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.
