Discerning singlet and triplet scalars at the electroweak phase transition and gravitational wave

In this article we examine the prospect of first order phase transition with a Y=0 real $SU(2)$ triplet extension of the Standard Model, which remains odd under $Z_2$, considering the observed Higgs boson mass, perturbative unitarity, dark matter constraints, etc. Especially we investigate the role of Higgs-triplet quartic coupling considering one- and two-loop beta functions and compare the results with the complex singlet extension case. It is observed that at the one-loop level, no solution can be found for both, demanding the Planck scale perturbativity. However, for a much lower scale of $10^4$ GeV, the singlet case predicts first order phase transition consistent with the observed Higgs boson mass. On the contrary, at the two-loop, both the scenarios foresee strongly first order phase transition consistent with the observed Higgs mass with upper bounds of 310, 909 GeV on the triplet and singlet masses, respectively. This puts the triplet in apparent contradiction with the observed dark matter relic bound and thus requires additional field for that. The preferred regions of the parameter space in both cases are identified by benchmark points, that predict the Gravitational Waves with detectable frequencies in the present and future experiments.