A no-go theorem in bumblebee vector-tensor cosmology

Bumblebee models, a class of vector-tensor theories in which a vector field acquires a nonzero vacuum expectation value that spontaneously breaks spacetime symmetries, are ubiquitous in the literature. By constructing the most general bumblebee action from all diffeomorphism-invariant marginal operators together with a general potential, aiming to cover all the bumblebee models studied in the literature, we perform a complete linear perturbation analysis on a spatially flat FLRW background. We show that for generic marginal couplings, the scalar sector propagates extra degrees of freedom beyond the single scalar expected for a massive vector. Enforcing the correct number of propagating modes in a cosmological setup forces degeneracy relations between the marginal couplings, which in turn completely fix the potential at the background level and render the remaining scalar infinitely strongly coupled already at linear order of perturbations. We establish a no-go theorem stating that the following conditions cannot be simultaneously satisfied: (i) the most general marginal action, (ii) a homogeneous and isotropic background, (iii) no extra propagating degrees of freedom around a spatially flat FLRW background, and (iv) healthy cosmological perturbations.