Effective Constraints of Loop Quantum Gravity Mikhail Kagan

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 1

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

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. 3

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

Effective Approximation. Summary. Classical Constraints & Poisson Algebra Constraint Operators & Commutation Relations Effective Constraints & Effective Poisson Algebra (differs from classical constraint algebra) Effective 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) Anomaly Issue (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 5

Source of Corrections. Densitized triad Basic Variables Ashtekar connection Scalar field Field momentum Diffeomorphism Constraint intact Hamiltonian constraint a(E) D(E) s(E)6

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

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

Effective Constraints. Hamiltonian. Curvature 0 Hamiltonian Inverse triad operator +higher curvature corrections 9

Effective Constraints. Inverse triad corrections. Asymptotics: Asymptotics 10

Effective Constraints. Inverse triad corrections. Generalization for and 11

Implications. Corrections to Newton’s potential. Corrected Poisson Equation k 2 ab a On Hubble scales: classically 0, 1 12

Implications. Inflation. (P = w r ) Corrected Raychaudhuri Equation e 1 e 2 e 3 _ where e 3 = -2 a p 2/a < 0 Large-scale Fourier Modes e 3 Two Classical Modes decaying (l+ < 0) const (l_=0) With Quantum Corrections decaying (l+ < 0) growing (l_≈ -e 3/n) (l_- mode describes measure of inhomogeneity) classically 0, 1 13

Implications. Inflation. Conformal time (changes by e 60) Effective corrections _ e 3 = -2 a p 2/a ~ Conservative bound (particle physics) Energy scale during Inflation Metric perturbation corrected by factor (Phys. Rev. Lett. 98, 031301, 2007) 14

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. • potentially observable predictions. 3. Indications that quantization ambiguities are restricted. 15