Analytical approximation of the scalar spectrum in the ultraslow-roll inflationary models

Abstract

The ultra-slow-roll (USR) inflationary models predict large-amplitude scalar perturbations at small scales which can lead to the primordial black hole production and scalar-induced gravitational waves. In general scalar perturbations in the USR models can only be obtained using numerical method because the usual slow-roll approximation breaks. In this work, we propose an analytical approach to estimate the scalar spectrum which is consistent with the numerical result. We find that the USR inflationary models predict a peak with power-law slopes in the scalar spectrum and energy spectrum of gravitational waves, and we derive the expression of the spectral indexes in terms of the inflationary potential. In turn, the inflationary potential near the USR regime can be reconstructed from the negative spectral index of the gravitational wave energy spectrum.

Scalar Induced Gravitational Waves Review

“…[77]; -Induced GW spectral features: There are cases where the induced GW spectrum may be investigated semi-analytically. These are for example, the low frequency tail [74,78,79], the UV tail [80,81] and the log-normal peak in the primordial spectrum [82]. Furthermore, the primordial spectrum may also present oscillatory features which are captured into the induced GW spectrum [83][84][85][86][87].…”

“…Unfortunately, we are not aware of a general analytical formula for the time integrals (80) unless b ∈ Z, in which case the Bessel functions reduce to spherical Bessel functions and can be written in terms of transcendental functions. This is for example the case of radiation domination b = 0 (w = 1/3), where the general kernel can be found in Ref.…”

“…[63]. Nevertheless, we can integrate (80) in two relevant limits, x 1 (subhorizon) and x 1 (superhorizon).…”

“…This corresponds to the case when all modes of interest entered the horizon much before reheating, that is k k rh . Then, we can safely send the upper integration limit in (80) to infinity, i.e., x = kτ 1. This is because when we eventually match our solution at reheating to a free GW propagating in a radiation dominated universe, the upper limit would be x rh = kτ rh ∼ k/k rh 1.…”

“…This is because when we eventually match our solution at reheating to a free GW propagating in a radiation dominated universe, the upper limit would be x rh = kτ rh ∼ k/k rh 1. The integrals (80) for x → ∞ are derived by Gervois and Navelet [233] and given in Appendix G. Here we just give the form of (79) after integration, which reads…”