Effective Constraints in Loop Quantum Gravity: Implications for Cosmological Structure Formation

1. Effective Constraints of Loop Quantum GravityMikhail KaganInstitute for Gravitational Physics and Geometry,Pennsylvania State Universityin collaboration withM. Bojowald, G. Hossain,(IGPG, Penn State)H.H.Hernandez, A. Skirzewski(Max-Planck-Institut für Gravitationsphysik,Albert-Einstein-Institut, Potsdam, Germany

2. Outline • Motivation • Effective approximation. Overview. • 3. Effective constraints. • 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 & [ , ] Quantum variables: expectation values, spreads, deformations, etc. Effective Theory classically well behaved expressions classically diverging expressions Classical Expressions Classical Expressions Correction Functions x Truncation Expectation Values

5. Effective Approximation. Summary. effective approximation quantization Classical Constraints & Poisson Algebra Constraint Operators & Commutation Relations Effective Constraints & Effective Poisson Algebra (differs from classical constraint algebra) Anomaly Issue 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) (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. Densitized triad Ashtekar connection Basic Variables Scalar field Field momentum Diffeomorphism Constraint intact Hamiltonian constraint a(E) D(E) s(E)

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

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

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

10. Effective Constraints.Inverse triad corrections. Asymptotics:

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

12. Implications. Corrections to Newton’s potential. Corrected Poisson Equation k2 a ab On Hubble scales: 0, 1 classically

13. Implications.Inflation. Corrected Raychaudhuri Equation assume perfect fluid (P = wr) e1 e2 e3 Large-scale Fourier Modes _ wheree3 = -2ap2/a < 0 e3 With Quantum Corrections e3 Two Classical Modes growing (l_≈ -e3/n) decaying (l+< 0) decaying (l+< 0) const (l_=0) 1 0, classically (l_- modedescribes measure of inhomogeneity)

14. Implications. Inflation. Conformal time Effective corrections _ e3 = -2ap2/a ~ (changes by e60) Conservative bound (particle physics) Energy scale during Inflation Metric perturbation corrected by factor (Phys. Rev. Lett.98, 031301, 2007)

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