- 1. Second Order Perturbations During Inﬂation Beyond Slow-roll Ian Huston Astronomy Unit, Queen Mary, University of London IH, K.A.Malik, arXiv:1103.0912 and 0907.2917 (JCAP 0909:019) Software available at http://pyﬂation.ianhuston.net
- 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. 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. 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. 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. Faucher-Gigure et al., Science 2008
- 7. perturbations Long review: Malik & Wands 0809.4944 Short technical review: Malik & Matravers 0804.3276
- 8. Separate quantities into background and perturbation.
- 9. ϕ(η, x) = ϕ0(η) + δϕ(η, x) 1 + δϕ2(η, x) 2
- 10. ϕ(η, x) = ϕ0(η) + δϕ1(η, x) 1 + δϕ2(η, x) 2 + ...
- 11. Gauge Choice Required Background split not covariant Many possible descriptions Should give same physical answers! ⇒ Use Gauge Invariant Variables
- 12. Gauge Choice Required Background split not covariant Many possible descriptions Should give same physical answers! ⇒ Use Gauge Invariant Variables
- 13. First order transformation xµ → xµ + ξ µ between gauges µ ξ1 = (α1, β1,i + γ1) i + T1 = T1 + £ξ1 T0 ⇓ δϕ1 = δϕ1 + ϕ0α1
- 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. 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. 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. Choosing a gauge Longitudinal: zero shear Comoving: zero 3-velocity Flat: zero curvature Uniform density: zero energy density ...
- 18. Example for Flat gauge: Metric transformation: ψ1 = ψ1 − Hα1 Flat gauge: α1 = ψ1 /H Scalar transformation: δϕ1 = δϕ1 + ϕ0 α1 Result ψ1 δϕ1 ﬂat = δϕ1 + ϕ0 H Sasaki 1986, Mukhanov 1988
- 19. Well-known gauge invariant variables ζ = ψ1 + H δρ1 Curvature perturbation on uniform ρ0 density hypersurfaces R = ψ1 − H(v1 + B1 ) Curvature perturbation on comoving hypersurfaces Ψ = ψ1 − H(B1 − E1 ) Curvature perturbation on zero shear hypersurfaces (longitudinal gauge)
- 20. 1 ϕ(η, x) = ϕ0(η) + δϕ1(η, x) + δϕ2(η, x) 2 Increasing complexity at second order: Terms quadratic in ﬁrst order quantities Coupling of diﬀerent perturbation types “True” second order quantities still decouple
- 21. δGµν = 8πGδTµν ⇓ Eqs of Motion
- 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. 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. 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. 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 diﬀerent phenomena: First Order (linear) =⇒ ζ 2 Power Spectrum ζ 3 Non-Gaussianity Second Order =⇒ Vorticity Other non-linear eﬀects
- 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. results Second Order Perturbations During Inﬂation Beyond Slow-roll, Huston & Malik, arXiv:1103.0912 2nd order equations: Malik, arXiv:astro-ph/0610864
- 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. 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. 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. 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. 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. 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. 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. 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. 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. code(): First Order Numerical Reviews: Salopek et al. PRD40 1753, Martin & Ringeval a-ph/0605367
- 38. Download at http://pyﬂation.ianhuston.net Papers: arXiv:1103.0912, 0907.2917 Uses Python & Numpy with compiled parts Source calculation is parallelisable Code is Open Source
- 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. 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. 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. Progress Single ﬁeld slow roll Single ﬁeld full equation Multi-ﬁeld calculation (underway)
- 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. 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. 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. 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. Other interesting non-linear processes Vorticity generation (Half-day Vorticity meeting in RAS 14th July) Magnetic ﬁeld generation 2nd order Gravitational waves
- 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. 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 ) 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