Loop Quantum Cosmology corrections to in fl ationary models

1. Loop Quantum Cosmology corrections to inflationarymodels Michał Artymowski Loop Quantum Cosmology holonomycorrections to inflationarymodels University of Warsaw In colaborationwith L. Szulc and Z. Lalak Warszawa 15-06-2009 Michał Artymowski, University of Warsaw 2009-06-15

2. Outline Loop Quantum Cosmology corrections to inflationarymodels Phenomenology of loopcorrections • Backgroudevolutioninthe FRW cosmologywiththeholonomyloopcorrection • Initialconditions for theacceleratedexpansion • Evolution of thescalarmetricperturbations • Effectivecomovingsonichorizon • Thepower spectrum of theinitial energy densityperturbations Michał Artymowski, University of Warsaw 2009-06-15

3. Theloopcorrection and Ashtekarvariables Loop Quantum Cosmology corrections to inflationarymodels Ashtekarvariablesinthe FRW Universe: The hamiltonian givesus Friedmann equations Theparallel transport aroundtheloopchangesthevector. If we wouldshrink theloop to thesmallestpossiblesize we wouldgettheelementarycorrection Michał Artymowski, University of Warsaw 2009-06-15

4. Quantum of length Loop Quantum Cosmology corrections to inflationarymodels By consideringtheloop quantum gravitymodifications to the we get , whereisthe quantum of length. Theholonomyloopcorrectiondoes not changes p! Thisis an extremelyimportantvariable! No specificvalue of j chosen by nature! Critical (maximal) energy density of theUniverse For and big values of j we have Michał Artymowski, University of Warsaw 2009-06-15

5. Friedmann equations Loop Quantum Cosmology corrections to inflationarymodels Friedmann equations normal FRW Effectivevariables Michał Artymowski, University of Warsaw 2009-06-15

6. Initialconditions for inflation Loop Quantum Cosmology corrections to inflationarymodels Acceleratedexpansionalwaysappearswhen Initialconditions for theacceleratedexpansion much more natural thaninthe standard FRW but we have to counte-foldsfromthe moment of anyway Michał Artymowski, University of Warsaw 2009-06-15

7. Perturbationsin LQC Loop Quantum Cosmology corrections to inflationarymodels We canwrite Hamilton equations for , where We consider k 0 so perturbationsarefunctions of time only. We do not haveanyanisotropicpressure and fromperturbated Friedmann equations we have For theadiabaticperturbations we obtain Thisequationisalmoustidenticalwiththe one fromthe standard FRW. Perturbationsarefrozenoutsidetheeffectivesonichorizon. Michał Artymowski, University of Warsaw 2009-06-15

8. Theeffectivespeed of sound Loop Quantum Cosmology corrections to inflationarymodels In LQC theeffectivespeed of soundbecomesinfinite for No conservedinformationleftoverfromthe period Theeffective Big Bang scenario! is not a physicalvelocity!!! Michał Artymowski, University of Warsaw 2009-06-15

9. Thepower spectrum Loop Quantum Cosmology corrections to inflationarymodels Equations for theinflaton and it’sperturbationare not changed by theloopcorrection whereistheinflatonperturbation For theslow-rollapproximation , Thepower spectrum of thecurvature perturbationsisthenin form of Fromthe COBE normalization we know, that if we want to avoidfinetuning we need to have Thisgivesusthe limit for j. For big values of j Thespectralindexischanged by the LQC to Michał Artymowski, University of Warsaw 2009-06-15

10. Conclusions Loop Quantum Cosmology corrections to inflationarymodels • FRW Universeinthelow energy limit for LQC • No informationaboutthescalarperturbationscan cross the energy regimeuntouched • We can limit the quantum of length by the COBE normalization • More models fit the data for strong LQC holonomyeffects Michał Artymowski, University of Warsaw 2009-06-15