Collisionless damping of the gravitational instability in fuzzy dark matter: spectral shape and quantum-to-thermal crossover

We present a quantum-kinetic linear theory of the gravitational instability in the context of fuzzy dark matter universe. Starting from the Wigner transport equation, we apply Landau's approach to the linearized Wigner--Poisson system and derive a kinetic dispersion relation that incorporates quantum effects exactly by introducing the plasma dispersion function. The growth rate as a function of wavenumber is characterized by a dimensionless quantum-to-thermal ratio alpha = k qJ/k J, where k qJ and k J represent the quantum and thermal Jeans wavenumbers, respectively. We derive an analytic expression for the spectral slope at the cutoff wavenumber, revealing that the spectral shape undergoes a sharp transition across alpha about 0.5. This implies a crossover from a thermally dominated kinetic regime, in which collisionless damping occurs via phase mixing and Landau resonance, to a regime dominated by quantum pressure. By applying these results to fuzzy dark matter, we show that the cutoff scale and its spectral shape depend sensitively on both the particle mass and the initial velocity dispersion, suggesting a method for simultaneously constraining these parameters through observations of the matter power spectrum. This framework provides a theoretical basis for future studies on the transition from early-phase thermal states to the formation of Bose-Einstein condensates in galactic structures.