# Constraints on primordial curvature perturbations from primordial black hole dark matter and secondary gravitational waves

## TL;DR

This paper derives analytical formulas for secondary gravitational waves and uses observational constraints on primordial black holes to set limits on primordial curvature perturbations, aiding early universe physics studies.

## Contribution

It introduces a piecewise power law parametrization for the primordial curvature spectrum and links PBH constraints to gravitational wave observations for the first time.

## Key findings

- Derived analytical formulas for gravitational wave energy density evolution.
- Established upper bounds on primordial curvature spectrum from PBH constraints.
- Discussed future gravitational wave detection prospects for testing the model.

## Abstract

Primordial black holes and secondary gravitational waves can be used to probe the small scale physics at very early time. For secondary gravitational waves produced after the horizon reentry, we derive an analytical formula for the time integral of the source and analytical behavior of the time dependence of the energy density of induced gravitational waves is obtained. By proposing a piecewise power law parametrization for the power spectrum of primordial curvature perturbations, we use the observational constraints on primordial black hole dark matter to obtain an upper bound on the power spectrum, and discuss the test of the model with future space based gravitational wave antenna.