  While large scale primordial non-Gaussianity is strongly constrained by
present-day data, there are no such constraints at Mpc scales. Here we
investigate the effect of significant small-scale primordial non-Gaussianity on
structure formation and the galaxy formation process with collisionless
simulations: specifically, we explore four different types of
non-Gaussianities. All of these prescriptions lead to a distinct and
potentially detectable feature in the matter power spectrum around the
non-linear scale. The feature might have interesting consequences for the $S_8$
tension. We then show in particular that a negatively-skewed distribution of
the potential random field, hence positively skewed in terms of overdensities,
with $f_{\rm NL}$ of the order of 1000 at these scales, implies that typical
galaxy-sized halos reach half of their present-day mass at an earlier stage and
have a quieter merging history at $z<3$ than in the Gaussian case. Their
environment between 0.5 and 4 virial radii at $z=0$ is less dense than in the
Gaussian case. This quieter history and less dense environment has potentially
interesting consequences in terms of the formation of bulges and bars.
Moreover, we show that the two most massive subhalos around their host tend to
display an interesting anti-correlation of velocities, indicative of kinematic
coherence. All these hints will need to be statistically confirmed in
larger-box simulations with scale-dependent non-Gaussian initial conditions,
followed by hydrodynamical zoom-in simulations to explore the detailed
consequences of small-scale non-Gaussianities on galaxy formation.
