Superinflation in Loop Quantum Cosmology Nelson Nunes ITP , - PowerPoint PPT Presentation

Superinflation in Loop Quantum Cosmology Nelson Nunes ITP , University of Heidelberg • Slow-roll from LQC k = 0 and k = 1 • Superinflation in LQC • Scalar and tensor power spectrum Lidsey, Mulryne, Nunes, Tavakol (2004) Mulryne, Nunes, Tavakol, Lidsey (2004) Mulryne, Nunes (2006) Copeland, Mulryne, Nunes, Shaeri (2007) and (2008)

1. Loop Quantum Gravity Strongest candidate to a quantum theory of gravity that is non-perturbative and background independent. Based on Ashtekar’s variables which bring GR into the form of a gauge theory. • Densitized triad E a i and E a i E b i = q ab q • SU(2) connection A i a = Γ i a − γK i a Γ i a - spin connection; K i a - extrinsic curvature; γ - Barbero-Immirzi parameter. Quantization proceeds by using as basic variables holonomies, � τ i A i e a dt h e = exp a ˙ e along curves e , and fluxes, � τ i E a i n a d 2 y F = S in spacial surfaces S . Flux operators have a discrete spectrum.

2. Loop Quantum Cosmology Focuses on minisuperspace settings with finite degrees of freedom ( = homogeneous and isotropic spacetimes). 1. Inverse triad corrections: Based on the modification of the inverse scale factor below a critical scale a ∗ . 2. Holonomy corrections: Loops on which holonomies are computed have a non-vanishing minimum area. Leads to a ρ 2 modification in the Friedmann equation. These corrections lead to interesting applications: • Resolution of the initial singularity; • Increase of the viability of the onset of inflation; • Avoidance of a big crunch and oscillatory universes;

3. Key features of loop quantization A i a = c ω i a , c = γ ˙ a { c, p } = 8 πG p = a 2 , E a i = p e a i , 3 γ E a j E b π 2 √ 1 φ k F i H = 8 πG ǫ ijk √ ab + √ + det E V ( φ ) det E 2 det E We want to write this Hamiltonian in terms of holonomies h ( λ ) = exp( λ c τ i ) 1. Write Hamiltonian in terms of positive powers of the connection. This can be done in several different ways ⇒ ambiguity parameter ℓ 2. Write the connection in terms of holonomies. Need to take the trace over representation j of su(2) ⇒ ambiguity parameter j ⇒ critical scale a ∗ ;

3. Key features of loop quantization (cont.) 3. Curvature component obtained by considering holonomies around closed square loop. Area is shrunk to the minimum eigenvalue of the area µ 2 a 2 = ∆ [ ⇒ holonomy corrections ]; operator ∆ ≈ ℓ 2 ⇒ λ → ¯ µ and ¯ pl 4. Quantization proceeds by promoting triads and holonomies to operators (` a la LQG); √ 5. Find eigenvalues of inverse triad operators such as E ai E bi / det E √ and 1 / det E ; Spectrum of eigenvalues can be approximated by continuous 6. correction functions S ( a ) and D l,j ( a ) [ inverse triad corrections ]; 7. Finally, Hamiltonian looks like this: + D l,j a − 3 π 2 8 πG S a sin 2 (¯ H = − 3 µ c ) 2 + a 3 V ( φ ) φ γ 2 ¯ µ 2 8. p = { p, H} ˙ ⇒ Friedmann equation

4. Inverse volume operator d ( a ) = a − 3 Classically: a ∗ ∝ √ j ℓ pl � 2 � a d l,j ( a ) = D l ( q ) a − 3 LQC: where q = , a ∗ 1.4 semiclassical phase 1.2 D ( q ) ≈ D ⋆ a n for a ≪ a ∗ , 1 0.8 D 3/4 0.6 classical phase for a ≫ a ∗ , D ( q ) ≈ 1 0.4 0.2 0 0 1 2 3 4 5 (a/a * ) 2

5. Modified semi-classical equations 1. Modified Friedmann equation � ˙ � � � 2 ˙ φ 2 − S 2 a = S 1 H 2 ≡ D + V ( φ ) a 2 a 3 2

5. Modified semi-classical equations 1. Modified Friedmann equation � ˙ � � � 2 ˙ φ 2 − S 2 a = S 1 H 2 ≡ D + V ( φ ) a 2 a 3 2 2. Modified Klein-Gordon equation � � φ + 3 ˙ a 1 − 1 d ln D φ + D dV ¨ ˙ dφ = 0 a 3 d ln a When d ln D/d ln a > 3 : antifriction in expanding Universe and friction in contracting universe.

5. Modified semi-classical equations 1. Modified Friedmann equation � ˙ � � � 2 ˙ φ 2 − S 2 a = S 1 H 2 ≡ D + V ( φ ) a 2 a 3 2 2. Modified Klein-Gordon equation � � φ + 3 ˙ a 1 − 1 d ln D φ + D dV ¨ ˙ dφ = 0 a 3 d ln a When d ln D/d ln a > 3 : antifriction in expanding Universe and friction in contracting universe. 3. Variation of the Hubble rate H = − S ˙ φ 2 � � � � 1 − 1 d ln D d ln a − 1 d ln S + S d ln S 1 − d ln S S 2 1 ˙ d ln aV + a 2 2 D 6 6 d ln a 6 d ln a Superinflation for n + r = d ln D/d ln a + d ln S/d ln a > 6 .

6. Consequences for inflation ( k = 0 ) V( φ ) slow−roll inflation superinflation, k = 0 φ Tsujikawa and Singh (2003) 1. Super-inflation is brief; init exp( − q 15 / 4 2. φ t ∝ ˙ φ init q − 6 init ) , independent of j ; 3. φ t < 2 . 4 ℓ − 1 pl if Hubble bound ( 1 /H > ℓ pl ) is satisfied ⇒ not enough slow-roll inflation!

