  Methods and concepts for the study of phase transitions mediated by a
time-dependent order-parameter field in curved spacetimes are discussed. A
practical example is the derivation of an effective (quasi-)potential for the
description of `slow-roll' inflation in the early universe. We first summarize
our early results on viewing the symmetry behavior of constant background
fields in curved but static spacetimes as finite size effect, and the use of
derivative expansions for constructing effective actions for slowly-varying
background fields. We then introduce the notion of dynamical finite size effect
to explain how an exponential expansion of the scale factor imparts a finite
size to the system and how the symmetry behavior in de Sitter space can be
understood qualitatively in this light. We reason why the exponential inflation
can be described equivalently by a scale transformation, thus rendering this
special class of dynamics as effectively static. Finally we show how, in this
view, one can treat the class of `slow-roll' inflation as a dynamic
perturbation off the effectively static class of exponential inflation and
understand it as a dynamical critical phenomenon in cosmology.
