Linking Electron Density with Elevated Star Formation Activity from $z=0$ to $z=10$

The interstellar medium (ISM) in high-redshift galaxies exhibits significantly higher electron densities ($n_{\rm e}$) than in the local universe. To investigate the origin of this trend, we analyze a sample of 9590 centrally star-forming galaxies with stellar masses greater than $10^9\,M_\odot$ at redshifts $0.01 < z < 0.04$, selected from the Dark Energy Spectroscopic Instrument (DESI) Data Release 1. We derive electron densities from the [S II] $\lambda\lambda6716,6731$ doublet, measuring values of $n_{\rm e} = 30$-$400~{\rm cm^{-3}}$ at $z \approx 0$. We find a tight correlation between $n_{\rm e}$ and the star formation rate surface density ($\Sigma_{\rm SFR}$), which is well described by a broken power law. Above a threshold of $\log(\Sigma_{\rm SFR} / M_\odot\,{\rm yr^{-1}\,kpc^{-2}}) \ge -1.46$, the relation follows $n_{\rm e} = (233 \pm 13)\,\Sigma_{\rm SFR}^{0.49 \pm 0.02}$. Below this threshold, $n_{\rm e}$ remains approximately constant at $44 \pm 3~{\rm cm^{-3}}$. Remarkably, this relation remains consistent with measurements of galaxies at $z = 0.9$-$10.2$. By converting the observed redshift evolution of $\Sigma_{\rm SFR}$ into $n_{\rm e}$ evolution through our $n_{\rm e}$-$\Sigma_{\rm SFR}$ relation, we obtain $n_{\rm e} = 40(1+z)^{1.4}~{\rm cm^{-3}}$, consistent with previous direct observations. The $n_{\rm e}$-$\Sigma_{\rm SFR}$ relation likely arises because the high $\Sigma_{\rm SFR}$, fueled by dense cold gas or elevated efficiency, enhances radiative and mechanical feedback and produces dense ionized gas whose electron densities are further regulated by ambient pressure. We conclude that the redshift evolution of $n_{\rm e}$ primarily reflects the evolution of cold gas density and star formation activity over cosmic time.