  Dark matter coupled solely gravitationally can be produced through the decay
of primordial black holes in the early universe. If the dark matter is lighter
than the initial black hole temperature, it could be warm enough to be subject
to structure formation constraints. In this paper we perform a more precise
determination of these constraints. We first evaluate the dark matter
phase-space distribution, without relying on the instantaneous decay
approximation. We then interface this phase-space distribution with the
Boltzmann code CLASS to extract the corresponding matter power spectrum, which
we find to match closely those of warm dark matter models, albeit with a
different dark matter mass. This mapping allows us to extract constraints from
Lyman-$\alpha$ data without the need to perform hydrodynamical simulations. We
robustly rule out the possibility, consistent with previous analytic estimates,
of primordial black holes having come to dominate the energy density of the
universe and simultaneously given rise to all the DM through their decay.
Consequences and implications for dark radiation and leptogenesis are also
briefly discussed.
