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 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, $\rho \propto r^{-12\mathrm{/}7}$. Within this cusp, the circular orbit period $P$ at each radius is related to the enclosed mass $M$ by $P=t_0(M\mathrm{/}{M}_c{)}^{2\mathrm{/}3}$ 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 $\gamma =-12\mathrm{/}7$ should describe the prompt cusps formed during the collapse of generic peaks, independent of any symmetry assumption. Further work is needed to investigate whether additional factors are required to explain the slightly flatter exponent, $\gamma \approx -1.5$, found in high-resolution numerical simulations of peak collapse.