Cosmological Perturbations and Numerical Simulations
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The document discusses cosmological perturbations and numerical simulations, focusing on the mathematical treatment of perturbations in a Friedmann-Robertson-Walker (FRW) metric. It reviews various approaches to analyzing these perturbations, including gauge choices and non-Gaussian signals, and mentions simulation results that inform the understanding of inflationary models. Overall, the work presents a framework for investigating the second-order perturbations needed for accurate predictions in cosmological contexts.
![Perturbed FRW metric
g00 = −a2(1 + 2φ1) ,
g0i = a2B1i ,
gij = a2 [δij + 2C1ij ] .](https://image.slidesharecdn.com/qmul2010-100325081600-phpapp01/85/Cosmological-Perturbations-and-Numerical-Simulations-10-320.jpg)
Cosmological Perturbations and Numerical Simulations
- 1. Cosmological Perturbations and Numerical Simulations Ian Huston Astronomy Unit 24th March 2010 arXiv:0907.2917, JCAP 0909:019
- 2. perturbations Long review: Malik & Wands 0809.4944 Short technical review: Malik & Matravers 0804.3276
- 4. T(η, xi) = T0(η) + δT(η, xi) ∞ n i δT(η, x ) = δTn(η, xi) n=1 n! 1 ϕ = ϕ0 + δϕ1 + δϕ2 + . . . 2
- 5. T(η, xi) = T0(η) + δT(η, xi) ∞ n i δT(η, x ) = δTn(η, xi) n=1 n! 1 ϕ = ϕ0 + δϕ1 + δϕ2 + . . . 2
- 6. Gauges Background split not covariant Many possible descriptions Should give same physical answers!
- 9. First order transformation µ ξ1 = (α1, β1,i + γ1) i ⇓ δϕ1 = δϕ1 + ϕ0α1
- 10. Perturbed FRW metric g00 = −a2(1 + 2φ1) , g0i = a2B1i , gij = a2 [δij + 2C1ij ] .
- 11. Choosing a gauge Longitudinal: zero shear Comoving: zero 3-velocity Flat: zero curvature Uniform density: zero energy density ...
- 12. δGµν = 8πGδTµν ⇓ Eqs of Motion
- 13. non-Gaussianity Some reviews: Chen 1002.1416, Senatore et al. 0905.3746
- 14. Sim 1 Simulations from Ligouri et al, PRD (2007)
- 15. Sim 2 Simulations from Ligouri et al, PRD (2007)
- 16. Gaussian fields: All information in ζ(k1 )ζ(k2 ) = (2π)3 δ 3 (k1 + k2 )Pζ (k1 ) , where ζ is curvature perturbation on uniform density hypersurfaces. ζ(k1 )ζ(k2 )ζ(k3 ) = 0 , ζ 4 (ki ) = ζ(k1 )ζ(k2 ) ζ(k3 )ζ(k4 ) + ζ(k2 )ζ(k3 ) ζ(k4 )ζ(k1 ) + ζ(k1 )ζ(k3 ) ζ(k2 )ζ(k4 ) .
- 17. Bispectrum: ζ(k1 )ζ(k2 )ζ(k3 ) = (2π)3 δ 3 (k1 +k2 +k3 )B(k1 , k2 , k3 ) Local (squeezed) Equilateral
- 18. B(k1, k2, k3) fNLF (x2, x3) , xi = ki/k1 , 1 − x2 ≤ x3 ≤ x2 . Local 1 Higher Deriv. 1 0.9 x2 0.9 x2 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 8 1 6 0.75 4 F x2 , x3 0.5 F x2 , x3 2 0.25 0 0 0.2 0.4 0.2 0.6 0.4 0.8 0.6 1 0.8 1 x3 x3 Babich et al. astro-ph/0405356
- 19. WMAP7 bounds (95% CL) loc −10 < fNL < 74 loc fNL > 1 rules out ALL single field inflationary models.
- 20. WMAP7 bounds (95% CL) loc −10 < fNL < 74 loc fNL > 1 rules out ALL single field inflationary models.
