New constraints on direct collapse black hole formation in the early
  Universe
  Direct collapse black holes (DCBH) have been proposed as a solution to the
challenge of assembling supermassive black holes by $z>6$ to explain the bright
quasars observed at this epoch. The formation of a DCBH seed with $\rm
M_{BH}\sim10^{4-5}\ \rm M_{\odot}$ requires a pristine atomic-cooling halo to
be illuminated by an external radiation field that is sufficiently strong to
entirely suppress H$_{2}$ cooling in the halo. Many previous studies have
attempted to constrain the critical specific intensity that is likely required
to suppress H$_{2}$ cooling, denoted as $J_{\rm crit}$. However, these studies
have typically assumed that the incident external radiation field can be
modeled with a black-body spectrum. Under this assumption, it is possible to
derive a {unique} value for $J_{\rm crit}$ that depends only on the temperature
of the black-body. In this study we consider a more realistic spectral energy
distribution (SED) for the external source of radiation that depends entirely
on its star formation history and age. The rate of destruction of the species
responsible for suppressing molecular hydrogen cooling depends on the detailed
shape of the SED. Therefore the value of $J_{\rm crit}$ is tied to the shape of
the incident SED of the neighbouring galaxy. We fit a parametric form to the
rates of destruction of H$_2$ and H$^-$ that permit direct collapse. Owing to
this, we find that $J_{\rm crit}$ is not a fixed threshold but can lie anywhere
in the range $J_{\rm crit} \sim 0.5$--$10^{3}$, depending on the details of the
source stellar population.
