Gauss-Bonnet inflation with a constant rate of roll

We consider the constant-roll condition in the model of the inflaton nonminimal coupling to the Gauss-Bonnet term. By assuming the first Gauss-Bonnet flow parameter $\delta_1$ is a constant, we discuss the constant-roll inflation with constant $\epsilon_1$, constant $\epsilon_2$ and constant $\eta_H$, respectively. Using the Bessel function approximation, we get the analytical expressions for the scalar and tensor power spectrum and derive the scalar spectral index $n_{\mathcal{R}}$ and the tensor to scalar ratio $r$ to the first order of $\epsilon_1$. By using the Planck 2018 observations constraint on $n_{\mathcal{R}}$ and $r$, we obtain some feasible parameter space and show the result on the $n_{\mathcal{R}}-r$ region. The scalar potential is also reconstructed in some spectral cases.


I. INTRODUCTION
Inflation in the early Universe has become a well established part of modern cosmology, it naturally generates the density perturbations that become the seeds of large-scale structure and temperature anisotropies of the cosmic microwave background(CMB). The predictions have been confirmed by numerous observations, such as the WMAP [1] and Planck space missions [2]. Recently, the Planck 2018 data constrain the scalar spectral index and the tensor-to-scalar ratio to be n R = 0.9649 ± 0.0042 at 68% confidence level and r 0.002 < 0.10 at 95% confidence level [2], which will narrow down some inflationary models.
In the conventional inflationary models, it is usually assumed that the scalar field rolls down slowly, or more precisely to assumed that the flow parameters satisfy the conditions ǫ H ≪ 1 and η H ≪ 1. This so called slow-roll inflation leads to a nearly scale-invariant spectrum of density perturbations and are consistent with the latest observations. Recently, a new type of inflation called constant-roll inflation has been put forward in [3], where one of the flow parameters η H is assumed to be a constant, and not necessarily small. Such class of models can also produce a scale-invariant spectrum and is compatible with the observational constraints without the assumption of slow-roll approximation. The constant-roll condition with small η H is compatible with the slow-roll inflation, and a special value η H = 3 amounts to the ultra-slow-roll inflation [4,5], in which the potential is very flat and the inflaton almost stops rolling.The constant-roll inflationary models are discussed in more details in [6][7][8], and extended to modified theories of gravity [9][10][11][12][13][14] or two fields case [15], and using other constant-roll conditions [16][17][18]. Other developments are appeared in [19][20][21].
Since the inflation occurs in the early Universe which is believed to be described by quantum gravity, so it is interest to discuss inflation in the framework of quantum gravity theories, such as string theory. It is known that when discuss the effective action in the early universe, the correction terms of higher orders in the curvature coming from superstrings may play a significant role, and the simplest of such correction is the Gauss-Bonnet (GB) term in the low-energy effective action of the heterotic string [22,23]. There are many works discussing accelerating cosmology with the GB correction in four and higher dimensions [24][25][26][27], and with nonminimal coupled to the inflaton [28][29][30] or to the Higgs [31][32][33][34][35].
In this work, we shall discuss the constant-roll inflation in the model with the inflation field φ nonminimal couple to the GB term. The presence of such coupling will generate a new degree of freedom, so here we consider a special condition that the first GB flow parameter δ 1 is a constant. Assuming that one of the flow parameters, ǫ 1 , ǫ 2 or η H is constant, we derive the analytical expressions of the scalar spectral index n R and the tensor to scalar ratio r for each case. Combine with the Planck 2018 observations, we obtain some feasible parameter spaces, and find that in some cases, the chosen of GB flow parameter δ 1 > 1 can also produce a nearly scale-invariant scalar power spectrum, which is different from the slow-roll inflation. At last, we also reconstruct the potential in some spectral cases.
The outline of this paper is as follows: In the next section, we briefly review the inflationary model with nonminimal coupling to the Gauss-Bonnet term. In Section 3,4 and 5, we focus on the inflationary models with constant ǫ 1 , constant ǫ 2 or with constant η H , respectively, derive the formalism for the scalar spectral index and the tensor-to-scalar ratio of each model and constrain the parameter space with the Planck 2018 data. In Section 6, we try to reconstruct the potential in some spectral cases. And the final section is devoted to summary.

