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 (SNR) thresholds and false-alarm rates from state-of-the-art detection pipelines. The bias introduced by approximating the matched filter SNR with the optimal SNR 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).

中文翻译：

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引力波选择效应的快速配方


选择效应的精确建模是引力波天文学成功的关键因素。检测概率在统计群体研究（它进入分层贝叶斯似然）和天体物理建模（其中它用于将群体合成代码的预测转换为可观察的分布）中都起着至关重要的作用。我们回顾了最常用的近似值，扩展了它们，并提出了一些简单实现的方法。其中包括以所谓的 Marcum Q 函数编写的捕获多个探测器和噪声实现的封闭式表达式，以及信噪比 (SNR) 阈值和误报率之间的即用型映射来自最先进的检测管道。通过用最佳 SNR 近似匹配滤波器 SNR 引入的偏差是不对称的：名义上低于阈值的源比高于阈值的源更容易被检测到。在检测管道中使用分析考虑因素和软件注入，我们确认在估计选择函数时包括噪声实现会引入几个％的平均变化。这种效应与处于可探测性边缘的大型目录和特定源子群最相关（例如高红移）。