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

The knowledge about the black hole mass function (BHMF) and its evolution would help to understand the origin of the BHs and how BH binaries formed at different stages of the history of the Universe. We demonstrate the ability of 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 the 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 as $\alpha$. 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 find that only one thousand of GW events registered by ET ($\sim1\%$ amount of its yearly detection rate) could accurately infer the $\alpha$ with a precision of $\alpha\sim0.1$. Furthermore, we investigate the validity of our method to recover a scenario where $\alpha$ evolves with redshift as $\alpha(z) = \alpha_0 + \alpha_1\frac{z}{1+z}$. Taking a thousand of GW events and using $d_L$ as the redshift estimator, our tests show that one could infer the value of evolving parameter $\alpha_1$ accurately with the uncertainty level of $\sim0.5$. Our numerical tests verify the reliability of our method. The uncertainty levels of the inferred parameters can be trusted directly for the several sets of the parameter we assumed, yet shouldn't be treated as a universal level for the general case.