Inflation with Derivative Self-interaction and Coupling to Gravity

We consider a subclass of Horndeski theories for studying cosmic inflation. In particular, we investigate models of inflation in which the derivative self-interaction of the scalar field and the non-minimal derivative coupling to gravity are present in the action and equally important during inflation. In order to control contributions of each term as well as to approach the single-term limit, we introduce a special relation between the derivative interaction and the coupling to gravity. By calculating observable quantities including the power spectra and spectral tilts of scalar and tensor perturbation modes, and the tensor-to-scalar ratio, we found that the tensor-to-scalar ratio is suppressed by a factor of $(1+1/\gamma)$, where $\gamma$ reflects the strength of the derivative self-interaction of the inflaton field with respect to the derivative coupling gravity. We placed observational constraints on the chaotic and natural inflation models and showed that the models are consistent with the current observational data mainly due to the suppressed tensor-to-scalar ratio.


I. INTRODUCTION
Inflation in the early universe is a successful paradigm for explaining the cosmological problems including the horizon, flatness and monopole problems [1][2][3][4]. Driven by a slowly rolling scalar field (or "inflaton"), inflation also generates the primordial density perturbations necessary for the formation of large scale structures in the universe [5][6][7][8][9][10]. The constraints on inflationary models, in particular, sufficient amount of inflation and cosmic microwave background (CMB) temperature anisotropy measurements favor a nearly flat potential during inflation [11]. In addition to the nearly flat potential, the conventional scalar-field action also consists of a canonical kinetic term [12,13]. Arising naturally from particle physics, inflationary models with noncanonical kinetic terms have received much attention over the past few decades as they can reconcile the simplest realization of inflation with the current observational data and leave their distinct signatures in the cosmological observations [14][15][16][17][18].
It is demonstrated that special combinations of higher-order kinetic terms in the action give rise to the equations of motion that contain no higher than the second-order derivatives [19,20]. Applications of such extended scenarios provide a unified framework upon which one can construct or embed new models of inflation.
The Horndeski theories as a generalization -or an extension -of the scalar-tensor theories of gravity have the most general higher derivative extensions and its dynamics is governed by second-order equations of motion [21].
A broad spectrum of single-field models of inflation with the second-order equations of motion are constructed from Eq. (1) and the associated cosmological perturbations are well established in Refs. [23,24]. Thus far, the successful proposals of inflationary models within this framework often employ special combinations of the independent functions of the scalar field and its derivatives: K(φ, X) and G 3,4,5 (φ, X). This is mainly due to the fact that it is nontrivial how the background as well as perturbations evolve when multiple terms are present in the action. For example, the inflationary models with the non-minimal derivative coupling to gravity [25][26][27][28][29] have focused on a case where all terms except the L 3 are present in the action. On the other hand, the inflationary models with the derivative self-interactions of the scalar field (i.e., G-inflation) [18,23,30,31] (see Ref. [30] for the potential driven G-inflation scenarios) concentrate on the presence of L 3 by omitting L 5 in Eq. (1). However, although the general formulas are well established in Refs. [23,24], little attention has been paid to the case where all terms in Eq. (1) are present in the action and equally important during inflation.
Our aim for the present work is to investigate the potential driven single-field models of slow-roll inflation in which all terms in Eq. (1) are present and equally important during inflation. In addition, by introducing a special relation, we show how they could approach the single term limit with respect to model parameters. We focus in particular on the following setup: where V (φ) is the inflaton potential, c 1 and c 3 are the model parameters, and ξ(φ) is the coupling function of φ. In fact, aforementioned two classes of inflationary models; namely, inflation with the derivative selfinteraction of the scalar field and inflation with the non-minimal derivative coupling to gravity, are combined into one setup if c 1 = 0 and c 3 = 0 in Eq. (3). We examine the observational consistency of chaotic inflation with V (φ) = λ(κφ) n /(nκ 4 ) and natural inflation with V (φ) = Λ 4 [1 + cos(φ/f )] for our setup Eq. (3). This is because theoretical predictions of these models for standard single-field inflation with Einstein gravity are disfavored by the current observational data [11].
This paper is organized as follows: in Sec. II, after deriving the background equations of motion for our setup, we introduce a special relation between the L 3 and L 5 of Eq. (1) that allows us to control contributions of each term during inflation. In Sec. III, following Ref. [23], we obtain the observable quantities including the power spectra (P S , P T ) and spectral tilts of the scalar and tensor perturbation modes (n S , n T ), and the tensor-to-scalar ratio r. Our results of observational constraints on the chaotic and natural inflation models are presented in Sec. IV and we conclude our work in Sec. V.

