The bumblebee field excitations in a cosmological braneworld

Abstract

We investigate the effects of the spacetime curvature and extra dimensions on the excitations of the Lorentz violating bumblebee field $B_\mu$. %The self-interacting quadratic potential leads to a spontaneous breaking of the Lorentz symmetry. By assuming the bumblebee field living in a \revision{five dimensional} $AdS_{5}$ bulk, we found an exponential suppression of the bumblebee self-interacting constant $\lambda$ and of the bumblebee vacuum expectation value (VEV) \revision{$b_{M}$} along the \revision{fifth} extra dimension. The fluctuations of the bumblebee \revision{field} upon the VEV can be decomposed into transverse and longitudinal modes with respect to $b_{M}$. For a spacelike $b_{M}$ along the extra dimension and assuming a \revision{thin} FRW 3-brane embedded in the $AdS_{5}$, the transverse mode is localized on the brane. The bulk geometry leads to a propagating and unbound \revision{longitudinal mode along the extra dimension}. On the brane, the cosmological expansion leads to the decay of the longitudinal mode in time, as $\Delta t \approx H^{-1}$.