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Leading order gravitational
backreactions in de Sitter spacetime
Bojan Losic
Theoretical Physics Institute
University of Alberta
IRGAC 2006, Barcelona
July 14, 2006
Outline
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Probing backreactions in a simple arena
Perturbation ansatz
Linearization instability
Quantum anomalies
De Sitter group invariance of fluctuations
Conclusions
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Based on gr-qc/0604122
(B.L. and W.G. Unruh)
de Sitter spacetime perturbations
•Trivial (constant) scalar field with constant potential
•Perturbation ansatz:
↔
de Sitter Spacetime
Leading order is
second order
Overbar denotes
`background`
Background metric
(closed) slicing
• Similarly perturb the scalar field
Constant
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Quantum perturbation
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
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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
Integrate both sides and
use Gauss’ theorem
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Zero if Killing eqn. holds
Variation of
Christoffel symbols
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
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What are the consequences of this constraint?
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
Recall
Curvature scalar, [length]-4
• We show that the anomalies present in the LS conditions for de Sitter are of
the form
Normal Killing
~ 0
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A number
Normal component
of Killing vector
component is odd
over space
Volume measure of
hypersurface
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
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Problems with de Sitter invariant
states
• Allen showed no SO(4,1) invariant states for massless scalar field:
Massless scalar field action with zero mode
• How are dynamics possible with such symmetric states?
• How do we understand the flat (Minkowski) limit?
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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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