Dusty Gas Accretion onto Massive Black Holes and Infrared Diagnosis of
  the Eddington Ratio
  Evidence for dust around supermassive black holes (SMBHs) in the early
Universe is strongly suggested by recent observations. However, the accretion
mechanism of SMBHs in dusty gas is not well understood yet. We investigate the
growth of intermediate-mass black-holes (IMBHs) of $\sim 10^{5}~M_{\odot}$ in
dusty clouds by using one-dimensional radiative-hydrodynamics simulations. We
find that the accretion of dusty gas onto IMBHs proceeds gently with small
fluctuations of the accretion rate, whereas that of pristine gas causes more
violent periodic bursts. At dust-to-gas mass ratios similar to the solar
neighborhood, the time averaged luminosity becomes smaller than that for
primordial gas by one order of magnitude and the time-averaged Eddington ratio
ranges from $\sim 10^{-4}$ to $\sim 10^{-2}$ in clouds with initial gas
densities of $n_{\rm H} = 10 - 1000~\rm cm^{-3}$. Our calculations show that
the effect of dust opacity alone is secondary compared to the radiation
pressure on dust in regulating the BH growth. We also derive spectral energy
distributions at IR bands by calculating dust thermal emission and show that
the flux ratio between $\lambda \lesssim 20~\rm \mu m$ and $\gtrsim 100~\rm \mu
m$ is closely related to the Eddington ratio. Thermal emission from hot dust
near the BH dominates only during the high accretion phase, producing higher
flux density at $\lesssim 20~\rm \mu m$. Therefore, we suggest that the
combinations of MIR observations by JWST and FIR observation by ALMA or Spitzer
can be used to estimate the Eddington ratio of massive BHs.
