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Second Order Perturbations During         Inflation Beyond Slow-roll                           Ian Huston           Astrono...
Outline    1        Perturbation Theory        1st and 2nd Order Perturbations, Gauge Invariance    2        Our Results  ...
Outline    1        Perturbation Theory        1st and 2nd Order Perturbations, Gauge Invariance    2        Our Results  ...
Outline    1        Perturbation Theory        1st and 2nd Order Perturbations, Gauge Invariance    2        Our Results  ...
Outline    1        Perturbation Theory        1st and 2nd Order Perturbations, Gauge Invariance    2        Our Results  ...
Faucher-Gigure et al., Science 2008
perturbations         Long review: Malik & Wands 0809.4944 Short technical review: Malik & Matravers 0804.3276
Separate quantities intobackground andperturbation.
ϕ(η, x) = ϕ0(η) + δϕ(η, x)                  1                + δϕ2(η, x)                  2
ϕ(η, x) = ϕ0(η) + δϕ1(η, x)                  1                + δϕ2(η, x)                  2                + ...
Gauge Choice Required   Background split not covariant   Many possible descriptions   Should give same physical answers!⇒ ...
Gauge Choice Required   Background split not covariant   Many possible descriptions   Should give same physical answers!⇒ ...
First order transformation                                 xµ → xµ + ξ µbetween gauges                 µ                ξ1...
Perturbed flat FRW metric at first order  g00 = −a2 (1 + 2φ1 )  g0i = a2 (B1,i − S1i )  gij = a2 (1 − 2ψ)γij + 2E1,ij + 2F1(...
Perturbed flat FRW metric at first order  g00 = −a2 (1 + 2φ1 )  g0i = a2 (B1,i − S1i )  gij = a2 (1 − 2ψ)γij + 2E1,ij + 2F1(...
Perturbed flat FRW metric at first order         g00 = −a2 (1 + 2φ1 )          g0i = a2 (B1,i )          gij = a2 (1 − 2ψ)δi...
Choosing a gauge   Longitudinal: zero shear   Comoving: zero 3-velocity   Flat: zero curvature   Uniform density: zero ene...
Example for Flat gauge:Metric transformation:               ψ1 = ψ1 − Hα1Flat gauge:                              α1 = ψ1 ...
Well-known gauge invariant variables       ζ = ψ1 + H δρ1 Curvature perturbation on uniform                  ρ0           ...
1  ϕ(η, x) = ϕ0(η) + δϕ1(η, x) + δϕ2(η, x)                               2Increasing complexity at second order:     Terms...
δGµν = 8πGδTµν       ⇓ Eqs of Motion
ϕ0 + 2Hϕ0 + a2 V,ϕ = 0        δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1                              =0        δϕ2 + 2Hδϕ2 + k 2 δ...
ϕ0 + 2Hϕ0 + a2 V,ϕ = 0        δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1                              =0        δϕ2 + 2Hδϕ2 + k 2 δ...
ϕ0 + 2Hϕ0 + a2 V,ϕ = 0        δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1                              =0        δϕ2 + 2Hδϕ2 + k 2 δ...
What have perturbations ever done for us?Can use curvature perturbation ζ (conserved on large scales)to link observations ...
Other Approaches:   δN formalism   Lyth, Malik, Sasaki a-ph/0411220, etc.   In-In formalism   Maldacena a-ph/0210603, etc....
resultsSecond Order Perturbations During Inflation Beyond Slow-roll,             Huston & Malik, arXiv:1103.0912     2nd or...
Bump Potential                     1                       ϕ − ϕb             Vb (ϕ) = m2 ϕ2 1 + c sech                   ...
Breaking Slow Roll             2             0            −2       ηV            −4            −6        Step Potential   ...
First Order Power Spectrum                  10−2                                          Full Bump Potential             ...
First Order Power Spectrum                         ×10−5                   3.1      −1/2      k 3/2 |δϕ1 |/MPL            ...
Source term S                δϕ2 (k i ) + 2Hδϕ2 (k i ) + Mδϕ2 (k i ) = S(k i )              10−1                          ...
Second order perturbation δϕ2                                        Full Bump Potential                                  ...
