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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