Prompt cusp formation from the gravitational collapse of peaks in the initial cosmological density field

Understand

I present an analytic model for the early post-collapse evolution of a spherical density peak on the coherence scale of the initial fluctuations in a universe filled with collisionless and pressure-free "dust".

• On a time-scale which is short compared to the peak's collapse time $t_0$, its inner regions settle into an equilibrium cusp with a power-law density profile, $\rho\propto r^{-12/7}$.
• Within this cusp, the circular orbit period $P$ at each radius is related to the enclosed mass $M$ by $P = t_0 (M/M_c)^{2/3}$ where $M_c$ is a suitably defined characteristic mass for the initial peak.
• The relaxation mechanism which produces this cusp gives insight into those which are active in high-resolution simulations of first halo formation in Cold or Warm Dark Matter universes, and, indeed, a simple argument suggests that the same power-law index $\gamma=-12/7$ should describe the prompt cusps formed during the collapse of generic peaks, independent of any symmetry assumption.