Delayed Feedback in High-$z$ Starbursts Revealed by Lyman-$\alpha$ Profiles and Metal Line Diagnostics
Lyman-$ \alpha $ emission, which couples strongly 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 $, however, often lack contextual comparisons with more direct outflow diagnostics from down-the-barrel metal absorption lines and driving-source properties from metal emission lines. Here, we make such checks by taking advantage of the lensing magnification provided by galaxy clusters for 573 Lyman-$ \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, their blueshifted velocities indicate outflows, while their line ratios point to absorption by a clumpy medium. We demonstrate a significant positive correlation between absorption-line outflow velocity and Lyman-$ \alpha $ line dispersion, confirming that Lyman-$ \alpha $ profile morphology encodes useful kinematic information about outflowing gas. We further find that the equivalent width of high-ionization metal emission lines are anticorrelated with measured outflow velocity, suggesting younger stellar populations produce slower outflows, and providing a critical empirical constraint on feedback timescales in high-$ z $ starbursts. Fitting model Lyman-$ \alpha $ profiles based on simple expanding shell geometry to those observed, we find that such models successfully reproduce some aspects of the data, yet often yield unphysical parameters -- calling for caution when inferring outflow