L.N. Granda , D. F. Jimenez
Departamento de Fisica, Universidad del Valle
A.A. 25360, Cali, Colombialuis.firstname.lastname@example.org@correounivalle.edu.co
The linear and quadratic perturbations for a scalar-tensor model with non-minimal coupling to curvature, coupling to the Gauss-Bonnet invariant and non-minimal kinetic coupling to the Einstein tensor are developed. The quadratic action for the scalar and tensor perturbations is constructed and the power spectra for the primordial scalar and tensor fluctuations are given. A consistency relation that is useful to discriminate the model from the standard inflation with canonical scalar field was found. For some power-law potentials it is shown that the Introduction of additional interactions, given by non-minimal, kinetic and Gauss-Bonnet couplings, can lower the tensor-to-scalar ratio to values that are consistent with latest observational constraints, and the problem of large fields in 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 theory gives by now the most likely scenario for the early universe, since it provides the explanation to flatness, horizon and monopole problems, among others, for the standard hot Bing Bang cosmology [8, 9, 10, 11, 12, 13]. In other words, the inflation can 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. Besides that, the quantum fluctuations during inflation could provide the seeds for the large scale structure and the observed CMB anisotropies [14, 15, 16, 17, 18, 19, 20]. In particular, inflation allows as to understand how the scale-invariant power spectrum can be generated, though it does not predict an exact scale invariant but nearly scale invariant power spectrum. The deviation from scale invariance is connected with the microphysics description of the inflationary theory which is still incomplete.
The simplest and most studied model of inflation consists of minimally-coupled scalar field 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 [21, 22, 23], kinetic inflation , vector inflation [25, 26, 27], inflaton potential in supergravity [28, 29, 30], string theory inspired inflation [31, 32, 33], Dirac-Born-Infeld inflation model [34, 35, 36, 37], α-attractor models originated in supergravity [38, 39, 40, 41]. Apart from the DBI models of inflation, another class of ghost-free models has been recently considered, named ”Galileon” models [42, 43]. In spite of the higher derivative nature of these models, the gravitational and scalar field equations contain derivatives no higher than two. The effect of these Galileon terms is mostly reflected in the modification of the kinetic term compared to the standard canonical scalar field, which in turn can improve (or relax) the physical constraints on the potential. In the case of the Higgs potential, for instance, one of the effects of the higher derivative terms is the reduction of the self coupling of the Higgs boson, so that the spectra of primordial density perturbations are consistent with the present observational data [44, 45] (which is not possible within the standard canonical scalar field inflation with Higgs potential). Different aspects aspects of Galilean-inflation have been considered in [44, 45, 46, 47, 48, 49]. A particular and important case belonging to the above class of models is the scalar field with kinetic coupling to the Einstein tensor [50, 51, 52, 53, 54] whose application in the context of inflationary cosmology has been analyzed in [55, 56, 57].
In the present paper we consider a scalar-tensor model with non-minimal coupling to scalar curvature, non-minimal kinetic coupling to the Einstein tensor and coupling of the scalar field to the Gauss-Bonnet 4-dimensional invariant, to study the slow-roll inflation and the observable magnitudes, the scalar espectral index and the tensor-to-scalar ratio, derived from it. For studies of inflation with GB coupling see, for instance
[58, 59, 60, 61, 62, 63, 64].
The interaction terms presented in this model have direct correspondence with terms presented in Galileon theories  and also give rise to second order gravitational and scalar field equations.
In the appendixes we develop in detail the linear and quadratical perturbations for all the interaction terms of the model and deduce the second order action for the scalar and tensor perturbations.
In appendix A we present the basic formulas for the first order perturbations, needed for the model, in the Newtonian gauge. In appendix B we deduce the gravitational and scalar field equations in a general background. In appendix C and D we give the first order perturbations of the field equations in the Newtonian gauge. In Appendix E we give the details for constructing the second order action using the Xpand tool [syril], and
in appendix F we give a detailed description of the slow-roll mechanism for the minimally coupled scalar field.
