1. in collaboration with M. Bojowald, G. Hossain, S. Shankaranarayanan
Cosmological perturbation theory in canonical variables Mikhail Kagan Penn State Abington and Institute for Gravitation and the Cosmos, Penn State
in collaboration with M. Bojowald, G. Hossain, S. Shankaranarayanan
Thursday, November 15, 2018Thursday, November 15, 2018

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.
Matter
Gravity
Perturbation theory
Metric:
scalar perturbations:
Matter:

4. Covariant Formulation. Gauge Issues.
Hom & Iso Universe
preferable coordinate system
Perturbations
no (obvious) preferable
coordinate system
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
Perturbed
Einstein’s Equations
Klein-Gordon Equation

7. Canonical Formulation. Constraints.
Metric ds2=gabdxadxb=-N2dt2+qab(dxa+Nadt)(dxb+Nbdt)
space-time splitting
Action (Hamiltonian+Gauss+Vector) + Poisson Brackets
constraints symplectic structure
Hamiltonian constraint
Gauss’ Constraint is solved explicitly. F is curvature of Ashtekar connection. Variables are on the next slide.
Diffeomorphism Constraint
where
Immirzi parameter

8. Canonical Formulation. Basic variables.
Matter
Poisson brackets
Scalar field
Field momentum
Gravity
Triad
Spin connection
Extrinsic curvature
Ashtekar connection
V0 will only appear in the basic variables and their symplectic structure, but not EoM
average quantities

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

10. Canonical Formulation. Gauge Issues.
Gauge transformations (parameterized by )
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. ? Gauge invariance Hamilton’s equations Constraint equations
gauge invariant by construction
Constraint equations
gauge invariant iff constraints first class
Consistency (unknown functions Ψ, Φ, φGI)
Pert Friedmann
Pert S-T Einstein
Pert Raychaudhuri (x2)
Pert K-G
constraint
1 function and
2 equations
Ensured automatically in the covariant formulation. Is not necessarily the case in the canonical one.
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. Strategy. Effective approximation Classical Theory
allows to extract predictions of the underlying quantum theory without going into consideration of quantum states
Where do corrections come from?
Strategy.
Classical Theory
Classical Constraints & { , }PB
Quantization
Quantum Operators & [ , ]
Effective Theory
Quantum variables:
expectation values, spreads, deformations, etc.
Truncation
Expectation Values
Classical Expressions
classically well
behaved expressions
classically diverging
expressions
Classical Expressions x
Correction
Functions
(inverse triad corrections)

13. Where do corrections come from?
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 ν s
Correction functions
depend only on triad (not extrinsic curvature),
depend on triad algebraically (no spatial derivatives),
(originate from unperturbed expression).
(iii) in the perturbed context, depend on and
only in the combination

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

15. Anomaly cancellation conditions . Counterterms.
Relax assumptions
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.
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.
3. Mention that the system is extremely fragile.