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Numerical resolution effects on simulations of massive black hole seeds

We have performed high-resolution numerical simulations with the hydrodynamical AMR code Enzo to investigate the formation of massive seed black holes in a sample of six dark matter haloes above the atomic cooling threshold. The aim of this study is to illustrate the effects of varying the maximum refinement level on the final object formed. The virial temperatures of the simulated haloes range from $\rm{T} \sim 10000\ \rm{K} - 16000\ \rm{K}$ and they have virial masses in the range $\rm{M} \sim 2 \times 10^7 \rm{M_{\odot}}$ to $\rm{M} \sim 7 \times 10^7 \rm{M_{\odot}}$ at $z \sim 15$. The outcome of our six fiducial simulations is both generic and robust. A rotationally supported, marginally gravitationally stable, disk forms with an exponential profile. The mass and scale length of this disk depends strongly on the maximum refinement level used. Varying the maximum refinement level by factors between 1 / 64 to 256 times the fiducial level illustrates the care that must be taken in interpreting the results. The lower resolution simulations show tentative evidence that the gas may become rotationally supported out to 20 pc while the highest resolution simulations show only weak evidence of rotational support due to the shorter dynamical times for which the simulation runs. The higher resolution simulations do, however, point to fragmentation at small scales of the order of $\sim 100$ AU. In the highest resolution simulations a central object of a few times $10^2\ \rm{M_{\odot}}$ forms with multiple strongly bound, Jeans unstable, clumps of $\sim 10\ \rm{M_{\odot}}$ and radii of 10 - 20 AU suggesting the formation of dense star clusters in these haloes.