Leading order gravitational backreactions in de Sitter spacetime

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

2. Outline • Probing backreactions in a simple arena • Perturbation ansatz • Linearization instability • Quantum anomalies • De Sitter group invariance of fluctuations • Conclusions Based on gr-qc/0604122 (B.L. and W.G. Unruh)

3. de Sitter spacetime perturbations • Trivial (constant) scalar field with constant potential ↔ de Sitter Spacetime • Perturbation ansatz: Leading order is second order Overbardenotes `background` Background metric (closed) slicing • Similarly perturb the scalar field Constant Quantum perturbation

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

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

6. 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?

7. Recall 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: 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 Normal Killing component is odd overspace ~ 0 Volume measure of hypersurface A number Normal component of Killing vector

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

9. 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

10. 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!).