The Black Hole Mass Function: From Stellar to Supermassive

The overall purpose of this thesis is to establish the black hole (BH) mass function, from stellar to supermassive. We divide this work into two: 1) the stellar black hole mass function covering the mass range $m_{\bullet} \sim 5 - 150 \ M_{\odot}$; 2) the supermassive BH mass function spanning $M_{\bullet} \sim 10^{6 - 10} \ M_{\odot}$; the intermediate BH mass function $(M_{\bullet} - 10^{3}-10^{5} \ M_{\odot})$ will be a byproduct of the continuity equation approach used to track supermassive BH growth and allow for the stitching together of a complete BH mass function. We incorporate a wide range of disciplines including: galaxy evolution, stellar evolution, and black hole evolution. We seek to create a self-contained, self-consistent, model that spans from the present day to $z\sim10$, which can be used to make estimations of future observational predictions specifically in regards to gravitational wave instruments (Laser Interferometer Space Antenna, Deci-hertz Interferometer, and Einstein Telescope) and electromagnetic detections (James Webb Space Telescope and \textit{Athena}). Regarding the stellar BH mass function, we mainly consider the standard, and likely dominant, production channel of stellar mass BHs constituted by isolated single/binary star evolution. 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 ${\rm d}^2N/{\rm d}V{\rm d}\log m_\bullet$ 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 normalisation 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 predominantly 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. We highlight that our results can provide a firm theoretical basis for a physically-motivated light seed distribution at high redshift, to be implemented in semi-analytic and numerical models of BH formation and evolution. In terms of the supermassive BH mass function, we consider two main mechanisms to grow the central BH, that are expected to cooperate in the high-redshift star-forming progenitors of local massive galaxies. The first is the gaseous dynamical friction process, that can cause the migration toward the nuclear regions of stellar-mass BHs originated during the intense bursts of star formation in the gas-rich host progenitor galaxy, and the buildup of a central heavy BH seed $M_\bullet\sim 10^{3-5}\, M_\odot$ within short timescales $\lesssim$ some $10^7$ yr. The second mechanism is the standard Eddington-type gas disk accretion onto the heavy BH seed, through which the central BH can become (super)massive $M_\bullet\sim 10^{6-10}\, M_\odot$ within the typical star-formation duration $\lesssim 1$ Gyr of the host. We validate our semi-empirical approach by reproducing the observed redshift-dependent bolometric AGN luminosity functions and Eddington ratio distributions, and the relationship between the star-formation and the bolometric luminosity of the accreting central BH. We then derive the relic (super)massive BH mass function at different redshifts via a generalised continuity equation approach, and compare it with present observational estimates. Finally, we reconstruct the overall BH mass function from the stellar to the (super)massive regime, over more than ten orders of magnitudes in BH mass. Overall we have found that the number of black holes within the observable Universe (a sphere of diameter around 90 billion light years) at present time is about 40 billion billions (i.e. about $40 \times 10^{18}$).