Inner cusps of the first dark matter haloes: Formation and survival in a cosmological context

We use very high resolution cosmological zoom simulations to follow the early evolution of twelve first-generation haloes formed from gaussian initial conditions with scale-free power spectra truncated on small scales by a gaussian in wavenumber. Initial collapse occurs with a diverse range of sheet- or filament-like caustic morphologies, but in almost all cases it gives rise to a numerically converged density cusp with $\rho = Ar^{-3/2}$ and total mass comparable to that of the corresponding peak in the initial linear density field. The constant $A$ can be estimated to within about 10 per cent from the properties of this peak. This outcome agrees with earlier work on the first haloes in cold and warm dark matter universes. Within central cusps, the velocity dispersion is close to isotropic, and the equidensity surfaces tend to align with those of the main body of the halo at larger radii. As haloes grow, their cusps are often (but not always) overlaid with additional material at intermediate radii to produce profiles more similar to the Einasto or NFW forms typical of more massive haloes. Nevertheless, to the extent that we can resolve them, cusps survive at the smallest radii. Major mergers can disturb them, but the effect is quite weak in the cases that we study. The cusps extend down to the resolution limits of our simulations, which are typically a factor of several larger than the cores that would be produced by phase-space conservation if the initial power spectrum cutoff arises from free streaming.