Quick recipes for gravitational-wave selection effects

TL;DR

This paper reviews and extends methods for modeling gravitational-wave detection probabilities, providing practical recipes and analyzing biases, crucial for accurate astrophysical population studies and detection pipelines.

Contribution

It introduces new analytical expressions and practical tools for modeling selection effects, including noise realizations, enhancing accuracy in gravitational-wave astronomy.

Findings

Including noise realizations affects detection probability estimates by a few percent.

Biases from approximating SNR can favor detection of marginal sources.

The methods are validated with software injections and analytical considerations.

Abstract

Accurate modeling of selection effects is a key ingredient to the success of gravitational-wave astronomy. The detection probability plays a crucial role in both statistical population studies, where it enters the hierarchical Bayesian likelihood, and astrophysical modeling, where it is used to convert predictions from population-synthesis codes into observable distributions. We review the most commonly used approximations, extend them, and present some recipes for a straightforward implementation. These include a closed-form expression capturing both multiple detectors and noise realizations written in terms of the so-called Marcum Q-function and a ready-to-use mapping between signal-to-noise ratio thresholds and false-alarm rates from state-of-the-art detection pipelines. The bias introduced by approximating the matched filter signal-to-noise ratio with the optimal signal-to-noise ratio is not symmetric: sources that are nominally below threshold are more likely to be detected than sources above threshold are to be missed. Using both analytical considerations and software injections in detection pipelines, we confirm that including noise realizations when estimating the selection function introduces an average variation of a few %. This effect is most relevant for large catalogs and specific subpopulations of sources at the edge of detectability (e.g. high redshifts).