  Galaxy clustering contains information on cosmology, galaxy evolution, and
the relationship between galaxies and their dark matter hosts. On small scales,
the detailed kinematics of galaxies within their host halos determines the
galaxy clustering. In this paper, we investigate the dependence of the central
and satellite galaxy kinematics on $\boldsymbol{\theta}$, the intrinsic host
halo properties (mass, spin, concentration), cosmology ($\Omega_{\textrm{m}}$,
$\sigma_8$), and baryonic feedback from active galactic nuclei and supernovae
($A_{\rm AGN1}$, $A_{\rm AGN2}$, $A_{\rm SN1}$, $A_{\rm SN2}$). We utilize
2,000 hydrodynamic simulations in CAMELS run using IllustrisTNG and SIMBA
galaxy formation models. Focusing on central and satellite galaxies with
$M>10^9M_\ast$, we apply neural density estimation (NDE) with normalizing flows
to estimate their $p(\Delta r|\boldsymbol{\theta})$ and $p(\Delta
v|\boldsymbol{\theta})$, where $\Delta r$ and $\Delta v$ are the magnitudes of
the halo-centric spatial and velocity offsets. With NDE, we accurately capture
the dependence of galaxy kinematics on each component of $\boldsymbol{\theta}$.
For central galaxies, we identify significant spatial and velocity biases
dependent on halo mass, concentration, and spin. For satellite distributions,
we find significant deviations from an NFW profile and evidence that they
consist of distinct orbiting and infalling populations. However, we find no
significant dependence on $\boldsymbol{\theta}$ besides a weak dependence on
host halo spin. For both central and satellite galaxies, there is no
significant dependence on cosmological parameters and baryonic feedback. These
results provide key insights for improving the current halo occupation
distribution (HOD) models. This work is the first in a series that will
re-examine and develop HOD frameworks for improved modeling of galaxy
clustering at smaller scales.
