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

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

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.

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

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

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

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

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

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

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

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

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

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

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.