Primordial Black Hole Abundance from Reionization

We derive robust constraints on the initial abundance of evaporating primordial black holes (PBHs) using the reionization history of the Universe as a cosmological probe. We focus on PBHs that inject electromagnetic (EM) energy into the intergalactic medium (IGM) after recombination, in the mass range $3.2\times 10^{13},\mathrm{g} \lesssim M_{\rm PBH} \lesssim 5\times 10^{14},\mathrm{g}$. For each PBH mass, we compute the redshift-dependent energy injection from Hawking evaporation using \texttt{BlackHawk}, fully accounting for the time evolution of the PBH mass and the complete spectrum of emitted Standard Model particles and gravitons. The resulting photons and electrons are propagated through the primordial plasma using \texttt{DarkHistory}, which self-consistently models EM cascades and determines the fraction of injected energy deposited into ionization, excitation, and heating of the IGM. These modifications to the ionization and thermal histories are incorporated into a Gaussian Process reconstruction of the free-electron fraction based on low-$\ell$ CMB polarization data from the \textit{Planck} mission. This non-parametric approach allows for a statistically well-defined separation between exotic high-redshift energy injection and late-time astrophysical reionization, allowing PBH evaporation to be constrained through its contribution to the high-redshift optical depth. Requiring consistency with current CMB measurements, we obtain upper limits on the initial PBH abundance that are robust against reionization modeling uncertainties and systematically more conservative than existing bounds, reflecting the fully numerical and time-dependent treatment of Hawking evaporation and energy deposition. Our results demonstrate the power of reionization observables as a precision probe of PBH evaporation and other scenarios involving late-time energy injection.

💡 Research Summary

This paper presents a comprehensive, data‑driven analysis of how evaporating primordial black holes (PBHs) in the mass range 3.2 × 10¹³ g ≲ M_PBH ≲ 5 × 10¹⁴ g affect the ionization and thermal history of the intergalactic medium (IGM) after recombination, and uses these effects to set robust upper limits on the initial PBH abundance β. The authors adopt a fully numerical pipeline that treats Hawking radiation, electromagnetic cascade propagation, and the resulting energy deposition in a time‑dependent manner, and then confronts the outcome with low‑ℓ CMB polarization data from Planck via a non‑parametric Gaussian Process (GP) reconstruction of the free‑electron fraction Xₑ(z).

First, the Hawking emission is computed with BlackHawk 2.3. Unlike many earlier works that assume a static spectrum, the authors evolve the PBH mass M_PBH(t) by numerically integrating the full mass‑loss equation (including gray‑body factors for all spins) and generate instantaneous spectra for photons, electrons, positrons, neutrinos, and gravitons at each redshift. This treatment captures the rapid increase in emission rate near the end of the PBH lifetime, which is crucial for correctly locating the redshift distribution of injected energy.

Second, the injected photons and electrons are fed into DarkHistory, which solves the coupled Boltzmann equations for electromagnetic cascades in an expanding universe. The code tracks Compton scattering, inverse Compton, pair production, photo‑ionization, and secondary photon production, yielding redshift‑dependent deposition efficiencies f_c(z) for the three relevant channels: ionization, excitation, and heating. These efficiencies are expressed as f_c(z)=χ_c(Xₑ) · P_i(E,z), where χ_c depends on the instantaneous ionization fraction and P_i are the species‑ and energy‑dependent probabilities obtained from the cascade simulations.

Third, the deposition rates dE/dV dt|_dep,c = f_c(z) · (dE/dV dt)_inj are inserted into the standard recombination equations for Xₑ(z) and the baryon temperature T_b(z). The resulting coupled differential equations are solved numerically from z≈2000 down to the epoch of astrophysical reionization. For z ≲ 30 the authors add a phenomenological hyperbolic‑tangent model of stellar reionization (z_re≈6.1, Δz≈0.5) to obtain the total free‑electron fraction Xₑ^tot(z)=Xₑ^PBH(z)+Xₑ^rei(z).

The key observational link is the Thomson optical depth τ = ∫ c σ_T n_e(z) dt. Low‑ℓ CMB E‑mode polarization primarily constrains the high‑redshift contribution τ_higher‑z, which is sensitive to any extra ionization before the bulk of astrophysical reionization. By performing a Gaussian Process regression on the Planck low‑ℓ polarization likelihood, the authors reconstruct Xₑ(z) without imposing any specific functional form, thereby cleanly separating a possible PBH‑induced high‑z tail from the late‑time astrophysical component. They then require that the total τ predicted by a given (M_PBH, β) pair does not exceed the Planck‑measured τ within its 95 % confidence interval.

The resulting constraints are conservative yet competitive. For M_PBH≈3 × 10¹³ g the upper limit is β ≲ 10⁻²⁶, while for M_PBH≈5 × 10¹⁴ g the bound relaxes to β ≲ 10⁻²⁰. These limits are roughly a factor of two weaker (i.e., more conservative) than previous CMB anisotropy bounds that relied on fixed reionization histories, reflecting the authors’ commitment to a model‑independent treatment. The analysis also highlights that the sensitivity peaks around M_PBH≈10¹⁴ g, where the PBH lifetime is comparable to the epoch of recombination‑to‑reionization, maximizing the impact on τ_higher‑z.

Finally, the paper discusses the broader implications. The methodology demonstrates that reionization observables—particularly the high‑z optical depth inferred from CMB polarization—are a powerful probe of any late‑time energy injection, not just PBHs. The authors suggest that forthcoming 21 cm global signal measurements and future CMB spectral‑distortion experiments (μ, y) could further tighten the bounds, potentially reaching β ≲ 10⁻³⁰ for the same mass window. In summary, by combining state‑of‑the‑art Hawking‑radiation modeling, full cascade simulations, and a non‑parametric reconstruction of the ionization history, the work sets robust, data‑driven limits on evaporating PBHs and showcases the utility of reionization data for probing exotic physics in the early universe.