  Master equations are commonly employed in cosmology to model the effect of
additional degrees of freedom, treated as an "environment", onto a given
"system". However, they rely on assumptions that are not necessarily satisfied
in cosmology, where the environment may be out of equilibrium and the
background is dynamical. In this work, we apply the master-equation program to
a model that is exactly solvable, and which consists of two linearly coupled
scalar fields evolving on a cosmological background. The light field plays the
role of the system and the heavy field is the environment. By comparing the
exact solution to the output of the master equation, we can critically assess
its performance. We find that the master equation exhibits a set of "spurious"
terms that explicitly depend on the initial conditions, and which arise as a
consequence of working on a dynamical background. Although they cancel out in
the perturbative limit of the theory (i.e. at leading orders in the interaction
strength), they spoil resummation. However, when those terms are removed, the
master equation performs impressively well to reproduce the power spectra and
the amount of the decoherence of the light field, even in the strongly
decohered regime. We conclude that master equations are able to perform
late-time resummation, even though the system is far from the Markovian limit,
provided spurious contributions are suppressed.
