Phase transitions and gravitational wave tests of pseudo-Goldstone dark matter in the softly broken $U(1)$ scalar singlet model

We study phase transitions in a softly broken $U(1)$ complex singlet scalar model in which the dark matter is the pseudoscalar 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.

Figure 1

Phase diagram and thermal evolution of VEVs in the considered model. Left panel: Thermal evolution of the field to $T=0$ (dot) is shown by the black line. The scalar fields undergo a two-step phase transition from the completely symmetric phase (white) to the intermediate phase (red), where only the singlet has a VEV, to the electroweak vacuum (yellow), where both Higgs and the singlet have VEVs. Right panel: The two phase transitions. Evolution of the VEVs of the Higgs boson (green) and the complex singlet (red) with temperature. Critical temperatures are marked by thin vertical lines.