  We perform one-dimensional radiation hydrodynamical simulations to solve
accretion flows onto massive black holes (BHs) with a very high rate. Assuming
that photon trapping limits the luminosity emerging from the central region to
$L\lesssim L_{\rm Edd}$, Inayoshi, Haiman & Ostriker (2016) have shown that an
accretion flow settles to a "hyper-Eddington" solution, with a steady and
isothermal ($T\simeq 8000$ K) Bondi profile reaching $\gtrsim 5000$ times the
Eddington accretion rate $\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/c^2$. Here we
address the possibility that gas accreting with finite angular momentum forms a
bright nuclear accretion disc, with a luminosity exceeding the Eddington limit
($1\lesssim L/L_{\rm Edd}\lesssim 100$). Combining our simulations with an
analytic model, we find that a transition to steady hyper-Eddington accretion
still occurs, as long as the luminosity remains below $L/L_{\rm Edd}\lesssim
35~(M_{\rm BH}/10^4~M_\odot)^{3/2}(n_\infty/10^5~{\rm
cm^{-3}})(T_\infty/10^4~{\rm K})^{-3/2}(r_{\star}/10^{14}~{\rm cm})^{-1/2}$,
where $n_\infty$ and $T_\infty$ are the density and temperature of the ambient
gas, and $r_\star$ is the radius of the photosphere, at which radiation
emerges. If the luminosity exceeds this value, accretion becomes episodic. Our
results can be accurately recovered in a toy model of an optically thick
spherical shell, driven by radiation force into a collapsing medium. When the
central source is dimmer than the above critical value, the expansion of the
shell is halted and reversed by ram pressure of the collapsing medium, and by
shell's weight. Our results imply that rapid, unimpeded hyper-Eddington
accretion is possible even if the luminosity of the central source far exceeds
the Eddington limit, and can be either steady or strongly episodic.
