  The first dark matter halos form by direct collapse from peaks in the matter
density field, and evidence from numerical simulations and other analyses
suggests that the dense inner regions of these objects largely persist today.
These halos would be the densest dark matter structures in the Universe, and
their abundance can probe processes that leave imprints on the primordial
density field, such as inflation or an early matter-dominated era. They can
also probe dark matter through its free-streaming scale. The first halos are
qualitatively different from halos that form by hierarchical clustering, as
evidenced by their $\rho\propto r^{-3/2}$ inner density profiles. In this work,
we present and tune models that predict the density profiles of these halos
from properties of the density peaks from which they collapsed. These models
predict the coefficient $A$ of the $\rho=Ar^{-3/2}$ small-radius asymptote of
the density profile along with the maximum circular velocity $v_\mathrm{max}$
and associated radius $r_\mathrm{max}$. These models are universal; they can be
applied to any cosmology, and we confirm this by validating them using six
$N$-body simulations carried out in wildly disparate cosmological scenarios. We
find that these models can even predict the full population of halos with
reasonable accuracy in scenarios with narrowly supported power spectra,
although for broader power spectra, an understanding of the impact of halo
mergers is needed. With their connection to the primordial density field
established, the first dark matter halos will serve as probes of the early
Universe and the nature of dark matter.
