Double-Peaked Ly$\alpha$ Emission during Reionization Requires Nearby Voids and a Favorable Local Ionizing Background

Several Lyman-alpha (Ly$\alpha$) emitters deep into the reionization era exhibit double-peaked Ly$\alpha$ emission profiles, raising the question of how the intergalactic medium can transmit photons blueward of the Ly$\alpha$ resonance at such high redshifts. To investigate this, we compute Ly$\alpha$ transmission along sightlines originating from galaxies in the Cosmic Dawn III simulation and identify cases that closely reproduce the observed double-peaked emission. In these cases, the sightlines intersect highly underdense voids located a few comoving megaparsecs from the source galaxy. These voids allow photons emitted blueward of Ly$\alpha$ to redshift through resonance without scattering while traversing them. The low opacity arises because the neutral hydrogen density scales with the square of the underlying gas density under ionization equilibrium, making sufficiently underdense regions with $\lesssim30~\%$ of cosmic mean density highly transmissive. Such voids naturally occur in the fluctuating cosmic density field, even in the vicinity of galaxies, and can also be associated with transmissive spikes in the Ly$\alpha$ forest. We find that the global probability of observing double-peaked emission is $\sim3\times10^{-3}$ during reionization at an 80\% global ionization fraction, while no cases are found at 60\% ionization. We also find that this probability depends sensitively on the local ionizing background intensity, increasing by $\sim10^{4}$ for a tenfold increase in intensity. These results suggest that the fraction of double-peaked Ly$\alpha$ emission in high-$z$ galaxies can serve as a sensitive probe of the ionizing background during the late stages of cosmic reionization.