Deciphering Spatially Resolved Lyman-Alpha Profiles in Reionization Analogs: The Sunburst Arc at Cosmic Noon

The hydrogen Lyman-alpha (Ly\(\alpha\)) emission line, the brightest spectral feature of a photoionized gas, is considered an indirect tracer of the escape of Lyman continuum (LyC) photons, particularly when the intergalactic medium is too opaque for direct detection. However, resonant scattering complicates interpreting the empirical properties of Ly\(\alpha\) photons, necessitating radiative transfer simulations to capture their strong coupling with underlying gas kinematics. In this study, we leverage the exceptional spatial resolution from strong gravitational lensing to investigate the connection between Ly\(\alpha\) line profiles and LyC leakage on scales of a few 100 pc in the Sunburst Arc galaxy at \(z\sim2.37\). New optical echelle spectra obtained using Magellan MIKE show that both the LyC leaking and non-leaking regions exhibit a classic double-peak Ly\(\alpha\) feature with an enhanced red peak, indicating outflows at multiple locations in the galaxy. Both regions also show a central Gaussian peak atop the double peaks, indicating directly escaped Ly\(\alpha\) photons independent of LyC leakage. We introduce a machine learning-based method for emulating Ly\(\alpha\) simulations to quantify intrinsic dynamics (\(\sigma_{\mathrm{int}}\)), neutral hydrogen column density (\(N_{\mathrm{HI}}\)), outflow velocity (\(v_{\mathrm{exp}}\)), and effective temperature (\(T\)) across continuous parameter spaces. By comparing the spatially and spectrally resolved Ly\(\alpha\) lines in Sunburst, we argue that the directly escaped Ly\(\alpha\) photons originate in a volume-filling, warm ionized medium spanning \(\sim1\) kpc, while the LyC leakage is confined to regions of \(\lesssim200\) pc. These sub-kpc variations in Ly\(\alpha\) profiles highlight the complexity of interpreting integrated properties in the presence of inhomogeneous mixtures of gas and young stars, emphasizing the need for spatially and spectrally resolved observations of distant galaxies.