Dynamical analysis of attractor behavior in constant roll inflation

Abstract

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 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. *

Constant roll and primordial black holes

“…The canonical constant-roll inflation with parameter β ≈ 0.015 generates red-tilted spectrum compatible with CMB constraints [14,15] (see also [21]). On the other hand, the dynamics of the inflaton with a different range of the constant-roll parameter has been also extensively studied recently [22][23][24].…”

On the Constant-Roll Inflation with Large and Small ηH

“…For the model (18) with η H > 3, the inflation climbs up instead of rolling down the potential and the constant-roll inflationary solution is not an transactor [12]. Furthermore, as shown in [12], in the model (18) with η H > 3/2, the curvature perturbation grows on both the subhorizon and superhorizon scales, but the curvature perturbation decreases on the subhorizon scales and is frozen on the superhorizon scales in both the model (18) with η H < 1 and the model (22) as shown in Figures 1 and 2, so this duality is false because the behaviors of the background and the curvature perturbations are totally different for the constant-roll inflation with large and small η H . In Figure 1, we also see that the analytical power spectrum (12) for the scalar perturbation approximates the exact result very well even when η H > 3 and the curvature perturbations grow on superhorizon scales.…”

“…However, for the constant-roll inflation with η H > 1, the slow-roll parameter H decreases with time and is small during inflation, so we can still use the standard method of Bessel function approximation to calculate the power spectra. Neglecting the contribution from H , it was found there exists a duality between the ultra-slow-roll inflation and the slow-roll inflation [20][21][22], i.e., if we replace η H byη H = 3 − η H , we get the same result for the scalar spectral tilt. Recall that the observational data constrained η H to be small [23][24][25]; these results are in conflict with the duality relation, so it is necessary to revisit the observational constraint to include the constraint on the ultra-slow inflation.…”

“…If the duality holds, then the model (18) with η H > 3 is dual to the model (22) with η H < 0. However, the two potentials (18) and (22) and the background evolutions are totally different, and˙ H < 0 for the model (18), whereas˙ H > 0 for the model (22). For the model (18) with η H > 3, the inflation climbs up instead of rolling down the potential and the constant-roll inflationary solution is not an transactor [12].…”

“…Due to the growth of the curvature perturbations on superhorizon scales for the constant-roll model (18) with η H > 3/2, the scalar power spectrum (12) should be evaluated at the end of inflation instead of the horizon crossing [18,[46][47][48][49]. This brings a new problem on the evaluation of the power spectrum, because inflation does not end naturally and some unknown mechanisms need to be introduced to end inflation in the model (18), but this problem does not exist in the model (22). For the model (20), because no inflation happens if η H > 1, the duality is inapplicable to this model.…”