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 $T_H<T_c={\lambda }^{1/4}v/2.419$ where $\lambda$ is the quartic coupling and $v$ is the tree-level vacuum expectation value and the Goldstone mode acquires a radiatively generated mass ${\mathcal{M}}_{\pi }^2\propto {\lambda }^{1/4}H$. 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 ${\sigma }_{0c}\simeq 0.61H/{\lambda }^{1/4}$. In the strict $N\to \infty$ 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.

• Received 16 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.023509