) k + k v k = 0 . (F.28)which allows the quantization of the mode function in complete analogy with thequantization of (massless) scalar field on Minkowski background. Then, the choice ofvacuum as the minimum energy state and the positivity of the normalization conditionfor the fluctuations v k [10, 12, 92, 93] leads to the unique plane-wave solution v k = 1 √ k e − ikτ . (F.29)This solution can be used as a boundary condition (at k >> aH ) for the generalsolution of Eq. (F.27). Assuming µ constant for slowly varying slow-roll parameters,the general solution to the equation (F.27) is given by v k = √− τ (cid:104) c k H (1) µ ( − kτ ) + c k H (2) µ ( − kτ ) (cid:105) (F.30)where H (1) µ and H (2) µ are the Hankel functions of the first and second kind respectively.These functions have the following asymptotic behavior H (1) µ ( x >> (cid:39) (cid:114) πx e i ( x − π µ − π ) (F.31) H (2) µ ( x >> (cid:39) (cid:114) πx e − i ( x − π µ − π ) (F.32)47aking x = − kτ , if x >> k >> aH which corresponds to sub horizon scales.Then imposing (F.29) as the boundary condition at − kτ >> 1, it is found that c k = √ π e i π ( µ + ) , c k = 0and the general solution takes the form v k = √ π e i π ( µ + ) √− τ H (1) µ ( − kτ ) (F.33)On the other hand, on super horizon scales where k << aH ( x << H (1) µ ( x ) = (cid:114) π e − i π µ − Γ( µ )Γ( ) x − µ (F.34)and replacing in (F.33) we find the solution v k = e i π ( µ − ) µ − Γ( µ )Γ( ) 1 √ √− τ ( − kτ ) − µ (F.35)To evaluate the power spectra we find from (F.23) and (F.25) z (cid:48) z = (1 − δ + (cid:15) ) (cid:15) − τ (F.36)for slowly varying slow roll parameters one finds z ∝ τ − µ (F.37)where µ = 32 + 2 (cid:15) − δ. Assuming µ (cid:39) / µ − and Γ( µ ) in (F.35) gives v k = 1 √ e iπ/ √− τ ( − kτ ) − µ . (F.38)Then in the super horizon regime ξ k = v k z ∝ τ k − µ = k − − (cid:15) + δ (F.39)48epending only on k , which agrees with the solution ξ k = const. on super horizonscales (see (F.6)). Then for the power spectra we find P ξ = k (2 π ) | ξ k | ∝ k δ − (cid:15) (F.40)and the scalar spectral index is given by n s − d ln P ( ξ ) d ln k = 2 δ − (cid:15) (F.41)where the scale invariance corresponds to n s = 1. Acknowledgments This work was supported by Universidad del Valle under project CI 71074 and byCOLCIENCIAS Grant No. 110671250405. DFJ acknowledges support from COL-CIENCIAS, Colombia. References [1] . A. R. Ade et al., Planck Collaboration (Planck 2013 results. XXII. Constraintson inflation), Astron. and Astrophys. (2014) A22; arXiv:1303.5082 [astro-ph.CO][2] P. A. R. Ade et al., Planck Collaboration (Planck 2015 results. XX. 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