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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
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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
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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
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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
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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
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Faucher-Gigure et al., Science 2008
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perturbations
Long review: Malik & Wands 0809.4944
Short technical review: Malik & Matravers 0804.3276
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Separate quantities intobackground andperturbation.
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ϕ(η, x) = ϕ0(η) + δϕ(η, x)
+1
2δϕ2(η, x)
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ϕ(η, x) = ϕ0(η) + δϕ1(η, x)
+1
2δϕ2(η, x)
+ . . .
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Gauge Choice Required
Background split not covariant
Many possible descriptions
Should give same physical answers!
⇒ Use Gauge Invariant Variables
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Gauge Choice Required
Background split not covariant
Many possible descriptions
Should give same physical answers!
⇒ Use Gauge Invariant Variables
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First order transformationbetween gauges
xµ → xµ + ξµ
ξµ1 = (α1, βi
1, + γi1)
+
T̃1 = T1 +£ξ1T0
⇓δ̃ϕ1 = δϕ1 + ϕ′0α1
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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
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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
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Perturbed flat FRW metric at first order
g00 = −a2(1 + 2φ1)
g0i = a2(B1,i)
gij = a2
[(1− 2ψ)δij + 2E1,ij
]
Bardeen 1980
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Choosing a gauge
Longitudinal: zero shear
Comoving: zero 3-velocity
Flat: zero curvature
Uniform density: zero energy density
. . .
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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
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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)
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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
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δGµν = 8πGδTµν
⇓Eqs of Motion
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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.
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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.
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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.
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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
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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
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results
Second Order Perturbations During Inflation Beyond Slow-roll,Huston & Malik, arXiv:1103.0912
2nd order equations: Malik, arXiv:astro-ph/0610864
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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.
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Breaking Slow Roll
5354555657Nend −N
−8
−6
−4
−2
0
2η V
Step Potential
Bump Potential
Standard Quadratic Potential
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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
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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
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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
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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
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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
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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
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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
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code():First Order Numerical Reviews: Salopek et al. PRD40 1753,
Martin & Ringeval a-ph/0605367
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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
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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)
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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
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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
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Progress
2� Single field slow roll
2� Single field full equation
2 Multi-field calculation (underway)
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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
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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
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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
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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
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Other interesting non-linear processes
Vorticity generation(Half-day Vorticity meeting in RAS 14th July)
Magnetic field generation
2nd order Gravitational waves
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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)
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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