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

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

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

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

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

2δϕ2(η, x)

ϕ(η, x) = ϕ0(η) + δϕ1(η, x)

+1

2δϕ2(η, x)

+ . . .

Gauge Choice Required

Background split not covariant

Many possible descriptions

Should give same physical answers!

⇒ Use Gauge Invariant Variables

Gauge Choice Required

Background split not covariant

Many possible descriptions

Should give same physical answers!

⇒ Use Gauge Invariant Variables

First order transformationbetween gauges

xµ → xµ + ξµ

ξµ1 = (α1, βi

1, + γi1)

+

T̃1 = T1 +£ξ1T0

⇓δ̃ϕ1 = δϕ1 + ϕ′0α1

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 − 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: ψ̃1 = ψ1 −Hα1

Flat gauge: α1 = ψ1/H

Scalar transformation: δ̃ϕ1 = δϕ1 + ϕ′0α1

Result

δϕ1flat = δϕ1 + ϕ′0ψ1

HSasaki 1986, Mukhanov 1988

Well-known gauge invariant variables

ζ = ψ1 +H δρ1ρ′0

Curvature perturbation on uniform

density hypersurfaces

R = ψ1 −H(v1 +B1) Curvature perturbation on comovinghypersurfaces

Ψ = ψ1 −H(B1 − E ′1) Curvature perturbation on zero shear

hypersurfaces (longitudinal gauge)

ϕ(η, x) = ϕ0(η) + δϕ1(η, x) +1

2δϕ2(η, x)

Increasing 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

ϕ′′0 + 2Hϕ′0 + a2V,ϕ = 0

δϕ′′1 + 2Hδϕ′1 + k2δϕ1 + a2M1δϕ1

= 0

δϕ′′2 + 2Hδϕ′2 + k2δϕ2 + a2M2δϕ2

= S(δϕ1, δϕ′1)

Malik, JCAP 0703 (2007) 004, arXiv:astro-ph/0610864.

ϕ′′0 + 2Hϕ′0 + a2V,ϕ = 0

δϕ′′1 + 2Hδϕ′1 + k2δϕ1 + a2M1δϕ1

= 0

δϕ′′2 + 2Hδϕ′2 + k2δϕ2 + a2M2δϕ2

= S(δϕ1, δϕ′1)

Malik, JCAP 0703 (2007) 004, arXiv:astro-ph/0610864.

ϕ′′0 + 2Hϕ′0 + a2V,ϕ = 0

δϕ′′1 + 2Hδϕ′1 + k2δϕ1 + a2M1δϕ1

= 0

δϕ′′2 + 2Hδϕ′2 + k2δϕ2 + a2M2δϕ2

= S(δϕ1, δϕ′1)

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) =⇒ 〈ζ2〉 Power Spectrum

Second Order =⇒〈ζ3〉 Non-Gaussianity

Vorticity

Other non-linear effects

Other Approaches:

δN formalismLyth, Malik, Sasaki a-ph/0411220, etc.

In-In formalismMaldacena a-ph/0210603, etc.

Moment transport equationsMulryne, Seery, Wesley 0909.2256, 1008.3159

Generalised Slow RollStewart a-ph/0110322, Adshead et al. 1102.3435

results

Second Order Perturbations During Inflation Beyond Slow-roll,Huston & Malik, arXiv:1103.0912

2nd order equations: Malik, arXiv:astro-ph/0610864

Bump Potential

Vb(ϕ) =1

2m2ϕ2

[1 + c sech

(ϕ− ϕb

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

5354555657Nend −N

−8

−6

−4

−2

0

2η V

Step Potential

Bump Potential

Standard Quadratic Potential

First Order Power Spectrum

0102030405060Nend −N

10−5

10−4

10−3

10−2k

3/2|δϕ

1|/M

−1/2

PL

Full Bump Potential

Half Bump Potential

Zero Bump Potential

First Order Power Spectrum

5354555657Nend −N

2.7

2.8

2.9

3.0

3.1

k3/2|δϕ

1|/M

−1/2

PL

×10−5

Full Bump Potential

Half Bump Potential

Zero Bump Potential

Source term S

δϕ′′2(ki) + 2Hδϕ′

2(ki) +Mδϕ2(k

i) = S(ki)