7. Bouncing Universe in k = +1 I / III (a) II / IV ln( ρ ) ln(a) (b) ln( ρ ) ln(a) φ 2 > V Field redshifts more rapidly than curvature term provided ˙ ( w > − 1 / 3 ). As the field moves up the potential this condition becomes more difficult to satisfy and is eventually broken. Slow-roll inflation follows.

8. Bouncing Universe in k = +1 , with self interacting potential V( φ ) slow−roll inflation superinflation, k = 1 φ 1 1 φ 2 t ∝ ˙ q 3 / 2 φ init init a 3 ∗ ⇒ larger for lower j ⇒ more e -folds.

9. The story so far... 1. Flat geometry • φ does not move high enough; • φ t independent of quantization parameter j .

9. The story so far... 1. Flat geometry • φ does not move high enough; • φ t independent of quantization parameter j . 2. Positively curved geometry • Allows oscillatory Universe; • For massless scalar field cycles are symmetric and consequently ever lasting; • Presence of a self interaction potential breaks symmetry and establishes initial conditions for inflation; • Low j results into more inflation.

Can superinflation during the semi-classical phase replace standard slow-roll inflation?

Can superinflation during the semi-classical phase replace standard slow-roll inflation? • Does it solve the flatness and horizon problems? • Does it give rise to a scale invariant spectrum of curvature perturbations? • Is the spectrum of gravitational waves compactible with current bounds?

10. Inflation inflation e.g. Slow-roll inflation with scalar field(s). Structure originates from quantum fluctuations of the field(s).

11. Superinflation s u p e r i n f l a t i o n e.g. Ekpyrotic/cyclic universe, phantom field.

11. Superinflation s u p e r i n f l a t i o n e.g. Ekpyrotic/cyclic universe, phantom field, LQC effects.

12. Inflation, and the horizon problem ln (1/aH) inflation standard cosmology −1 k 0 k −1 −1 k e a e a 0 ln a

13. Superinflation, and the horizon problem ln (1/aH) superinflation standard cosmology −1 k 0 k −1 −1 k e a e a 0 ln a

14. Number of e -folds and the horizon problem Requirement that the scale entering the horizon today exited N e -folds before the end of inflation: � M Pl � 1 / 4 � a end H end � � � ρ end = 68 − 1 − 1 ln 2 ln 3 ln a N H N H end ρ reh � � � � a end H end a end 1. In standard inflation: ln ≈ ln ≡ N ≈ 60 a N H N a N 2. In LQC with a = ( − τ ) p and p ≪ 1 � a N � 1 /p � a end H end � = ln τ N = − 1 ln = ln pN a N H N τ end a end N ≈ − 60 p Number of e -folds of super-inflation required to solve the horizon problem can be of only a few.

15. Scaling solution (inverse triad corrections) ˙ φ 2 / (2 DV ) ≈ cnst. Scaling solution ⇔ Lidsey (2004) V = V 0 φ β a = ( − τ ) p 1.16 1.14 2 α p = 1.12 2¯ ǫ − (2 + r ) α φ 2 / (2 DV ) 1.1 � 2 � V ,φ ǫ = 1 D ¯ 1.08 - ˙ 2 S V 1.06 V = V 0 φ β 1.04 1.02 −0.5 −0.45 −0.4 −0.35 −0.3 ln a/a ∗ D ∝ a n , S ∝ a r . β = 4¯ ǫ/ ( n − r ) α > 0 , α = 1 − n/ 6 , Scaling solution is stable attractor for ¯ ǫ > 3 α 2 or β > ( n − 6) /n ∼ O (1) .

16. Perturbation equations Define effective action that gives background equations of motion � φ ′ 2 2 Da 2 − δ ij � � dτ d 3 x a 4 S = a 2 ∂ i φ∂ j φ − V Perturb field φ = φ b + δφ .

16. Perturbation equations Define effective action that gives background equations of motion � φ ′ 2 2 Da 2 − δ ij � � dτ d 3 x a 4 S = a 2 ∂ i φ∂ j φ − V Perturb field φ = φ b + δφ . √ Define u = aδφ/ D and expand in plane waves: d 3 k � � � a † a k + ω ∗ e − i k . x u ( τ, x) = ˆ ω k ( τ )ˆ k ( τ )ˆ − k (2 π ) 3 / 2

16. Perturbation equations Define effective action that gives background equations of motion � φ ′ 2 2 Da 2 − δ ij � � dτ d 3 x a 4 S = a 2 ∂ i φ∂ j φ − V Perturb field φ = φ b + δφ . √ Define u = aδφ/ D and expand in plane waves: d 3 k � � � a † a k + ω ∗ e − i k . x u ( τ, x) = ˆ ω k ( τ )ˆ k ( τ )ˆ − k (2 π ) 3 / 2 Obtain equation of motion D ∗ A n ( − τ ) np k 2 + m 2 eff τ 2 � � ω ′′ k + ω k = 0 τ 2 where for the scaling solution eff τ 2 = − 2 + (3 − 2 n ) p + 1 m 2 2(6 − 2 n − n 2 ) p 2

17. General solution General normalised solution is: √ π � − τ H (1) ω k ( τ ) = | ν | ( x ) 2 | 2 + np | √ 9 − 12 p + 8 np − 12 p 2 − 4 p 2 n + 2 n 2 p 2 � Dk x ∝ aH , ν = − 2 + np √ D Define, by analogy with standard inflation, effective horizon aH or effective √ wavenumber Dk .