Primordial black holes as a novel probe of primordial gravitational waves. II: Detailed analysis

TL;DR

This paper develops a detailed theoretical framework to constrain primordial gravitational waves by analyzing the overproduction of primordial black holes caused by second-order scalar perturbations induced by tensor modes.

Contribution

It provides a comprehensive derivation of the induced scalar perturbations and an approximate PDF for density fluctuations, refining previous simple estimates.

Findings

New upper bounds on primordial tensor perturbations from PBH abundance

Derived an approximate PDF for induced density perturbations

Compared constraints with existing bounds from BBN, CMB, LIGO/Virgo, and pulsar timing

Abstract

Recently we have proposed a novel method to probe primordial gravitational waves from upper bounds on the abundance of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations generated in the early Universe is fairly large, they induce substantial scalar perturbations due to their second-order effects. If these induced scalar perturbations are too large when they reenter the horizon, then PBHs are overproduced, their abundance exceeding observational upper limits. That is, primordial tensor perturbations on superhorizon scales can be constrained from the absence of PBHs. In our recent paper we have only shown simple estimations of these new constraints, and hence in this paper, we present detailed derivations, solving the Einstein equations for scalar perturbations induced at second order in tensor perturbations. We also derive an approximate formula for the probability density function of induced density perturbations, necessary to relate the abundance of PBHs to the primordial tensor power spectrum, assuming primordial tensor perturbations follow Gaussian distributions. Our new upper bounds from PBHs are compared with other existing bounds obtained from big bang nucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timing arrays.