Cosmological Perturbation Theory in Canonical Variables

1. Cosmological perturbation theoryin canonical variablesMikhail KaganPenn State Abingtonand Institute for Gravitation and the Cosmos, Penn State in collaboration withM. Bojowald, G. Hossain,S. Shankaranarayanan

2. Outline • Classical perturbation theory. Gauge invariant variables, • gauge invariant equations of motion (EoM) • a) Covariant formulation • b) Canonical formulation • 2. Quantum corrected perturbation theory • 3. Summary

3. Covariant Formulation. Action and metric. Action Matter Gravity Metric: Matter: scalar perturbations: Perturbation theory

4. Covariant Formulation. Gauge Issues. Hom & Iso Universe preferable coordinate system no (obvious) preferable coordinate system Perturbations Gauge freedom Fictitious perturbation modes (do not describe real inhomogeneities, reflect only properties of coordinate system used )

5. Covariant Formulation. Gauge transformations. Gauge (coordinate) transformation: Metric: Matter: Gauge invariant variables

6. Covariant Formulation. Equations of motion. Friedmann equation Raychaudhuri equation Klein-Gordon equation Background Einstein’s Equations Perturbed Klein-Gordon Equation

7. Canonical Formulation. Constraints. space-time splitting Action(Hamiltonian+Gauss+Vector) + Poisson Brackets constraints symplectic structure Immirzi parameter Metricds2=gabdxadxb=-N2dt2+qab(dxa+Nadt)(dxb+Nbdt) Hamiltonian constraint Diffeomorphism Constraint where

8. Canonical Formulation.Basic variables. Poisson brackets Scalar field Field momentum Triad Ashtekar connection Spin connection Extrinsic curvature average quantities Matter Gravity

9. Canonical Formulation.Classical EoM. Constraint equations BG Friedmann Pert S-T Einstein Pert Friedmann Dynamical equations Pert K-G BG K-G BG Raychaudhuri Pert Raychaudhuri Pert Raychaudhuri with

10. Gauge transformations (parameterized by ) Canonical Formulation.Gauge Issues. Phase space function Lagrange multipliers (direct gauge transformations not available in canonical formulation) Gauge transform both sides Obtain transformations of N and Na In general Lemma

11. Pert Friedmann Pert S-T Einstein Pert Raychaudhuri (x2) Pert K-G ? Gauge invariance Hamilton’s equations gauge invariant by construction Constraint equations gauge invariant iff constraints first class Consistency(unknown functions Ψ, Φ, φGI) constraint 1 function and 2 equations Eliminate 2 functions Diag. Pert. Raychaudhuri consistent if constraint preserved under evolution not independent Both consistency and gauge invariance are guaranteed if the constraints are of first class

12. Classical Constraints & { , }PB Classical Theory Quantization Quantum Operators & [ , ] Quantum variables: expectation values, spreads, deformations, etc. Effective Theory Classical Expressions Correction Functions x classically well behaved expressions classically diverging expressions Classical Expressions Effective approximation allows to extract predictions of the underlying quantum theory without going into consideration of quantum states Where do corrections come from? Strategy. Truncation Expectation Values (inverse triad corrections)

13. (iii) in the perturbed context, depend on and only in the combination (originate from unperturbed expression). Effective approximation allows to extract predictions of the underlying quantum theory without going into consideration of quantum states Where do corrections come from? Diffeomorphism Constraint intact Hamiltonian constraint a s ν Correction functions • depend only on triad (not extrinsic curvature), • depend on triad algebraically (no spatial derivatives),

14. Anomaly cancellation conditions . Unperturbed case. (iii) in the perturbed context, depend on and only in the combination (originate from unperturbed expression). has zero density weight Correction functions • depend only on triad (not extrinsic curvature), • depend on triad algebraically (no spatial derivatives), Cannot be satisfied with the assumptions above

15. Anomaly cancellation conditions . Counterterms. 2. Introduce counterterms in the perturbed Hamiltonian constraint insert terms, of the same structure as already there, multiplied by background dependent coefficients: 3. All counterterm coefficients → 0, as → 0. Neglect terms quadratic in counterterm coefficients. • Relax assumptions 4. Diffeomorphism constraint remains unchanged. 5. Impose conditions on the coefficients by requiring anomaly cancellation.

16. Gauge Invariant Equations of Motion. BG Friedmann BG Raychaudhuri BG Klein-Gordon

17. Gauge Invariant Equations of Motion. Pert Friedmann Pert S-T Einstein Pert Raychaudhuri Pert K_G (f, f1, f3, g1 – counterterm coefficients fixed by anomaly free conditions)

18. Summary. • Canonical perturbation theory • Gauge cannot be fixed prior to deriving EoM • cannot be anomaly free • 3. Non-trivial deformation of effective constraints • 4. Anomaly free version of quantum corrected constraints • 5. Gauge invariant equations of motion • 6. Effects below Planck density. Non-conservation of power.