Redshift Evolution of the Electron Density in the ISM at $z\sim 0-9$ Uncovered with JWST/NIRSpec Spectra and Line-Spread Function Determinations

We present electron densities $n_{\rm e}$ in the inter-stellar medium (ISM) of star-forming galaxies at $z=4-9$ observed by the JWST/NIRSpec GLASS, ERO, and CEERS programs. We carefully evaluate line-spread functions of the NIRSpec instrument as a function of wavelength with the calibration data of a planetary nebula taken onboard, and obtain secure [OII]$\lambda\lambda$3726,3729 doublet fluxes for 14 galaxies at $z=4.02-8.68$ falling on the star-formation main sequence with the NIRSpec high and medium resolution spectra. We thus derive the electron densities of singly-ionized oxygen nebulae with the standard $n_{\rm e}$ indicator of [OII] doublet, and find that the electron densities of the $z=4-9$ galaxies are $n_{\rm e}\gtrsim 300$ cm$^{-3}$ significantly higher than those of low-$z$ galaxies at a given stellar mass, star-formation rate (SFR), and specific SFR. Interestingly, typical electron densities of singly ionized nebulae increase from $z=0$ to $z=1-3$ and $z=4-9$, which is approximated by the evolutionary relation of $n_{\rm e}\propto(1+z)^{p}$ with $p\sim 1-2$. Although it is not obvious that the ISM property of $n_{\rm e}$ is influenced by global galaxy properties, these results may suggest that nebula densities of high-$z$ galaxies are generally high due to the compact morphologies of high-$z$ galaxies evolving by $r_{\rm e}$ approximately proportional to $(1+z)^{-1}$ ($r_{\rm vir} \propto (1+z)^{-1}$) for a given stellar (halo) mass whose inverse square corresponds to the $p\sim 2$ evolutionary relation. The $p\sim 1-2$ evolutionary relation can be explained by a combination of the compact morphology and the reduction of $n_{\rm e}$ due to the high electron temperature of the high-$z$ metal poor nebulae.