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 _ { H } = 10 - 1000 ~ { } 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 ~ { } \mu m and \gtrsim 100 ~ { } \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 ~ { } \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 .