Perturbed recombination from inhomogeneous photon injection and application to accreting primordial black holes

Exotic electromagnetic energy injection in the early Universe may alter cosmological recombination, and ultimately cosmic microwave background (CMB) anisotropies. Moreover, if energy injection is inhomogeneous, it may induce a spatially-varying ionization fraction, and non-Gaussianity in the CMB. The observability of these signals, however, is contingent upon how far the injected particles propagate and deposit their energy into the primordial plasma, relative to the characteristic scale of energy injection fluctuations. In this study we inspect the spatial properties of energy deposition and perturbed recombination resulting from an inhomogeneous energy injection of sub-10 MeV photons, relevant to accreting primordial black holes (PBHs). We develop a novel Monte-Carlo radiation transport code accounting for all relevant photon interactions in this energy range, and including secondary electron energy deposition efficiency through a new analytic approximation. For a specified injected photon spectrum, the code outputs an injection-to-deposition Green's function depending on time and distance from the injection point. Combining this output with a linearized solution of the perturbed recombination problem, we derive time- and scale-dependent deposition-to-ionization Green's functions. We apply this general framework to accreting PBHs, whose luminosity is strongly spatially modulated by supersonic relative velocities between cold dark matter and baryons. We find that the resulting spatial fluctuations of the free-electron fraction are of the same magnitude as its mean deviation from standard recombination, from which current CMB power spectra constraints are derived. This work suggests that the sensitivity to accreting PBHs might be substantially improved by propagating these inhomogeneities to CMB anisotropy power spectra and non-Gaussian statistics, which we study in subsequent papers.