Loop Quantum Cosmology holonomy corrections to inflationary models

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1 Michał Artymowski Loop Quantum Cosmology holonomy corrections to inflationary models University of Warsaw With colaboration with L. Szulc and Z. Lalak Oxford Michał Artymowski, University of Warsaw

2 Outline Fenomenology of loop corrections Backgroud evolution in the FRW cosmology with the holonomy loop correction The evolution of the scalar metric perturbations The ective comoving sonic horizon The power spectrum of the initial energy density perturbations Michał Artymowski, University of Warsaw

3 The loop correction and Ashtekar variables. Ashtekar variables in the FRW Universe: c = γ a p = a { p} c, = The hamiltonian 3 N H = p c + H gives us Friedmann equations mat γ γ 3 The parallel transport around the loop changes the vector. If we would shrink the loop to the smallest possible size we would get the elementary correction Michał Artymowski, University of Warsaw

4 Quantum of length By considering the loop quantum gravity modifications to the c sin(cl j / p) l p j l j, where is the quantum of length. The holonomy loop correction does not changes p! c we get l j l pl(j(j + 1)) 1/ 4 This is an extremely important variable! No specific value of j chosen by nature! j = 1 3,1,,... ρ 4 3 Mpl = Critical (maximal) energy density of the Universe γ l j(j+ 1) j For and big values of j we have ~ 1/j 8πG = 1 ρ Michał Artymowski, University of Warsaw

5 Friedmann equations. H Friedmann equations H ρ ρ 1 3 ρ = 1 ρ = ( ρ + P) 1 ρ ρ/ρ 0 normal FRW Effective variables P ρ 3 ρ ρ 1 3 ρ = = P 1 ρ ρ ρ ρ Michał Artymowski, University of Warsaw

6 Perturbations ds = N dt + p(d x) We can write Hamilton equations for, where p = a (1 + Ψ) N = (1 Φ) We consider k 0 so perturbations are functions of time only. We do not have any anisotropic pressure and from perturbated friedmann equations we have. 3H Ψ+ 3H Φ = Ψ 1 Ψ = - δρ... Ψ + 4HΨ+ 3H Ψ = 1 δp For the adiabatic perturbations we obtain.. Ψ+ (4 + 3c s.. ) H Ψ+[H+ 3H (1+ c s )]Ψ = 0 This equation is almoust identical with the one from the standard FRW. Perturbations are frozen outside the ective sonic horizon. Michał Artymowski, University of Warsaw

7 The ective speed of sound In LQC the ective speed of ρ sound becomes infinite for ρ = cs = δ P δρ = c s (ρ+ P) /ρ - 1- ρ/ρ No conserved information. left over from the H > 0 period The ective Big Bang scenario! c s is not a physical velocity!!! Michał Artymowski, University of Warsaw

8 The power spectrum Equations for the inflaton and it s perturbation are not changed by the loop correction... δφ+ 3Hδφ+ [V'' + k /a ]δφ 0 where δφ is the inflaton perturbation 1 (1- ρ/ρ V' ) V For the slow roll approximation, ε = V = V(1 V/ρ ) The power spectrum of the curvature perturbations is then in form of From the COBE normalisation we know, that if we want to avoid fine tuning we need to have ρ > (10 16 GeV) 4 This gives us the limit for j. For big values of j The spectral index n s ( k) -1 = η -6ε ρ < 4 1 Mpl/j j 10 is changed by the LQC to ns loop ( k) -1 = ns ( k) -1 ( 1-V / ρ ) Michał Artymowski, University of Warsaw

9 Conclusions The FRW Universe in the low energy limit for LQC No information about the scalar perturbations can oss the energy regime untouched ρ We can limit the quantum of length by the COBE normalisation More models fit s the data for strong LQC holonomy efects Michał Artymowski, University of Warsaw

10 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw

11 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw

12 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw

13 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw

14 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw

15 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw