The supermassive black hole mass - S\'ersic index relations for bulges
  and elliptical galaxies
  Scaling relations between supermassive black hole mass, M_BH, and host galaxy
properties are a powerful instrument for studying their coevolution. A complete
picture involving all of the black hole scaling relations, in which each
relation is consistent with the others, is necessary to fully understand the
black hole-galaxy connection. The relation between M_BH and the central light
concentration of the surrounding bulge, quantified by the S\'ersic index n, may
be one of the simplest and strongest such relations, requiring only
uncalibrated galaxy images. We have conducted a census of literature S\'ersic
index measurements for a sample of 54 local galaxies with directly measured
M_BH values. We find a clear M_BH - n relation, despite an appreciable level of
scatter due to the heterogeneity of the data. Given the current M_BH - L_sph
and the L_sph - n relations, we have additionally derived the expected M_BH - n
relations, which are marginally consistent at the 2 sigma level with the
observed relations. Elliptical galaxies and the bulges of disc galaxies are
each expected to follow two distinct bent M_BH - n relations due to the
S\'ersic/core-S\'ersic divide. For the same central light concentration, we
predict that M_BH in the S\'ersic bulges of disc galaxies are an order
magnitude higher than in S\'ersic elliptical galaxies if they follow the same
M_BH - L_sph relation.
