Rigorous Analytic Solution to the Gravitational-wave Overlapping Event Rates

Abstract In the era of the next-generation gravitational-wave detectors, signal overlaps will become prevalent due to the 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 of 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.