  We analyze deep X-ray, optical and mid-infrared Spitzer observations of the
CDF-N/GOODS-N region to study 13 submillimeter-detected galaxies (SMGs) with
spectroscopic redshifts (median z=2.2). We find a correlation between the
estimated stellar and X-ray luminosity, implying that masses of the black holes
may be related to the stellar masses of their host galaxies. Although the
submillimeter emission implies that these galaxies are undergoing an epoch of
intense star-formation, the Spitzer data reveal a massive stellar population
already in place. These stellar masses are then compared to previously
published black hole mass estimates derived from the X-ray luminosities under
the assumption of Eddington-limit accretion. We find that the black hole masses
for our high-redshift sample are approximately 1-2 orders of magnitude smaller
than galaxies of comparable stellar mass in the local Universe. Although our
estimates of black hole masses will increase if the accretion is sub-Eddington,
and our stellar masses will decrease if we assume a much younger stellar
population or a different initial mass function, we find that only through a
combination of effects is it possible to shift the high redshift galaxies such
that they lie on the local relation. This suggests that the black holes need to
grow substantially between z=2.2 and the present-day, with much of the black
hole growth occurring after the current obscured, far-infrared luminous phase
of activity which is likely associated with the formation of the spheroid. This
interpretation supports a scenario where SMGs pass through a subsequent
accretion-dominated phase, where they would appear as optically bright quasars.
(ABRIDGED)
