  We investigate cosmological phase transitions in various composite Higgs
models consisting of four-dimensional asymptotically-free gauge field theories.
Each model may lead to a confinement-deconfinement transition and a phase
transition associated with the spontaneous breaking of a global symmetry that
realizes the Standard Model Higgs field as a pseudo-Nambu-Goldstone boson.
Based on the argument of universality, we discuss the order of the phase
transition associated with the global symmetry breaking by studying the
renormalization group flow of the corresponding linear sigma model at finite
temperature, which is calculated by utilizing the $\epsilon$-expansion
technique at the one-loop order. Our analysis indicates that some composite
Higgs models accommodate phenomenologically interesting first-order phase
transitions. We also explore the confinement-deconfinement transition in a
UV-completed composite Higgs model based on a $Sp(2N_c)$ gauge theory. It is
found that the first-order phase transition is favored when the number of
degrees of freedom for the $Sp(2N_c)$ gauge field is much larger than that of
matter fields in the fundamental representation of $Sp(2N_c)$. We comment on
the gravitational wave signal generated by the confinement-deconfinement
transition and its detectability at future observations. Our discussions
motivate further studies on phase transitions in composite Higgs models with
the use of lattice simulations.
