Neutrino self-interactions in post-reionization era: Lyman-$\alpha$, 21-cm and cross-spectra

Neutrino self-interactions delay the onset of free-streaming in the early universe, leaving distinct, scale-dependent signatures on the matter power spectrum. We investigate these signatures in post-reionization 21-cm intensity mapping and the Lyman-$\alpha$ (Ly$\alpha$) forest at redshifts $z \sim 2$--$3.5$, and forecast the constraints achievable with upcoming surveys using Fisher matrix analysis. Modeling neutrino self-interactions through an effective four-fermion parameterization with coupling $G_{\rm eff}$, we compute modifications to the Ly$\alpha$ and 21-cm auto- and cross-power spectra for both strongly interacting (SI$_\nu$, $\log_{10}G_{\mathrm{eff}} = -1.77$) and moderately interacting (MI$_\nu$, $\log_{10}G_{\mathrm{eff}} = -5$) scenarios. We then combine these with forecasts for a representative next-generation cosmic microwave background (CMB) mission to evaluate the capabilities of SKA1-Mid and PUMA. We find that the Ly$\alpha$--21-cm cross-correlation provides a systematics-resilient probe of the interaction signal, and decisively breaks the degeneracy between the primordial scalar power spectrum amplitude ($A_s$) and $G_{\rm eff}$ that limits CMB only analysis, particularly for the SI$_\nu$ mode. Furthermore, the CMB+PUMA combination emerges as the optimal survey configuration for both regimes, reaching 1$\sigma$ constraints of $\mathcal{O}(10^{-3})$ on $\sigma(\log_{10}G_{\rm eff})$ for the SI$_\nu$ mode and $\mathcal{O}(10^{-2})$ for the MI$_\nu$ mode. Compared to the CMB-only baseline, this represents an improvement of approximately one order of magnitude for the SI$_\nu$ mode, and nearly two orders of magnitude for the MI$_\nu$ mode. We show that this conclusion holds uniformly over the full range of coupling strengths from $\log_{10}G_{\rm eff} = -6$ to $-1.77$.