The expressions for the primordial density fluctuations in terms of the slow-roll parameters and the corresponding power spectra were found. We have found a consistency relation that is useful to discriminate the model from the standard inflation with canonical scalar field. The latest observational data disfavor monomial-type models V∝ϕn with n≥2 in the minimally coupled scalar field. With the Introduction of additional interactions like the non-minimal coupling, kinetic coupling and Gauss-Bonnet coupling (GB), it is shown that the tensor-to-scalar ratio can be lowered to values that are consistent with latest observational constraints [2, 3]. This is sown in the case of quadratic potential with non-minimal and kinetic coupling, quadratic potential with kinetic and GB coupling and the general power-law potential with GB coupling.
The paper is organized as follows. In the next section we introduce the model, the background field equations and define the slow-roll parameters. In section 3 we use quadratic action for the scalar and tensor perturbations (details are given in the appendix) to evaluate the primordial power spectra. In section 4 we work some explicit models. Some discussion is given in section 5.
2 The model and background equations
We consider the scalar-tensor model with non-minimal coupling of the scalar field to curvature, non-minimal kinetic coupling of the scalar field to the Einstein’s tensor and coupling of the scalar field to the Gauss-Bonnet (GB) 4-dimensional invariant
where Gμν is the Einstein’s tensor, G is the GB 4-dimensional invariant given by
One remarkable characteristic of this model is that it yields second-order field equations and can avoid Ostrogradski instabilities. Using the general results of Appendix B, expanded on the flat FRW background
one finds the following equations
where (′) denotes derivative with respect to the scalar field. Related to the different terms in the action (2.1) we define the following slow-roll parameters
The slow-roll conditions in this model are satisfied if all these parameters are much smaller than one, and will be used in the next section.
From the cosmological equations (2.5) and (2.6) and using the parameters (2.8)-(2.11) we can write the following expressions for ˙ϕ2 and V
where we used
It is also useful to define the variable Y from Eq. (2.13) as
where it follows that Y=O(ε). Notice that for the simplest case of minimally coupled scalar field (F=1/κ2, F1=F2=0), the Eqs. (2.12) and (2.13) give the standard equations
Under the slow-roll conditions ¨ϕ<<3H˙ϕ and ℓi,ki,Δi<<1, it follows from (2.5)-(2.7)
showing that the potential V gives the dominant contribution to the Hubble parameter, while Eqs. (2.17) and (2.18) determine the dynamics of the scalar field in the slow-roll approximation. The number of e-folds can be determined from
where ϕI and ϕE are the values of the scalar field at the beginning and end of inflation respectively, and the expression for ˙ϕ was taken from (2.18). The criteria for choosing the initial values will be discussed below.
3 Quadratic action for the scalar and tensor perturbations
After the computation of the second order perturbations we are able to write the second order action for the scalar perturbations as follows
And the sound speed of scalar perturbations is given by
The conditions for avoidance of ghost and Laplacian instabilities as seen from the action (3.1) are
We can rewrite GT, FT, Θ and Σ in terms of the slow-roll parameters (2.8)-(2.11) and using Eqs. (2.13) and (2.14), as follows
The expressions for GS and c2S in terms of the slow roll parameters can be written as
Notice that in general GS=FO(ε) and c2S=1+O(ε). Also in absence of the kinetic coupling it follows that c2S=1+O(ε2). Keeping first order terms in slow-roll parameters, the expressions for GS y c2S reduce to
To normalize the tensor perturbations we perform the change of variables [kobayashi] (see (F.7))
Then, under the approximation of slowly varying cS and up to first-order in slow-roll variables we find the following expression for ~z′′/~z
This expression reduces to the one of the canonical scalar field given in Appendix I, Eq. (F.24), in the case ℓ0=Δ0=0 where cS=1 and ϵ1=2(ϵ0−δ), with δ defined in (F.22). In what follows the reasoning is similar to the simplest case, corresponding to minimally-coupled scalar field, which is analyzed in detail in Appendix I. So on sub-horizon scales when the k2 term dominates in Eq. (3.21) we choose the same Bunch-Davies vacuum solution defined for the scalar field, which leads to
Note that from the expression
in the approximation of slowly varying cS and ϵ0 one can integrate the last equation to obtain
Then in the limit ϵ→0 for de Sitter expansion it follows that
In this last case and neglecting the slow-roll parameters (in this limit cS=1) we can write from (3.23)
which allows the integration of Eq. (3.21), giving the known solution for the scalar perturbations in a de Sitter background.
