Leading order gravitational backreactions in de Sitter spacetime
Bojan LosicTheoretical Physics InstituteTheoretical Physics Institute
University of AlbertaUniversity 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/0604122(B.L. and W.G. Unruh)
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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 perturbationConstant
Overbar denotes
`background`
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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
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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
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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?
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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:
RecallCurvature 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
~ 0Normal Killing component is odd over space
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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
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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
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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!).
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