Wave dark matter halo formation in the presence of a neutrino vacuum condensate

We investigate the formation and relaxation of dark matter halos composed of ultralight bosonic particles evolving in the presence of a neutrino vacuum condensate. The bosonic component is described as wave dark matter and is evolved by numerically solving the Schrödinger-Poisson system. The neutrino condensate, arising from flavor mixing in quantum field theory, is included as an additional gravitational source in the Poisson equation, while its dynamical evolution and possible direct couplings to the bosonic field are neglected. We compare the final halo configurations with the standard wave-dark-matter case. For ultraviolet cutoff values of order a few eV, the final halos remain close to the standard prediction: a solitonic core forms at the center and is surrounded by an outer NFW-like envelope. The main differences appear near the transition between the core and the external halo, with relative deviations up to about 13%. Larger cutoff values increase the central density and make the solitonic core more compact. For Λ = 500 eV, the system does not reach a fully virialized halo within a Hubble time. These results suggest that a neutrino condensate can coexist with wave dark matter for physically motivated cutoff values, while leaving possible signatures in the core-halo transition region.