Hidden universality in the merger rate distribution in the primordial black hole scenario

TL;DR

This paper explores the merger rates of primordial black hole binaries with a broad mass distribution, revealing a universal property that can help test the primordial black hole scenario.

Contribution

It derives a general formula for the merger time distribution in the PBH mass plane and identifies a universal quantity that is nearly constant across different mass functions.

Findings

The universal quantity $oldsymbol{ extit{ extbf{ extalpha}}}$ is approximately between 0.97 and 1.05 for all binary masses.

The merger rate distribution $oldsymbol{ extit{ extbf{ extcal R}}}(m_1,m_2)$ exhibits universality in the PBH scenario.

The results provide a potential observational test for primordial black hole models.

Abstract

It has been proposed that primordial black holes (PBHs) form binaries in the radiation dominated era. Once formed, some fraction of them may merge within the age of the Universe by gravitational radiation reaction. We investigate the merger rate of the PBH binaries when the PBHs have a distribution of masses around ${\cal O}(10) M_\odot$, which is a generalization of the previous studies where the PBHs are assumed to have the same mass. After deriving a formula for the merger time probability distribution in the PBH mass plane, we evaluate it under two different approximations. We identify a quantity constructed from the mass-distribution of the merger rate density per unit cosmic time and comoving volume $\mathcal{R}(m_1,m_2)$, $\alpha = -{(m_1+m_2)}^2\partial^2 \ln\mathcal{R}/\partial m_1\partial m_2 $, which universally satisfies $0.97 \lesssim \alpha \lesssim 1.05$ for all binary masses independently of the PBH mass function. This result suggests that the measurement of this quantity is useful for testing the PBH scenario.