Rigorous analytic solution to the gravitational-wave overlapping event rates

TL;DR

This paper develops a rigorous analytical framework to estimate the rates and statistical properties of overlapping gravitational-wave signals, aiding future data analysis in high-rate detection scenarios.

Contribution

It provides the first analytical expressions for the expectation and variance of overlapping event counts, including extensions to complex scenarios and a proof of the beta distribution for event time differences.

Findings

Derived analytical formulas for overlapping event expectations and variances.

Proved the beta distribution describes time differences between events.

Framework applicable to complex multi-signal overlap scenarios.

Abstract

In the era of the next-generation gravitational-wave detectors, signal overlaps will become prevalent due to high detection rate and long signal duration, posing significant challenges to data analysis. While effective algorithms are being developed, there still lacks an integrated understanding on the statistical properties for the population of overlapping compact-binary-coalescence signals. For the first time, in order to aid rapid and robust estimation, we rigorously derive and establish analytical expressions for the expectation and variance for the number of overlapping events. This framework is highly extensible, allowing analytical calculation for more complicated scenarios, such as multi-signal overlaps, overlaps between different types of sources, and source-dependent thresholds. We also mathematically prove that the time difference between events in a single observation run is described by the beta distribution, offering an analytical prior reference for Bayesian analysis.