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.
