# *f*(*R*) constant-roll inflation

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## Abstract

The previously introduced class of two-parametric phenomenological inflationary models in general relativity in which the slow-roll assumption is replaced by the more general, constant-roll condition is generalized to the case of *f*(*R*) gravity. A simple constant-roll condition is defined in the original Jordan frame, and exact expressions for a scalaron potential in the Einstein frame, for a function *f*(*R*) (in the parametric form) and for inflationary dynamics are obtained. The region of the model parameters permitted by the latest observational constraints on the scalar spectral index and the tensor-to-scalar ratio of primordial metric perturbations generated during inflation is determined.

## 1 Introduction

The constant-roll inflation is a two-parametric class of phenomenological inflationary model which satisfies the assumption of constant rate of the inflaton [1, 2, 3]. The assumption is a generalization of the standard slow-roll inflation with an approximately flat inflaton potential, and so-called ultra-slow-roll inflation [4, 5, 6], in which the potential is constant for an extended period, and the curvature perturbation grows on superhorizon scales. The attempt of such a generalization first proposed in [1], and the inflaton potential was constructed so that it satisfies the constant-roll condition approximately. Later, it was clarified in [2] that there exists a potential that satisfies the constant-roll condition exactly. In addition, the model possesses the exact solution that is an attractor for inflationary dynamics. It is also elucidated that the curvature perturbation is conserved on superhorizon scales. Not only does the constant-roll inflation serve theoretically interesting framework, it is also viable with the most recent observational data. In [3], we showed that the model can satisfy the latest observational constraint on the spectral index of the curvature power spectrum and the tensor-to-scalar ratio.

This constant-roll construction refers to inflationary models in General Relativity (GR) where gravity is not modified but a new scalar field has to be introduced. On the other hand, in the opposite limit one can construct inflationary models without new scalar fields, by changing the gravity sector only, as typified by the \(R+R^2\) model [7] and its *f*(*R*) gravity modifications [8, 9, 10, 11, 12]. This purely geometrical approach is equivalent to introducing a scalar degree of freedom (dubbed a scalaron in [7]), which can be explicitly seen by performing a conformal transformation from the Jordan frame to the Einstein frame. Viable inflationary models in *f*(*R*) gravity are slow-rolling, too. Since the present level of accuracy of astronomical observations make it interesting to go beyond the slow-roll approximation, in this paper we construct a new constant-roll inflationary model in the framework of *f*(*R*) gravity. In contrast to previous works [1, 2, 3] where the constant-roll condition was effectively imposed in the Einstein frame, since inflation in GR was considered, we impose a new constant-roll condition in the original Jordan frame where the form of equations is simpler in fact; see e.g. Eq. (11) below.

The rest of the paper is organized as follows. In Sect. 2, we review *f*(*R*) gravity focusing on its Jordan and Einstein frame description. In Sect. 3, we introduce a novel constant-roll condition in the Jordan frame and derive exact solutions for the potential and the Hubble parameter in the Einstein frame. In Sect. 4, we derive a parametric expression of *f*(*R*), and explore the inflationary dynamics in the Jordan frame. In Sect. 5, we consider the spectral parameters for the inflationary power spectra. We use them in Sect. 6 to show the model possesses an available parameter region. We conclude in Sect. 7. In the appendix, two alternative derivations of the parametric expression for the constant-roll *f*(*R*) function are presented, with the latter of them using the Jordan frame only.

## 2 *f*(*R*) gravity

*f*(

*R*) gravity and the relation between the Einstein and Jordan frames (see e.g. [13] for a more extensive review and the list of references). We consider the action

*f*(

*R*) is specified in the Jordan frame, the scalaron \(\phi \) and the potential \(V(\phi )\) in the Einstein frame are given by the above definition. Conversely, once the potential is specified in the Einstein frame, the Ricci scalar and the function

*f*(

*R*) in the Jordan frame are given by

*f*(

*R*) gravity the master first-order equation for \(H_\mathrm{J}\) in the original Jordan frame considered as a function of the Ricci scalar \(R_\mathrm{J}\) has even a simpler form, which can be obtained as follows. We represent \(\dot{F}\) as

## 3 *f*(*R*) constant-roll potential

*f*(

*R*) gravity:

*not*conformally dual to the former one used in GR. Of course, such generalization can be produced in many ways. We have chosen just the form (12) for the constant-roll condition in

*f*(

*R*) gravity from reasons of simplicity and aesthetic elegance.

