Boosting the optical depth to Thomson scattering with primordial black hole evaporation at high redshift

Technical Summary: Boosting the Optical Depth to Thomson Scattering with Primordial Black Hole Evaporation at High Redshift

Problem Statement
Current cosmological data, specifically Baryon Acoustic Oscillation (BAO) measurements from DESI DR2 and Cosmic Microwave Background (CMB) data from Planck, ACT, and SPT, exhibit a mild tension when interpreted within the standard $\Lambda$CDM model. This discrepancy manifests in two related ways: a "matter density deficit," where the non-relativistic matter density inferred from CMB data exceeds that from BAO, and a "CMB lensing excess," where the measured lensing power exceeds the prediction of the combined CMB+BAO fit.

Recent works (Sailer et al. 2025; Jhaveri et al. 2025) suggested that excluding low-$\ell$ ($\ell < 30$) CMB polarization data reduces this tension and allows the optical depth to Thomson scattering, $\tau$, to rise to $\simeq 0.09$. However, such a high $\tau$ produced by standard reionization at $z \lesssim 10$ is strongly disfavored by the observed "reionization bump" in the large-scale polarization power spectrum ($\ell \sim 5\text{--}10$) and spectroscopic constraints from high-redshift quasars, which prefer $\tau \simeq 0.05$. The authors investigate whether a high-redshift contribution to $\tau$ (at $z > 10$) from exotic processes could boost the total optical depth to $\sim 0.09$ without violating low-$\ell$ constraints, thereby alleviating the BAO–CMB tension.

Methodology
The authors model high-redshift ionization driven by Hawking radiation from a monochromatic population of Primordial Black Holes (PBHs).

• Physical Model: They utilize the PBH parameterization from Poulin et al. and Stöcker et al., implemented in the CLASS code. The model assumes low-mass PBHs ($10^{13}\text{--}10^{15}$ g) where evaporation dominates over accretion. Key assumptions include a monochromatic mass distribution, Schwarzschild black holes, and "on-the-spot" energy deposition. The free parameters are the PBH mass ($M_{\text{PBH}}$) and the fraction of dark matter they constitute ($f_{\text{PBH}}$).
• Ionization History: Unlike previous reconstructions that truncate at $z \sim 30$ or use principal component analysis (PCA) which discards poorly constrained high-$z$ modes, this study forward-models the physical ionization history ($X_e(z)$) extending from recombination up to $z \sim 300$.
• Data and Likelihoods: The analysis employs Bayesian parameter inference (MCMC) using the Cobaya framework.
  • Baseline: Planck 2018 (PR3) CMB power spectra (high-$\ell$ plik lite TT/TE/EE, low-$\ell$ Commander TT, and low-$\ell$ SimAll EE).
  • Extended Datasets: Combinations including ACT DR6, SPT-3G D1 spectra, CMB lensing reconstructions (Planck PR4, ACT DR6, SPT-3G), and DESI DR2 BAO data.
• Strategy: The study retains high-$\ell$ TT/TE/EE data, which are sensitive to the specific imprint of high-redshift free electrons (signals appearing at $\ell > 30$), rather than discarding them as some previous high-$\tau$ studies have done. The reionization redshift ($z_{\text{reio}}$) is varied jointly with the PBH parameters.

Key Results

1. Limited Boost in Optical Depth: The inclusion of PBH evaporation does not significantly increase the preferred value of $\tau$. The posterior mean shifts from $\tau = 0.055^{+0.005}_{-0.008}$ (standard $\Lambda$CDM) to $\tau = 0.063^{+0.007}_{-0.009}$. The 95% confidence upper limit increases only to $0.079$. This boost of $\Delta\tau \simeq 0.008$ is well short of the $\Delta\tau \simeq 0.03$ required to fully resolve the BAO–CMB tension.
2. Mechanism of Compensation: The failure to achieve a large $\tau$ is due to a compensatory shift in the standard reionization redshift ($z_{\text{reio}}$). As the PBH signal adds free electrons at high redshift, the fit lowers $z_{\text{reio}}$ to maintain consistency with the low-$\ell$ EE "reionization bump" data. This adjustment cancels roughly half of the optical depth supplied by the PBHs.
3. Impact on Cosmological Parameters:
   • Matter Density Deficit: The deficit is only marginally reduced, from $1.68\sigma$ to $1.57\sigma$. The shift in the matter density parameter $\Omega_m$ is negligible ($5.7\%$ of the difference between Planck and DESI means).
   • Lensing Excess: The lensing excess remains largely unresolved. While the amplitude parameter $A_s$ increases slightly to compensate for the optical depth (keeping $A_s e^{-2\tau}$ constant), the predicted lensing power increases by only $\sim 1.5\%$. The lensing amplitude parameter $A_{\text{lens}}$ shifts from $1.081$ to $1.068$, reducing the tension from $3.2\sigma$ to $3.1\sigma$.
4. Role of High-$\ell$ Data: The high-$\ell$ polarization data ($\ell > 30$) are crucial. They constrain the specific shape of the ionization history introduced by PBHs. The PBH signal introduces structure at multipoles well above the reionization bump that a simple shift in $z_{\text{reio}}$ cannot reproduce. This structure drives an upward push on $\omega_c$ (the physical cold dark matter density) that counteracts the downward push expected from increased lensing power, resulting in a net negligible shift in $\omega_c$ and $\Omega_m$.
5. Prior Sensitivity: The observed shift in $\tau$ is largely driven by the volume of the parameter space opened by the one-sided extension (PBHs can only add electrons, not remove them) rather than a genuine preference in the data. The maximum-likelihood point remains near the standard $\Lambda$CDM value ($\tau \simeq 0.054$). Switching to log-uniform priors on $f_{\text{PBH}}$ reduces the shift to $\Delta\tau \simeq 0.003$.

Significance and Conclusions
The paper concludes that boosting $\tau$ via Hawking radiation from a monochromatic population of PBHs is not a viable solution to the BAO–CMB tension. The CMB data do not significantly prefer these additional signals, and the resulting boost in $\tau$ is insufficient to eliminate the matter density deficit or the lensing excess.

The authors identify two primary reasons for this failure:

1. Degeneracy with $z_{\text{reio}}$: When $z_{\text{reio}}$ is allowed to vary, it adjusts to compensate for the low-$\ell$ signal of the PBHs, reducing the net effective optical depth.
2. High-$\ell$ Constraints: The specific ionization history of PBHs leaves imprints at high multipoles ($\ell > 30$) that constrain the model and prevent the necessary shifts in cosmological parameters (specifically $\omega_c$) that would be required to resolve the tension.

The authors note that while their analysis is specific to monochromatic PBHs, the general mechanism—where high-redshift ionization introduces new signals at small scales that must fit high-$\ell$ data—likely applies to other proposed sources (e.g., Population III.1 stars). They suggest that resolving the tension may require mechanisms that lower the observed large-scale polarization power (e.g., suppressed primordial power or cosmic birefringence) or modifications to pre-recombination physics, rather than simply increasing $\tau$ through high-redshift ionization.