Formation of defects associated with both spontaneous and explicit symmetry breaking

TL;DR

This paper investigates the formation of cosmic strings and domain walls resulting from spontaneous and explicit symmetry breaking of a $U(1)$ symmetry, using classical field simulations to estimate conditions for defect formation in the early universe.

Contribution

It provides the first detailed analysis of defect formation when explicit symmetry breaking occurs before spontaneous breaking, with quantitative estimates for different symmetry breaking scenarios.

Findings

Cosmic strings attached to domain walls form when explicit breaking is much smaller than spontaneous breaking.

Domain walls inevitably form at high temperatures, regardless of explicit breaking magnitude.

Cosmic strings also form under certain conditions when explicit breaking is smaller than spontaneous breaking.

Abstract

We discuss formation of cosmic strings associated with a spontaneously broken approximate $U(1)$ symmetry by performing classical field-theoretical simulations. An original $U(1)$ symmetry is explicitly broken down to its subgroup $Z_N$ even before spontaneous breaking takes place. We estimate the ratio of explicit breaking to that of spontaneous breaking for which topological defects for $N=1$ and $N=2$ are formed. For $N=1$, a cosmic string attached to a single domain wall can be formed when the amount of the explicit breaking is three orders of magnitude smaller than that of the spontaneous breaking. For $N=2$, no matter how large the explicit breaking is, domain walls are inevitably formed as long as the temperature of the Universe is high enough to restore $Z_2$ symmetry. In that case, cosmic strings are also inevitably formed as long as the amount of the explicit breaking is smaller than that of the spontaneous breaking.