# Phase Transitions and Gravitational Wave Tests of Pseudo-Goldstone Dark Matter in the Softly Broken U(1) Scalar Singlet Model

## TL;DR

This paper investigates phase transitions in a softly broken U(1) scalar singlet model to assess the potential of gravitational wave detection as a test, finding the relevant transition is always second order and thus unlikely to produce detectable gravitational waves.

## Contribution

It demonstrates that the phase transition in the model is always second order, ruling out gravitational wave signals as a test for this dark matter scenario.

## Key findings

- The phase transition is always of second order.
- Detection of a stochastic gravitational wave background would rule out this model.
- The model cannot produce detectable gravitational waves from the phase transition.

## Abstract

We study phase transitions in a softly broken $U(1)$ complex singlet scalar model in which the dark matter is the pseudo-scalar part of a singlet whose direct detection coupling to matter is strongly suppressed. Our aim is to find ways to test this model with the stochastic gravitational wave background from the scalar phase transition. We find that the phase transition which induces vacuum expectation values for both the Higgs boson and the singlet - necessary to provide a realistic dark matter candidate - is always of the second order. If the stochastic gravitational wave background characteristic to a first order phase transition will be discovered by interferometers, the soft breaking of $U(1)$ cannot be the explanation to the suppressed dark matter-baryon coupling, providing a conclusive negative test for this class of singlet models.