Evolution of the star formation rate surface density main sequence. Insights from a semi-analytic simulation since $z = 12

Recent high-redshift ($z>4$) spatially resolved observations with the James Webb Space Telesescope have shown the evolution of the star formation rate (SFR) surface density ($\Sigma_{\rm SFR}$) and its main sequence in the $\Sigma_{\rm SFR}$-$M_*$ diagram ($\Sigma_{\rm SFR}{\rm MS}$). The $\Sigma_{\rm SFR}{\rm MS}$\ is already observed at cosmic morning ($z\sim7.5$). The use of $\Sigma_{\rm SFR}$\ is physically motivated because it is normalized by the area in which the star formation occurs, and this indirectly considers the gas density. The $\Sigma_{\rm SFR}$-$M_*$ diagram has been shown to complement the widely used (specific) SFR-$M_*$, particularly when selecting passive galaxies. We establish the $\Sigma_{\rm SFR}$\ evolution since $z=12$ in the framework of the L-Galaxies2020 semi-analytical model (SAM), and we interpret recent observations. We estimated $\Sigma_{\rm SFR}$(-$M_*$) and the cosmic star formation rate density (CSFRD) for the simulated galaxy population and for the subsamples, which were divided into stellar mass bins in the given redshift. The simulated $\Sigma_{\rm SFR}$\ decreases by $\sim3.5$ dex from $z=12$ to $z=0$. We show that galaxies with different stellar masses have different paths of $\Sigma_{\rm SFR}$\ evolution. We find that $\Sigma_{\rm SFR}{\rm MS}$\ is already observed at $z\sim11$. The simulated $\Sigma_{\rm SFR}{\rm MS}$\ agrees with the observed one at $z=0, 1, 2, 5$, and $7.5$ and with individual galaxies at $z>10$. We show that the highest $\Sigma_{\rm SFR}{\rm MS}$\ slope of $0.709\pm0.005$ is at $z\sim3$ and decreases to $\sim0.085\pm0.003$ at $z=0$. This is mostly driven by a rapid decrease in SFR with an additional size increase for the most massive galaxies in this redshift range. This coincides with the dominance of the most massive galaxies in the CSFRD from the SAM.