  The two-dimensional $\sigma$-model with the de Sitter target space has a
local canonical description in the north pole diamond of the Penrose diagram in
the cosmological gauge. The left and right moving modes on the embedded base
space with the topology of a cylinder are entangled among themselves and
interact with the time-dependent components of the metric of the de Sitter
space. Firstly we address the issue of the existence of the untangled
oscillator representation and the description of the nonequilibrium dynamics of
the untangled modes. We show that the untangled oscillators can be obtained
from the entangled operators by applying a set of Bogoliubov transformations
that satisfy a set of constraints that result from the requirement that the
partial evolution generator be diagonal. Secondly, we determine the
nonequilibrium dynamics of the untangled modes in the Non-Equilibrium Thermo
Field Dynamics formalism. In this setting, the thermal modes are represented as
thermal doublet oscillators that satisfy partial evolution equations of
Heisenberg-type. We use these equations to compute the local free one-body
propagator of an arbitrary mode between two times. Thirdly, we discuss the
field representation of the thermal modes. We show that there is a set of
thermal doublet fields that satisfy the equal time canonical commutation
relations, are solutions to the $\sigma$-model equations of motion and can be
decomposed in terms of thermal doublet oscillators. Finally, we construct a
local partial evolution functional of Hamilton-like form for the thermal
doublet fields.
