  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.
