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Inhomogeneities in Loop Cosmology: A Study of Cosmological Structure Formation

This article explores the evolution of inhomogeneities in Loop Quantum Cosmology to elucidate cosmological structure formation and observable results. It covers the Lagrangean and Canonical formulations, quantization, correction functions, effective equations, and implications. The study aims to test the robustness of results obtained from homogeneous and isotropic models. It provides insights into how inhomogeneities impact the cosmos.

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Inhomogeneities in Loop Cosmology: A Study of Cosmological Structure Formation

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  1. Inhomogeneities in Loop CosmologyMikhail KaganInstitute for Gravitational Physics and Geometry,Pennsylvania State Universityin collaboration withM. Bojowald, P. Singh(IGPG, Penn State)H.H.Hernandez, A. Skirzewski(Max-Planck-Institut für Gravitationsphysik,Albert-Einstein-Institut, Potsdam, Germany)

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

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

  8. 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 identification

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

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

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

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

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

  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.

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