Taking into account the slow-roll parameters and using (3.26) we can rewrite the Eq. (3.21) in the form
where we have expanded up to first order in slow-roll parameters. The general solution of Eq. (3.30) for constant μs (slowly varying slow-roll parameters) is
and after matching the boundary condition related with the choosing of the Bunch-Davies vacuum (3.24) we find the solution
using the asymptotic behavior of H(1)μs(x) at x>>1, we find at super horizon scales (cSk<<aH)
To evaluate the power spectra we use the relationship
where we used (3.26) for aH, and for the last equality we have expanded up to first order in slow-roll parameters, resulting in
Assuming again the approximation of slowly varying slow-roll parameters we can Integrate this equation to find
which gives, in the super horizon regime, for the amplitude of the scalar perturbations the following expression
where the τs dependence disappears as expected from the solution (3.33) in super horizon scales (csk<<aH). The power spectra for the scalar perturbations takes the following k-dependence
It is worth noticing that the slow-roll parameter k0, related to the kinetic coupling, do not appear in the above expression for the scalar spectral index. This is because k0 appears only in second order terms (or higher) in the expressions for GS and FS (see (3.13) and (3.14)).
The second order action for the tensor perturbations takes the form
where GT and FT are defined in (E.2) and (E.3) (in terms of the slow-roll variables in (3.18) and (3.19)) and the velocity of tensor perturbations is given by
As in the case of scalar perturbations, in order to canonically normalize the tensor perturbations the following variables are used 
leading to the quadratic action
which gives the equation
Or for the corresponding Fourier modes
which is of the same nature as the equation for the scalar perturbations, and therefore the perturbations hij 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 for the scalar perturbations.
To this end we write the expression for z′′T/zT, up to first order in slow-roll parameters, as follows
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
where the tensor e(k)ij describe the polarization states of the tensor perturbations for the k-mode, and
At super horizon scales (cTk<<aH) the tensor modes (3.47) have the same functional form for the asymptotic behavior as the scalar modes (3.33), and therefore we can write power spectrum for tensor perturbations as
where h(k)ij=v(k)ij/zT, and the sum over the polarization states must be taken into account.
Then, the tensor spectral index will be given by
An important quantity is the relative contribution to the power spectra of tensor and scalar perturbations, defined as the tensor/scalar ratio r
For the scalar perturbations, using (3.38), we can write the power spectra as
and all magnitudes are evaluated at the moment of horizon exit when csk=aH (kτs=−1). For ~z we used (3.18) with a=cSk/H. In analogous way we can write the power spectra for tensor perturbations as
Noticing that AT/AS≃1 when evaluated at the limit ϵ0,ℓ0,Δ0,...<<1, as follows from (3.35) and (3.48), we can write the tensor/scalar ratio as follows
taking into account the expressions for GT,FT,GS,FS given in (3.9), (3.10), (LABEL:slr8e) and (LABEL:slr8f), up to first order, and using the condition ϵ0,ℓ0,k0,Δ0<<1, then we can see that cT≃cS≃1 (in fact in the limit ℓ0→0, cS=1 independently of the values of ϵ0 and Δ0) and we can make the approximation
which is a modified consistency relation due to the non-minimal and GB couplings. In the limit ℓ0,Δ0→0 it gives the expected consistency relation for the standard inflation
with nT=−2ϵ0. Taking into account the non-minimal and GB couplings we find the deviation from the standard consistency relation in the form
with nT given by (3.50). Thus, the consistency relation is still valid in the case of non-minimal coupling, and if there is an observable appreciable deviation from the standard consistency relation, it can reveal the effect of an interaction beyond the the simple canonical scalar field
or even non-minimally coupled scalar field models of inflation.
