The renormalized expectation value of the energy-momentum tensor for a scal
ar field with any mass m and curvature coupling xi is studied for an arbitr
ary 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 adiaba
tic state that is infrared finite. Specifically, when m(2)+xiR>0, (T-ab) ap
proaches the Bunch-Davies de Sitter invariant value at late times, independ
ently of the initial state. When m = xi = 0, it approaches instead the de S
itter invariant Allen-Folacci value. When m = 0 and xi greater than or equa
l to0 we show that the state independent asymptotic value of the energy-mom
entum tensor is proportional to the conserved geometrical tensor H-(3)(ab),
which is related to the behavior of the quantum effective action of the sc
alar field under global Weyl rescaling. This relationship serves to general
ize the definition of the trace anomaly in the infrared for massless, nonco
nformal fields. In the case m(2) + xiR = 0, but m and xi separately differe
nt from zero, (T-ab) grows linearly with cosmic time at late times. For mos
t values of m(2) and xi in the tachyonic cases, m(2) + xiR<0, (T-ab) grows
exponentially at late cosmic times for all physically admissible initial st
ates.