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 $ε\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$_{\rm DM}^{0.3}$).

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