The dynamics of a cosmological (de)confinement phase transition is studied in nearly conformally invariant field theories, where confinement is predominantly spontaneously generated and associated with a light dilaton field. We show how the leading contribution to the transition rate can be computed within the dilaton effective theory. In the context of Composite Higgs theories, we demonstrate that a simple scenario involving two renormalization-group fixed points can make the transition proceed much more rapidly than in the minimal scenario, thereby avoiding excessive dilution of matter abundances generated before the transition. The implications for gravitational wave phenomenology are discussed. In general, we find that more (less) rapid phase transitions are associated with weaker (stronger) gravitational wave signals. The various possible features of the strongly coupled composite Higgs phase transition discussed here can be concretely modeled at weak coupling within the AdS/CFT dual Randall-Sundrum extra-dimensional description, which offers important insights into the nature of the transition and its theoretical control. These aspects will be presented in a companion paper.
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