# Effects of the surrounding primordial black holes on the merger rate of primordial black hole binaries

## TL;DR

This paper presents an analytic model for calculating the merger rate of primordial black hole binaries, considering effects of surrounding black holes and density perturbations, revealing mass function independence in key parameters.

## Contribution

It introduces a formalism that accounts for torques from surrounding black holes and density perturbations, providing insights into merger rate calculations independent of the mass function.

## Key findings

- The parameter α is found to be 36/37, independent of the mass function.
- The ratio of merger rates considering surrounding torques to nearest black hole effects is mass-independent.
- The formalism enables more accurate predictions of primordial black hole binary mergers.

## Abstract

We develop an analytic formalism for computing the merger rate of primordial black hole binaries with a general mass function by taking into account the torques by the surrounding primordial black holes and linear density perturbations. We find that $\alpha=-(m_{i}+m_{j})^2\partial^{2} \ln {\mathcal R}(m_{i},m_{j})/\partial m_{i} \partial m_{j}=36/37$ is independent of the mass function. Moreover, the ratio of the merger rate density of primordial black hole binaries by taking into account the torques by the surrounding primordial black holes to by the nearest primordial black hole is independent of the masses of binaries.