Redshift Evolution of the Ratio of Supermassive Black Hole Mass to Stellar Mass

We run and analyze a suite of high-redshift zoom-in cosmological simulations with varying supernova feedback and supermassive black hole (SMBH) accretion prescriptions to study the joint evolution of stellar and SMBH mass in high-redshift galaxies down to $z=10$. The simulations reproduce the observed high-$z$ $M_{\mathrm{BH}}/M_{\star}$ relation if super-Eddington accretion is allowed prior to the final self-regulated phase. To extend the evolution to lower redshift, we model subsequent black hole and host growth using analytic halo assembly histories combined with a redshift-dependent effective Eddington duty cycle, $f_{\rm duty}=0.0004(1+z)^3$, calibrated to observations at $z\le6$, with conservative uncertainties at higher redshift. Within this framework, $M_{\mathrm{BH}}/M_{\star}$ exhibits a broad peak at $z\sim7$--10, reaching a few percent up to $\sim30\%$, followed by a steady, approximately power-law decline toward $z=0$. The model predicts $M_{\mathrm{BH}}/M_{\star}\sim(0.002,0.003,0.006,0.016,0.071,0.156)$ at $z=(0,1,2,3,5,10)$, consistent with available observations. This evolution is driven by rapid SMBH growth at high redshift, with effective mass e-folding times shorter than those of stellar mass, while at later times galaxy growth dominates, leading to the decline in $M_{\mathrm{BH}}/M_{\star}$. These results demonstrate that the emergence of a high-redshift peak and subsequent decline is robust despite uncertainties in the duty-cycle normalization.