The SINFONI Black Hole Survey: The black hole fundamental plane revisited and the paths of (co)evolution of supermassive black holes and bulges

We investigate the correlations between the black hole (BH) mass M_BH, the velocity dispersion σ, the bulge mass M_Bu, the bulge average spherical density ${\rho }_h$, and its spherical half-mass radius r_h, constructing a database of 97 galaxies (31 core ellipticals, 17 power-law ellipticals, 30 classical bulges, and 19 pseudobulges) by joining 72 galaxies from the literature to 25 galaxies observed during our recent SINFONI BH survey. For the first time we discuss the full error covariance matrix. We analyze the well-known M_BH–σ and M_BH–M_Bu relations and establish the existence of statistically significant correlations between M_Bu and r_h and anticorrelations between M_Bu and ${\rho }_h$. We establish five significant bivariate correlations (M_BH–σ–ρ_h, M_BH–σ–r_h, M_BH–M_Bu–σ, M_BH–M_Bu–ρ_h, M_BH–M_Bu–r_h) that predict M_BH of 77 core and power-law ellipticals and classical bulges with measured and intrinsic scatter as small as $\approx 0.36$ dex and $\approx 0.33$ dex, respectively, or 0.26 dex when the subsample of 45 galaxies defined by Kormendy and Ho is considered. In contrast, pseudobulges have systematically lower M_BH but approach the predictions of all of the above relations at spherical densities ${\rho }_h⩾{10}^{10}{M}_{\odot }{kpc}^{-3}$ or scale lengths $r_h⩽1kpc$. These findings fit in a scenario of coevolution of BH and classical-bulge masses, where core ellipticals are the product of dry mergers of power-law bulges and power-law ellipticals and bulges the result of (early) gas-rich mergers and of disk galaxies. In contrast, the (secular) growth of BHs is decoupled from the growth of their pseudobulge hosts, except when (gas) densities are high enough to trigger the feedback mechanism responsible for the existence of the correlations between M_BH and galaxy structural parameters.