  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.
