Effect of cosmic ray/X-ray ionization on supermassive black hole formation

We study the effects of external ionization by cosmic rays (CRs) and X-rays on the thermal evolution of primordial clouds under strong far-ultraviolet (FUV) radiation. A strong FUV radiation field photodissociates H₂ and quenches its cooling. Even in such an environment, a massive cloud with virial temperature >rsim10⁴K can contract isothermally at 8000K by hydrogen Lyman α cooling. This cloud collapses monolithically without fragmentation, and a supermassive star (>rsim10⁵ M_⊙) is believed to form at the centre, which eventually evolves into a supermassive black hole (SMBH). However, candidates of FUV sources, including star-forming galaxies, are probably sources of strong CRs and X-rays too. We find that external ionization promotes H₂ production and elevates the threshold FUV intensity J_crit needed for SMBH formation for CR energy density U_CR > rsim 10^-14 ergcm^-3 or X-ray intensity J_X > rsim 10^-24 ergs^-1cm^-2sr^-1Hz^-1 at 1keV. The critical FUV flux increases as J_crit∝U_CR^1/2 (∝J_X^1/2) in the high CR (respectively X-ray) limit. With the same value of FUV intensity at the Lyman limit (13.6eV), the H^- photodissociation rate, with a threshold of 0.755eV, increases and the H₂ abundance decreases with decreasing effective temperature of the FUV sources T_*. A lower value of T_* thus results in a lower critical FUV flux J_crit at the Lyman limit. Using an empirical relation between the intensities of FUV and CRs/X-rays from nearby star-forming galaxies, we find that the external ionization effect remarkably enhances the critical FUV flux for sources with T_* as high as 10⁵K and composed of stars with mass ≲100 M_⊙ to a level that is not realized in any halo. This indicates that to induce SMBH formation the FUV sources must be either Population II/I galaxies with low brightness temperature (T_*∼ 10⁴K), Population III galaxies (T_*∼ 10⁵K) with a very top-heavy initial mass function or Population III galaxies too young to harbour sources of CRs/X-rays, for example supernova remnants or high-mass X-ray binaries.