Cosmological horizon modes and linear response in de Sitter spacetime

Abstract

Linearized fluctuations of quantized matter fields and the spacetime geometry around de Sitter space are considered in the case that the matter fields are conformally invariant. Taking the unperturbed state of the matter to be the de Sitter invariant Bunch-Davies state, the linear variation of the stress tensor about its self-consistent mean value serves as a source for fluctuations in the geometry through the semiclassical Einstein equations. This linear response framework is used to investigate both the importance of quantum backreaction and the validity of the semiclassical approximation in cosmology. The full variation of the stress tensor $\ensuremath{\delta}⟨T^{a}{}_{b}⟩$ contains two kinds of terms: (1) those that depend explicitly upon the linearized metric variation $\ensuremath{\delta}{g}_{cd}$ through the $⟨[T^{a}{}_{b},{T}^{cd}]⟩$ causal response function; and (2) state dependent variations, independent of $\ensuremath{\delta}{g}_{cd}$. For perturbations of the first kind, the criterion for the validity of the semiclassical approximation in de Sitter space is satisfied for fluctuations on all scales well below the Planck scale. The perturbations of the second kind contain additional massless scalar degrees of freedom associated with changes of state of the fields on the cosmological horizon scale. These scalar degrees of freedom arise necessarily from the local auxiliary field form of the effective action associated with the trace anomaly, are potentially large on the horizon scale, and therefore can lead to substantial nonlinear quantum backreaction effects in cosmology.