Primordial black hole evaporation and dark matter production. II. Interplay with the freeze-in or freeze-out mechanism

We study how the evaporation of primordial black holes (PBHs) can affect the production of dark matter (DM) particles through thermal processes. We consider fermionic DM interacting with Standard Model particles via a spin-1 mediator in the context of a freeze-out or freeze-in mechanism. We show that when PBHs evaporate after dominating the Universe’s energy density, PBHs act as a source of DM and continuously inject entropy into the visible sector that can affect the thermal production in three qualitatively different ways. We compute the annihilation cross sections which account for the interactions between and within the PBH produced and thermally produced DM populations, and establish a set of Boltzmann equations which we solve to obtain the correct relic abundance in those different regimes and confront the results with a set of different cosmological constraints. We provide analytic formulas to calculate the relic abundance for the freeze-out and freeze-in mechanism in a PBH dominated early Universe. We identify regions of the parameter space where the PBHs dilute the relic density and thermalization occurs. Furthermore, we have made our code that numerically solves the Boltzmann equations publicly available.

Figure 1

Example of processes leading the production of thermalized DM particles.

Figure 2

PBH energy fraction $\beta$ as a function of the PBH mass leading to the observed relic abundance $\mathrm{\Omega }{h}^2=0.11$ for different values of the DM mass (in GeV).

Figure 3

Schematic representation of the case where the PBH energy density (blue line) transiently dominates over the SM energy density (orange line). See the main text for a detailed description of the different regimes.

Figure 4

Maximum PBH mass (left panel, brown curve) and the corresponding minimum evaporation temperature (right panel, brown curve) leading to the correct relic abundance of DM particles when it is exclusively produced through PBH evaporation (no mediator, and no BH spin). See the description of the different curves in the text. The grey-shaded area corresponds to the region where PBHs can only be produced below the critical density in order to not overclose the universe. The green-shaded area corresponds to regions where PBHs are able to dominate the energy density of the universe and evaporate without overclosing the Universe. The blue-dashed line denotes the contour where the PBHs evaporation temperature equals the DM mass.

Figure 5

Comparison of cross section for PBH produced DM to annihilate into SM particles as a function of temperature. The blue (green) dashed line shows the cross section computed using the phase-space distributions of PBH-emitted particles in the geometric optic limit (Boltzmann distribution) for $m_{\mathrm{DM}}={10}^2\mathrm{GeV}$, $m_X={10}^3\mathrm{GeV}$, $g_V=g_D=0.73$.

Figure 6

The relic density as a function of the initial PBH mass for the FO mechanism with ${\beta }^{\prime }={10}^{-10}$, $m_X=10\mathrm{GeV}$, $m_{\mathrm{DM}}=1\mathrm{GeV}$, $\mathrm{Br}(X\to \mathrm{DM})=0.5$ and $a^{*}=0$ for thermally averaged cross section values ${\log }_{10}(\sigma v/[{\mathrm{GeV}}^{-2}])=-8.5,-9.0,-9.5$ shown in red, blue and green respectively. The grey dashed line indicates the observed relic density value. The grey dot-dashed lines indicate the separation of regimes IV, III and II.

Figure 7

Two-dimensional scan over the PBH fraction $\beta$ and mass $M_{\mathrm{BH}}$ for a mediator mass $m_X=10\mathrm{GeV}$ and a dark matter mass $m_{\mathrm{DM}}=1\mathrm{GeV}$, and $\mathrm{Br}(X\to \mathrm{DM})=0.5$. The color map indicates the value of the nonrelativistic cross section of DM annihilation leading to the correct relic abundance in the freeze-out case. See the main text for a description of the different constraints.

Figure 8

The left (right) plot shows the value of the relic density as a function of the initial PBH mass for the FI mechanism with ${\beta }^{\prime }={10}^{-10}$, $m_X=10\mathrm{GeV}$, $m_{\mathrm{DM}}={10}^{-3.0}\mathrm{GeV}$ ($m_{\mathrm{DM}}={10}^{-1.0}\mathrm{GeV}$) and $\mathrm{Br}(X\to \mathrm{DM})=0.5$, $a^{*}=0$ for thermally averaged cross section values of ${10}^{-41}$, ${10}^{-42}$, ${10}^{-43}$ and ${10}^{-44}{\mathrm{GeV}}^{-2}$ shown in red, blue, yellow and green (purple, pink, cyan and orange) respectively. The dotted portions of the different lines indicate regions where the DM is hot and would disrupt structure formation. The grey dashed line indicates the observed relic density value. The grey dot-dashed lines indicate the separation of regimes IV, III and II.

Figure 9

Top (bottom) left shows the time evolution of the energy density of the radiation, PBHs, thermally produced DM, PBH produced DM and mediator is shown in blue, black, green, red and purple respectively for $M_{\mathrm{PBH}}={10}^5\mathrm{g}$ ($M_{\mathrm{PBH}}={10}^8\mathrm{g}$). The top (bottom) right shows the thermal cross section multiplied by the number density of thermal DM particles divided by the Hubble expansion rate as a function of inverse temperature. The grey dashed line indicates the $\sigma N_{TH}/H=1$.

Figure 10

Two-dimensional plots showing the correlations between ${\log }_{10}(M_{\mathrm{BH}}^{\mathrm{in}}/[\mathrm{GeV}])$, ${\log }_{10}({\beta }^{\prime })$ and ${\log }_{10}(\sigma v/[{\mathrm{GeV}}^{-2}])$. The darker blue regions shows higher posterior probabilities and the lighter blue regions show regions with lower posterior probability. The regions of the parameter space, $\overrightarrow{p}$, consistent at the one (two) sigma level with the observed relic abundance are within the solid (dashed blue) contours.

Figure 11

Two-dimensional scan over the PBH fraction $\beta$ and mass $M_{\mathrm{BH}}$ for a mediator mass $m_X=1\mathrm{TeV}$, a dark matter mass $m_{\mathrm{DM}}=1\mathrm{MeV}$, and $\mathrm{Br}(X\to \mathrm{SM})={10}^{-7}$. The color map indicates the value of the nonrelativistic cross section of DM annihilation leading to the correct relic abundance in the freeze-in case. See the main text for a description of the different constraints.

Figure 12

The relic density as a function of the mediator mass, $m_X$, for the FI mechanism with ${\beta }^{\prime }={10}^{-10}$, $m_{\mathrm{DM}}={10}^{-3.0}\mathrm{GeV}$ and $\mathrm{Br}(X\to \mathrm{DM})=0.5$, $a^{*}=0$ and $\sigma v={10}^{-41.0}{\mathrm{GeV}}^{-2}$ for initial PBH mass of ${\log }_{10}(M_{\mathrm{BH}}^{\mathrm{in}}/[\mathrm{g}])=5.0$, 6.0, 7.0, 8.0 shown in red, cyan, green and purple respectively. The colored dashed lines indicate when DM has thermalized and the freeze-in mechanism transitions to the freeze-out mechanism. The grey dashed line stands for the central value of the DM relic abundance, as measured by the Planck collaboration [42].

Figure 13

The relic density as a function of the PBH spin, $a^{*}$, for the FI mechanism with ${\beta }^{\prime }={10}^{-10}$, $m_{\mathrm{DM}}={10}^{-3.0}\mathrm{GeV}$ and $\mathrm{Br}(X\to \mathrm{DM})=0.5$ and ${\log }_{10}(\sigma v/[{\mathrm{GeV}}^{-2}])=-42.0$ for initial PBH mass of various initial PBH masses.