Self-tuning kinetic gravity braiding: Cosmological dynamics, shift symmetry, and the tadpole

We study the self-tuning subclass of kinetic gravity braiding and obtain robust predictions on self-tuning and dynamics in the tadpole-free shift symmetric sector of the theory. In particular, we show inevitability of cosmic acceleration, prove the dynamical stability of this late-time asymptotic state, and derive ghost and gradient stability constraints on the self-tuning vacuum. We discuss the results concretely in the context of generalized cubic covariant Galileon theory and an exponential kinetic gravity braiding.