On the occurrence rate of short and long GRBs

There is mounting evidence for the binary nature of the progenitors of gamma-ray bursts (GRBs). For a long GRB, the induced gravitational collapse (IGC) paradigm proposes as progenitor, or "in-state", a tight binary system composed of a carbon-oxygen core (CO$_{\rm core}$) undergoing a supernova (SN) explosion which triggers hypercritical accretion onto a neutron star (NS) companion. For a short GRB, a NS-NS merger is traditionally adopted as the progenitor. We divide long and short GRBs into two sub-classes, depending whether or not a black hole (BH) is formed in the merger or in the hypercritical accretion process exceeding the NS critical mass. For long bursts, when no BH is formed we have the sub-class of X-ray flashes (XRFs), with isotropic energy $E_{iso}\lesssim10^{52}$ erg and rest-frame spectral peak energy $E_{p,i}\lesssim200$~keV. When a BH is formed we have the sub-class of binary-driven hypernovae (BdHNe), with $E_{iso}\gtrsim10^{52}$ erg and $E_{p,i}\gtrsim200$~keV. In analogy, short bursts are similarly divided into two sub-classes. When no BH is formed, short gamma-ray flashes (S-GRFs) occur, with $E_{iso}\lesssim10^{52}$ erg and $E_{p,i}\lesssim2$~MeV. When a BH is formed, the authentic short GRBs (S-GRBs) occur, with $E_{iso}\gtrsim10^{52}$ erg and $E_{p,i}\gtrsim2$~MeV. We give examples and observational signatures of these four sub-classes and their rate of occurrence. From their respective rates it is possible that "in-states" of S-GRFs and S-GRBs originate from the "out-states" of XRFs. We indicate two additional progenitor systems: white dwarf-NS and BH-NS. These systems have hybrid features between long and short bursts. In the case of S-GRBs and BdHNe evidence is given of the coincidence of the onset of the high energy GeV emission with the birth of a Kerr BH.