Anisotropic dispersion relation of ultralight Bose gases in modified Newtonian dynamics

We investigate the dispersion relation of collective modes in ultralight Bose gases under Modified Newtonian Dynamics (MOND). Starting from the coupled Gross-Pitaevskii and MOND Poisson equations, we derive an anisotropic dispersion relation that depends on the angle between the perturbation wavevector and the background gravitational field. This anisotropy arises directly from the nonlinear structure of the MOND field equation and leads to a direction-dependent Jeans instability, with critical wavelengths and masses varying with orientation. Our results provide a distinctive signature of MOND in a quantum astrophysical context and suggest that ultralight Bose gases can serve as novel probes of modified gravity.