  The electroweak (EW) phase transition in the early Universe might be
supercooled due to the presence of the classical scale invariance involving
Beyond the Standard Model (BSM) sectors and the supercooling could persist down
till a later epoch around which the QCD chiral phase transition is supposed to
take place. Since this supercooling period keeps masslessness for all the six
SM quarks, it has simply been argued that the QCD phase transition is the first
order, and so is the EW one. However, not only the QCD coupling but also the
top Yukawa and the Higgs quartic couplings get strong at around the QCD scale
due to the renormalization group running, hence this scenario is potentially
subject to a rigorous nonperturbative analysis. In this work, we employ the
ladder Schwinger-Dyson (LSD) analysis based on the Cornwall-Jackiw-Tomboulis
formalism at the two-loop level in such a gauge-Higgs-Yukawa system. We show
that the chiral broken QCD vacuum emerges with the nonperturbative top
condensate and the lightness of all six quarks is guaranteed due to the
accidental U(1) axial symmetry presented in the top-Higgs sector. We employ a
quark-meson model-like description in the mean field approximation to address
the impact on the EW phase transition arising due to the top quark condensation
at the QCD phase transition epoch. In the model, the LSD results are encoded to
constrain the model parameter space. We then observe the cosmological phase
transition of the first-order type and discuss the induced gravitational wave
(GW) productions. We find that in addition to the conventional GW signals
sourced from an expected BSM at around or over the TeV scale, the dynamical
topponium-Higgs system can yield another power spectrum sensitive to the BBO,
LISA, and DECIGO, etc.
