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 collaboration with L. Szulc and Z. Lalak Pennstate Michał Artymowski, University of Warsaw
2 Outline Phenomenology of loop corrections Introduction to cosmology and inflation Background evolution in the FRW cosmology with the holonomy loop correction Evolution of the scalar metric perturbations Effective comoving sonic horizon Power spectrum of the initial energy density perturbations Michał Artymowski, University of Warsaw
3 Clasical cosmology and inflation Let us consider the FRW Universe with k=0. From Einstein equations one obtains Friedmann equations a&& H = = - ( + 3P) H = - ( + P) 3 a 6 & To solve these equations we need to know the barotropic parameter ω, where P=ω, or we need to know the equation of motion for matter fields Inflation appears when a && > a 0 and the most popular scenarios assume ω We need P rather unnatural equation of state! Ht Then a( t) e and the comoving Hubble horizon deeases rapidly ah Michał Artymowski, University of Warsaw
4 Inflaton field The most popular model: The early Universe is dominated by the scalar field φ = φ& + V(φ) P = φ& - V(φ) to get ω we need φ << V(φ) & φ && + 3Hφ & + V = 0 Equation of motion for the inflaton Michał Artymowski, University of Warsaw
5 Initial perturbations Now we work with the inflaton field φ(t) and its perturbation δφ(x,t) ( t) P( t) P ( o) ( o) r ( t) + δ( x, t) r ( t) + δp( x, t) g g oo ij r r ( t) goo( t) + δgoo( x, t) = -( + Φ( x, t)) r r ( t) g ( t) + δg ( x, t) = a ( - Ψ( x, t))δ ij ij ij δg μν = δt μν We do not have any anisotropic pressure. Δ... 3H Ψ+ 3H Ψ + δ Ψ+ 4 H Ψ+ a Ψ = - Φ = Ψ ( 3H + H& ) Ψ = δp After Fourier transformation we have Ψ+ (4 + 3c ) H Ψ+[H+ 3H (+ c ) - k c / a ]Ψ = s s s For kcs<<ah we have the solution Ψ const Perturbations are frozen outside the comoving sonic horizon Michał Artymowski, University of Warsaw
6 Power spectra For the strong slow rollapproximation... δφ evolves like the masless scalar field δφ+ 3Hδφ+ k δφ/a = 0 Conformal time: a(η)dη=dt Ht For a( t) e we obtain η = - ah ηh δφ = e ik i kη - k - η From the equation of motion for δφ(k,η) we get ( ) Then we define the Power spectrum of δφ by ( k,η) = H k η + 4π And finally ( ) H = Ψ + δφ φ& δφ In slow roll approximation 3 k δφ ( k,η) = δφ( k, η) π Mukhanov Sasaki variable. This is something like 4 H 4π φ& V 4π we can not measure that! Michał Artymowski, University of Warsaw ε δ
7 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
8 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 + )) / 4 This is an extremely important variable! No specific value of j chosen by nature! j = 3,,, Mpl = Critical (maximal) energy density of the Universe γ l j(j+ ) j For and big values of j we have ~ /j 8πG = Michał Artymowski, University of Warsaw
9 Friedmann equations. H Friedmann equations H 3 = = ( + P) / 0 normal FRW Effective variables P 3 3 = = P Michał Artymowski, University of Warsaw
10 Perturbations in LQC ds = N dt + p(d x) We can write Hamilton equations for, where p = a ( - Ψ) N = (+ 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 Φ = Ψ. 3H Ψ+ 3H Ψ = - δ... Ψ + 4HΨ+ ( 3H + H& ) Ψ = δp For the adiabatic perturbations we obtain.. Ψ+ (4 + 3c s.. ) H Ψ+[H+ 3H (+ 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
11 The ective speed of sound In LQC the ective speed of sound becomes infinite for = cs = δ P δ = c s (+ P) / - - / 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
12 The power spectrum Equations for the inflaton and it s perturbation are not changed by the loop correction... δφ+ 3Hδφ+ k δφ/a = 0 where δφ is the inflaton perturbation (- / V' ) V For the slow roll approximation, ε = V = V( V/ ) 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 > (0 6 GeV) 4 This gives us the limit for j. For big values of j The spectral index n s ( k) - = η -6ε < 4 Mpl/j j 0 is changed by the LQC to ns loop ( k) - = ns ( k) - ( -V / ) Michał Artymowski, University of Warsaw
13 Conclusions FRW Universe in the low energy limit for LQC No information about the scalar perturbations can oss the energy regime untouched 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
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
16 Nazwa rozdziału Loop Quantum Cosmology corrections in the inflationary models Michał Artymowski, University of Warsaw
Loop Quantum Cosmology holonomy corrections to inflationary models
Michał Artymowski Loop Quantum Cosmology holonomy corrections to inflationary models University of Warsaw With colaboration with L. Szulc and Z. Lalak Oxford 4 09 008 Michał Artymowski, University of Warsaw
More informationarxiv: v1 [gr-qc] 1 Jul 2008
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