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In the singlet Majoron model, we study cosmological phase transitions (PTs)and their resulting gravitational waves (GWs), in the two-field phase space,without freezing any of the field directions. We first calculate the effectivepotential, at one loop and at finite temperature, of the Standard Model Higgsdoublet together with one extra Higgs singlet. We make use of the publicavailable Python package CosmoTransitions to simulate the two-dimensionalcosmological PTs and evaluate the gravitational waves generated by first-orderPTs. With the full 2D simulation, we are able not only to confirm the PTs'properties previously discussed in the literature, but also we find newpatterns, such as strong first-order PTs tunneling from a vacuum located on oneaxis to another vacuum located on the second axis. The two-field phase spaceanalysis presents a richer panel of cosmological PT patterns compared toanalysis with a single-field approximation. The PTGW amplitudes turn out to beout of the reach for the space-borne gravitational wave interferometers such asLISA, DECIGO, BBO, TAIJI and TianQin when constraints from colliders physicsare taken into account.

Two-field cosmological phase transitions and gravitational waves in the singlet Majoron model