2025MNRAS.544.2316K

Lyman-$\alpha$ damping wings towards quasars provide a unique probe of the global volume-averaged neutral hydrogen (H I) fraction $\langle x_\mathrm{HI} \rangle$. Cosmic variance in the intergalactic medium (IGM), however, is a major source of stochasticity since the local neutral environment around a quasar varies significantly even when $\langle x_\mathrm{HI} \rangle$ is fixed. We show that the IGM damping wing carries additional information about this local ionization topology, unexploited by current analysis frameworks. We introduce two new physically motivated summary statistics encoding the local information about the H I distribution in the IGM before it is altered by the quasar's ionizing radiation, encompassing (1) the H I column density, weighted by a Lorentzian profile mimicking the frequency dependence of the Lyman-$\alpha$ cross-section, and (2) the quasar's distance to the first neutral patch. This description, when combined with the quasar lifetime as a third parameter, reduces the IGM transmission scatter in the damping wing region of the spectrum to $\lesssim 1~{{\ \rm per\ cent}}$ across physical parameter space. We introduce a simple procedure for generating synthetic H I sightlines around quasars and demonstrate that the resulting damping wing profiles are statistically indistinguishable from a realistic reionization topology. This opens the door for optimally extracting the damping wing's salient local information in a model-independent fashion. In the context of a specific reionization model, measurements of these local parameters can be translated into constraints on the global timing of reionization, but additionally provide information about the reionization topology, hitherto unused. A marginally modified version of our framework can also be employed for damping wings towards galaxies.