II. INFLATION WITH A GAUSS-BONNET COUPLING
In this section, we shall review the inflation with the GB coupling, and present the mode equations of scalar and tensor perturbations. Consider the following action with the inflation field φ coupled to the GB term where V (φ) is the potential for the scalar field φ, R 2 GB = R µνρσ R µνρσ − 4R µν R µν + R 2 is the GB term, and ξ(φ) is the coupling functional. The coefficient ω in the kinetic term can be chosen the values ±1 to ensure conventional inflationary behaviour [28,29]. We work in Planckian units, = c = 8πG = 1.
In the Friedmann-Robertson-Walker homogeneous universe, the background equations can be written as where a dot represents a derivative with respect to cosmic time and (...) ,φ denotes a derivative with respect to the field φ.
In standard inflation, it is useful to define a series of flow parameters, such as the Hubble flow parameters furthermore, we also introduce the horizon flow parameters In the presence of the GB coupling, the new degrees of freedom suggest to defining another hierarchy of flow parameters [29] For the slow-roll inflation, these flow parameters should satisfy the slow-roll conditions |ǫ i | ≪ 1 and |δ i | ≪ 1. However, in the constant-roll inflation, such conditions are not necessary to satisfy, we only need 0 < ǫ 1 < 1 to achieve a significative inflation.
At linear order in perturbation theory, the Fourier modes of curvature perturbations where a prime represents the derivative with respect to conformal time τ . z R and the sound speed c R can be expressed in terms of the horizon and GB flow parameters as work we consider a special condition that the first GB flow parameter δ 1 is also a constant, then from the definition of GB flow parameters (7) we have δ i = 0(i ≥ 2).
In the following, we first discuss the case with ǫ 1 = constant. From the relations d dτ and assuming that ǫ 1 is a constant, one can express the factor aH in (11) and (16) then can be approximated as a constant. Therefore the general solution to the mode equation Choosing c 1 = 1 and c 2 = 0, the usual Minkowski vacuum state is recovered in the asymptotic past. Then on super-horizon scales, c R k ≪ aH, the power spectrum of the scalar perturbation is with the scalar spectral index is Using the same procedure as in the case of scalar perturbations, we get the power spectrum of tensor perturbations with and the tensor spectral index is Combing Eq. (21) and (23), we obtain the tensor to scalar ratio Below we will assume that the first GB flow parameter δ 1 is a constant which contains two The n R − r region predicted by the model with ǫ 1 = constant and δ 1 = 0 are show in Fig.1, where the contours are the marginalized joint 68% and 95% confidence level regions for n R and r at the pivot scale k * = 0.002 Mpc −1 from the Planck 2018 TT,TE,EE+lowE+lensing data. We could see that the curve is ruled out by the observations, so we didn't interest in this case.
If δ 1 =constant but not zero, the expression of scalar spectral index is the same as in the previous case and the tensor-to-scalar ratio is more complex Combine these expressions with the observational constraints from Planck 2018, one can obtain the constraints on ǫ 1 and δ 1 . However, for a significative inflation we haveä > 0, which is equivalent to ǫ 1 < 1, so only the parameter region satisfies ǫ 1 < 1 is reasonable. We then obtain two regions of parameter space, and show them in Fig  index depends only on ǫ 1 , so for the same chosen of ǫ 1 , the curves look the same. However, different from that panel, in this time the n R − r dots go along the curves from bottom to top as δ 1 increase.
We could see that for the choices of parameter space in Fig.2, the inflation prediction is in agreement with the observational constraints. However, the constant-roll inflation with ǫ 1 = constant is no natural end, hence an additional mechanism is required to stop it.
