  Light scalar fields such as axions and string moduli can play an important
role in early-universe cosmology. However, many factors can significantly
impact their late-time cosmological abundances. For example, in cases where the
potentials for these fields are generated dynamically --- such as during
cosmological mass-generating phase transitions --- the duration of the time
interval required for these potentials to fully develop can have significant
repercussions. Likewise, in scenarios with multiple scalars, mixing amongst the
fields can also give rise to an effective timescale that modifies the resulting
late-time abundances. Previous studies have focused on the effects of either
the first or the second timescale in isolation. In this paper, by contrast, we
examine the new features that arise from the interplay between these two
timescales when both mixing and time-dependent phase transitions are introduced
together. First, we find that the effects of these timescales can conspire to
alter not only the total late-time abundance of the system --- often by many
orders of magnitude --- but also its distribution across the different fields.
Second, we find that these effects can produce large parametric resonances
which render the energy densities of the fields highly sensitive to the degree
of mixing as well as the duration of the time interval over which the phase
transition unfolds. Finally, we find that these effects can even give rise to a
"re-overdamping" phenomenon which causes the total energy density of the system
to behave in novel ways that differ from those exhibited by pure dark matter or
vacuum energy. All of these features therefore give rise to new possibilities
for early-universe phenomenology and cosmological evolution. They also
highlight the importance of taking into account the time dependence associated
with phase transitions in cosmological settings.
