Constant-roll Inflation Research Articles

Constant-roll approach to noncanonical inflation

The scenario of constant-roll inflation is studied where the inflaton is a scalar field with modified kinetic term, known as non-canonical scalar field. This modification leads to some changes in the slow-roll parameters, and also by taking the second slow-roll parameter as a constant, the differential equation for the Hubble parameter is altered as well. Assuming $\eta=\beta$ and reconsidering the perturbation equations makes it clear that there should be some modified terms in the scalar spectral index and amplitude of scalar perturbations. After finding the exact solution, the main perturbation parameters are obtained at the horizon crossing time. Then by plotting the $r-n_s$ diagram it is shown that for some specific values of $\beta$ one could find a good agreement with observational data. In addition, considering the attractor behavior of the model leads to this result that this feature could be appropriately satisfied.

Dynamical analysis of attractor behavior in constant roll inflation

There has been considerable recent interest in a new class of non-slow roll inflationary solutions known as constant roll inflation. Constant roll solutions are a generalization of the ultra-slow roll (USR) solution, where the first Hubble slow roll parameter ϵ is small, but the second Hubble slow roll parameter η is not. While it is known that the USR solutions represent dynamical transients, there has been some disagreement in literature about whether or not large-η constant roll solutions are attractors or are also a class of transient solutions. In this paper we show that the large-η constant roll solutions do in fact represent transient solutions by performing stability analysis on the exact analytic (large-η) constant roll solutions.

Primordial black hole formation with non-Gaussian curvature perturbations

In the context of transient constant-roll inflation near a local maximum, we derive the non-perturbative field redefinition that relates a Gaussian random field with the true non-Gaussian curvature perturbation. Our analysis shows the emergence of a new critical amplitude ζ*, corresponding to perturbations that prevent the inflaton from overshooting the local maximum, thus becoming trapped in the false minimum of the potential. For potentials with a mild curvature at the local maximum (and thus small non-Gaussianity), we recover the known perturbative field redefinition. We apply these results to the formation of primordial black holes, and discuss the cases for which ζ* is smaller or of the same order than the critical value for collapse of spherically symmetric overdensities. In the latter case, we present a simple potential for which the power spectrum needs an amplitude 10 times smaller that in the Gaussian case for producing a sizeable amount of primordial black holes.

Constant-roll inflation driven by a scalar field with nonminimal derivative coupling

In this work, we study constant-roll inflation driven by a scalar field with nonminimal derivative coupling to gravity, via the Einstein tensor. This model contains a free parameter, [Formula: see text], which quantifies the nonminimal derivative coupling and a parameter [Formula: see text] which characterizes the constant-roll condition. In this scenario, using the Hamilton–Jacobi-like formalism, an ansatz for the Hubble parameter (as a function of the scalar field) and some restrictions on the model parameters, we found new exact solutions for the inflaton potential which include power-law, de Sitter, quadratic hilltop and natural inflation, among others. Additionally, a phase-space analysis was performed and it is shown that the exact solutions associated to natural inflation and a “cosh-type” potential, are attractors.

Observational constraints on constant roll inflation

Constant-roll inflation was recently introduced by Motohashi, Starobinsky and Yokoyama as a phenomenological way to parametrize deviations from the slow-roll scenarios. In this paper, we investigate the dynamics of both the background and the perturbations in this model, without making any slow-roll assumptions. The perturbation spectra are computed with an efficient and accurate novel method that allowed us to quickly scan the parameter space of constant-roll inflation. We derive the constraints on the model parameters from the cosmic microwave background anisotropy measurements provided by the joint analysis of the Planck Collaboration and the BICEP2/Keck Array data.

Inflation with Gauss-Bonnet coupling

We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation $\delta_1=2\lambda\epsilon_1$ between the two slow-roll parameters $\delta_1$ and $\epsilon_1$. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function $\xi(\phi)$ and the potential $V(\phi)$, and it leads to the relation $r=16(1-\lambda)\epsilon_1$ that reduces the tensor-to-scalar ratio $r$ by a factor of $1-\lambda$. For the constant-roll inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor-to-scalar ratio to the first order of $\epsilon_1$ by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of $1-\lambda+\lambda\tilde \eta$. Comparing the derived $n_s$-$r$ with the observations, we obtain the constraints on the model parameters $\tilde\eta$ and $\lambda$.

