A No-Go Theorem for Direct Collapse Black Holes Without a Strong
  Ultraviolet Background
  Explaining the existence of supermassive black holes (SMBHs) larger than
$\sim 10^9 M_\odot$ at redshifts $z >\sim 6$ remains an open theoretical
question. One possibility is that gas collapsing rapidly in pristine atomic
cooling halos ($T_{\rm vir} >\sim 10^4 \rm{K}$) produces $10^4-10^6 M_\odot$
black holes. Previous studies have shown that the formation of such a black
hole requires a strong UV background to prevent molecular hydrogen cooling and
gas fragmentation. Recently it has been proposed that a high UV background may
not be required for halos that accrete material extremely rapidly or for halos
where gas cooling is delayed due to a high baryon-dark matter streaming
velocity. In this work, we point out that building up a halo with $T_{\rm vir}
>\sim 10^4 \rm{K}$ before molecular cooling becomes efficient is not sufficient
for forming a direct collapse black hole (DCBH). Though molecular hydrogen
formation may be delayed, it will eventually form at high densities leading to
efficient cooling and fragmentation. The only obvious way that molecular
cooling could be avoided in the absence of strong UV radiation, is for gas to
reach high enough density to cause collisional dissociation of molecular
hydrogen ($\sim 10^4 ~ {\rm cm}^{-3}$) before cooling occurs. However, we argue
that the minimum core entropy, set by the entropy of the intergalactic medium
(IGM) when it decouples from the CMB, prevents this from occurring for
realistic halo masses. This is confirmed by hydrodynamical cosmological
simulations without radiative cooling. We explain the maximum density versus
halo mass in these simulations with simple entropy arguments. The low densities
found suggest that DCBH formation indeed requires a strong UV background.
