  We argue that the velocity dispersions and masses of galactic bulges and
spheroids are byproducts of the feedback that regulates rapid black hole growth
in protogalaxies. We suggest that the feedback energy liberated by accretion
must pass through the accreting material, in an energy-conserving flux close-in
and a momentum-conserving flux further out. If the inflowing gas dominates the
gravitational potential outside the Bondi radius, feedback from
Eddington-limited accretion drives the density profile of the gas to that of a
singular isothermal sphere. We find that the velocity dispersion associated
with the isothermal potential, sigma, increases with time as the black hole
mass M grows, in such a way that M is proportional to sigma^4. The coefficient
of this proportionality depends on the radius at which the flow switches from
energy conserving to momentum conserving, and gives the observed M-sigma
relation if the transition occurs at ~100 Schwarzschild radii. We associate
this transition with radiative cooling and show that bremsstrahlung, strongly
boosted by inverse Compton scattering in a two-temperature (T_p >> T_e) plasma,
leads to a transition at the desired radius.
  According to this picture, bulge masses M_b are insensitive to the virial
masses of their dark matter haloes, but correlate linearly with black hole
mass. Our analytic model also explains the M_b-sigma (Faber-Jackson) relation
as a relic of black hole accretion. The model naturally explains why the
M-sigma relation has less scatter than either the M-M_b (Magorrian) or the
Faber-Jackson relation. It suggests that the M-sigma relation could extend down
to very low velocity dispersions, and predicts that the relation should not
evolve with redshift.
