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

I present an analytic model for the early post-collapse evolution of a spherical density peak on coherence scale initial fluctuations in universe filled with collisionless and pressure-free "dust". On time-scale which is short compared to peak's collapse time $t_0$, its inner regions settle into equilibrium cusp power-law profile, $\rho\propto r^{-12/7}$. Within this cusp, circular orbit period $P$ at each radius related enclosed mass $M$ by $P = t_0 (M/M_c)^{2/3}$ where $M_c$ suitably defined characteristic peak. The relaxation mechanism produces gives insight those are active high-resolution simulations first halo formation Cold or Warm Dark Matter universes, and, indeed, simple argument suggests that same index $\gamma=-12/7$ should describe prompt cusps formed during generic peaks, independent any symmetry assumption. Further work needed investigate whether additional factors required explain slightly flatter exponent, $\gamma\approx -1.5$, found numerical collapse.