Second order perturbation δϕ2                        ×10−7                                Full Bump Potential             ...
Features Inside and Outside the Horizon               10−5                   Sub-Horizon Bump                             ...
Features Inside and Outside the Horizon                      1.04                                          Sub-Horizon Bum...
code():First Order Numerical Reviews: Salopek et al. PRD40 1753,                               Martin & Ringeval a-ph/0605...
Download at http://pyflation.ianhuston.netPapers: arXiv:1103.0912, 0907.2917    Uses Python & Numpy with compiled parts    ...
Pyflation uses Python       Quick and easy development       Boost performance using Cython or linking C/Fortran libs      ...
Pyflation is Open Source  Pyflation is released under the (modified) BSD-license.  Benefits of open source code for scientific ...
Pyflation is parallelisable           k j qj δϕ1(q i)δϕ1(k i − q i)d3q      Numerically intensive calculation      Can be e...
Progress      Single field slow roll      Single field full equation      Multi-field calculation (underway)
Implementation  Four Stages:    1 Run background system to find end of inflation    2 Run first order system for range of wav...
Future Plans      Three-point function of δϕ      Using Green’s function solution from Seery, Malik, Lyth      arXiv:0802....
Future Plans      Three-point function of δϕ      Using Green’s function solution from Seery, Malik, Lyth      arXiv:0802....
Future Plans      Three-point function of δϕ      Using Green’s function solution from Seery, Malik, Lyth      arXiv:0802....
Other interesting non-linear processes      Vorticity generation      (Half-day Vorticity meeting in RAS 14th July)      M...
Summary    Perturbation theory extends beyond linear order    New phenomena and observables at higher    orders    Second ...
i            i     2      i      2        8πG                                          2 8πG              iδϕ2 (k ) + 2Hδϕ...
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Second Order Perturbations During Inflation Beyond Slow-roll

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This is a talk I gave at the University of Sussex in June 2011. It outlines the newly released numerical code Pyflation and the results published in arXiv:1103.0912.

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Second Order Perturbations During Inflation Beyond Slow-roll

  1. 1. Second Order Perturbations During Inflation Beyond Slow-roll Ian Huston Astronomy Unit, Queen Mary, University of LondonIH, K.A.Malik, arXiv:1103.0912 and 0907.2917 (JCAP 0909:019) Software available at http://pyflation.ianhuston.net
  2. 2. Outline 1 Perturbation Theory 1st and 2nd Order Perturbations, Gauge Invariance 2 Our Results Source term and Second Order results for feature models 3 Our Code Implementation, properties and future goals
  3. 3. Outline 1 Perturbation Theory 1st and 2nd Order Perturbations, Gauge Invariance 2 Our Results Source term and Second Order results for feature models 3 Our Code Implementation, properties and future goals
  4. 4. Outline 1 Perturbation Theory 1st and 2nd Order Perturbations, Gauge Invariance 2 Our Results Source term and Second Order results for feature models 3 Our Code Implementation, properties and future goals
  5. 5. Outline 1 Perturbation Theory 1st and 2nd Order Perturbations, Gauge Invariance 2 Our Results Source term and Second Order results for feature models 3 Our Code Implementation, properties and future goals
  6. 6. Faucher-Gigure et al., Science 2008
  7. 7. perturbations Long review: Malik & Wands 0809.4944 Short technical review: Malik & Matravers 0804.3276
  8. 8. Separate quantities intobackground andperturbation.
  9. 9. ϕ(η, x) = ϕ0(η) + δϕ(η, x) 1 + δϕ2(η, x) 2
  10. 10. ϕ(η, x) = ϕ0(η) + δϕ1(η, x) 1 + δϕ2(η, x) 2 + ...
