G-inflation: From the intermediate, logamediate and exponential models

The intermediate, logamediate and exponential inflationary models in the context of Galileon inflation or G-inflation are studied. By assuming a coupling of the form $G(\phi,X)\propto\phi^{\nu}\,X^{n}$ in the action, we obtain different analytical solutions from the background cosmological perturbations assuming the slow-roll approximation. General conditions required for these models of G-inflation to be realizable are determined and discussed. In general, we analyze the condition of inflation and also we use recent astronomical and cosmological observations for constraining the parameters appearing in these G-inflationary models.


I. INTRODUCTION
It is well known that the inflationary epoch [1][2][3][4][5][6][7] provides more than the mechanism for solving the problems of the hot big bang model (flatness, horizon etc). In this sense, one of the achievements of the inflationary universe is to provide the primordial curvature perturbations, which seed the observed cosmic microwave background (CMB) temperature anisotropies [8][9][10][11][12][13][14][15] and the structure formation of the universe, that are generated from vacuum fluctuations of the scalar field which drives the accelerated expansion [16][17][18][19][20][21]. One can test the inflationary paradigm by comparing the theoretical predictions for various models of inflation with current astrophysical and cosmological observations, in particular those that come from the CMB temperature anisotropies. In doing so, the predictions of representative inflationary models, given on the n s − r plane, are compared with the allowed contour plots from the observational data. In this context, the BICEP2/Keck-Array collaboration [22] published new more precise data regarding the CMB temperature anisotropies, improving the upper bound on the tensor-to-scalar ratio to be r 0.05 < 0.07 (95% CL) in comparison to latest data of Planck [15], for which r 0.002 < 0.11 (95% CL).
On the other hand, in the context of exact inflationary solutions, one of the more interesting are found by using an exponential potential for the inflaton, yielding a power-law evolution of the scale factor in cosmic time, i.e., a(t) ∝ t p , where p > 1 [23]. Another exact solution corresponds to de Sitter inflation in which the effective potential is a constant [2].
We also have an exact solution for an inverse power-law potential. Here, the inflationary stage can be described by the intermediate inflation model, in which the scale factor has the following dependence on cosmic time [24][25][26].
where A and f are constant parameters, satisfying the conditions A > 0 and 0 < f < 1.
This intermediate expansion law becomes slower than de Sitter inflation, but faster than power-law inflation instead. In addition, a generalized inflation model is provided by the model of logamediate inflation, in which the scale factor evolves as [27] a(t) = exp B ln(t) λ , here, B and λ are dimensionless constant parameters such that B > 0 and λ > 1. Note that for the special case λ = 1 and B = p, the logamediate inflation model reduces to power-law inflation with an exponential potential [23].
Originally, these inflationary models were studied as exact solutions of background evolution. However, the slow-roll formalism provides a better analysis regarding the dynamics of primordial perturbations. In practice, these models are completely ruled out by current observational data [15] in the standard canonical inflationary scenario. In particular, for the intermediate inflation model, it was found that for the special case f = 2/3, the scalar spectral index becomes n s = 1, corresponding to the Harrison-Zel'dovich spectrum, being not supported by current data. Also, an observational consequence is that for both inflationary models, the tensor-to-scalar ratio r, becomes significantly r = 0, but this ratio is always r > 0.1, as it was shown in [26,27]. If we go further the standard cold scenario, e.g., in the warm inflation scenario, both intermediate and logamediate models may be reconciled with current observations available at that time [28][29][30][31].
Instead of considering the parametrization of the scale factor as function the cosmic time, alternatively the authors in Ref. [32] introduced an explicit expression for the Hubble rate.
Here, they studied a Hubble parameter having an exponential dependence on cosmic time of the form where α denotes the value of the Hubble rate when cosmic time tends to zero and β is a constant parameter, such that β > 0. On the contrary of the intermediate and logamediate inflation models, this exponential Hubble rate has the novelty of addressing the end of inflation [32]. Nevertheless, regarding the predictions for this model on the n s − r plane, the trajectory lie outside the 95 % CL region, being completely ruled out by current observations.
