Non-Cold Dark Matter from Memory-Burdened Primordial Black Holes

Non-cold dark matter particles can arise from the evaporation of primordial black holes (PBHs). In this paper, we further investigate how the memory-burden effect, which delays the full evaporation of black holes, affects the Lyman-$\alpha$ bound on such non-cold dark matter (NCDM) particles. We mainly focus on scenarios in which PBHs have fully evaporated by today, undergoing a semi-classical evaporation phase followed by a memory-burden dominated phase. In this framework, PBH evaporation generically leads to two distinct dark-matter populations with different velocity dispersions, which can imprint observable signatures on the matter power spectrum. We compute the resulting NCDM phase-space distribution and its impact on small-scale overdensities using the $\texttt{BlackHawk}$ and $\texttt{CLASS}$ codes. This is then used to reinterpret Lyman-$\alpha$ forest constraints for thermal warm dark matter, deriving both a velocity-dispersion-based and a matter-power-spectrum-based estimate. In particular, we discuss how we obtain constraints on scenarios in which NCDM particles constitute only a fraction of the total relic dark matter. Finally, we discuss the viable parameter space as a function of dark matter masses, PBH initial conditions, and memory-burden parameters. We show that even subdominant NCDM components from PBH evaporation can be constrained, and confirm that NCDM can only account for all of the dark matter in the absence of PBH domination, as in the semi-classical case.