Spontaneous symmetry breaking in inflationary cosmology: on the fate of Goldstone Bosons

TL;DR

In inflationary cosmology, spontaneous symmetry breaking does not produce massless Goldstone bosons due to infrared effects, leading to massive modes and a first-order phase transition influenced by the Hawking temperature.

Contribution

This paper demonstrates that in de Sitter space, Goldstone bosons acquire a mass and the symmetry breaking transition is first order, contrasting with standard expectations.

Findings

Goldstone modes gain a radiatively generated mass in de Sitter space

Spontaneous symmetry breaking occurs for finite N below a critical Hawking temperature

The transition is first order with a specific jump in the order parameter

Abstract

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 \emph{do not} imply the existence of \emph{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 \emph{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 $\mathcal{M}^2_\pi \propto \lambda^{1/4} H$. 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 $\sigma_{0c} \simeq 0.61\, {H}/{\lambda^{1/4}}$. In the strict $N\rightarrow \infty$ the symmetry cannot be spontaneously broken. Furthermore, the lack of kinematic thresholds imply that the Goldstone modes \emph{decay} into Goldstone and Higgs modes by emission and absorption of superhorizon quanta.