- 21. One way of getting local fNL ζ(x) = ζL (x) + 3 fNL ζL (x) 5 loc 2 ∆T 1 loc − ζ, fNL > 0 T 5 ⇓ ∆T < ∆TL
- 22. Sim 1: fNL = 1000 Simulations from Ligouri et al, PRD (2007)
- 23. Sim 2: fNL = 0 Simulations from Ligouri et al, PRD (2007)
- 24. code(): Paper: Huston & Malik 0907.2917, JCAP 2nd order equations: Malik astro-ph/0610864, JCAP
- 25. Approaches: δN formalism Moment transport equations Field Equations
- 26. 1 ϕ = ϕ0 + δϕ1 + δϕ2 2
- 27. i i 2 i 2 8πG 2 8πG i δϕ2 (k ) + 2Hδϕ2 (k ) + k δϕ2 (k ) + a V,ϕϕ + 2ϕ0 V,ϕ + (ϕ0 ) V0 δϕ2 (k ) H H 1 3 3 3 i i i 16πG i i 2 i i + d pd qδ (k − p − q ) Xδϕ1 (p )δϕ1 (q ) + ϕ0 a V,ϕϕ δϕ1 (p )δϕ1 (q ) (2π)3 H 8πG 2 2 i i i i + ϕ0 2a V,ϕ ϕ0 δϕ1 (p )δϕ1 (q ) + ϕ0 Xδϕ1 (p )δϕ1 (q ) H 4πG 2 ϕ X 0 i i i i i −2 Xδϕ1 (k − q )δϕ1 (q ) + ϕ0 δϕ1 (p )δϕ1 (q ) H H 4πG i i 2 8πG i i + ϕ0 δϕ1 (p )δϕ1 (q ) + a V,ϕϕϕ + ϕ0 V,ϕϕ δϕ1 (p )δϕ1 (q ) H H 1 3 3 3 i i i 8πG pk q k i i i + d pd qδ (k − p − q ) 2 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q ) (2π)3 H q2 2 16πG i i 4πG 2 ϕ 0 pi qj kj ki +p δϕ1 (p )ϕ0 δϕ1 (q ) + p q l − ϕ δϕ (ki − q i )ϕ δϕ (q i ) l 0 1 0 1 H H H k2 X 4πG 2 p q l p q m + p2 q 2 l m i i i +2 ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q ) H H k2 q 2 4πG q 2 + pl q l i i l i i + 4X δϕ1 (p )δϕ1 (q ) − ϕ0 pl q δϕ1 (p )δϕ1 (q ) H k2 4πG pl q l pm q m 2 ϕ 0 i i i i + Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q ) HH p2 q 2 ϕ0 pl q l + p2 2 i i q 2 + pl q l i i + 8πG q δϕ1 (p )δϕ1 (q ) − δϕ1 (p )δϕ1 (q ) H k2 k2 4πG 2 kj k pi pj i i i i + 2 Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) = 0 H k2 p2
- 28. Single field slow roll Single field full equation Multi-field calculation
- 29. δϕ1(q i)δϕ1(k i − q i)d3q
- 30. code(): 1000+ k modes python & numpy parallel
- 31. Four potentials ×10−9 3.0 V (ϕ) = 1 m2 ϕ2 2 2.8 V (ϕ) = 1 λϕ4 4 2 V (ϕ) =σϕ 3 2.6 V (ϕ) = U0 + 1 m2 ϕ2 2 0 PR1 2.4 2 2.2 2.0 1.8 −61 10 10−60 10−59 10−58 k/MPL
- 32. Source term 10−1 V (ϕ) = 1 m2 ϕ2 2 V (ϕ) = 1 λϕ4 4 2 10−5 V (ϕ) =σϕ 3 V (ϕ) = U0 + 1 m2 ϕ2 2 0 |S| 10−9 10−13 10−17 0 10 20 30 40 50 60 N − N init
- 33. Second order perturbation ×10−95 4 3 2 1 √1 k 2 δϕ2 0 3 2π −1 −2 −3 −4 64 63 62 61 Nend − N
- 34. Future Plans: Full single field equation Multi field equation Vector & Vorticity similarities Rework code for efficiency
- 35. Summary: Perturbations seed structure 2nd order needed for fNL Numerically intensive calculation
- 36. kmax IA (k) = dq 3 δϕ1 (q i )δϕ1 (k i − q i ) = 2π dq q 2 δϕ1 (q i )A(k i , q i ) , kmin √ √ √ √ πα2 kmax − k + kmax k + kmax + kmax IA (k) = − 3k 3 log √ + log √ √ 18k k kmin + k + kmin √ π kmin + − arctan √ 2 k − kmin − kmax 3k 2 + 8kmax 2 k + kmax − kmax − k + 14kkmax k + kmax + kmax − k + kmin 3k 2 + 8kmin 2 k + kmin + k − kmin + 14kkmin k + kmin − k − kmin .
- 37. 10−6 10−7 rel 10−8 10−9 k ∈ K1 −10 k ∈ K2 10 k ∈ K3 −61 10 10−60 10−59 10−58 10−57 k/MPL K1 = 1.9 × 10−5 , 0.039 Mpc−1 , ∆k = 3.8 × 10−5 Mpc−1 , K2 = 5.71 × 10−5 , 0.12 Mpc−1 , ∆k = 1.2 × 10−4 Mpc−1 , K3 = 9.52 × 10−5 , 0.39 Mpc−1 , ∆k = 3.8 × 10−4 Mpc−1 .