ISO(4,1) symmetry in the EFT of inflation

In DBI inflation the cubic action is a particular linear combination of the two, otherwise independent, cubic operators (pi) over dot(3) and (pi) over dot(partial derivative(i)pi)(2). We show that in the Effective Field Theory (EFT) of inflation this is a consequence of an approximate 5D Poincare symmetry, ISO(4,1), non-linearly realized by the Goldstone pi. This symmetry uniquely fixes, at lowest order in derivatives, all correlation functions in terms of the speed of sound c(s). In the limit c(s) -> 1, the ISO(4,1) symmetry reduces to the Galilean symmetry acting on pi. On the other hand, we point out that the non-linear realization of SO(4,2), the isometry group of 5D AdS space, does not fix the cubic action in terms of c(s).