To be submitted to The Astrophysical Journal Formation Rates of Black Hole Accretion Disk Gamma-Ray Bursts

The cosmological origin of at least an appreciable fraction of classical gamma-ray bursts (GRBs) is now supported by redshift measurements for a half-dozen faint host galaxies. Still, the nature of the central engine (or engines) that provide the burst energy remains unclear. While many models have been proposed, those currently favored are all based upon the formation of and/or rapid accretion into stellar-mass black holes. Here we discuss a variety of such scenarios and estimate the probability of each. Population synthesis calculations are carried out using a Monte Carlo approach in which the many uncertain parameters intrinsic to such calculations are varied. We estimate the event rate for each class of model as well as the propagation distances for those having significant delay between formation and burst production, i.e., double neutron star (DNS) mergers and black hole-neutron star (BH/NS) mergers. One conclusion is a 1-2 order of magnitude decrease in the rate of DNS and BH/NS mergers compared to that previously calculated using invalid assumptions about common envelope evolution. Other major uncertainties in the event rates and propagation distances include the history of star formation in the universe, the masses of the galaxies in which merging compact objects are born, and the radii of the hydrogen-stripped cores of massive stars. For reasonable assumptions regarding each, we calculate a daily event rate in the universe for (1) merging neutron stars: ~100 day-1; (2) neutron star-black hole mergers: ~450 day-1; (3) collapsars: ~104 day-1; (4) helium star black hole mergers: ~1000 day-1; and (5) white dwarf-black hole mergers: ~20 day-1. The range of uncertainty in these numbers, however, is very large, typically 2-3 orders of magnitude. These rates must additionally be multiplied by any relevant beaming factor (fΩ < 1) and sampling fraction (if the entire universal set of models is not being observed). Depending upon the mass of the host galaxy, one-half of the DNS mergers will happen within 60 kpc (for a galaxy with a mass comparable to that of the Milky Way) to 5 Mpc (for a galaxy with negligible mass) from the Galactic center. The same numbers characterize BH/NS mergers. Because of the delay time, neutron star and black hole mergers will happen at a redshift 0.5-0.8 times that of the other classes of models. Information is still lacking regarding the hosts of short, hard bursts, but we suggest that they are due to DNS and BH/NS mergers and thus will ultimately be determined to lie outside of galaxies and at a closer mean distance than long complex bursts (which we attribute to collapsars). In the absence of a galactic site, the distance to these bursts may be difficult to determine.