1 Effective Constraints of Loop Quantum Gravity Mikhail Kagan Institute for Gravitational Physics and Geometry, Pennsylvania State University in collaboration with M. Bojowald, G. Hossain, (IGPG, Penn State) H.H.Hernandez, A. Skirzewski (Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany

2 Outline 1. Motivation 2. Effective approximation. Overview. 3. Effective constraints. Implications. Implications. 5. Summary

3 Motivation. Test semi-classical limit of LQG. Evolution of inhomogeneities is expected to explain cosmological structure formation and lead to observable results. Effective approximation allows to extract predictions of the underlying quantum theory without going into consideration of quantum states.

4 Effective Approximation. Strategy. Classical Theory Classical Constraints & {, } PB Quantization Quantum Operators & [, ] Effective Theory Quantum variables: expectation values, spreads, deformations, etc. Truncation Expectation Values classically diverging expressions Classical Expressions x CorrectionFunctions classically well behaved expressions

5 Effective Approximation. Summary. Classical Constraints & Poisson Algebra Constraint Operators & Commutation Relations Anomaly Issue (differs from classical constraint algebra) Effective Equations of Motion (Bojowald, Hernandez, MK, Singh, Skirzewski Phys. Rev. D, 74, , 2006; Phys. Rev. Lett. 98, , 2007 for scalar mode in longitudinal gauge) Effective 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 found

6 Source of Corrections. Basic Variables Diffeomorphism Constraint Hamiltonian constraint Densitized triad Ashtekar connection intact Scalar field Field momentum  D 

7 Effective Constraints.Lattice formulation. Effective Constraints. Lattice formulation. ( scalar mode/longitudinal gauge, Bojowald, Hernandez, MK, Skirzewski, 2007) -labels v I J K Fluxes ( integrated over S v,I )Holonomies ( integrated over e v,I ) Basic operators:

8 Effective Constraints.Types of corrections. Effective Constraints. Types of corrections. 1.Discretization corrections. 2.Holonomy corrections (higher curvature corrections). 3. Inverse triad corrections.

Effective Constraints.Hamiltonian. Effective Constraints. Hamiltonian. Curvature Hamiltonian 0 Inverse triad operator +higher curvature corrections

Effective Constraints.Inverse triad corrections. Effective Constraints. Inverse triad corrections. Asymptotics Asymptotics:

Effective Constraints.Inverse triad corrections. Effective Constraints. Inverse triad corrections. Generalization for and

12 Implications. Corrections to Newton’s potential. On Hubble scales: Corrected Poisson Equation   k2k2k2k2 classically0,1

13 Implications.Inflation. Implications. Inflation. classically0,1 Large-scale Fourier Modes   Two Classical Modes decaying (   < 0) const (  =0) With Quantum Corrections decaying (   < 0) growing (    )  mode   mode  describes measure of inhomogeneity) (P = w     where    p 2 /  < 0 _ Corrected Raychaudhuri Equation

14 Implications. Inflation. Effective corrections    p 2 /  ~ _ Conservative bound (particle physics) Metric perturbation corrected by factor Conformal time (changes by e 60 ) () ( Phys. Rev. Lett. 98, , 2007 ) Energy scale during Inflation

Summary. 1.There is a consistent set of corrected constraints which are first class. 2.Cosmology: can formulate equations of motion in terms of gauge invariant variables.can formulate equations of motion in terms of gauge invariant variables. potentially observable predictions.potentially observable predictions. 3.Indications that quantization ambiguities are restricted.