Lyman-$\alpha$ Forest Signatures of Mixed Fuzzy and Cold Dark Matter

We investigate Lyman-alpha forest flux statistics in mixed fuzzy dark matter (FDM) and cold dark matter (CDM) cosmologies using the Fluctuating Gunn-Peterson Approximation (FGPA) applied to hybrid Schr\"odinger-Poisson and N-body simulations. We evolve the dark matter distribution from z = 120 to z = 2 for an axion mass ( m_22 = 0.01) and FDM fraction (f_A = 0.1), and compare two realizations with identical initial conditions: one evolved with a particle-only approximation and one with full wave--mechanical dynamics. We find that, despite near-degeneracy in the nonlinear matter power spectrum, the corresponding Ly {\alpha} flux power spectra differ at the 10 percent level on intermediate scales. This discrepancy arises from a strong suppression of small-scale velocity power in the Schr\"odinger--Poisson evolution, which is not captured by N-body treatments with matched initial transfer functions. As a result, the flux statistics cannot be fully characterized by the matter power spectrum alone, but depend sensitively on the dynamical evolution of the velocity field. These results demonstrate that wave-mechanical effects in FDM leave distinct kinematic imprints in Ly {\alpha} observables beyond those associated with initial-condition suppression. While our analysis is based on an idealized FGPA framework, it isolates a mechanism by which mixed dark matter models can break degeneracies present in standard structure-based probes, motivating further investigation with full hydrodynamical simulations.