Delayed Feedback in High-$z$ Starbursts Revealed by Lyman-$\alpha$ Profiles and Metal Line Diagnostics

Lyman-$\alpha$ emission, which owing to its resonant nature strongly couples the emergent line profile to gas kinematics, is a key observable for probing outflows from star-forming galaxies in the early universe. Inferences of outflow properties from Lyman-$\alpha$, however, often lack contextual comparisons with more direct outflow diagnostics from down-the-barrel metal absorption lines and driving-source properties from metal emission lines. Here, we make such checks by taking advantage of the lensing magnification provided by galaxy clusters for 338 Lyman-$\alpha$ sources observed with the Multi-Unit Spectroscopic Explorer (MUSE). Using metal emission lines to measure systemic redshifts, we confirm that the Lyman-$\alpha$ profiles are consistent with outflowing gas: single peaks redshifted relative to, or double peaks straddling, the systemic redshift. In cases where metal absorption lines are detected, blueshifted velocities indicate outflows, while line ratios point to absorption by a clumpy medium. We find systematic differences in both metal absorption and emission lines associated with single- versus double-peaked Lyman-$\alpha$ profiles, such that the latter are preferentially associated with weaker and narrower metal absorption profiles, but stronger emission lines indicating younger stellar ages ($\lesssim4\,$Myr for double-peaked Lyman-$\alpha$ vs $\gtrsim10\,$Myr for single-peaked Lyman-$\alpha$). Double-peaked Lyman-$\alpha$ profiles may therefore reflect weaker feedback in extremely young starbursts due to the delayed onset of core-collapse supernovae. Fitting model Lyman-$\alpha$ profiles based on simple expanding shell geometry to those observed, we find that such models successfully reproduce the data, yet systematically overestimate systemic redshifts and yield unphysical parameters -- calling for caution when inferring outflow properties from such models.