1. Inhomogeneities in Loop Cosmology Mikhail Kagan Institute for Gravitational Physics and Geometry, Pennsylvania State University in collaboration with M. Bojowald, P. Singh (IGPG, Penn State) H.H.Hernandez, A. Skirzewski (Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany)
Thursday, November 15, 2018Thursday, November 15, 2018

2. Outline Motivation Classical description Canonical formulation
a) Quantization
b) Correction functions
c) Effective Equations
Implications
Summary

3. Motivation. Test robustness of results of homogeneous and isotropic
Loop Quantum Cosmology.
Evolution of inhomogeneities is expected to explain cosmological structure formation and lead observable
results.

4. Lagrangean Formulation. Background metric.
Action
Matter
Gravity
homogeneity
isotropy
Friedman equation
Klein-Gordon equation
Raychaudhuri equation

5. Lagrangean Formulation. Perturbations.
perturbed
metric
(scalar mode, longitudinal gauge)
Einstein Equations
Klein-Gordon Equation

6. Canonical Formulation. Basic variables.
Matter
Scalar field
Field momentum
Poisson brackets
Gravity
(densitized) Triad
Ashtekar connection
Spin connection
Extrinsic curvature
average quantities
Immirzi parameter

7. Canonical Formulation. Constraints.
Hamiltonian
Diffeomorphism
(vector)
Gravity
Matter
Total

8. 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 identification

9. Canonical Formulation. Constraints.
Hamiltonian
Diffeomorphism
(vector)
Gravity
Matter
Total

10. Quantization. Correction functions.
Sources of corrections: inverse powers of triad
Modified constraints: a b 2 D s
Typical behavior of
correction functions: D

11. Quantization. Effective EoM.
a'p a ab
Pert Friedmann
2a'p a
Pert S-T Einstein
2a'p 3a
4a'pb ab
a'p a
b - 1
5a'p
2a''p2
Pert Raychaudhuri
D''p2 D
D'p 2D s
Pert K_G
classically 0, 1

12. Implications. Newton’s potential.
Pert S-T Einstein
Pert Friedmann
assume perfect fluid
Corrected Poisson Equation ab a k2
Length Scale
a'p
a3b k2
as a(p)~1+c(lP/p)n, (c, n>0) 2 _
so |a'p|=n(a -1)~(lP/p)n
Green’s Function k
Within one Hubble Radius k
a'p _
classically 0, 1

13. Implications. Power spectrum.
BG, Pert Raychaudhuri
BG, Pert Friedmann
assume perfect fluid
(P = wr) e3 e1 e2
where e3 = -2ap2/a < 0 _
Large-scale Fourier Modes e3
Two Classical Modes
decaying (l+ < 0)
const (l_=0)
With Quantum Corrections
decaying (l+ < 0)
growing (l_≈ -e3/n > 0)
(l_- mode describes measure of inhomogeneity)
classically 0, 1

14. Summary. Formalism for canonical treatment of inhomogeneities.
Now correction functions depend on p(x).
Effective equations for cosmological perturbations.
Quantum corrections arise on large scales:
a) Newton’s potential is modified by a factor smaller than one,
which can be interpreted as small repulsive quantum contribution.
b) Cosmological modes evolve differently,
resulting in non-conservation of curvature perturbations.
5. Results can be generalized to describe vector & tensor modes.