  Quenching of star formation in the central galaxies of cosmological halos is
thought to result from energy released as gas accretes onto a supermassive
black hole. The same energy source also appears to lower the central density
and raise the cooling time of baryonic atmospheres in massive halos, thereby
limiting both star formation and black hole growth, by lifting the baryons in
those halos to greater altitudes. One predicted signature of that feedback
mechanism is a nearly linear relationship between the central black hole's mass
(MBH) and the original binding energy of the halo's baryons. We present the
increasingly strong observational evidence supporting a such a relationship,
showing that it extends up to halos of mass Mhalo ~10^14 MSun. We then compare
current observational constraints on the MBH--Mhalo relation with numerical
simulations, finding that black hole masses in IllustrisTNG appear to exceed
those constraints at Mhalo < 10^13 MSun and that black hole masses in EAGLE
fall short of observations at Mhalo ~ 10^14 MSun. A closer look at IllustrisTNG
shows that quenching of star formation and suppression of black hole growth do
indeed coincide with black hole energy input that lifts the halo's baryons.
However, IllustrisTNG does not reproduce the observed MBH--Mhalo relation
because its black holes gain mass primarily through accretion that does not
contribute to baryon lifting. We suggest adjustments to some of the parameters
in the IllustrisTNG feedback algorithm that may allow the resulting black hole
masses to reflect the inherent links between black hole growth, baryon lifting,
and star formation among the massive galaxies in those simulations.
