Probability distribution function for inclinations of merging compact binaries detected by gravitational wave interferometers

Abstract

We analytically discuss probability distribution functions (PDFs) for inclinations of merging compact binaries whose gravitational waves are coherently detected by a network of ground-based interferometers. These PDFs would be useful for studying the possibility of (1) simultaneously detecting electromagnetic signals (such as gamma-ray bursts) associated with binary mergers and (2) statistically constraining the related theoretical models using actual observational data frommulti-messenger astronomy. Our approach is similar to Schutz (2011), but we explicitly include the dependence of the polarization angles of the binaries, based on the concise formulation given in Cutler & Flanagan (1994). We find that the overall profiles of the PDFs are similar for any network of second-generation detectors (Advanced-LIGO, Advanced-Virgo, KAGRA and LIGO-India). For example, 5.1 per cent of detected binaries would have inclination angle less than 10° with at most 0.1 per cent differences between the potential networks. A perturbative expression is also provided for generating the PDFs with a small number of parameters given by directional averages of the quantity ε that characterizes the asymmetry of network sensitivities to two incoming orthogonal polarization modes.