Fluctuations in the high-redshift Lyman-Werner background: close halo pairs as the origin of supermassive black holes

The earliest generation of stars and black holes must have established an early 'Lyman-Werner' background (LWB) at high redshift, prior to the epoch of reionization. Because of the long mean free path of photons with energies hv <13.6 eV, the LWB was nearly uniform. However, some variation in the LWB is expected due to the discrete nature of the sources, and their highly clustered spatial distribution. In this paper, we compute the probability distribution function (PDF) of the LW flux that irradiates dark matter (DM) haloes collapsing at high redshift (z approximate to 10). Our model accounts for (i) the clustering of DM haloes, (ii) Poisson fluctuations in the number of corresponding star-forming galaxies and (iii) scatter in the LW luminosity produced by haloes of a given mass ( calibrated using local observations). We find that >99 per cent of the DM haloes are illuminated by an LW flux within a factor of 2 of the global mean value. However, a small fraction, similar to 10(-8) to 10(-6), of DM haloes with virial temperatures T-vir greater than or similar to 10(4) K have a close luminous neighbour within less than or similar to 10 kpc, and are exposed to an LW flux exceeding the global mean by a factor of >20, or to J(21,LW) > 10(3) (in units of 10(-21) erg s(-1) Hz(-1) sr(-1) cm(-2)). This large LW flux can photodissociate H-2 molecules in the gas collapsing due to atomic cooling in these haloes, and prevent its further cooling and fragmentation. Such close halo pairs therefore provide possible sites in which primordial gas clouds collapse directly into massive black holes (M-BH approximate to 10(4-6)M(circle dot)), and subsequently grow into supermassive (M-BH greater than or similar to 10(9)M(circle dot)) black holes by z approximate to 6.