  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.