II. SETUP FOR POTENTIAL-DRIVEN SLOW-ROLL INFLATION
For our setup given in Eq. (3), the action Eq. (1) reduces to where c 1 = 0 and c 3 = 0 constants. Varying this action with respect to metric g µν yields the Einstein equation where the energy-momentum tensors T φ µν , T c1 µν , and T c3 µν are given by respectively. Using the Bianchi identity ∇ µ G µν = 0 and the conservation law ∇ µ T µν = 0, we get from Eq. (5) which, as a consequence, yields a evolution equation for the scalar field.
In a spatially flat Friedman-Robertson-Walker universe with metric Under the slow-roll conditions, the terms containing c 1 and c 3 in Eq. (11) are much smaller than V (φ) during inflation hence Eqs. (11) and (15) reduce to where A ≡ −3c 1 (1 + 1/γ)H 2 after taking Eq. (16) into account. When A 1, the friction term significantly enhances hence it is regarded as the high friction limit, see Refs. [25][26][27][28] for the further details. On the other hand, the standard slow-roll inflation with Einstein gravity is realized when |A| 1. Thus, terms with c 1 and c 3 play an important role when |A| 1. We derive potential based slow-roll parameters using Eq. (17) The amount of inflation is quantified by the number N of e-folds, which reads where φ e is the scalar-field value at the end of inflation and is to be estimated by solving 1 (φ e ) ≡ 1.
In this section, we discuss the linear perturbation analyses for scalar and tensor modes in the flat FRW background and our discussion mainly follows Ref. [23] as the most general perturbation analysis for the Horndeski theories is carried out there in great detail. The perturbed metric in the Arnowitt-Deser-Misner formalism [32] is given by where N , N i , and γ ij are the lapse function, the shift function, and the metric for the three-dimensional space, respectively, and are given by Here, α, β, and ζ denote scalar perturbations while h ij is a tensor perturbation satisfying the traceless and transverse conditions; h ii = 0 = h ij,j . The scalar field is decomposed into a background and inhomogeneous parts, e.g., φ(t, x) = φ(t) + δφ(t, x), and we employ the uniform field gauge with δφ(t, x) = 0.

A. Tensor perturbations
Let us first consider the tensor perturbations. Substituting the perturbed metric into the action Eq. (4) and then expanding the action to the second order in h ij , one can obtain the quadratic action as [23] S (2) where In order to avoid from the ghost and gradient instabilities, the G T > 0 and F T > 0 conditions must be satisfied in Eq. (23). In terms of slow-roll parameters, we get the quadratic action Eq. (23) is rewritten as where Here, the prime denotes the derivative with respect to the conformal time τ which relates the physical time t via adτ = c T dt. Each perturbation mode crosses the sound horizon when we obtain a wave equation for the tensor perturbation modes where The exact solution to Eq. (30) can be obtained by adopting the Bunch-Davies vacuum for the initial fluctuation modes at c T k|τ | 1 and assuming constant slow-roll parameters during inflation. The solution therefore reads The power spectra of the tensor modes can be calculated with Eq. (32) on the large scale c T k|τ | 1 as The tensor spectral index at the time of horizon crossing is computed as

B. Scalar perturbations
Next, let us discuss the scalar perturbations by setting h ij = 0 in Eq. (22). Substituting the perturbed metric into the action in Eq. (4) and then expanding to the second order in ζ, one can also obtain the action [23] where Here, F S > 0 and G S > 0 are also necessary for avoiding the ghost and gradient instabilities. By introducing where the conformal time τ is related to the physical time t via adτ = c S dt and each perturbation mode crosses the sound horizon when k 2 = z S /z S ∼ 1/τ 2 . The effective sounds speed c 2 S is expressed as .
In terms of the slow-roll parameters, we obtain where only leading order contribution is collected. By employing the canonical quantization we arrive at a equation for the scalar perturbation modes where After adopting the Bunch-Davies vacuum for the initial fluctuation modes, the solution to Eq. (44) is given by The power spectra of the scalar modes can be computed with Eq. (46) on the large scale c S k|τ | 1 as The spectral index of the scalar perturbation modes at the time of horizon crossing is obtained as Consequently, the tensor-to-scalar ratio becomes One can notice here that quantities n S , n T , and r are suppressed by a factor of (1 + A), which is not surprising because such suppression was previously discussed in Refs. [25][26][27][28]. However, what we found as a result of our computation is the additional factor (1 + 1/γ) of suppression in the A 1 limit, where A ≡ −3c 1 (1 + 1/γ)H 2 .
Therefore, the presence of the derivative self-interaction and the non-minimal derivative coupling to gravity terms is responsible for this additional suppression factor. The results of standard slow-roll inflation with Einstein gravity n S − 1 = 2η V − 6 V , n T = −2 V , and r = 16 V is recovered in the |A| 1 limit.

