Evolution of baryon density perturbation in a relativistic MOND model based on Lorentz-violating vector field

A candidate for relativistic MOND with successful cosmology was proposed recently by using a Lorentz-violating vector field in Einstein's gravity. We show that the dynamic nature of the vector field makes it challenging to realize the MOND. Only in the stationary limit, thus excluding cosmological situations, one can achieve the MOND limit. We study the evolution of density perturbations in the baryon-vector field system using both post-Newtonian approximation and relativistic perturbation theory. Our results show that, considering the dynamic nature of the vector field, the faster growth of structures expected in the low-acceleration MOND regime is not recovered in the relativistic MOND.