Electroweak phase transition with composite Higgs models: calculability,
  gravitational waves and collider searches
  We study the strong first order electroweak phase transition (SFOEWPT) with
the $SO(6)/SO(5)$ composite Higgs model, whose scalar sector contains one Higgs
doublet and one real singlet. Six benchmark models are built with fermion
embeddings in $\textbf{1}$, $\textbf{6}$, and $\textbf{15}$ of $SO(6)$. We show
that SFOEWPT cannot be triggered under the minimal Higgs potential hypothesis,
which assumes the scalar potential is dominated by the form factors from the
lightest composite resonances. To get a SFOEWPT, the contributions from local
operators induced by physics above the cutoff scale are needed. We take the
$\textbf{6}+\textbf{6}$ model as an example to investigate the gravitational
waves prediction and the related collider phenomenology.
