  Theoretical models struggle to reproduce dynamically cold disks with
significant rotation-to-dispersion support($V_{\rm{rot}}/\sigma$) observed in
star-forming galaxies in the early Universe, at redshift $z>4$. We aim to
explore the possible emergence of dynamically cold disks in cosmological
simulations and to understand if different kinematic tracers can help reconcile
the tension between theory and observations. We use 3218 galaxies from the
SERRA suite of zoom-in simulations, with $8<\log(M_*/M_{\odot})<10.3$ and
SFR$<128\,M_{\odot}{yr}^{-1}$, within $4<z<9$ range. We generate hyper-spectral
data cubes for 6436 synthetic observations of H$\alpha$ and [CII]. We find that
the choice of kinematic tracer strongly influences gas velocity dispersion
estimates. When using H$\alpha$ ([CII]) synthetic observations, we observe a
strong (mild) correlation between $\sigma$ and $M_*$. Such a difference arises
mostly for $M_*>10^9\,M_{\odot}$ galaxies, for which
$\sigma_{H\alpha}>2\sigma_{CII}$ for a significant fraction of the sample.
Regardless of the tracer, our predictions suggest the existence of massive
($M_*>10^{10}M_{\odot}$) galaxies with $V_{rot}/\sigma>10$ at $z>4$,
maintaining cold disks for >10 orbital periods (200Myr). Furthermore, we do not
find any significant redshift dependence for $V_{rot}/\sigma$ ratio in our
sample. Our simulations predict the existence of dynamically cold disks in the
early Universe. However, different tracers are sensitive to different kinematic
properties. While [CII] effectively traces the thin, gaseous disk of galaxies,
H$\alpha$ includes the contribution from ionized gas beyond the disk,
characterized by prevalent vertical or radial motions that may be associated
with outflows. The presence of H$\alpha$ halos could be a signature of such
galactic outflows. This emphasizes the importance of combining ALMA and
JWST/NIRspec studies of high-z galaxies.
