Searching for Age-Dependent Lensing Wakes: A Phenomenological Probe of Metric Relaxation in Dissociative Cluster Mergers

Dissociative galaxy-cluster mergers provide a sensitive test of the dynamical relation between baryonic plasma, lensing morphology, and post-merger evolution. We introduce a phenomenological viscoelastic response model to constrain a finite metric-relaxation timescale. In this framework, a merger-induced lensing wake is generated during the core-interpenetration phase and subsequently decays over a characteristic time $\tau_{\rm relax}$. Applying this framework to a sample of well-characterized dissociative mergers, we find that the Bullet Cluster lies near the peak of the predicted visibility window, $t_{\rm age} \sim t_{\rm inj} \sim \tau_{\rm relax} \sim 150\,{\rm Myr}$, whereas older systems exhibit the expected decay in residual lensing signatures. Furthermore, we introduce convergence-map morphology estimators to quantify elongation and merger-axis alignment, directly applying them to a free-form lensing reconstruction of the Bullet Cluster. Because global morphology estimators are strongly affected by projection effects and reconstruction noise, we introduce a targeted smooth-cone estimator that measures the radial slope of the convergence field along the merger axis. Applied to a recent JWST-based convergence map of the Bullet Cluster, the estimator yields a significant negative slope ($p < 5 \times 10^{-4}$), consistent with a relaxing wake. While distinguishing this viscoelastic signature from complex CDM/SIDM dynamics will require homogeneous lensing samples and forward-modelled simulations, the definitive observational prediction of this framework is the joint presence of a negative axis-aligned convergence slope and an age-dependent lensing--gas separation controlled by $\tau_{\rm relax}$.