We investigate the correlations between the black hole mass M BH , the velocity dispersion σ, the bulge mass M Bu , the bulge average spherical density ρ 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, 19 pseudo bulges) by joining 72 galaxies from the literature to 25 galaxies observed during our recent SINFONI black hole survey. For the first time we discuss the full error covariance matrix. We analyse 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 anti-correlations between M Bu and ρ h . We establish five significant bivariate correlationsthat predict M BH of 77 core and power-law ellipticals and classical bulges with measured and intrinsic scatter as small as ≈ 0.36 dex and ≈ 0.33 dex respectively, or 0.26 dex when the subsample of 45 galaxies defined by Kormendy & Ho (2013) is considered. In contrast, pseudo bulges have systematically lower M BH , but approach the predictions of all the above relations at spherical densities ρ h ≥ 10 10 M /kpc 3 or scale lengths r h ≤ 1 kpc. These findings fit in a scenario of co-evolution of BH and classical-bulge masses, where core ellipticals are the product of dry mergers of power-law bulges and power-law Es 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 pseudo bulge 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.

We present evidence for a strong correlation between the concentration of bulges and the mass of their central supermassive black hole (M bh ) -more concentrated bulges have more massive black holes. Using C re (1/3) from Trujillo, Graham, & Caon (2001b) as a measure of bulge concentration, we find that log(M bh /M ⊙ ) = 6.81(±0.95)C re (1/3) + 5.03 ± 0.41. This correlation is shown to be marginally stronger (Spearman's r s = 0.91) than the relationship between the logarithm of the stellar velocity dispersion and log M bh (Spearman's r s = 0.86), and has comparable, or less, scatter (0.31 dex in log M bh , which decreases to 0.19 dex when we use only those galaxies whose supermassive black hole's radius of influence is resolved and remove one well understood outlying data point). It would appear that the central black hole mass can be estimated from surface photometry alone, without the expensive addition of velocity dispersion determinations.