Large- η constant-roll inflation is never an attractor

Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $\eta$ is larger than unity. In Ultra-slow Roll Inflation, the large-$\eta$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, recent papers have concluded that Constant-Roll Inflation represents a new class of non-slow roll attractor solutions. In this paper we show that these attractor solutions are actually the usual slow roll attractor, disguised by a parameter duality, and that the large-$\eta$ solutions, as in the case of ultra-slow roll, represent a dynamical transient.

Constant-roll tachyon inflation and observational constraints

For the constant-roll tachyon inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts and the tensor to scalar ratio to the first order of ϵ1 by using the method of Bessel function approximation. The derived ns-r results are compared with the observations, we find that only the constant-roll inflation with ηH being a constant is consistent with the observations and observations constrain the constant-roll inflation to be slow-roll inflation. The tachyon potential is also reconstructed for the constant-roll inflation which is consistent with the observations.

Constant-roll (quasi-)linear inflation

In constant-roll inflation, the scalar field that drives the accelerated expansion of the Universe is rolling down its potential at a constant rate. Within this framework, we highlight the relations between the Hubble slow-roll parameters and the potential ones, studying in detail the case of a single-field Coleman-Weinberg model characterised by a non-minimal coupling of the inflaton to gravity. With respect to the exact constant-roll predictions, we find that assuming an approximate slow-roll behaviour yields a difference of Δ r = 0.001 in the tensor-to-scalar ratio prediction. Such a discrepancy is in principle testable by future satellite missions. As for the scalar spectral index ns, we find that the existing 2-σ bound constrains the value of the non-minimal coupling to ξϕ ∼ 0.29–0.31 in the model under consideration.

Tachyon constant-roll inflation

The constant-roll inflation is studied where the inflaton is taken as a tachyon field. Since in this approach the second slow-roll parameter is assumed to be of order one instead of being small, then the perturbation parameters will be considered again. The results are compared with observational data, and it is confirmed that the model could stand as a proper candidate for inflation.

The observational constraint on constant-roll inflation

We discuss the constant-roll inflation with constant $\epsilon_2$ and constant $\bar\eta$. By using the method of Bessel function approximation, the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor to scalar ratio are derived up to the first order of $\epsilon_1$. The model with constant $\epsilon_2$ is ruled out by the observations at the $3\sigma$ confidence level, and the model with constant $\bar\eta$ is consistent with the observations at the $1\sigma$ confidence level. The potential for the model with constant $\bar\eta$ is also obtained from the Hamilton-Jacobi equation. Although the observations constrain the constant-roll inflation to be slow-roll inflation, the $n_s-r$ results from the constant-roll inflation are not the same as those from the slow-roll inflation even when $\bar\eta\sim 0.01$.

On the constant-roll inflation

The primordial power spectra of scalar and tensor perturbations during slow-roll inflation are usually calculated with the method of Bessel function approximation. For constant-roll or ultra slow-roll inflation, the method of Bessel function approximation may be invalid. We compare the numerical results with the analytical results derived from the Bessel function approximation, and we find that they differ significantly on super-horizon scales if the constant slow-roll parameter ηH is not small. More accurate method is needed for calculating the primordial power spectrum for constant-roll inflation.

Systematics of constant roll inflation

We study constant roll inflation systematically. This is a regime, in which the slow roll approximation can be violated. It has long been thought that this approximation is necessary for agreement with observations. However, recently it was understood that there can be inflationary models with a constant, and not necessarily small, rate of roll that are both stable and compatible with the observational constraint ns ≈ 1. We investigate systematically the condition for such a constant-roll regime. In the process, we find a whole new class of inflationary models, in addition to the known solutions. We show that the new models are stable under scalar perturbations. Finally, we find a part of their parameter space, in which they produce a nearly scale-invariant scalar power spectrum, as needed for observational viability.

New perspectives on constant-roll inflation

We study constant-roll inflation using the β-function formalism. We show that the constant rate of the inflaton roll is translated into a first order differential equation for the β-function which can be solved easily. The solutions to this equation correspond to the usual constant-roll models. We then construct, by perturbing these exact solutions, more general classes of models that satisfy the constant-roll equation asymptotically. In the case of an asymptotic power law solution, these corrections naturally provide an end to the inflationary phase. Interestingly, while from a theoretical point of view (in particular in terms of the holographic interpretation) these models are intrinsically different from standard slow-roll inflation, they may have phenomenological predictions in good agreement with present cosmological data.