On the other hand, going beyond the standard canonical inflation scenario, a noncanonical inflation model, whose Lagrangian contains higher derivative terms, has become of a special interest from the theoretical and observational points of view, yielding a large or small amount of non-Gaussianities and a non-trivial speed of sound. A special class of such a models, dubbed Galileon inflation models or G-inflation, were inspired by theories exhibiting "Galilean" symmetry, ∂ µ φ → ∂ µ φ + b µ [33]. Interestingly, the field equations derived from such a theories still contain derivatives up to second order, avoiding ghosts [33].
In the framework of modified gravity theories having extra degrees of freedom, the action for linearized gravitational waves (GWs) reads where M * is an effective Planck mass which would depend on the particular theory under consideration, and h A are the amplitudes of the polarization states of the perturbations h µν around the Minkowski space. The quantity c T corresponds to the speed of the GW, which can be parameterized more convenient as c 2 T = 1 + α T . By combing the gravitational wave event GW170817 [50], observed by the LIGO/Virgo collaboration, and the gamma-ray burst GRB 170817A [51], it has been possible to strongly constrain the speed of GWs, determining that GWs propagate at the speed of light with |α T | 10 −15 [52]. As a direct consequence for Horndeski's theory, is that a large model space of this theory has been eliminated. Specifically, all the terms that lead to non-minimal kinetic couplings are ruled out, leaving this theory constructed only with k-essence, cubic Galileon and non-minimally coupling sectors, in which the Lagrangian density can be written as [52,53] In [54], the authors explored the viability of considering the intermediate inflation model in the framework of G-inflation, with a cubic Galileon term of the form G(φ, X) φ ∝ X n φ.
Interestingly, it was found the compatibility of this model with Planck 2015 data. However, regarding the discussion in [25,26] on how to evaluate correctly the observable in intermediate inflation, the authors in [54] did not evaluate carefully such an observables, yielding a misleading constraint on the parameter space of the model assuming that intermediate inflation ends.
The main goal of the present article is to explore the observational consequences of studying the intermediate, logamediate and exponential Hubble inflation models in the framework of the cubic Galileon and how these models are modified with the coupling G(φ, X). In doing so, we consider a coupling of the form G(φ, X) ∝ φ ν X n , which generalizes the cases G(φ, X) ∝ φX and G(φ, X) ∝ X n already studied in Refs. [44] and [54], respectively. We will show that, for each inflation model studied, there exist a region in the space of param-eters for which its predictions lie inside the allowed region from BICEP2/Keck-Array data, resurrecting these inflationary models. In addition, we will show that the allowed region in the space of parameters becomes different than the obtained in the case of intermediate model [54].
We have organized this article as follows. In the next section, we present a brief review of G-inflation. In sections III, IV, and V we study the background and perturbative dynamics of our concrete inflationary models under the slow-roll approximation. Contact between the predictions of the model and observations will be done by computing the power spectrum, the scalar spectral index as well as the tensor-to-scalar ratio. We summarize our findings and present our conclusions in Section VI. We chose units so that c = = 8πG = 1.

II. G-INFLATION
In this section we give a brief review on the background dynamics and the cosmological perturbations in the model of G-inflation. Our starting point, is the 4-dimensional action in the framework of the Galilean model given by Here, the quantity g 4 corresponds to the determinant of the space-time metric g µν , R denotes the Ricci scalar and X = −g µν ∂ µ φ∂ ν φ/2. The scalar field is denoted by φ and the quantities K and G are arbitrary functions of X and φ.
By assuming a spatially flat Friedmann Robertson Walker (FRW) metric and a homogeneous scalar field φ = φ(t), then the modified Friedmann equations can be written as and where H =˙a a corresponds to Hubble rate and a denotes the scale factor. In the following, we will consider that the dots denote differentiation with respect to cosmic time and the notation K X denotes K X = ∂K/∂X, while K XX corresponds to K XX = ∂ 2 K/∂X 2 , and G φ From variation of the action (5) with respect to the scalar field we have In the specific cases in which the functions K = X − V (φ) (with V (φ) being the effective potential for the scalar field) and G = 0, General Relativity (GR) is recovered.
