Cosmology of Supercooled Universe
  First-order phase transitions (FOPTs) are ubiquitous in physics beyond the
Standard Model (SM). Recently, models with no dimensionful parameters in the
tree-level action have been attracting much attention because they can predict
a very strong FOPT with ultra-supercooling. In this paper, we study the
cosmological signatures of such a supercooling model. As a concrete model, we
consider the SM with two additional real scalars $\phi$ and $S$, which can
realize the electroweak symmetry breaking via Coleman-Weinberg mechanism. One
of the additional scalars $S$ can naturally become a Dark Matter (DM) candidate
due to the $Z_2^{}$ symmetry of the action. We study the FOPT of this model and
calculate the Gravitational Wave (GW) signals and the thermal relic abundance
of $S$. As a result, we find that (i) the GW peak amplitude can reach $\sim
10^{-10}$ around the frequency $f\sim 10^{-3}~$Hz for some model parameters and
(ii) the scalar mixing coupling $\lambda_{\phi S}^{}$ is constrained to be
$0.8\lesssim \lambda_{\phi S}^{}\lesssim 1$ in order to achieve both FOPT and
DM relic abundance.