^{1}In particular, in the case of the \(R+R^2\) inflationary model, it reduces to

*f*(

*R*) function, substituting the constant-roll condition (12) to (3) and integrating it, we obtain a very simple and elegant relation which has to be satisfied for all models in this class at all times:

Below we shall show that one can construct an inflationary model that satisfies the constant-roll condition (15), and has an exact solution for inflationary evolution. Further, we shall clarify that the model has a parameter region that satisfies the latest observational constraint on spectral parameters of inflationary power spectra.

Following [2], we employ the Hamiltonian–Jacobi formalism and regard \(H_\mathrm{E}=H_\mathrm{E}(\phi )\), assuming that \(t_\mathrm{E} = t_\mathrm{E}(\phi )\) is a single-valued function, or \(\mathrm{d}\phi /\mathrm{d}t_\mathrm{E}\ne 0\). When \(\mathrm{d}\phi /\mathrm{d}t_\mathrm{E}=0\), the Hamiltonian–Jacobi formalism breaks down, and the stochastic effect becomes dominant. It should be avoided that the inflaton passes such a point during inflation. If the breakdown is located before inflation, there is no problem to rely on the Hamiltonian–Jacobi formalism. We will check this point later on.

*M*(mass dimension 1) and \(\gamma \) (dimensionless). Using redefinition of

*M*and \(\phi \), we can always normalize \(\gamma \). Therefore, without loss of generality, we consider only \(\gamma =0,\pm 1\) for the following. On the other hand the amplitude of

*M*is determined by the CMB normalization. Below we work in the unit where \(M=1\).

For \(\gamma =0\), the potential is given by a single exponential function. On the other hand, for \(\beta \approx 0\) or \(-3\) the potential is mainly described by a single exponential function with a constant, which is of our target. However, we do not consider the case \(\beta < -3\) and \(\gamma = -1\) as the potential is always negative.

## 4 *f*(*R*) constant-roll dynamics

Reasons why parameter regions except \(\beta \lesssim 0, \gamma =-1\) are excluded

\(\gamma =+1\) | \(\gamma =-1\) | \(\gamma =0\) | |
---|---|---|---|

\(\beta \lesssim -3\) | Inhomogeneity | Always \(R,V<0\) | \(r = 8 (1-n_s)\) |

\(\beta \gtrsim -3\) | Inhomogeneity | \(r\ge 21.3\) for \(R_\mathrm{J}\ge 0\) | \(r = 8 (1-n_s)\) |

\(\beta \lesssim 0\) | Inhomogeneity | Viable, Fig. 3 | \(r = 8 (1-n_s)\) |

\(\beta \gtrsim 0\) | \(r\ge 9.48\) for \(\frac{\mathrm{d}R_\mathrm{J}}{\mathrm{d}t_\mathrm{E}}\le 0\) | Inhomogeneity | \(r = 8 (1-n_s)\) |

*f*(

*R*):

*f*(

*R*) are depicted in Fig. 2 for the case \(\beta =-0.02, \gamma =-1\), for which \(4 \le \phi \le 4.8\) amounts to \(3.4 \le R_\mathrm{J}/10^2 \le 6.6\). The relative error increases as \(\phi \) or \(R_\mathrm{J}\) decreases, and reaches \(5,10\%\) at \(R_\mathrm{J}/10^{-2}=1.4\), 0.88, respectively.

## 5 Inflationary power spectra

Now we check the spectral parameters of inflationary power spectra and compare them with observational constraint to find viable parameter set \((\beta ,\gamma )\). First, the power spectrum of scalar (curvature) and tensor perturbations can be calculated in the Jordan frame directly, e.g. as was quantitatively correctly done in [17] for the model [7] using the \(\delta N\) formalism. Second, the calculation in the Einstein frame leads to the same result since the constant modes of scalar (curvature) and tensor perturbations are not affected by a generic (inhomogeneous) conformal transformation after the end of inflation; see e.g. [18, 19] for more details, and [20] for more general invariance under disformal transformation. The subtle point is that though the value of the power spectrum is the same in both frames, it refers to slightly different inverse scales \(k_\mathrm{E}\) and \(k_\mathrm{J}\). However, corrections to the power spectra of scalar and tensor perturbations following from this difference are proportional to \(|n_s-1|\) and \(|n_t|\) correspondingly. In particular, they would be absent for the exactly scale-invariant spectra. Thus, they can be neglected in the leading order of the slow-roll approximation.

