The galaxy's gas content regulated by the dark matter halo mass results
  in a super-linear M$_{\rm BH}$-M$_{\star}$ relation
  Supermassive black holes (SMBHs) are tightly correlated with their hosts but
the origin of such connection remains elusive. To explore the cosmic build-up
of this scaling relation, we present an empirically-motivated model that tracks
galaxy and SMBH growth down to z=0. Starting from a random mass seed
distribution at z=10, we assume that each galaxy evolves on the star-forming
"main sequence" (MS) and each BH follows the recently-derived stellar mass
(M$_{\star}$) dependent ratio between BH accretion rate and star formation
rate, going as BHAR/SFR$\propto$M$_{\star}^{0.73[+0.22,-0.29]}$. Our simple
recipe naturally describes the BH-galaxy build-up in two stages. At first, the
SMBH lags behind the host that evolves along the MS. Later, as the galaxy grows
in M$_{\star}$, our M$_{\star}$-dependent BHAR/SFR induces a super-linear BH
growth, as M$_{\rm BH}$$\propto$M$_{\star}^{1.7}$. According to this formalism,
smaller BH seeds increase their relative mass faster and earlier than bigger BH
seeds, at fixed M$_{\star}$, thus setting along a gradually tighter M$_{\rm
BH}$-M$_{\star}$ locus towards higher M$_{\star}$. Assuming reasonable values
of the radiative efficiency $\epsilon \sim$0.1, our empirical trend agrees with
both high-redshift model predictions and intrinsic M$_{\rm BH}$-M$_{\star}$
relations of local BHs. We speculate that the observed non-linear BH-galaxy
build-up is reflected in a twofold behavior with dark matter halo mass (M$_{\rm
DM}$), displaying a clear turnover at M$_{\rm
DM}\sim$2$\times$10$^{12}$M$_{\odot}$. While Supernovae-driven feedback
suppresses BH growth in smaller halos (BHAR/SFR$\propto$M$_{\rm DM}^{1.6}$),
above the M$_{\rm DM}$ threshold cold gas inflows possibly fuel both BH
accretion and star formation in a similar fashion (BHAR/SFR$\propto$M$_{\rm
DM}^{0.3}$).