IV. OBSERVATIONAL CONSTRAINTS ON EXPLICIT MODELS
In the presence of both the non-minimal derivative coupling to gravity and the inflaton derivative selfinteraction, we put observational constraints on (A) chaotic inflation [2,33] and (B) natural inflation [34,35] in this section. In the framework of standard single-field inflationary models with Einstein gravity, these models are disfavored by the current Planck 2018 plus BK14 data [11] due to their predictions of large tensor-to-scalar ratio. However, we showed in the previous section that the tensor-to-scalar ratio is significantly suppressed for our model. Thus, based on the information of the n S and r given in Eqs. (48) and (49), we examine the observational bounds of each model in the following subsections.
The scalar-field potential for the chaotic inflation model [2,33] is given by where λ = m 2 κ 2 for a quadratic n = 2 potential. A shape of the coupling function ξ(φ) during inflation can be determined from Eqs. (16) with a use of above potential. From Eq. (20), the number of e-folds becomes where The scalar field value at the end of inflation φ e can be estimated by solving (φ e ) = 1. In our case, we get 2(κφ e ) 2 [n + δ(κφ e ) n ] = n 3 .
From Eqs. (48)-(49), we obtain the spectral index and the tensor-to-scalar ratio as In order to examine the observational consistency of the model, we often express n S and r as fuctions of N .
For that purpose, we solve Eqs. (51) and (53) for φ and φ e . Let us assume for computational simplicity that the δ 1 is a small parameter during inflation and then expand φ to the leading order in δ as follows After plugging Eq. (56) into Eq. (54), we finally express the observable quantities in terms of N as follows In Fig. 1, together with the observational data, we present the theoretical predictions of chaotic inflation with the quadratic potential. The background shaded regions show the 1σ(darker orange) and 2σ(lighter orange) contours of the observational data by Planck TT, TE, EE + lowE + lensing + BK14 + BAO [11]. The red points indicate the δ → 0 limit or correspond to results of the standard single-field chaotic inflation models with and 2σ(lighter) contours of the observational data [11]. The inset plot shows a closer look on the N = 60 solid line. δ max 3.5727 × 10 −2 , 3.0094 × 10 −2 , and 2.6008 × 10 −2 for N = 50, 60, and 70, respectively. These maximum values give us the blue points in Fig. 1. For example, the N = 60 blue point at (n S , r) (0.966946, 0.0468) corresponds to the δ max = 3.0094 × 10 −2 value. Thus, the δ varies between zero and δ max . In the figure, the δ increases from red to blue points and significantly reduces the r value; its impact on n S is a slight increment, see the plot embedded in Fig. 1.  Fig. 1 as they would reside outside of the 2σ contour hence disfavored by the current data [11]. This is why we plot only n = 2 potential in Fig. 1.
As can be seen in Fig. 1, chaotic inflation with the quadratic potential is consistent with the current observational data for N = 60 and N = 70 e-folds, whereas N = 50 e-folds appears to be disfavored because its prediction residing outside of the 2σ contour. After taking r 0.002 < 0.065 into account from Ref. [11] (95% CL by Planck TT, TE, EE + lowE + lensing + BK14 + BAO), a lower limit of the δ can be tightly constrained. For N = 60 e-folds, for example, the favored δ range by observation is given as: 1.7618×10 −2 δ obs < 3.0094×10 −2 .
Using the δ values favored by observation together with the normalized value for the primordial scalar perturbation P S 2.09 × 10 −9 for k * = 0.05Mpc −1 [11], we can find the relation between the γ and M parameters from Eq. (47). By substituting Eq. (50) into Eq. (47), we obtain δ obs 12n 4 π 2 where δ obs is the numerical value of δ in the range favored by the observational data and (κφ) 60 is the value of κφ at N = 60. After putting Eq. (56) into Eq. (59) and setting the n value, we basically have two parameters: M and γ, to tune the correct amplitude of the primordial scalar perturbation. Furthermore, Fig. 2 shows that the scale M approaches to the Planck scale M p = κ −1 in the γ 1 regime.
This regime corresponds to the case where the effects of the derivative self-interaction G 3 (φ, X) becomes stronger than that of the non-minimal derivative coupling to gravity G 5 (φ, X) during inflation. If we impose κM 1 to avoid the quantum gravity, γ 10 −4 seems to be more favored. Thus, too small values ( as small as γ < 10 −4 ) may violate the quantum gravity constraint, see Ref. [35] for further details. The scale M is much smaller than the Planck scale in both the γ ∼ 1 and γ 1 regimes and evolves differently in each regime. As can be seen in the figure, the evolution is κM ∼ (1 + 1/γ) in the γ ∼ 1 regime and nearly a constant for the γ 1 regime, where the G 5 (φ, X) term plays in an important role during inflation.
Thus, for chaotic inflation with the quadratic potential, we emphasize that the γ 1 limit or the G-inflation scenario, may suffer from the quantum gravity constraints, whereas no such issues are evident in the γ ∼ O(1) regime or for inflationary models with the non-minimal derivative coupling to gravity.