  11. 11. Gauge Choice Required Background split not covariant Many possible descriptions Should give same physical answers!⇒ Use Gauge Invariant Variables
  12. 12. Gauge Choice Required Background split not covariant Many possible descriptions Should give same physical answers!⇒ Use Gauge Invariant Variables
  13. 13. First order transformation xµ → xµ + ξ µbetween gauges µ ξ1 = (α1, β1,i + γ1) i + T1 = T1 + £ξ1 T0 ⇓ δϕ1 = δϕ1 + ϕ0α1
  14. 14. Perturbed flat FRW metric at first order g00 = −a2 (1 + 2φ1 ) g0i = a2 (B1,i − S1i ) gij = a2 (1 − 2ψ)γij + 2E1,ij + 2F1(i,j) + h1ij Bardeen 1980
  15. 15. Perturbed flat FRW metric at first order g00 = −a2 (1 + 2φ1 ) g0i = a2 (B1,i − S1i ) gij = a2 (1 − 2ψ)γij + 2E1,ij + 2F1(i,j) + h1ij Bardeen 1980
  16. 16. Perturbed flat FRW metric at first order g00 = −a2 (1 + 2φ1 ) g0i = a2 (B1,i ) gij = a2 (1 − 2ψ)δij + 2E1,ij Bardeen 1980
  17. 17. Choosing a gauge Longitudinal: zero shear Comoving: zero 3-velocity Flat: zero curvature Uniform density: zero energy density ...
  18. 18. Example for Flat gauge:Metric transformation: ψ1 = ψ1 − Hα1Flat gauge: α1 = ψ1 /HScalar transformation: δϕ1 = δϕ1 + ϕ0 α1Result ψ1 δϕ1 flat = δϕ1 + ϕ0 H Sasaki 1986, Mukhanov 1988
  19. 19. Well-known gauge invariant variables ζ = ψ1 + H δρ1 Curvature perturbation on uniform ρ0 density hypersurfacesR = ψ1 − H(v1 + B1 ) Curvature perturbation on comoving hypersurfacesΨ = ψ1 − H(B1 − E1 ) Curvature perturbation on zero shear hypersurfaces (longitudinal gauge)
  20. 20. 1 ϕ(η, x) = ϕ0(η) + δϕ1(η, x) + δϕ2(η, x) 2Increasing complexity at second order: Terms quadratic in first order quantities Coupling of different perturbation types “True” second order quantities still decouple
  21. 21. δGµν = 8πGδTµν ⇓ Eqs of Motion
  22. 22. ϕ0 + 2Hϕ0 + a2 V,ϕ = 0 δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1 =0 δϕ2 + 2Hδϕ2 + k 2 δϕ2 + a2 M2 δϕ2 = S(δϕ1 , δϕ1 )Malik, JCAP 0703 (2007) 004, arXiv:astro-ph/0610864.
  23. 23. ϕ0 + 2Hϕ0 + a2 V,ϕ = 0 δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1 =0 δϕ2 + 2Hδϕ2 + k 2 δϕ2 + a2 M2 δϕ2 = S(δϕ1 , δϕ1 )Malik, JCAP 0703 (2007) 004, arXiv:astro-ph/0610864.
  24. 24. ϕ0 + 2Hϕ0 + a2 V,ϕ = 0 δϕ1 + 2Hδϕ1 + k 2 δϕ1 + a2 M1 δϕ1 =0 δϕ2 + 2Hδϕ2 + k 2 δϕ2 + a2 M2 δϕ2 = S(δϕ1 , δϕ1 )Malik, JCAP 0703 (2007) 004, arXiv:astro-ph/0610864.
  25. 25. What have perturbations ever done for us?Can use curvature perturbation ζ (conserved on large scales)to link observations with primordial origins.At different orders have different observables and differentphenomena: First Order (linear) =⇒ ζ 2 Power Spectrum ζ 3 Non-Gaussianity Second Order =⇒ Vorticity Other non-linear effects
  26. 26. 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 Generalised Slow Roll Stewart a-ph/0110322, Adshead et al. 1102.3435
  27. 27. resultsSecond Order Perturbations During Inflation Beyond Slow-roll, Huston & Malik, arXiv:1103.0912 2nd order equations: Malik, arXiv:astro-ph/0610864
  28. 28. Bump Potential 1 ϕ − ϕb Vb (ϕ) = m2 ϕ2 1 + c sech 2 d Chen et al. arXiv:0801.3295 etc. Transient breaking of slow roll around feature Asymptotes to quadratic potential away from feature Demonstrated step potential in paper Plots show result for WMAP pivot scale. X-axis is efolds remaining until end of inflation.
