Concentration, Ellipsoidal Collapse, and the Densest Dark Matter haloes
  The smallest dark matter haloes are the first objects to form in the
hierarchical structure formation of cold dark matter (CDM) cosmology and are
expected to be the densest and most fundamental building blocks of CDM
structures in our universe. Nevertheless, the physical characteristics of these
haloes have stayed illusive, as they remain well beyond the current resolution
of N-body simulations (at redshift zero). However, they dominate the
predictions (and uncertainty) in expected dark matter annihilation signal,
amongst other astrophysical observables. Using the conservation of total energy
and the ellipsoidal collapse framework, we can analytically find the mean and
scatter of concentration $c$ and 1-D velocity dispersion $\sigma_{\rm 1d}$ for
haloes of different virial mass $M_{200}$. Both $c$ and $\sigma_{\rm
1d}/M_{200}^{1/3}$ are in good agreement with numerical results within the
regime probed by simulations -- slowly decreasing functions of mass that
approach constant values at large masses. In particular, the predictions for
the 1-D velocity dispersion of cluster mass haloes are surprisingly robust as
the inverse heat capacity of cosmological haloes crosses zero at $M_{200} \sim
10^{14} M_\odot$. However, we find that current extrapolations from simulations
to smallest CDM haloes dramatically depend on the assumed profile (e.g. NFW vs.
Einasto) and fitting function, which is why theoretical considerations, such as
the one presented here, can significantly constrain the range of feasible
predictions.
