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Second Order Perturbations During    Inflation Beyond Slow-roll                       Ian Huston        Astronomy Unit, Que...
Faucher-Gigure et al., Science 2008
perturbations        Long review: Malik & Wands 0809.4944Short technical review: Malik & Matravers 0804.3276
Separate quantities intobackground andperturbation.
ϕ(t, x) = ϕ0(t) + δϕ1(t, x)                  1                + δϕ2(t, x)                  2                + ...
δGµν = 8πGδTµν       ⇓ Eqs of Motion
code():      Papers: 1103.0912, 0907.2917Software: http://pyflation.ianhuston.net  2nd order equations:   Malik astro-ph/0...
Non-linear processes:   Non-Gaussianity of CMB   Vorticity generation (See Adam’s poster)   Magnetic field generation   2nd...
Other Approaches:   δN formalism   Lyth, Malik, Sasaki a-ph/0411220, etc.   In-In formalism   Maldacena a-ph/0210603, etc....
pyflation():     python & numpy     parallel     open sourceFollowing Salopek et al. PRD40 1753, Martin &Ringeval a-ph/060...
Single field slow rollSingle field full equationMulti-field calculation
k j qj δϕ1(q i)δϕ1(k i − q i)d3q
Bump potential                              1                 ϕ − ϕb                      Vb (ϕ) = m2 ϕ2 1 + c sech       ...
Source term                    δϕ2 (k i ) + 2Hδϕ2 (k i ) + Mδϕ2 (k i ) = S(k i )              10−1                        ...
Second order perturbation                                                Full Bump Potential                              ...
Second order perturbation                       ×10−7                               Full Bump Potential                2.6...
Features Inside and Outside the Horizon              10−5                      Sub-Horizon Bump                           ...
Features Inside and Outside the Horizon                     1.04                                             Sub-Horizon B...
Future Plans     Three-point function of δϕ     Multi-field equation     Tensor & Vorticity similarities
Summary    Perturbations seed structure    Non-linear regime observationally    interesting    Numerically intensive calcu...
i            i     2      i     2       8πG                                                2 8πG               iδϕ2 (k ) +...
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Second Order Perturbations - National Astronomy Meeting 2011

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This is a short talk I gave at the National Astronomy Meeting 2011 in Llandudno, Wales.

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Second Order Perturbations - National Astronomy Meeting 2011

  1. 1. Second Order Perturbations During Inflation Beyond Slow-roll Ian Huston Astronomy Unit, Queen Mary, University of London In Collaboration with Karim Malik (QMUL) arXiv:1103.0912 and 0907.2917 (JCAP 0909:019) Software available at http://pyflation.ianhuston.net
  2. 2. Faucher-Gigure et al., Science 2008
  3. 3. perturbations Long review: Malik & Wands 0809.4944Short technical review: Malik & Matravers 0804.3276
  4. 4. Separate quantities intobackground andperturbation.
  5. 5. ϕ(t, x) = ϕ0(t) + δϕ1(t, x) 1 + δϕ2(t, x) 2 + ...
  6. 6. δGµν = 8πGδTµν ⇓ Eqs of Motion
  7. 7. code(): Papers: 1103.0912, 0907.2917Software: http://pyflation.ianhuston.net 2nd order equations: Malik astro-ph/0610864, JCAP
  8. 8. Non-linear processes: Non-Gaussianity of CMB Vorticity generation (See Adam’s poster) Magnetic field generation 2nd order Gravitational waves
  9. 9. Other Approaches: δN formalism Lyth, Malik, Sasaki a-ph/0411220, etc. In-In formalism Maldacena a-ph/0210603, etc. Moment transport equations Mulryne, Seery, Wesley 0909.2256, 1008.3159
  10. 10. pyflation(): python & numpy parallel open sourceFollowing Salopek et al. PRD40 1753, Martin &Ringeval a-ph/0605367
  11. 11. Single field slow rollSingle field full equationMulti-field calculation
  12. 12. k j qj δϕ1(q i)δϕ1(k i − q i)d3q
  13. 13. Bump potential 1 ϕ − ϕb Vb (ϕ) = m2 ϕ2 1 + c sech 2 d ×10−5 10−2 Full Bump Potential Half Bump Potential Zero Bump Potential 3.1 −1/2 −1/2 10−3 k 3/2 |δϕ1 |/MPL k 3/2 |δϕ1 |/MPL 3.0 2.9 10−4 2.8 Full Bump Potential Half Bump Potential Zero Bump Potential 10−5 2.7 60 50 40 30 20 10 0 57 56 55 54 53 Nend − N Nend − N
  14. 14. Source term δϕ2 (k i ) + 2Hδϕ2 (k i ) + Mδϕ2 (k i ) = S(k i ) 10−1 Full Bump Potential 10 −3 Half Bump Potential Zero Bump Potential 10−5 10−7 −2 |S|/MPL 10−9 10−11 10−13 10−15 60 50 40 30 20 10 0 Nend − N
  15. 15. Second order perturbation Full Bump Potential Half Bump Potential 10−5 Zero Bump Potential −2 |δϕ2 (k)|/MPL 10−7 10−9 60 50 40 30 20 10 0 Nend − N
  16. 16. Second order perturbation ×10−7 Full Bump Potential 2.60 Half Bump Potential Zero Bump Potential 2.55 2.50 −2 |δϕ2 (k)|/MPL 2.45 2.40 2.35 2.30 2.25 57 56 55 54 53 Nend − N
  17. 17. Features Inside and Outside the Horizon 10−5 Sub-Horizon Bump Super-Horizon Bump Standard Quadratic Potential 10−7 −2 |S|/MPL 10−9 10−11 10−13 61 60 59 58 57 56 55 54 Nend − N
  18. 18. Features Inside and Outside the Horizon 1.04 Sub-Horizon Bump 1.02 Super-Horizon Bump Standard Quadratic Potential 1.00 |δϕ2 (k)|/|δϕ2quad | 0.98 0.96 0.94 0.92 0.90 70 60 50 40 30 20 10 0 Nend − N
  19. 19. Future Plans Three-point function of δϕ Multi-field equation Tensor & Vorticity similarities
  20. 20. Summary Perturbations seed structure Non-linear regime observationally interesting Numerically intensive calculation Code available now (http://pyflation.ianhuston.net)
  21. 21. i i 2 i 2 8πG 2 8πG iδϕ2 (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

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