Leading order gravitational backreactions in de Sitter spacetime Bojan Losic Theoretical Physics Institute University of Alberta IRGAC 2006, Barcelona July 14, 2006

July 14, 2006 Outline Probing backreactions in a simple arena Perturbation ansatz Linearization instability Quantum anomalies De Sitter group invariance of fluctuations Conclusions Based on gr-qc/ (B.L. and W.G. Unruh)

July 14, 2006 de Sitter spacetime perturbations Trivial (constant) scalar field with constant potential ↔ de Sitter Spacetime Perturbation ansatz: Background metric Leading order is second order (closed) slicing Similarly perturb the scalar field Quantum perturbation Constant Overbar denotes `background`

July 14, 2006 Higher order equations Stress energy is quadratic in field → leading contribution in de Sitter spacetime at second order Defining the monomials (assuming Leibniz rule) we may write the leading order stress-energy as Background D’Alembertian Background covariant derivative Leading order Einstein equations are of the form

July 14, 2006 Linearization instability I Vary the Bianchi identity around the de Sitter background to obtain Lambda constant, so drops out of variation Now vary the Bianchi identity times a Killing vector of the de Sitter background: Zero if Killing eqn. holdsDe Sitter Killing vector ∫∫ Variation of Christoffel symbols Integrate both sides and use Gauss’ theorem

July 14, 2006 Linearization stability II The integral is independent of hypersurface and variation of metric. Thus get However we want the fluctuations to obey the Einstein equations Thus we get an integral constraint on the scalar field fluctuations: Linearization stability (LS) condition What are the consequences of this constraint?

July 14, 2006 Anomalies in the LS conditions Hollands, Wald, and others have worked out a notion of local and covariant nonlinear (interacting) quantum fields in curved space-time One can redefine products of fields consistent with locality and covariance in their sense: Recall Curvature scalar, [length] -2 Curvature scalar, [length] -4 We show that the anomalies present in the LS conditions for de Sitter are of the form A number Normal component of Killing vector Volume measure of hypersurface ~ 0 Normal Killing component is odd over space

July 14, 2006 LS conditions and SO(4,1) symmetry It turns out that the LS conditions form a Lie algebra But it also turns out that the Killing vectors form the same algebra The same structure constants holds LS condition Structure constants No quantum anomalies in commutator The LS conditions demand that all physical states are SO(4,1) invariant

July 14, 2006 Problems with de Sitter invariant states Allen showed no SO(4,1) invariant states for massless scalar field: How are dynamics possible with such symmetric states? How do we understand the flat (Minkowski) limit? Massless scalar field action with zero mode

July 14, 2006 Conclusion Linearization insatbilities in de Sitter spacetime imply nontrivial constraints on the quantum states of a scalar field in de Sitter spacetime. It turns out that the quantum states of a scalar field in de Sitter spacetime must, if consistently coupled to gravity to leading order, be de Sitter invariant (and not covariant!).

July 14, 2006