Anisotropic constant-roll Inflation

We study constant-roll inflation in the presence of a gauge field coupled to an inflaton. By imposing the constant anisotropy condition, we find new exact anisotropic constant-roll inflationary solutions which include anisotropic power-law inflation as a special case. We also numerically show that the new anisotropic solutions are attractors in the phase space.

Constant-roll inflation in F(R) gravity

In this paper, we study constant-roll inflation in gravity. We take two different approaches, one that relates gravity to well-known scalar models of constant-roll and a second that examines the constant-roll condition in gravity directly. With regard to the first approach, by using well-known techniques, we find the gravity that realizes a given constant-roll evolution in the scalar–tensor theory. We also perform a conformal transformation in the resulting gravity and find the Einstein frame counterpart theory. As we demonstrate, the resulting scalar potential is different in comparison to the original scalar constant-roll case, and the same applies for the corresponding observational indices. Moreover, we discuss how cosmological evolutions that can realize constant-roll to constant-roll era transitions in the scalar-tensor description can be realized by vacuum gravity. With regard to the second approach, we examine the effects of the constant-roll condition on the inflationary dynamics of vacuum gravity directly. We present in detail the formalism of constant-roll gravity inflationary dynamics and we discuss the inflationary indices for this case. We use two well-known gravities in order to illustrate our findings: the R2 model and a power-law gravity in vacuum. As we demonstrate, in both cases the parameter space is enlarged in comparison to the slow-roll counterparts of the models and, in effect, the models can also be compatible with the observational data. Finally, we briefly address the graceful exit issue.

Reheating in constant-roll F(R) gravity

In this work, we address the reheating issue in the context of F(R) gravity, for theories that the inflationary era does not obey the slow-roll condition but the constant-roll condition is assumed. As it is known, the reheating era takes place after the end of the inflationary era, so we investigate the implications of a constant-roll inflation era on the reheating era. We quantify our considerations by calculating the reheating temperature for the constant-roll R2 model and we compare to the standard reheating temperature in the context of F(R) gravity. As we demonstrate, the new reheating temperature may differ from the standard one, and in addition, we show how the reheating era may restrict the constant-roll era by constraining the constant-roll parameter. However, due to the observational constraints on the constant-roll parameter, no large deviations from the slow-roll picture occur.

Unification of constant-roll inflation and dark energy with logarithmic R2-corrected and exponential F(R) gravity

In this paper we investigate how to describe in a unified way a constant-roll inflationary era with a dark energy era, by using the theoretical framework of F(R) gravity. To this end, we introduce some classes of appropriately chosen F(R) gravity models, and we examine in detail how the unification of early and late-time acceleration eras can be achieved. We study in detail the inflationary era, and as we demonstrate it is possible to achieve a viable inflationary era, for which the spectral index of primordial curvature perturbations and the scalar-to-tensor ratio can be compatible with the latest observational data. In addition, the graceful exit issue is briefly discussed for a class of models. Finally, we discuss the dark energy oscillations issue, and we investigate which model from one of the classes we introduced, can produce oscillations with the smallest amplitude.

Inflation and bounce from classical and loop quantum cosmology imperfect fluids

We investigate how various inflationary and bouncing cosmologies can be realized by imperfect fluids with a generalized equation of state, in the context of both classical and loop quantum cosmologies. With regards to the inflationary cosmologies, we study the intermediate inflation scenario, the [Formula: see text] inflation scenario and two constant-roll inflation scenarios, and with regards to the bouncing cosmologies, we study the matter bounce scenario, the singular bounce and the super bounce scenario. Within the context of the classical cosmology, we calculate the spectral index of the power spectrum of primordial curvature perturbations, the scalar-to-tensor ratio and the running of the spectral index and we compare the resulting picture with the Planck data. As we demonstrate, partial compatibility with the observational data is achieved in the imperfect fluid description, however none of the above scenarios is in full agreement with data. This result shows that although it is possible to realize various cosmological scenarios using different theoretical frameworks, it is not guaranteed that all the theoretical descriptions are viable.

F(R) constant-roll inflation

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.