It is worth noticing that in the first-order formalism the kinetic-coupling related slow-roll parameter k does not appear in the spectral index for the scalar and tensor perturbations and is also absent in the tensor-to-scalar ratio, appearing only starting form the second order expansion in slow-roll parameters. Nevertheless, all the couplings are involved in the definition of the slow-roll parameters trough the field equations. Of special interest are the cases of monomial potentials V∝ϕn. These potentials are disfavored by the observational data for n≥2 in the minimally coupled model. As will be shown for some cases, with the GB and (or) kinetic coupling added, the spectral index and especially the scalar-to tensor ratio can be accommodated within the range of values obtained from the latest observational data.
4 Some explicit cases
First we consider the particular case of the non-minimal coupling ξϕ2 with quadratic potential and kinetic coupling with constant F1.
Using 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 coupling constants. For the model (4.1) the slow-roll parameters take the form
where ϕ is dimensionless (ϕ has been rescaled as κϕ→ϕ to measure it in units of Mp) and γ has dimension of mass−2 . Additionally, the scalar field at the end of inflation can be evaluated under the condition ϵ0(ϕE)=1.
Sitting ϵ0=1 in (4.2) it follows
From Eq. 2.19 it follows that the number of e-foldings can be evaluated as
giving an approximate relation between the values of the scalar field at the beginning and end of inflation as
So, assuming N=60 gives ϕI≈3.9ϕE. This will have sense only if the scalar spectral index and the tensor-to-scalar ratio behave properly. In fact from (4.2) and replacing in (3.39) and (3.55), we find (under the condition ξϕ2<<1 and m2γ>>ξ)
where we have used (4.6) for ϕI. Additional simplification can be made if we assume that the scalar field at the beginning of inflation is of the order of Mp (ϕ≃1). This can be achieved if m2γ=8N+2, as follows from (4.6), which gives
Thus, for 60 e-foldings we find ns≈0.98 and r≈0.067 (ξ=10−2). In this case the inflation begins with ϕI=Mp and ends with ϕE≈0.25Mp.
For the numerical analysis with the exact expressions, we assume N=60, m=10−6Mp. In fact from Eqs. (4.2) follows that the spectral index ns and the tensor-to-scalar ratio depend on the dimensionless combination m2γ. Fig. 1 shows the behavior of ns and r in the interval 102<m2γ<5×102, for ξ<0.1/6,0.2.
Figure 1: The behavior of the scalar spectral index ns and r as function of m2γ for some values of ξ. The horizontal lines correspond to the upper limit of the observational quotes from Planck 2015, with values ns=0.968±0.006 and r<0.1
The following example considers a model with kinetic and GB couplings
where the constant η has dimension of mass2 and ϕ is measured in units of Mp. The slow-roll parameters from , necessary to evaluate ns and r, take the form
And from (3.55) and (4.11) we find the expression for the tensor-to-scalar ratio as
For N=60 and taking m=10−6Mp we can find the behavior of ns and r in terms of the dimensionless parameter m2γ. In Fig. 2 we show the behavior of the scalar field at the beginning and end of inflation for 1<m2γ<5
Figure 2: The values of the scalar field at the beginning and end of inflation for 1<m2γ<5 and m2η=0,67,0.7 in units of M4p.
In Fig. 3 we show the corresponding behavior of ns and r.
Figure 3: The values ns and r in the interval 1<m2γ<1 for m2η=0,67,0.7 (in units of M4p). The horizontal line is the upper limit set by Planck 2015.
The following model considers the general power-law potential and non-minimal power-law functions for the GB and kinetic couplings