Past exam of the astrophysics course of the University of Oxford / 2015 / 1 / c / Solution

Treat the quoted distances as sharp isotropic detection radii in a homogeneous Euclidean population. With galaxy number density $n_g$, the number surveyed is $N_g=(4\pi /3)n_g{D}^3$. One event per year requires a per-galaxy merger rate $\mathcal{R}=1/N_g$ per year. ThusThe larger gravitational wave range gives enormous volume leverage: even a rare merger in a typical galaxy can yield frequent detections across the surveyed population. This supports promising detection prospects if astrophysical rates exceed these thresholds. The estimate is not a prediction of the rates themselves. Real sensitivity depends on sky direction, binary inclination, detector duty cycle and a signal-to-noise selection function; a quoted horizon distance is not automatically a volume-averaged range. Cosmological volume and source-time corrections also matter toward the largest quoted distance. The calculation follows the simple assumptions specified by the paper.