  The renormalized expectation value of the energy-momentum tensor for a scalar
field with any mass m and curvature coupling xi is studied for an arbitrary
homogeneous and isotropic physical initial state in de Sitter spacetime. We
prove quite generally that <T_{ab}> has a fixed point attractor behavior at
late times, which depends only on m and xi, for any fourth order adiabatic
state that is infrared finite. Specifically, when m^2 + xi R > 0, <T_{ab}>
approaches the Bunch-Davies de Sitter invariant value at late times,
independently of the initial state. When m = xi = 0, it approaches instead the
de Sitter invariant Allen-Folacci value. When m = 0 and xi \ge 0 we show that
this state independent asymptotic value of the energy-momentum tensor is
proportional to the conserved geometrical tensor (3)H_{ab}, which is related to
the behavior of the quantum effective action of the scalar field under global
Weyl rescaling. This relationship serves to generalize the definition of the
trace anomaly in the infrared for massless, non-conformal fields. In the case
m^2 + xi R = 0, but m and xi separately different from zero, <T_{ab}> grows
linearly with cosmic time at late times. For most values of m and xi in the
tachyonic cases, m^2 + xi R < 0, <T_{ab}> grows exponentially at late cosmic
times for all physically admissable initial states.