In order to study the model of G-inflation from different inflationary expansions, we will analyze the specific case in which the functions K(φ, X) and G(φ, X) are given by respectively. Here, the coupling g(φ) is a function that depends exclusively on the scalar field φ and the power n is such that n > 0. Also, in the following we will assume a power-law dependence on the scalar field for the coupling where the parameter γ and the power ν are both real, with γ > 0. Thus, the function G(φ, X) is defined as G(φ, X) = γ φ ν X n and then the Galilean term in the action is G(φ, X) φ ∝ φ ν X n φ. We mention that for the particular case in which ν = 0 i.e., g(φ) =const., and therefore the function G(φ, X) ∝ X n was already analyzed in Ref. [54] for the specific model of intermediate inflation.
Following Ref. [42], we will consider the model of G-inflation under the slow-roll approximation. In this sense, the effective potential dominates over the functions X, |G X Hφ 3 | and |G φ X|. Thus, under this approach, the Friedmann equation given by Eq. (6) can be approximated to By assuming the slow-roll approximation, we can introduce the set of slow-roll parameters for G-inflation, defined as [42] From the parameters defined above and combining with the Friedmann equations (6) and (7), the slow-roll parameter ε 1 can be rewritten as Now, from the functions K(φ, X) and G(φ, X) given by Eq.(9) and considering the slow-roll parameters from Eqs. (12) and (13), the equation of motion for the scalar field read as In the context of the slow-roll analysis, we are going to consider that the slow-roll parameters |ε 1 |, |ǫ 2 |, |ǫ 3 |, |ǫ 4 | ≪ 1, see Ref. [42]. Then, the slow-roll equation of motion for the scalar field, given by Eq. (14), can be approximated to where A is a function defined as A ≡ 3HφG X = 3n g(φ)X n−1 Hφ = 3n γ φ ν X n−1 Hφ.
From the slow-roll equation (15), we may distinguish two opposite limits. First, we have the limit |A| ≪ 1, which corresponds to the standard slow-roll equation in GR for the scalar field. However, when |A| ≫ 1, the Galileon term modifies the equation for the scalar field, and hence its dynamics. In this context, we are interested in the latter limit in which the Galileon effect changes the field dynamics. Then, by combining Eqs. (11) and (15), we find that the scalar field can be written as Note that this expression for φ(t) could be expressed explicitly in terms of the cosmic time t for any model and, in particular, for any scale factor a(t) or Hubble rate H(t).
From Eq.(16) we obtain that the function A can be rewritten as Here, we have used the Friedmann equation given by (11).
On the other hand, the analysis of the cosmological perturbations in G-inflation was developed in Refs. [39,42]. In the following, we briefly review the basic relations governing the dynamics of cosmological perturbations in the framework of G-inflation. In this context, the power spectrum of the primordial scalar perturbation P S in the slow-roll approximation can be written as [39,42] where the quantities q s and ε s are defined as and Here, we mention that the scalar propagation speed squared is given by c 2 s = εs qs . In this form, assuming the functions given by Eq.(9) and using the slow-roll parameter ǫ 3 , we find that the parameters q s and ε s are rewritten as From Eq.(18) and considering the above parameters, the scalar power spectrum in the slowroll approximation results [39,42] and the scalar propagation speed squared becomes c 2 s = 1+4A/3 1+2nA ≤ 1, where the power n is such that n ≥ 2/3. In the limit A ≫ 1, the scalar power spectrum, given by Eq. (21), becomes approximately Also, the scalar spectral index n S associated with the tilt of the power spectrum, is defined as n s = d ln P S /d ln k. Thus, from Eq. (21), the scalar spectral index under the slow-roll approximation can be written as [39,42] where ǫ and η are the standard slow-roll parameters, defined as respectively. Here, we observe that in the limit A → 0 (or equivalently g → 0), the scalar spectral index given by Eq.(23) coincides with the expression obtained in GR, where n s ≃ −6ǫ + 2η. In the limit |A| ≫ 1, where the Galileon term dominates the inflaton dynamics, the scalar index n S results On the other hand, the tensor power spectrum in the framework of G-inflation is similar to standard inflation in GR, where the amplitude of GWs have a tensor spectrum P G given by [39,42] P G = 2H 2 π 2 . In this sense, the tensor-to-scalar ratio, defined as r = P G /P S , in the framework of G-inflation under slow-roll approximation can be written as Again, we note that in the limit A → 0, the tensor-to-scalar ratio coincides with the expression obtained in standard inflation, where r ≃ 16ǫ. Now, by assuming the limit |A| ≫ 1, the tensor-to-scalar ratio r is approximated to In the following, we will study three different inflationary expansions; the intermediate, logamediate and exponential in the framework of G-inflation. In order to study these expansions we will assume the Galilean effect predominates over the standard inflation, i.e., in the limit |A| ≫ 1.