In this section, we assume that the second horizon flow parameter ǫ 2 is a constant. So from the relations (17) and using the definition of flow parameter (6), we obtain that to the first order approximation of ǫ 1 , the relation between aH and τ is [16,18] aH then, we obtain the power spectrum of the scalar perturbation and tensor perturbation with the scalar spectral index and the tensor-to-scalar ratio are In the following, we still assume that the first GB flow parameter δ 1 is a constant, which contains two cases: δ 1 = 0 or δ = 0.
A. δ 1 = 0 If δ 1 = 0, the model recovers to the case without GB coupling, and such case have been discussed in other works [16]. Here we list some useful results. To the first order approximation of ǫ 1 , the scalar spectral index can be approximated as and the tensor-to-scalar ratio Since ǫ 2 is a constant, we can integer the definition (6) and get the relation between ǫ 1 and where N is the e-folding number before the end of inflation, and we have use the condition If δ 1 =constant but not zero, to the first order of ǫ 1 , the expression of scalar spectral index is and the tensor-to-scalar ratio   Finally, the n R − r region for the parameters in region III of Fig.6 are show in Fig.9.
With δ 1 is taken as δ 1 = 0.505, 0.503, 0.501 from left to right in the left panel, and as ǫ 1 increase, the n R − r dots go along the curves to the right. In the right panel, ǫ 1 is taken as FIG. 9. The n R − r region predicted by the model for the parameter region III of Fig.6.

V. CONSTANT-ROLL INFLATION WITH CONSTANT η H
In this section, the second Hubble flow parameter η H = −Ḧ/(2HḢ) is assumed to be a constant. Combine with the Eq.(17), the relation between aH and τ to the first order approximation of ǫ 1 can be obtained as [18] aH ≈ − 1 Substitute this relation to the expressions of power spectrum, we get with the scalar spectral index and the tensor-to-scalar ratio are Similarly as in the previous section, we set the parameter δ 1 to be a constant and discuss two cases: δ 1 = 0 or δ = 0.
A. δ 1 = 0 If δ 1 = 0 and η H = constant, which is the original constant-roll model without GB coupling [3]. In this case, to the first order approximation of ǫ 1 , the scalar spectral index can be approximated as and the tensor-to-scalar ratio Since η H is a constant, from the definition of flow parameters (5) and (6), and the condition ǫ 1 (N = 0) = 1, we obtain the relation [18]  If δ 1 =constant but not zero, the scalar spectral index and the tensor-to-scalar ratio to the first order of ǫ 1 is obtained as and Using the relation (48) and setting the e-folds N = 60, we find four reasonable regions of parameter space, which are show in Fig.10. We can see that in this case of constant-roll inflation, the absolute of GB flow parameter δ 1 can be larger then one(region IV), which is different from the conventional slow-roll inflation with GB coupling.
The tensor-to-scalar ratio r versus the spectral index n R for the parameters region I of Similarly, the n R versus r for the parameters in region III of Fig.10 are show in Fig.13 In

VI. RECONSTRUCT THE POTENTIAL
In this section, we shall find the corresponding scalar potential V (φ) and the GB coupling ξ(φ) for the constant-roll inflation. If the GB flow parameter δ 1 is a constant, from the definition of flow parameters, we have δ 2 = 0, then the background equations (2)and (3) can be written as In the following, we will discuss the models with ǫ 1 =constant, ǫ 2 =constant and η H =constant, respectively.
In the model with ǫ 1 = constant, the case with δ 1 = 0 is ruled out by the observations, so we only interest in the case with δ 1 =constant but not zero. Substitute (51) into the definition of the flow parameter ǫ 1 we obtain a first-order differential equation of the Hubble parameter and get two solutions with c 1 is a integration constant. Substitute the Eqs.(54) and (51) into the Hamilton-Jacobi equation(52), we get the potential of the scalar field φ where the overall factor V 0 = c 2 1 /2 can be restricted by the amplitude of the primordial curvature perturbations.