B. Natural inflation
The scalar-field potential for natural inflation is given by [34,35] where Λ and f are model parameters having dimension of mass. 1 For the given potential, the coupling function ξ(φ) during inflation can be determined from Eq. (16). The number of e-folds is obtained as where By solving 1 (φ e ) = 1, we find the field value at the end of inflation From Eqs. (48)-(49), for the potential in Eq. (60), the observable quantities are calculated Here, we can express cos (φ/f ) in terms of N using Eq. (61). In the γ 1 limit, Eq. (64) reproduces the results of Ref. [26]. In Fig. 3,   (47), the CMB normalization by [11] at N = 60 corresponds to where κM = 4.2 × 10 −6 , the figure also shows that the f approaches to the Planck scale κf 1 (the horizontal red line) in the γ 1 regime. Thus, for the too large values of γ as large as γ 10 4 , inflationary models with non-minimal derivative coupling to gravity may not be able to avoid from the trans-Planckian masses by having f M pl . In the γ 1 regime, on the other hand, the model is not only consistent with the observational data but also respects the κf 1 and κM 1 constraints.

V. CONCLUSION
We have studied inflationary models with the non-minimal derivative coupling to gravity and the derivative self-interaction of the scalar field. After deriving the background equations of motion, we introduced the special relation Eq. (16) that holds during inflation. If we employ the relation, our model approaches to the single term limit with respect to the model parameters. Thus, we did not need to specify the coupling function ξ(φ) in the present study. In addition, the contributions of each term can be conveniently controlled by the γ parameter in Eq. (16), which reflects the strength of the inflaton derivative self-interactions with respect to the non-minimal derivative coupling to gravity.
The observable quantities including the power spectra for scalar and tensor perturbation modes, the spectral indices, and the tensor-to-scalar ratio are obtained in Eqs. (33), (34), (47), (48), and (49). We found as a result of our analytic computation that the observable quantities are suppressed by a factor of (1 + A), or (1 + 1/γ) in the A 1 limit. The suppression is mainly due to the presence of both the non-minimal derivative coupling to gravity and the inflaton derivative self-interaction terms.
We then placed observational constraints on the chaotic and natural inflation models using their theoretical predictions for the n S and r. Figs. 1 and 3 show that, for certain ranges of the model parameters, the both models are consistent with the current observational data [11] mainly due to the suppressed tensor-to-scalar ratio. In Figs. 2 and 4, the shaded regions illustrate the parameter spaces that give the correct amplitude of the scalar power spectrum P S for each inflation model and satisfy the associated f M pl and M M pl constraints. Although a broad range of the γ parameter is supported by the observational data [11], the values that are as small as γ 10 −4 for chaotic inflation and as large as γ 10 4 for natural inflation may suffer from avoiding the trans-Planckian mass and the quantum gravity constraints. There is no such issues apparent in the γ ∼ O(1) regime, where both the derivative coupling and self-interaction terms play equally important role during inflation, and both inflationary models fit well to the observational data [11].
The Planck 2018 result leads to the somewhat tighter constraint on the potential parameters: 4.8513 × 10 −10 λ = m 2 κ 2 5.1109 × 10 −7 for chaotic inflation with the quadratic potential and 4.2822 × 10 −3 κΛ 5.0631 × 10 −3 for natural inflation with the cosine potential. In addition, the observational bounds on the mass scale for natural inflation is constrained to be κM 4.2 × 10 −6 at 68% CL; therefore, the scale f is smaller than the Planck scale f M pl .
Based on our finding, we consider the background dynamics of the system including the post-inflationary evolution need to be further analyzed for the better understanding of the system Eq. (4). In addition, one would expect the introduction of new terms and a new mass scale M may produce non-Gaussian fluctuations larger than those in GR even for the same V (φ). We leave these as our future extensions to our present study.