Black Hole Mass Function and Its Evolution—The First Prediction for the Einstein Telescope

Knowledge of the black hole mass function (BHMF) and its evolution would help us understand the origin of BHs and how BH binaries formed at different stages in the history of the universe. We demonstrate the ability of a future third-generation gravitational-wave (GW) detector - the Einstein Telescope (ET) - to infer the slope of the BHMF and its evolution with redshift. We perform a Monte Carlo simulation of the measurements of chirp signals from binary BH systems (BBH) that could be detected by ET, including the BH masses and their luminosity distances (d_L). We use the mass of a primary black hole in each binary system to infer the BHMF as a power-law function with slope parameter α. Taking into account the bias that could be introduced by the uncertainty of measurements and by the selection effect, we carried out the numerical tests and found that only 1000 GW events registered by ET (∼1% of its yearly detection rate) could accurately infer the α with a precision of α ∼ 0.1. Furthermore, we investigate the validity of our method to recover a scenario where α evolves with redshift as α(z)= α₀ + α₁ {z}/{1+z}. Taking a thousand GW events and using d_L as the redshift estimator, our tests show that one could infer the value of evolving parameter α₁ accurately at the uncertainty level of ∼0.5. Our numerical tests verify the reliability of our method. The uncertainty levels of the inferred parameters can be trusted directly for several sets of the parameters we assumed, yet they should not be treated as general.