Second Order Perturbations During Inflation Beyond Slow-roll

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

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

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!⇒ Use Gauge Invariant Variables

First order transformation xµ → xµ + ξ µbetween gauges µ ξ1 = (α1, β1,i + γ1) i + T1 = T1 + £ξ1 T0 ⇓ δϕ1 = δϕ1 + ϕ0α1

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

Perturbed ﬂat FRW metric at ﬁrst order g00 = −a2 (1 + 2φ1 ) g0i = a2 (B1,i ) gij = a2 (1 − 2ψ)δij + 2E1,ij Bardeen 1980

Choosing a gauge Longitudinal: zero shear Comoving: zero 3-velocity Flat: zero curvature Uniform density: zero energy density ...

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

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)

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

δGµν = 8πGδTµν ⇓ Eqs of Motion

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.

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

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

resultsSecond Order Perturbations During Inﬂation Beyond Slow-roll, Huston & Malik, arXiv:1103.0912 2nd order equations: Malik, arXiv:astro-ph/0610864

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.

Breaking Slow Roll 2 0 −2 ηV −4 −6 Step Potential Bump Potential Standard Quadratic Potential −8 57 56 55 54 53 Nend − N

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

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

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

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

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

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

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

code():First Order Numerical Reviews: Salopek et al. PRD40 1753, Martin & Ringeval a-ph/0605367

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

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)

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

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

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 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

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

Other interesting non-linear processes Vorticity generation (Half-day Vorticity meeting in RAS 14th July) Magnetic ﬁeld generation 2nd order Gravitational waves

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)

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

Second Order Perturbations During Inflation Beyond Slow-roll

by Ian Huston on Jul 21, 2011

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Second Order Perturbations During Inflation Beyond Slow-roll — Presentation Transcript