III. INTERMEDIATE G-INFLATION.
Let us consider a scale factor that evolves according to Eq.(1) or commonly called intermediate expansion. Here, the Hubble rate is given by H(t) = Af t 1−f , and from Eq.(16), we find that the scalar field as a function of cosmic time becomes without loss of generality we can take φ(t = 0) = 0. Thus, the Hubble rate as function of the scalar field φ becomes where the constants k 1 and µ 1 are defined as , and µ 1 = 2n + 1 + ν 2n , respectively. From the Friedmann equation (11), the effective potential in terms of the scalar field can be written as which has an inverse power-law dependence on the scalar field, hence does not have a minimum.
The number of e-folds N between two different cosmic times t 1 and t 2 or, equivalently between two values of the inflaton field φ 1 and φ 2 , is given by Here, we have used Eq. (28).
In order to determine the beginning of inflationary phase, we find that dimensionless slow-roll parameter ε 1 = ε 1 (φ), is given by In this sense, the condition for inflation takes place is given by ε 1 <1 (or equivalentlÿ a > 0), then from Eq.(31) the scalar field is such that φ > 1−f Since inflation begins at the earliest possible scenario (see Fig.1), that is, when the slow-roll parameter ε 1 (φ = φ 1 ) = ε 1 (φ 1 ) = 1 (or equivalentlyä = 0), then the scalar field at the beginning inflation φ 1 results In order to satisfy the condition A ≫ 1, we write the parameter A in terms of the number of e-folds N as where A 0 = 3 n γ 2 n−1 2 n 3 n γ 2n−1 2n+1 and the scalar field φ(N) is defined as Here, we have used Eqs. (28), (30) and (32).
On the other hand, the scalar power spectrum P S in terms of the scalar field reads as where we have used Eqs. (22) and (28), respectively. Now, from Eqs. (30), (32) and (35), we can write the scalar power spectrum as function of the number of e-folds N in the form Similarly, the scalar spectral index n s can also be expressed in terms of the number N as Here, we noted that the scalar spectral index given by Eq.(37) coincides with the obtained in the standard intermediate inflation [26]. Thus, for the special case in which f = 2/3, the scalar spectral index n s = 1 (Harrison-Zel'dovich spectrum). In particular, assuming that the number of e-folds N = 60 and the spectral index n s = 0.967, we obtain that the value of the parameter f results f = 0.398 ≃ 0.4.
From Eq. (27), we find that the relationship between the tensor-to-scalar ratio r and the scalar spectral index n s results Here, we note that the consistency relation r(n s ) given by (38)  Here, we mention that the parameter A satisfies the condition A = 3ng(φ)X n−1 Hφ ≫ 1 as g(φ) ≫ (3nX n−1 Hφ) −1 . By assuming typically that H ∼ 10 −5 andφ ∼ 10 −5 , we find that the coupling g has a lower bound given by g(φ) ≫ 10 400 for n ∼ 40. This suggests that the coupling g(φ) must have a very large value as lower bound (googol 4 ). In particular for the N = 60, f = 0.4 and n = 38, and since that g(φ) = γφ ν , we find that for the case ν = 1, the lower limit is found to be γ ≫ 8 × 10 403 , while for ν = 0 (or equivalently g(φ) = const.) we have that γ ≫ 10 404 . Finally, for the case ν = −1 (or g(φ) ∝ φ −1 ), we found that γ ≫ 10 405 .
