# Measuring the delay time distribution of binary neutron stars. I. Through Scaling Relations of the Host Galaxies of Gravitational Wave Events

## TL;DR

This paper proposes a method to constrain the delay time distribution of binary neutron star mergers using host galaxy demographics from gravitational wave detections, requiring about a thousand galaxy observations.

## Contribution

It introduces a new approach leveraging galaxy mass and star formation history correlations to estimate DTD parameters from GW host galaxy data.

## Key findings

- Approximately 1,000 host galaxies are needed to constrain the DTD.
- Massive galaxies primarily encode the DTD information.
- Fewer galaxies (~100) suffice if the minimum delay time is fixed.

## Abstract

The delay time distribution of (DTD) of binary neutron stars (BNS) remains poorly constrained, mainly by the small known population of Galactic binaries, the properties of short gamma-ray burst host galaxies, and inferences from $r$-process enrichment. In the new era of BNS merger detections through gravitational waves (GW), a new route to the DTD is the demographics of the host galaxies, localized through associated electromagnetic counterparts. This approach takes advantage of the correlation between star formation history (SFH) and galaxy mass, such that the convolution of the SFH and DTD impacts the BNS merger rate as a function of galaxy mass. Here we quantify this approach for a power law DTD governed by two parameters: the power law index ($\Gamma$) and a minimum delay time ($t_{\rm min}$). Under the reasonable assumption that EM counterparts are likely only detectable in the local universe, accessible by the current generation of GW detectors, we study how many host galaxies at $z\sim 0$ are required to constrain the DTD parameters. We find that the DTD is mainly imprinted in the statistics of massive galaxies (stellar mass of $M_*\gtrsim 10^{10.5}$ M$_\odot$, comparable to the host galaxy of GW170817). Taking account of relevant uncertainties we find that $\mathcal{O}(10^3)$ host galaxies are required to constrain the DTD; for a fixed value of $t_{\rm min}$, as done in previous analyses of the DTD, $\mathcal{O}(10^2)$ host galaxies will suffice. Such a sample might become available within the next two decades, prior to the advent of third-generation GW detectors.