Dark matter halos and the M-\sigma relation for supermassive black holes
  We develop models of two-component spherical galaxies to establish scaling
relations linking the properties of spheroids at $z=0$ (total stellar masses,
effective radii $R_e$ and velocity dispersions within $R_e$) to the properties
of their dark-matter halos at both $z=0$ and higher redshifts. . Our main
motivation is the widely accepted idea that the accretion-driven growth of
supermassive black holes (SMBHs) in protogalaxies is limited by quasar-mode
feedback and gas blow-out. The SMBH masses, $M_{\rm{BH}}$, should then be
connected to the dark-matter potential wells at the redshift $z_{\rm{qso}}$ of
the blow-out. We specifically consider the example of a power-law dependence on
the maximum circular speed in a protogalactic dark-matter halo:
$M_{\rm{BH}}\propto V^4_{\rm{d,pk}}$, as could be expected if quasar-mode
feedback were momentum-driven. For halos with a given $V_{\rm{d,pk}}$ at a
given $z_{\rm{qso}}\ge 0$, our model scaling relations give a typical stellar
velocity dispersion $\sigma_{\rm{ap}}(R_e)$ at $z=0$. Thus, they transform a
theoretical $M_{\rm{BH}}$-$V_{\rm{d,pk}}$ relation into a prediction for an
observable $M_{\rm{BH}}$-$\sigma_{\rm{ap}}(R_e)$ relation. We find the latter
to be distinctly non-linear in log-log space. Its shape depends on the generic
redshift-evolution of halos in a {$\Lambda$}CDM cosmology and the systematic
variation of stellar-to-dark matter mass fraction at $z=0$, in addition to any
assumptions about the physics underlying the $M_{\rm{BH}}$-$V_{\rm{d,pk}}$
relation. Despite some clear limitations of the form we use for $M_{\rm{BH}}$
versus $V_{\rm{d,pk}}$, and even though we do not include any SMBH growth
through dry mergers at low redshift, our results for
$M_{\rm{BH}}$-$\sigma_{\rm{ap}}(R_e)$ compare well to data for local early
types if we take $z_{\rm{qso}} \sim$ 2-4.
