The dawn of disks: unveiling the turbulent ionised gas kinematics of the galaxy population at $z\sim4-6$ with JWST/NIRCam grism spectroscopy

Recent studies of gas kinematics at high redshift have reported disky systems which appear to challenge models of galaxy formation, but it is unclear whether they are representative of the underlying galaxy population. We present the first statistical sample of spatially resolved ionised gas kinematics at high redshift, comprised of $272$ H$\alpha$ emitters in GOODS-S and GOODS-N at redshifts $z\approx3.9-6.5$, observed with JWST/NIRCam slitless spectroscopy and imaging from JADES, FRESCO and CONGRESS. The sample probes two orders of magnitude in stellar mass ($\log (M_{\star}[\mathrm{M}_{\odot}])\approx8-10$) and star formation rate ($\text{SFR}\approx0.3-100\thinspace M_{\odot}/$yr), and is representative down to $\log(M_{\star}[\mathrm{M}_{\odot}])\approx 9$. Using a novel inference tool, $\texttt{geko}$, we model the grism data to measure morphological and kinematic properties of the ionised gas, as probed by H$\alpha$. Our results are consistent with a decrease of the rotational support $v/\sigma_0$\ and increase of the velocity dispersion $\sigma_0$ with redshift, with $\sigma_0\approx100$ km/s and $v/\sigma_0\approx1-2$ at $z\approx3.9-6.5$. We study the relations between $\sigma_0$, and $v/\sigma_0$, and different star formation tracers and find a large scatter and diversity, with the strongest correlations between $\sigma_0$ and SFR and SFR surface density. The fraction of rotationally supported systems ($v/\sigma_0>1$) slightly increases with cosmic time, from $(36\pm6)\%$ to $(41\pm6)\%$ from $z\sim 5.5$ to $z\sim 4.5$, for galaxies with masses $9<\log(M_{\star}[\mathrm{M}_{\odot}])<10$. Overall, disks do not dominate the turbulent high-redshift galaxy population in the mass range probed by this work. When placed in the context of studies up to cosmic noon, our results are consistent with a significant increase of disk-like systems with cosmic time.