The Black Hole Mass Function Across Cosmic Times I. Stellar Black Holes
  and Light Seed Distribution
  This is the first paper in a series aimed at modeling the black hole (BH)
mass function, from the stellar to the intermediate to the (super)massive
regime. In the present work we focus on stellar BHs and provide an ab-initio
computation of their mass function across cosmic times. Specifically, we
exploit the state-of-the-art stellar and binary evolutionary code
\texttt{SEVN}, and couple its outputs with redshift-dependent galaxy statistics
and empirical scaling relations involving galaxy metallicity, star-formation
rate and stellar mass. The resulting relic mass function ${\rm d}N/{\rm d}V{\rm
d}\log m_\bullet$ as a function of the BH mass $m_\bullet$ features a rather
flat shape up to $m_\bullet\approx 50\, M_\odot$ and then a log-normal decline
for larger masses, while its overall normalization at a given mass increases
with decreasing redshift. We highlight the contribution to the local mass
function from isolated stars evolving into BHs and from binary stellar systems
ending up in single or binary BHs. We also include the distortion on the mass
function induced by binary BH mergers, finding that it has a minor effect at
the high-mass end. We estimate a local stellar BH relic mass density of
$\rho_\bullet\approx 5\times 10^7\, M_\odot$ Mpc$^{-3}$, which exceeds by more
than two orders of magnitude that in supermassive BHs; this translates into an
energy density parameter $\Omega_\bullet\approx 4\times 10^{-4}$, implying that
the total mass in stellar BHs amounts to $\lesssim 1\%$ of the local baryonic
matter. We show how our mass function for merging BH binaries compares with the
recent estimates from gravitational wave observations by LIGO/Virgo, and
discuss the possible implications for dynamical formation of BH binaries in
dense environments like star clusters. [abridged]
