Precision of Hubble constant derived using black hole binary absolute distances and statistical redshift information

Abstract

Measured gravitational waveforms from black hole binary inspiral events directly determine absolute luminosity distances. To use these data for cosmology, it is necessary to independently obtain redshifts for the events, which may be difficult for those without electromagnetic counterparts. Here it is demonstrated that certainly in principle, and possibly in practice, clustering of galaxies allows extraction of the redshift information from a sample statistically for the purpose of estimating mean cosmological parameters, without identification of host galaxies for individual events. We extract mock galaxy samples from the 6th Data Release of the Sloan Digital Sky Survey resembling those that would be associated with inspiral events of stellar mass black holes falling into massive black holes at redshift z ~ 0.1 to 0.5. A simple statistical procedure is described to estimate a likelihood function for the Hubble constant H_0: each galaxy in a LISA error volume contributes linearly to the log likelihood for the source redshift, and the log likelihood for each source contributes linearly to that of H_0. This procedure is shown to provide an accurate and unbiased estimator of H_0. It is estimated that a precision better than one percent in H_0 may be possible if the rate of such events is sufficiently high, on the order of 20 to z = 0.5.