  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 effective
potential, at one loop and at finite temperature, of the Standard Model Higgs
doublet together with one extra Higgs singlet. We make use of the public
available Python package CosmoTransitions to simulate the two-dimensional
cosmological PTs and evaluate the gravitational waves generated by first-order
PTs. With the full 2D simulation, we are able not only to confirm the PTs'
properties previously discussed in the literature, but also we find new
patterns, such as strong first-order PTs tunneling from a vacuum located on one
axis to another vacuum located on the second axis. The two-field phase space
analysis presents a richer panel of cosmological PT patterns compared to
analysis with a single-field approximation. The PTGW amplitudes turn out to be
out of the reach for the space-borne gravitational wave interferometers such as
LISA, DECIGO, BBO, TAIJI and TianQin when constraints from colliders physics
are taken into account.
