The Galaxy’s Gas Content Regulated by the Dark Matter Halo Mass Results in a Superlinear M BH – M ⋆ Relation

Supermassive black holes (SMBHs) are tightly correlated with their hosts, but the origin of such connection remains elusive. To explore the cosmic buildup 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⋆) dependent ratio between BH accretion rate and star formation rate, going as $\mathrm{BHAR}/\mathrm{SFR}\propto {M}_{\star }^{0.73[+0.22,-0.29]}$. Our simple recipe naturally describes the BH–galaxy buildup in two stages. At first, the SMBH lags behind the host that evolves along the MS. Later, as the galaxy grows in M⋆, our M⋆-dependent BHAR/SFR induces a superlinear BH growth, as ${M}_{\mathrm{BH}}\propto {M}_{\star }^{1.7}$. According to this formalism, smaller BH seeds increase their relative mass faster and earlier than bigger BH seeds, at fixed M⋆, thus setting along a gradually tighter MBH–M⋆ locus toward higher M⋆. Assuming reasonable values of the radiative efficiency epsilon ∼ 0.1, our empirical trend agrees with both high-redshift model predictions and intrinsic MBH–M⋆ relations of local BHs. We speculate that the observed nonlinear BH–galaxy buildup is reflected in a twofold behavior with dark matter halo mass (MDM), displaying a clear turnover at MDM ∼ 2 × 1012 M⊙. While supernovae-driven feedback suppresses BH growth in smaller halos ($\mathrm{BHAR}/\mathrm{SFR}\propto {M}_{\mathrm{DM}}^{1.6}$), above the MDM threshold cold gas inflows possibly fuel both BH accretion and star formation in a similar fashion ($\mathrm{BHAR}/\mathrm{SFR}\propto {M}_{\mathrm{DM}}^{0.3}$).