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

Abstract

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-$α$ 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-$α$ 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.

Figures (11)

• Figure 1: Case of BH evaporation with $q = 0.5$ assuming an instantaneous transition ($\delta = 0$) without burst. Black contours represent constant lifetimes $\tau = t_{\rm mb}$ for PBHs subject to the MB effect. Gray contours indicate ratios of scale factors $a_{\rm mb}/a_{\rm q}$ of 10 and $10^{4}$, with $a_{\rm mb}$ the scale factor at full evaporation considering MB and $a_{\rm q}$ the one at the beginning of the MB phase. For low values of this ratio, one would expect comparable root-mean-squared velocities for particles emitted both in the MB and SC phases. The color bands show constraints from neutrino emissions, galactic gamma rays, CMB anisotropies and BBN as computed in Refs. Thoss:2024hsrChaudhuri:2025asm. See text for more details.
• Figure 2: Illustration of the change in the constraints coverage of the parameter space when varying $q$, fixing the time at which MB sets in, and $\delta$, fixing the smoothness of the transition. Left: Case of $q = 0.5$ assuming a smooth transition with e.g. $\delta \sim \mathcal{O}(0.1)$ without burst. Right: Case of BH evaporation with $q = 1$ and $\delta = 0$. The colored lines and contours are equivalent to the ones of Fig. \ref{['fig:lifetimes']}.
• Figure 3: Rescaled momentum distributions as a function of the comoving momentum ${\tt q}$. Continuous lines represent the NCDM distribution $\tilde{f}(q)$ for $k = 0$ in blue, in which case we recover the SC behaviour, and $k = 0.2$ in the case of burst in orange and of no burst in green. In the latter cases, we have set $q = 0.5$ or equivalently we consider that MB sets in at half mass. A rescaled Fermi-Dirac distribution is also shown for reference with a dashed gray line.
• Figure 4: Transfer function $T^2(\texttt{k})$ for $k=2$, $q=0.5$, $\beta=10^{-16}$, $M_{\rm F}=2\times10^4\,M_{\rm P}$, and $m_{\rm DM}$ chosen such that DM from PBH evaporation accounts for a fraction of $f_{\rm DM}=1$, 0.5, and 0.10 of the entire DM. In this scenario, only the MB phase of evaporation contributes to the NCDM component and the parameters were chosen to be close to the bound from the Lyman-$\alpha$ forest, shown in Figure \ref{['fig:mdmvsMF']}.
• Figure 5: Ratio of the lower mass bound on NCDM produced from PBH evaporation from velocity dispersion ($m_{\rm DM,\sigma}$) and from the area criterion ($m_{\mathrm{DM},\delta A}$). The ratio appears as the color gradient projected in the $(k,q)$ memory burden parameter space. Darker color indicates a larger discrepancy between the two criteria to evaluate the NCDM bound. This plot assumes that all the dark matter is produced from evaporating PBHs with $M_{\rm F}=10^2\,\rm{g}$. Note that the values for which the two criteria diverge significantly are not favoured by theory Dvali:2018xpyDvali:2021byy.