Cosmological implications of a shift symmetric Galileon field

A ghost-free metric formulation of the recently proposed covariant Galileon model [R. Banerjee and P. Mukherjee, arXiv: 1701.00971.] which retains the internal shift symmetry is constructed. This presents a new result because the covariant Galileon models introduced so far in the literature lack the full Galileon symmetry. We demonstrate that the same general procedure is applicable to all of the Galileon interaction terms and the resulting ghost-free Galileon action respecting internal shift symmetry cannot predict the late-time acceleration of the Universe. However, weakly breaking the shift symmetry in the covariant Galileon Lagrangian allows us to include an appropriate potential term which can explain late-time acceleration, in accordance with recent findings in the literature.