On leading order gravitational backreactions in de Sitter spacetime

TL;DR

This work analyzes the leading-order gravitational backreaction of nonlinear scalar fluctuations on a de Sitter background with $\Lambda$ from a constant scalar potential. By performing a second-order perturbative expansion and applying a carefully chosen gauge, the authors derive linearization stability (LS) constraints that must hold for the second-order solutions to exist, and they show these LS constraints are gauge-invariant and conserved. Using Hadamard renormalization in the Hollands–Wald framework, they demonstrate that quantum anomalies do not obstruct the gauge conditions or the LS constraints in de Sitter space, and that the LS generators correspond to de Sitter transformations. Consequently, the physical quantum states $|\Psi\rangle$ of both matter and metric fluctuations must be invariant under the de Sitter group $SO(4,1)$ at leading order. This strong invariance requirement extends beyond the vacuum state and has notable implications for the dynamics of fields in de Sitter and the flat-space limit $\Lambda\to 0$.

Abstract

Backreactions are considered in a de Sitter spacetime whose cosmological constant is generated by the potential of scalar field. The leading order gravitational effect of nonlinear matter fluctuations is analyzed and it is found that the initial value problem for the perturbed Einstein equations possesses linearization instabilities. We show that these linearization instabilities can be avoided by assuming strict de Sitter invariance of the quantum states of the linearized fluctuations. We furthermore show that quantum anomalies do not block the invariance requirement. This invariance constraint applies to the entire spectrum of states, from the vacuum to the excited states (should they exist), and is in that sense much stronger than the usual Poincare invariance requirement of the Minkowski vacuum alone. Thus to leading order in their effect on the gravitational field, the quantum states of the matter and metric fluctuations must be de Sitter invariant.