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