Strong First Order EWPT and Strong Gravitational Waves in $Z_3$-symmetric Singlet Scalar Extension

The nature of electroweak (EW) phase transition (PT) is of great importance. It may give a clue to the origin of baryon asymmetry if EWPT is strong first order. Although it is second order within the standard model (SM), a great many extensions of the SM are capable of altering the nature. Thus, gravitational wave (GW), which is supposed to be relics of strong first order PT, is a good complementary probe to new physics beyond SM (BSM). We in this paper elaborate the patterns of strong first order EWPT in the next to simplest extension to the SM Higgs sector, by introducing a $Z_3$-symmetric singlet scalar. We find that, in the $Z_3$-symmetric limit, the tree level barrier could lead to strong first order EWPT either via three or two-step PT. Moreover, they could produce two sources of GW, despite of the undetectability from the first-step strong first order PT for the near future GW experiments. But the other source with significant supercooling which then gives rise to $\alpha\sim{\cal O}(0.1)$ almost can be wholly covered by future space-based GW interferometers such as eLISA, DECIGO and BBO.