  29. 29. Breaking Slow Roll 2 0 −2 ηV −4 −6 Step Potential Bump Potential Standard Quadratic Potential −8 57 56 55 54 53 Nend − N
  30. 30. First Order Power Spectrum 10−2 Full Bump Potential Half Bump Potential Zero Bump Potential −1/2 10−3 k 3/2 |δϕ1 |/MPL 10−4 10−5 60 50 40 30 20 10 0 Nend − N
  31. 31. First Order Power Spectrum ×10−5 3.1 −1/2 k 3/2 |δϕ1 |/MPL 3.0 2.9 2.8 Full Bump Potential Half Bump Potential Zero Bump Potential 2.7 57 56 55 54 53 Nend − N
  32. 32. Source term S 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 −2 |S|/MPL 10−7 10−9 10−11 10−13 10−15 60 50 40 30 20 10 0 Nend − N
  33. 33. Second order perturbation δϕ2 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
  34. 34. Second order perturbation δϕ2 ×10−7 Full Bump Potential 2.60 Half Bump Potential 2.55 Zero Bump Potential −2 |δϕ2 (k)|/MPL 2.50 2.45 2.40 2.35 2.30 2.25 57 56 55 54 53 Nend − N
  35. 35. 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
  36. 36. Features Inside and Outside the Horizon 1.04 Sub-Horizon Bump 1.02 Super-Horizon Bump Standard Quadratic Potential |δϕ2 (k)|/|δϕ2quad | 1.00 0.98 0.96 0.94 0.92 0.90 70 60 50 40 30 20 10 0 Nend − N
  37. 37. code():First Order Numerical Reviews: Salopek et al. PRD40 1753, Martin & Ringeval a-ph/0605367
  38. 38. Download at http://pyflation.ianhuston.netPapers: arXiv:1103.0912, 0907.2917 Uses Python & Numpy with compiled parts Source calculation is parallelisable Code is Open Source
  39. 39. Pyflation uses Python Quick and easy development Boost performance using Cython or linking C/Fortran libs Open Source (can see implementation) One easy way to get started: Enthought Python Distribution http://www.enthought.com (free for academic use)
  40. 40. Pyflation is Open Source Pyflation is released under the (modified) BSD-license. Benefits of open source code for scientific projects: Source code is available for inspection and testing Code can be modified and re-used Guaranteed to remain freely accessible
  41. 41. Pyflation is parallelisable k j qj δϕ1(q i)δϕ1(k i − q i)d3q Numerically intensive calculation Can be easily parallelised by timestep Can also single out wavenumber of interest
  42. 42. Progress Single field slow roll Single field full equation Multi-field calculation (underway)
  43. 43. Implementation Four Stages: 1 Run background system to find end of inflation 2 Run first order system for range of wavemodes 3 Calculate source term convolution integral 4 Run second order system with source term Paper plots created with Results are saved in HDF5 files Matplotlib
  44. 44. Future Plans Three-point function of δϕ Using Green’s function solution from Seery, Malik, Lyth arXiv:0802.0588 Multi-field equation Could check δN predictions, ζ conservation etc. Tensor & Vorticity similarities Similar equations of motion in other non-linear processes
  45. 45. Future Plans Three-point function of δϕ Using Green’s function solution from Seery, Malik, Lyth arXiv:0802.0588 Multi-field equation Could check δN predictions, ζ conservation etc. Tensor & Vorticity similarities Similar equations of motion in other non-linear processes
  46. 46. Future Plans Three-point function of δϕ Using Green’s function solution from Seery, Malik, Lyth arXiv:0802.0588 Multi-field equation Could check δN predictions, ζ conservation etc. Tensor & Vorticity similarities Similar equations of motion in other non-linear processes
  47. 47. Other interesting non-linear processes Vorticity generation (Half-day Vorticity meeting in RAS 14th July) Magnetic field generation 2nd order Gravitational waves
  48. 48. Summary Perturbation theory extends beyond linear order New phenomena and observables at higher orders Second Order calculation intensive but possible Code available now (http://pyflation.ianhuston.net)
  49. 49. 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 ) H H 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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