  This paper extends the study of the quantum dissipative effects of a
cosmological scalar field by taking into account the cosmic expansion and
contraction. Cheung, Drewes, Kang and Kim calculated the effective action and
quantum dissipative effects of a cosmological scalar field. The analytic
expressions for the effective potential and damping coefficient were presented
using a simple scalar model with quartic interaction. Their work was done using
Minkowski-space propagators in loop diagrams. In this work we incorporate the
Hubble expansion and contraction of the comic background, and focus on the
thermal dynamics of a scalar field in a regime where the effective potential
changes slowly. We let the Hubble parameter, $H$, attain a small but non-zero
value and carry out calculations to first order in $H$. If we set $H=0$ all
results match those obtained previously in flat spacetime [1]. Interestingly we
have to integrate over the resonances, which in turn leads to an amplification
of the effects of a non-zero $H$. This is an intriguing phenomenon which cannot
be uncovered in flat spacetime. The implications on particle creations in the
early universe will be studied in a forthcoming work.
