Spontaneous symmetry breaking in inflationary cosmology: On the fate of Goldstone bosons

We argue that in an inflationary cosmology a consequence of the lack of time translational invariance is that spontaneous breaking of a continuous symmetry and Goldstone's theorem do not imply the existence of massless Goldstone modes. We study spontaneous symmetry breaking in an O(2) model, and implications for O(N) in de Sitter space-time. The Goldstone mode acquires a radiatively generated mass as a consequence of infrared divergences, and the continuous symmetry is spontaneously broken for any finite N; however there is a first order phase transition as a function of the Hawking temperature T-H = H/2 pi. For O(2) the symmetry is spontaneously broken for T-H < T-c = lambda(1/4)nu/2.419 where lambda is the quartic coupling and nu is the tree-level vacuum expectation value and the Goldstone mode acquires a radiatively generated mass M-pi(2) proportional to lambda H-1/4. The first order nature of the transition is a consequence of the strong infrared behavior of minimally coupled scalar fields in de Sitter space-time; the jump in the order parameter at T-H = T-c is sigma(0c) similar or equal to 0.6/H lambda(1/4). In the strict N -> infinity the symmetry cannot be spontaneously broken. Furthermore, the lack of kinematic thresholds imply that the Goldstone modes decay into Goldstone and Higgs modes by emission and absorption of superhorizon quanta.