The origin of the galaxy size-stellar metallicity relation: A semi-analytical perspective

Stellar metallicity encodes the integrated effects of gas inflow, star formation, and feedback-driven outflow, yet its connection to galaxy structure remains poorly understood. Using SDSS-IV MaNGA, we present the direct observational evidence that, at fixed stellar mass, smaller central galaxies are systematically more metal-rich, with a Spearman's rank correlation coefficient reaching $R_{\rm s}\approx -0.4$. The semi-analytical model L-GALAXIES reproduces this anti-correlation, albeit with a stronger amplitude ($R_{\rm s}\approx -0.8$). Within this framework, the trend cannot be explained by differences in gravitational potential depth or star formation history. Instead, smaller galaxies attain higher stellar metallicities because their elevated star formation efficiencies accelerate chemical enrichment, and, at fixed stellar mass, they inhabit less massive haloes, which makes their recycled inflows more metal-rich. The gas-regulator model demonstrates that star formation efficiency affects stellar metallicity when the system has not long remained in equilibrium, which is shown to be the case for central galaxies with $M_{\rm star}\lesssim 10^{10.5}\rm M_\odot$ in both L-GALAXIES and observation. The model also suggests a testable signature that, at fixed stellar mass, larger or lower-metallicity galaxies should inhabit more massive haloes than their smaller and higher-metallicity counterparts, providing a direct and testable imprint of the galaxy size-stellar metallicity relation on the galaxy-halo connection.