Attractor states and infrared scaling in de Sitter space

Abstract

The renormalized expectation value of the energy-momentum tensor for a scalar field with any mass m and curvature coupling \ensuremath{\xi} is studied for an arbitrary homogeneous and isotropic physical initial state in de Sitter spacetime. We prove quite generally that $〈{T}_{\mathrm{ab}}〉$ has a fixed point attractor behavior at late times, which depends only on m and \ensuremath{\xi}, for any fourth order adiabatic state that is infrared finite. Specifically, when ${m}^{2}+\ensuremath{\xi}R>0,$ $〈{T}_{\mathrm{ab}}〉$ approaches the Bunch-Davies de Sitter invariant value at late times, independently of the initial state. When $m=\ensuremath{\xi}=0,$ it approaches instead the de Sitter invariant Allen-Folacci value. When $m=0$ and $\ensuremath{\xi}>~0$ we show that the state independent asymptotic value of the energy-momentum tensor is proportional to the conserved geometrical tensor ${}^{(3)}{H}_{\mathrm{ab}},$ 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, nonconformal fields. In the case ${m}^{2}+\ensuremath{\xi}R=0,$ but m and \ensuremath{\xi} separately different from zero, $〈{T}_{\mathrm{ab}}〉$ grows linearly with cosmic time at late times. For most values of ${m}^{2}$ and \ensuremath{\xi} in the tachyonic cases, ${m}^{2}+\ensuremath{\xi}R<0,$ $〈{T}_{\mathrm{ab}}〉$ grows exponentially at late cosmic times for all physically admissible initial states.