Finally, let us focus on the tensor-to-scalar ratio *r* along with the evolution of \(R_\mathrm{J}\). As we mentioned in Sect. 4, we require \(R_\mathrm{J}>0\) and \(\mathrm{d}R_\mathrm{J}/\mathrm{d}t_\mathrm{E}<0\) during inflation. For the parameter set \(\beta \gtrsim -3\) with \(\gamma =-1\), \(R_\mathrm{J} \ge 0\) for \(\phi \ge \phi _r\) where \(\phi _r\) is defined by (29). Then it can be shown by checking \(\mathrm{d}r/\mathrm{d}\phi \) and \(\mathrm{d}^2r/\mathrm{d}\phi ^2\) that for the region \(\phi \ge \phi _r\) the minimum value of *r* is given at \(\phi =\phi _r\), which is given by (30). Therefore, so long as we consider the region where \(R_\mathrm{J} \ge 0\), we have \(r\ge 21.3\), which is much larger than the observationally allowed value. Likewise, for the parameter set \(\beta \gtrsim 0\) with \(\gamma >0\), by using (31) we can also show that \(\mathrm{d}R_\mathrm{J}/\mathrm{d}t_\mathrm{E}\le 0\) holds for \(\phi \le \frac{\sqrt{6}}{\beta -3} \log \left[ \frac{2(2-\beta )(3+\beta )}{3\gamma (1+\beta )} \right] \), and for this field position, \(r\ge \frac{16}{27} (4+\beta )^2 \ge 9.48\), which is also not acceptable.

We thus find that only allowed possibility is \(\beta \lesssim 0\) with \(\gamma =-1\). Indeed, parameter set \(-0.02 \lesssim \beta < 0 ,\gamma =-1\) satisfies the observational constraint on \((n_s,r)\). Other parameter regions are not feasible for various reasons, which are summarized in Table 1.

## 6 Observational constraints

As shown in Table 1, in the previous sections we checked that the parameter regions other than \(\beta \lesssim 0, \gamma =-1\) are excluded by various reasons. Now we show that the case \(\beta \lesssim 0, \gamma =-1\) can indeed satisfy the latest observational constraint.

We already see the typical behavior for this parameter set in Fig. 1. The potential is approximated by \(V \sim - e^{-\phi } + \mathrm{const}\) so long as \(|\beta \phi |\lesssim 1\). The inflaton rolls on the plateau of the potential at positive \(\phi \) region towards negative direction with \(\mathrm{d}\phi /\mathrm{d}t_\mathrm{E} < 0\). Before the inflaton reaches to \(\phi = \phi _c\) where \(V=0\), we need to cut the potential at some point \(\phi =\phi _0 > \phi _c\) to realize a graceful exit from inflation.

Indeed, this form of the potential with a long plateau is favored by the observational data. During the inflation on the plateau, *V* and \(R_\mathrm{J}\) remain positive, and the plateau is sufficiently long to produce a large number of e-folds \(N_\mathrm{J}\sim 50\) (see Fig. 1).

## 7 Conclusion

We have constructed a simple and natural generalization of the class of constant-roll inflationary models in GR to the case of *f*(*R*) gravity. The constant-roll condition (12) is introduced in the original Jordan frame. Using it, we derived the exact solutions for the Einstein-frame potential in (20), the parametric expression of \(f(R_\mathrm{J})\) in (26), as well as the inflationary evolution in the Einstein and Jordan frames. The functional form of \(f(R_\mathrm{J})\) is expressed parametrically, while for some special parameter values it is possible to write down \(f(R_\mathrm{J})\) explicitly as a function of \(R_\mathrm{J}\). We showed that the model has an interesting parameter region \(-0.1 \lesssim \beta < 0, \gamma =-1\) which satisfies the latest observational constraint on \((n_s,r)\) obtained by the Planck and BICEP2/Keck Array Collaborations.

## Footnotes

## Notes

### Acknowledgements

H.M. thanks the Research Center for the Early Universe (RESCEU), where part of this work was completed. He was also supported in part by MINECO Grant SEV-2014-0398, PROMETEO II/2014/050, Spanish Grant FPA2014-57816-P of the MINECO, and European UnionâĂŹs Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreements No. 690575 and 674896. A.S. acknowledges RESCEU for hospitality as a visiting professor. He was also partially supported by the grant RFBR 17-02-01008 and by the Scientific Programme P-7 (sub-programme 7B) of the Presidium of the Russian Academy of Sciences.

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