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

BAO and CMB data are somewhat discrepant when interpreted in the context of $Λ$cdm, discrepancies that show up as a `matter density deficit' and as a `CMB lensing excess'. One possible resolution is an increased optical depth to scattering off of free electrons in the post-recombination universe, $τ$, a possibility raised by Sailer et al. 2025 and Jhaveri et al. 2025. Since Planck measurements of the low-$\ell$ polarization `reionization bump' already constrain $τ$ from standard stellar-driven reionization at $z<10$, we investigate additional optical depth sourced by transient or partial reionization at higher redshift from exotic processes. For specificity, we explore the impact of Hawking radiation from a monochromatic spectrum of primordial black holes, retaining the high-$\ell$ $TT/TE/EE$ data that constrain such histories and varying the reionization redshift jointly. We find that the CMB data do not significantly prefer these additional signals: the boost is at most $Δτ\simeq 0.008$, well short of the $Δτ\simeq 0.03$ that would completely eliminate the moderate discrepancy. The matter density deficit and the lensing excess are not significantly eased: we explain why, tracing it to compensation from the reionization redshift and the residual PBH signal at $\ell > 30$.