How an improved implementation of H2 self-shielding influences the formation of massive stars and black holes

High redshift quasars at z>6 have masses up to ~$10^9$ M$_\odot$. One of the pathways to their formation includes direct collapse of gas, forming a supermassive star, precursor of the black hole seed. The conditions for direct collapse are more easily achievable in metal-free haloes, where atomic hydrogen cooling operates and molecular hydrogen (H2) formation is inhibited by a strong external UV flux. Above a certain value of UV flux (J_crit), the gas in a halo collapses isothermally at ~$10^4$ K and provides the conditions for supermassive star formation. However, H2 can self-shield, reducing the effect of photodissociation. So far, most numerical studies used the local Jeans length to calculate the column densities for self-shielding. We implement an improved method for the determination of column densities in 3D simulations and analyse its effect on the value of J_crit. This new method captures the gas geometry and velocity field and enables us to properly determine the direction-dependent self-shielding factor of H2 against photodissociating radiation. We find a value of J_crit that is a factor of two smaller than with the Jeans approach (~2000 J_21 vs. ~4000 J_21). The main reason for this difference is the strong directional dependence of the H2 column density. With this lower value of J_crit, the number of haloes exposed to a flux >J_crit is larger by more than an order of magnitude compared to previous studies. This may translate into a similar enhancement in the predicted number density of black hole seeds.