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Effective Constraints of Loop Quantum GravityEffective Constraints of Loop Quantum GravityMikhail KaganMikhail Kagan

Institute for Gravitational Physics and Geometry,Institute for Gravitational Physics and Geometry,Pennsylvania State UniversityPennsylvania State University

in collaboration within collaboration with

M. Bojowald, G. Hossain,M. Bojowald, G. Hossain,(IGPG, Penn State)(IGPG, Penn State)

H.H.Hernandez, A. SkirzewskiH.H.Hernandez, A. Skirzewski (Max-Planck-Institut f(Max-Planck-Institut für Gravitationsphysik,ür Gravitationsphysik,Albert-Einstein-Institut, Potsdam, GermanyAlbert-Einstein-Institut, Potsdam, Germany

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OutlineOutline

1.1. MotivationMotivation

2.2. Effective approximation. Overview.Effective approximation. Overview.

3. Effective constraints.3. Effective constraints.

• Implications.Implications. 5. Summary 5. Summary

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Motivation.Motivation. Test semi-classical limit of LQG.Test semi-classical limit of LQG.

Evolution of inhomogeneities is expected to explain Evolution of inhomogeneities is expected to explain cosmological cosmological structure formationstructure formation and lead to and lead to observable observable resultsresults..

Effective approximation allows to extract predictions of Effective approximation allows to extract predictions of the underlying quantum theory without going into the underlying quantum theory without going into consideration of quantum states.consideration of quantum states.

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Effective Approximation. Effective Approximation. Strategy.Strategy. Classical TheoryClassical Theory Classical Constraints & { , }Classical Constraints & { , }PBPB

QuantizationQuantization Quantum Operators & [ , ]Quantum Operators & [ , ]

Effective TheoryEffective Theory Quantum variables:Quantum variables:

expectation values, spreads, deformations, etc.expectation values, spreads, deformations, etc.

TruncationTruncation Expectation ValuesExpectation Values

classically divergingexpressions

Classical Classical ExpressionsExpressions

xCorrectionCorrectionFunctionsFunctions

Classical Classical ExpressionsExpressions

classically well behaved expressions

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Effective Approximation. Effective Approximation. Summary.Summary.

Classical Constraints & Poisson AlgebraClassical Constraints & Poisson Algebra

Constraint Operators & Commutation RelationsConstraint Operators & Commutation Relations

Anomaly IssueAnomaly Issue

(differs from classical constraint algebra)

Effective Equations of MotionEffective Equations of Motion(Bojowald, Hernandez, MK, Singh, Skirzewski

Phys. Rev. D, 74, 123512, 2006;Phys. Rev. Lett. 98, 031301, 2007

for scalar mode in longitudinal gauge)

Effective Constraints & Effective Poisson AlgebraEffective Constraints & Effective Poisson Algebra

(Non-anomalous algebra implies possibility of writing consistent system of equations of motion in terms of gauge invariant perturbation variables)

Non-trivial non-anomalous corrections foundNon-trivial non-anomalous corrections found

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Source of Corrections.Source of Corrections.

Basic VariablesBasic Variables

Diffeomorphism ConstraintDiffeomorphism Constraint

Hamiltonian constraintHamiltonian constraint

Densitized triad

Ashtekar connection

intactintact

Scalar field

Field momentum

DD

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Effective Constraints.Effective Constraints. Lattice formulation.Lattice formulation.(scalar mode/longitudinal gauge, Bojowald, Hernandez, MK, Skirzewski, 2007)

-labels

v

I

JK

FluxesFluxes

(integrated over Sv,I)

HolonomiesHolonomies

(integrated over ev,I)

Basic operators:Basic operators:

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Effective Constraints.Effective Constraints. Types of corrections.Types of corrections.

1.1. Discretization corrections.Discretization corrections.

2.2. Holonomy corrections Holonomy corrections (higher curvature corrections)(higher curvature corrections)..

3. Inverse triad corrections.3. Inverse triad corrections.

Effective Constraints.Effective Constraints. Hamiltonian.Hamiltonian.CurvatureCurvature

HamiltonianHamiltonian

00

Inverse triad operator

+higher curvature corrections

Effective Constraints.Effective Constraints. Inverse triad corrections.Inverse triad corrections.

AsymptoticsAsymptotics:

Effective Constraints.Effective Constraints. Inverse triad corrections.Inverse triad corrections.

GeneralizationGeneralization

for and

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Implications. Implications. Corrections to Newton’s potential.Corrections to Newton’s potential.

On Hubble scales:On Hubble scales:

Corrected Poisson EquationCorrected Poisson Equation

PHp 3

22

kk22

classicallyclassically 0,0, 11

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Implications.Implications. Inflation.Inflation.

classicallyclassically 0,0, 11

Large-scale Fourier ModesLarge-scale Fourier Modes

0//1 2 v vv

/4112

with,)(

Two Classical ModesTwo Classical Modes

decaying (decaying (< 0< 0)) const (const (=0=0))

With Quantum CorrectionsWith Quantum Corrections

decaying (decaying (< 0< 0)) growing (growing (≈≈))

modemodedescribes measure of inhomogeneity)

((P = wP = w

013 22 HwHw

wherewhere pp22//< 0< 0_

Corrected Raychaudhuri EquationCorrected Raychaudhuri Equation

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Implications.Implications. Inflation. Inflation.

Effective correctionsEffective corrections

pp22//

~ ~

_

Conservative bound (particle physics) Conservative bound (particle physics)

Metric perturbationMetric perturbation

corrected by factorcorrected by factor

Conformal timeConformal time

(changes by (changes by ee6060))

((Phys. Rev. Lett. 98, 031301, 2007))

Energy scale during InflationEnergy scale during Inflation

Summary.Summary.

1.1. There is a consistent set of corrected There is a consistent set of corrected constraints which are first class.constraints which are first class.

2.2. Cosmology: Cosmology: • can formulate equations of motion in terms can formulate equations of motion in terms

of gauge invariant variables.of gauge invariant variables.• potentially observable predictions.potentially observable predictions.

3.3. Indications that quantization ambiguities are Indications that quantization ambiguities are restricted.restricted.