In-depth analysis of the clustering of dark matter particles around primordial black holes. Part II. Analytical prescriptions for spikes

TL;DR

This work analyzes the coexistence of primordial black holes and thermally produced dark matter, showing that DM forms ultra-dense spikes around PBHs during the radiation era with four asymptotic post-collapse regimes characterized by slopes $\gamma = 3/4, 3/2$, and $9/4$. It develops fast, semi-analytic prescriptions—notably the kink and soft approximations—and a phase-diagram framework to predict spike densities and, crucially, the DM annihilation rate $\Gamma_{\rm BH}$ across parameter space, including self-annihilating DM. The authors derive saturation-density truncation, analytic scalings for $\Gamma_{\rm BH}$, and quantify the impact of annihilation on spike structure, achieving $\sim\pm 15\%$ accuracy relative to full numerics. Using this framework, they translate gamma-ray background and CMB angular-distortion data into stringent bounds on the PBH fraction $f_{\rm BH}$ and on WIMP properties, revealing a strong mutual exclusivity: PBHs cannot significantly co-exist with $s$-wave annihilating DM except in a narrow asteroid window. The results imply that even a small PBH admixture can severely constrain WIMP annihilation, and conversely, observed sub-solar PBHs would impose tight limits on WIMPs, with practical implications for upcoming PBH searches and CMB/gamma-ray analyses.

Abstract

Primordial black holes (PBHs) are very appealing dark matter (DM) candidates. It is highly plausible though, should they exist, that they would not make up all of the DM. Several studies showed that if the rest of DM is made of thermal particles, then these should accumulate around such PBHs, leading to the formation of very dense spikes in the radiation era. We contributed a detailed analytical study about this phenomenon, providing clear explanations as for the origin of scaling relations in the form of power-law density profiles with up to 3 different spectral indices, i.e. $3/4$, $3/2$, and $9/4$, and 4 asymptotic regimes. Here, we further derive an approximate analytical solution that enables fast numerical predictions for the density profiles of these spikes. We also address the specific case of self-annihilating DM species and derive new approximate analytical formulae. Our approximate density yields the correct annihilation rate within $\pm 15\%$ precision. We then focus on indirect detection in the cosmic microwave background and in extragalactic gamma-rays. We shed new and subtle light on how mutually exclusive PBHs and self-annihilating DM species can really be. In particular, the discovery of a population of sub-solar PBHs would set stringent constraints on the $s$-wave annihilation cross-section of these particles, a point so far missed in the literature.

Figures (13)

• Figure 1: The post-collapse DM density follows a power law whose radial index depends on radius $\tilde{r}$ and PBH mass $M_{\rm BH}$. The regions of prevalence for each particular index are plotted in this phase diagram with different colors. From the center of the DM halo to its outskirts, the slope is respectively equal to $3/4$, $3/2$ and, for heavy objects, $9/4$. This phase diagram corresponds to a DM mass $m_{\chi}$ of 1 TeV and a thermal decoupling parameter $x_{\rm kd}$ of $10^{4}$. The shaded gray vertical band lies below the Schwarzschild radius.
• Figure 2: In the left panel, the phase diagram of Fig. \ref{['fig:phase_diagram']} has been reproduced. Each thin gray line corresponds to a specific value of the post-collapse DM density which, from left to right, has been varied from $10^{12}$ down to $10^{-22} \; {\rm g \, cm^{-3}}$ by decrements of $10^{2}$. The critical points A, B$_{2}$ and B$_{1}$ lie along the solid black iso-density lines where the DM density is respectively equal to $7.28 \times 10^{6}$, $6.44 \times 10^{5}$ and $1.36 \times 10^{-19} \; {\rm g \, cm^{-3}}$. In the right panel, the DM density profiles have been plotted versus reduced radius $\tilde{r}$ for different values of the PBH mass. These are coded with color and line type as indicated. Each profile of the right panel corresponds to a particular horizontal line in the left panel. A few remarkable points are also featured in both panels and discussed in the text.
• Figure 3: The relative difference between the approximate post-collapse DM density and the exact numerical result is featured as a function of reduced radius $\tilde{r}$. The DM parameters are the same as for the previous figures. We have also considered the same PBH masses as in Fig. \ref{['fig:iso_rho_lines_and_DM_profiles']} with the same code for colors and line type. The left panel is devoted to the kink approximation. The sharp peaks which the curves exhibit correspond to changes in the asymptotic regime occuring at the very same points A, B$_{2}$, C, D and F as in Fig. \ref{['fig:iso_rho_lines_and_DM_profiles']}. The soft approximation is featured in the right panel. For each PBH mass, the colored vertical edges of the light-gray bands and their associated points indicate the radii below which the post-collapse DM density is larger than the saturation density $\rho_{\rm sat}$ taken at a redshift $z$ of $10^{3}$ for a thermal DM annihilation cross-section. To the left of these edges, the DM distribution has been erased by annihilation and the profiles have been flattened.
• Figure 4: Same panels as in Fig. \ref{['fig:iso_rho_lines_and_DM_profiles']} where DM annihilation is now included. In the left panel, the inner region of the phase diagram is replaced by a saturation plateau where the DM density is equal to the constant $\rho_{\rm sat}$. For a redshift $z$ of $10^{3}$ and a thermal DM annihilation cross-section, we find a value of $4.97 \times 10^{-9} \; {\rm g \, cm^{-3}}$. The solid black iso-density line that runs through points T and H corresponds to $\rho_{\rm sat}$, and features the border $\tilde{r}_{\rm sat}$ of the saturation plateau. The critical point B$_{1}$ lies along the solid black iso-density line where the DM density is equal to $1.36 \times 10^{-19} \; {\rm g \, cm^{-3}}$. In the right panel, the DM density profiles have been plotted versus reduced radius $\tilde{r}$ for different values of PBH mass. These are coded with color and line type as indicated. Each profile of the right panel corresponds to a particular horizontal line in the left panel. DM annihilation has erased the post-collapse profiles and flattened them into the saturation plateau A few remarkable points are also featured in both panels and discussed in the text.
• Figure 5: The DM annihilation rate of minispikes is plotted as a function of PBH mass. The solid line corresponds to the fiducial case as specified in the left panel, while the other curves feature each the effect on $\Gamma_{\rm BH}$ of modifying one of the parameters of the model. In the left panel, the redshift is decreased from $10^{3}$ (post-recombination epoch) to $0$ (today). The dotted short-dashed line shifts downward only at high PBH mass. The long-dashed curve shows how increasing the annihilation cross-section $\hbox{$\langle \sigma_{\rm ann} {v} \rangle$}$ by three orders of magnitude acts on $\Gamma_{\rm BH}$. In the right panel, decreasing the kinetic decoupling parameter $x_{\rm kd}$ from $10^{4}$ down to $10^{2}$ makes the annihilation rate grow only at low masses (dotted short-dashed). Varying the WIMP mass $m_{\chi}$ from $1 \; {\rm TeV}$ down to $10 \; {\rm GeV}$ induces an increase of $\Gamma_{\rm BH}$ at high mass but a decrease at low mass (long-dashed). All the curves indicate that the annihilation rate increases with $M_{\rm BH}$ as a power law with index $3$ at low mass and $1$ at high mass. These behaviors are further discussed in the text.