0102030405060Nend −N

10−15

10−13

10−11

10−9

10−7

10−5

10−3

10−1

|S|/M

−2

PL

Full Bump Potential

Half Bump Potential

Zero Bump Potential

Second order perturbation δϕ2

0102030405060Nend −N

10−9

10−7

10−5

|δϕ2(k

)|/M−

2P

L

Full Bump Potential

Half Bump Potential

Zero Bump Potential

Second order perturbation δϕ2

5354555657Nend −N

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

|δϕ2(k

)|/M−

2P

L×10−7

Full Bump Potential

Half Bump Potential

Zero Bump Potential

Features Inside and Outside the Horizon

5455565758596061Nend −N

10−13

10−11

10−9

10−7

10−5

|S|/M

−2

PL

Sub-Horizon Bump

Super-Horizon Bump

Standard Quadratic Potential

Features Inside and Outside the Horizon

010203040506070Nend −N

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04|δϕ

2(k

)|/|δϕ

2quad|

Sub-Horizon Bump

Super-Horizon Bump

Standard Quadratic Potential

code():First Order Numerical Reviews: Salopek et al. PRD40 1753,

Martin & Ringeval a-ph/0605367

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

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 Distributionhttp://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

∫kjqjδϕ1(q

i)δϕ1(ki − qi)d3q

Numerically intensive calculation

Can be easily parallelised by timestep

Can also single out wavenumber of interest

Progress

2� Single field slow roll

2� Single field full equation

2 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

Results are saved in HDF5 filesPaper plots created with

Matplotlib

Future Plans

Three-point function of δϕUsing Green’s function solution from Seery, Malik, Lyth

arXiv:0802.0588

Multi-field equationCould check δN predictions, ζ conservation etc.

Tensor & Vorticity similaritiesSimilar equations of motion in other non-linear processes

Future Plans

Three-point function of δϕUsing Green’s function solution from Seery, Malik, Lyth

arXiv:0802.0588

Multi-field equationCould check δN predictions, ζ conservation etc.

Tensor & Vorticity similaritiesSimilar equations of motion in other non-linear processes

Future Plans

Three-point function of δϕUsing Green’s function solution from Seery, Malik, Lyth

arXiv:0802.0588

Multi-field equationCould check δN predictions, ζ conservation etc.

Tensor & Vorticity similaritiesSimilar 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 higherorders

Second Order calculation intensive but possible

Code available now(http://pyflation.ianhuston.net)

δϕ′′2 (k

i) + 2Hδϕ′2(k

i) + k

2δϕ2(k

i) + a

2[V,ϕϕ +

8πG

H

(2ϕ′0V,ϕ + (ϕ

′0)

2 8πG

HV0

)]δϕ2(k

i)

+1

(2π)3

∫d3pd

3qδ

3(ki − pi − qi)

{16πG

H

[Xδϕ

′1(p

i)δϕ1(q

i) + ϕ

′0a

2V,ϕϕδϕ1(p

i)δϕ1(q

i)]

+

(8πG

H

)2ϕ′0

[2a

2V,ϕϕ

′0δϕ1(p

i)δϕ1(q

i) + ϕ

′0Xδϕ1(p

i)δϕ1(q

i)]

−2

(4πG

H

)2 ϕ′0XH

[Xδϕ1(k

i − qi)δϕ1(qi) + ϕ

′0δϕ1(p

i)δϕ′1(q

i)]

+4πG

Hϕ′0δϕ′1(p

i)δϕ′1(q

i) + a

2[V,ϕϕϕ +

8πG

Hϕ′0V,ϕϕ

]δϕ1(p

i)δϕ1(q

i)

}

+1

(2π)3

∫d3pd

3qδ

3(ki − pi − qi)

{2

(8πG

H

)pkq

k

q2δϕ′1(p

i)(Xδϕ1(q

i) + ϕ

′0δϕ′1(q

i))

+p2 16πG

Hδϕ1(p

i)ϕ′0δϕ1(q

i) +

(4πG

H

)2 ϕ′0H

[ plql − piqjkjki

k2

ϕ′0δϕ1(ki − qi)ϕ′0δϕ1(q

i)

]

+2X

H

(4πG

H

)2 plqlpmqm + p2q2

k2q2

[ϕ′0δϕ1(p

i)(Xδϕ1(q

i) + ϕ

′0δϕ′1(q

i)) ]

+4πG

H

[4X

q2 + plql

k2

(δϕ′1(p

i)δϕ1(q

i))− ϕ′0plq

lδϕ1(p

i)δϕ1(q

i)

]

+

(4πG

H

)2 ϕ′0H

[plq

lpmqm

p2q2

(Xδϕ1(p

i) + ϕ

′0δϕ′1(p

i)) (Xδϕ1(q

i) + ϕ

′0δϕ′1(q

i)) ]

+ϕ′0H

[8πG

plql + p2

k2q2δϕ1(p

i)δϕ1(q

i) −

q2 + plql

k2δϕ′1(p

i)δϕ′1(q

i)

+

(4πG

H

)2 kjkik2

(2pipj

p2

(Xδϕ1(p

i) + ϕ

′0δϕ′1(p

i))Xδϕ1(q

i)

)]}= 0