For the GB coupling ξ(φ), we substitute the result of Hubble parameter (54) into the definition of GB flow parameter δ 1 = 4ξH and get a differential equation of ξ(φ), the solutions are where ξ 0 is an integration constant represents the GB term with out coupling to φ, which is no contribution to the result, and is always set to zero. In addition, the coefficient ω in the expressions can be chosen the values ±1 to ensure the results H(φ),V (φ),ξ(φ) are real functions and generate reasonable inflation. For instance, if δ 1 − 2ǫ 1 + δ 1 ǫ 1 > 0, ω should be chosen −1.
The relation δ 1 ≪ ǫ 1 < 1 is likely to happen near the end of inflation, then the background equation (51) is approximated as combine with the definition of ǫ 1 , we get the relation which is the same as the standard slow-roll inflation without GB coupling. Substitute (63) into the definition of the flow parameter ǫ 2 , Eq.(58), we obtain the differential equation of the Hubble parameter and the solution is with c 1 and c 2 are integration constants. The potential of the scalar field can be obtained by substitute the Eqs.(63) and (64) into (52), which can be written as with the overall factor V 0 = c 2 2 /2 . Substitute the result of Hubble parameter (64) into the definition of GB flow parameter δ 1 = 4ξH, then we get the solution of GB coupling In this section, we reconstruct the potential for the constant-roll inflation with η H and δ 1 are constants. If δ 1 = 0, the model recover to the case without GB coupling, which is discussed in other reference [3]. Form the background equation, we get the relation Using the same method as in the previous subsections and combine the definition of η H we get the Hubble parameter and the scalar potential as functions of φ with c 1 and c 2 are integration constants. The results are consistent with other reference [3].
Similarly, for the case with δ 1 = 0, the differential equation to the Hubble parameter is no analytic solutions. And we discuss the approximate expressions in two special cases: If ǫ 1 ≪ δ 1 , combine the background equation (57) and (52) with c 1 and c 2 are integration constants and V 0 = c 2 2 /2. Substitute the Hubble parameter (71) into the definition of GB flow parameter δ 1 = 4ξH, then the solution of GB coupling can be written as for the + case of (71), and for the − case is 2. δ 1 ≪ ǫ 1 If δ 1 ≪ ǫ 1 < 1, using the same method, combine the relation (63) with the definition of parameter η H (68), we can get the solutions of the Hubble parameter H(φ) then substitute it into the background equation (52), we get the scalar potential V (φ) and the GB coupling ξ(φ) can be obtained by integer the definition of δ 1 VII. SUMMARY In this paper, we discuss the constant-roll inflation in the model with the inflation field φ nonminimal couple to the GB term. Since the presence of a new degree of freedom from GB coupling ξ(φ), we consider a special case that the first GB flow parameter δ 1 is a constant.
Then we discuss the constant-roll inflation with constant ǫ 1 , constant ǫ 2 and constant η H , respectively. Using the Bessel function approximation, we present the mode equations of scalar and tensor perturbations and get the analytical expressions for the scalar and tensor power spectrum to the first order of ǫ 1 . We derive the scalar spectral index and the tensor to scalar ratio and constraint the parameter space by using the Planck 2018 results. We first assume that ǫ 1 is a constant, the case with δ 1 = 0 is ruled out by the observations, and in the case that δ 1 = constant but not zero, we obtain two regions of parameter space and show the results on the n R − r region. Second, in the assumption with ǫ 2 = constant, δ 1 = 0 is the constant-roll inflation without GB coupling, which is also ruled out by the observations. If δ 1 =constant but not zero, we obtain three regions of parameter space.
Finally, we assume that η H = constant, the case with δ 1 = 0 is the original constant-roll inflation discussed in [3]. Here we update the constraint from Planck 2018 data, which is −0.0163 < η H < −0.0006. In the case with δ 1 = constant but not zero, we obtain four feasible regions of parameter space, and find that in some region, the chosen of GB flow parameter δ 1 > 1 can also produce a nearly scale-invariant scalar power spectrum, which is different from the conventional slow-roll inflation with GB coupling. In addition, we also find the analytical expressions for corresponding scalar potential of the constant-roll inflation and the nonminimum coupling ξ(φ) in some cases.