In Fig.1 n > 38, the model is well supported by the data. Also, we noted that when n ≫ 1, then the tensor-to-scalar ratio r ∼ 0.

IV. LOGAMEDIATE G-INFLATION
Now, we consider the situation in which the scale factor evolves according to logamediate inflation, given by Eq. (2). Here, the Hubble rate H(t) becomes H(t) = Bλ (ln t) λ t , and from Eq.(16), we find that the scalar field φ(t) results By assuming the slow-roll equation (11), we have that the effective potential in terms of the scalar field is given by where the constants V 0 , k 2 and µ 2 are defined as , and µ 2 = 2n + 1 + ν 2n + 1 , respectively. For the logamediate expansion in the context of G-inflation, the number of e-folds N between two different values of the scalar field φ 1 and φ 2 is written as Here, we have used Eq. (39).
As before, we write A(N) in order to satisfy the condition A ≫ 1. Thus, we have that where the field φ(N) and the function Ξ(N) are defined as φ(N) = 2n + 1 + ν 2n For the dimensionless slow-roll parameter ε 1 in the logamediate G-inflation, we have that and in order to get an inflationary scenario (ε 1 <1), we have that the scalar field φ > As before, if the inflationary stage begins at the earliest possible epoch, where the slow-roll parameter ε 1 (φ = φ 1 ) = 1, then we obtain that the field φ 1 is given by On the other hand, as before we find that the scalar power spectrum P S as function of the number of e-folds reads as Here, we have considered Eqs. (22), (43) and (41). Now, from Eqs. (25), (41) and (43), we find that the scalar spectral index n s is related to the number of e-folds N through the following expression Note that this expression for the scalar spectral index coincides with the obtained from logamediate inflation in GR [27]. In a similar fashion as we did before, we find that the consistency relation r = r(n s ) is given by (1 − n s ).
As in the previous case of intermediate G-inflation, we noted that the relation r = r(n s ) given by (46) strongly depends on the power n, when we make the comparison with the results of r(n s ) in the standard logamediate model in the framework of GR. In this sense, the dependence on the power n is crucial in order for the theoretical predictions of the model to enter in the allowed regions of the contour plot in the r − n s plane. We also note that, for large values of the power n such that n ≫ 1, the tensor-to-scalar ratio r tends to zero.
From BICEP2/Keck-Array data, we have that r < 0.07, then we find a lower bound for the power n, given by n > 15480(1 − n s ) 2 . In particular, considering that the scalar spectral index takes the value n s = 0.967, the lower limit for the power n yields n > 17.
For this Hubble rate, the number of e-folds N between two different values of the scalar field φ 1 and φ 2 results A as function of the number of e-folding N can be written as where the scalar field φ(N) reads as Unlike the intermediate and logamediate inflation models, this Hubble rate addresses the end of the accelerated expansion. In this sense, considering that inflation ends when ε 1 = 1, where the slow-roll parameter ε 1 is given by we have that the scalar field at the end of inflation, given by ε 1 (φ = φ 2 ) = 1, becomes Also, from the condition for inflation to occur in which ε 1 <1, then the scalar field becomes As before, we can express the the amplitude of the scalar power spectrum P S in terms of the number of e-folding N as and the scalar spectral index n s (N) results n s (N) = 1 − 3 N +1 . Also, we find that the consistency relation r = r(n s ) in this scenario can be written as (1 − n s ).
