Dark-matter haloes and the M-σ 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 haloes 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-BH, should then be connected to the dark-matter potential wells at the redshift z(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-BH proportional to V-d,pk(4), as could be expected if quasar-mode feedback were momentum-driven. For haloes with a given V-d,V-pk at a given z(qso) >= 0, our model scaling relations give a typical stellar velocity dispersion sigma(ap)(R-e) at z = 0. Thus, they transform a theoretical M-BH-V-d,V-pk relation into a prediction for an observable M-BH-sigma(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 haloes in a Lambda cold dark matter 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-BH-V-d,V-pk relation. Despite some clear limitations of the form we use for M-BH versus V-d,V-pk, and even though we do not include any SMBH growth through dry mergers at low redshift, our results for M-BH-sigma(ap)(R-e) compare well to data for local early types if we take z(qso) similar to 2-4.