As in the previous models of intermediate and logamediate, we observed that the consistency relation r = r(n s ) given by (51) also strongly depends on the power n. As before, the introduction of the power n in the model is fundamental in order to the theoretical predictions of this model enter in the allowed region of the contour plot in the r − n s plane from [22]. Assuming the BICEP2/Keck-Array, for which r < 0.07, we obtain a lower bound for the power n, given by n > 6880(1 − n s ) 2 . In particular assuming that the scalar spectral index n s is given by n s = 0.967, we find that the lower bound for the power n corresponds to n > 7.
In addition, from the the amplitude of the scalar power spectrum given by eq.(50), we can find a constraint for the parameter β, appearing in the Hubble rate, for several values of n when the number of e-folds N and the observational value of the power spectrum P S are given. Thus, particularly for the values P S = 2.2 × 10 −9 and N = 60, for the case when the power n takes the value n = 8, we found the value β = 2.9 × 10 −13 . As in the previous models, we can find a lower bound for the parameter γ from the condition A ≫ 1 given by Eq. (49). In particular, for the values N = 60, β = 2.9 × 10 −13 , α = 10 −3 and n = 8, we obtain that for the case in which ν = 1 (g(φ) ∝ φ), the lower bound is γ ≫ 3 × 10 181 , while for ν = 0 (or g(φ) = constant) we have that γ ≫ 6 × 10 183 . Finally, for the specific case in which ν = −1 (or g(φ) ∝ φ −1 ), we obtain that γ ≫ 10 186 . As in the previous models, from the two-dimensional marginalized constraints on the r − n s plane, this model becomes well supported by the Planck data when the power n satisfies n > 7 (figure not shown) and then the model works.

VI. CONCLUSIONS
In this paper we have investigated the intermediate, logamediate and exponential inflation in the framework of a Galilean action with a coupling of the form G(φ, X) ∝ φ ν X n . For a flat FRW universe, we have found solutions to the background and perturvative dynamics for each of these expansion laws under the slow-roll approximation. In particular, we have obtained explicit expressions for the corresponding scalar field, effective potential, number of e-folding as well as for the scalar power spectrum, scalar spectral index and tensor-to-scalar ratio. In order to bring about some analytical solutions, we have considered that the Galileon effect dominates over the standard inflation, in which the parameter A = 3HφG X satisfies the condition A ≫ 1. In this context, we have found analytic expressions for the constraints on the r−n s plane, and for all these G-inflation models we have obtained that the consistency relation r = r(n s ) depends on the power n which is crucial in order to the corresponding theoretical predictions enter on the two-dimensional marginalized constraints imposed by current BICEP2/Keck-Array data. In this sense, we have established that the inflationary models of intermediate, logamediate and exponential in the framework of G-inflation are well supported by the data, as could be seen from Figs.(1) and (2). In particular for the intermediate G-inflation, from the r − n s plane, we have found a lower bound for the power n, given by n > 38. For the logamediate model we have obtained that n > 17 and finally, for the exponential model we have got n > 7 as lower limit. Also, we have found that for values of n ≫ 1, the tensor-to-scalar ratio r → 0. Also, from the amplitude of the scalar power spectrum P S (N) and the scalar spectral index n s (N) as function of the number of e-folds, we have found constraints on the several parameters appearing in our models. Besides, considering that the Galileon effect dominates over GR given by the condition A ≫ 1, we have found a very large value as a lower limit for the parameter γ. The reason for this is due that typically H ∼φ ∼ 10 −5 ≪ 1, then from the condition A ≫ 1 suggesting g(φ) ≫ (3nX n−1 Hφ) −1 , thus we have found that g(φ) ≫ (3nX n−1 Hφ) −1 ∼ O(10 400 ), e.g.
In this work, we have determined that the intermediate, logamediate and exponential models in the context of G-inflation, are less restricted than those in the framework of standard GR, due to the modification in the action by the Galilean term G(φ, X) φ ∝ φ ν X n φ.
Finally, in this paper we have not addressed a mechanism to bring intermediate and logamediate G-inflation to an end and therefore to a study the mechanism of reheating, see Refs. [28,55]. We hope to return to this point for these models of G-inflation in the near future.