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This is a short talk I gave at the National Astronomy Meeting 2011 in Llandudno, Wales.
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Second Order Perturbations DuringInflation Beyond Slow-roll
Ian Huston
Astronomy Unit, Queen Mary, University of London
In Collaboration with Karim Malik (QMUL)
arXiv:1103.0912 and 0907.2917 (JCAP 0909:019)Software available at http://pyflation.ianhuston.net
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(t, x) = ϕ0(t) + δϕ1(t, x)
+1
2δϕ2(t, x)
+ . . .
δGµν = 8πGδTµν
⇓Eqs of Motion
code():
Papers: 1103.0912, 0907.2917
Software: http://pyflation.ianhuston.net
2nd order equations: Malik astro-ph/0610864, JCAP
Non-linear processes:
Non-Gaussianity of CMB
Vorticity generation (See Adam’s poster)
Magnetic field generation
2nd order Gravitational waves
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
pyflation():
python & numpy
parallel
open source
Following Salopek et al. PRD40 1753, Martin &Ringeval a-ph/0605367
2� Single field slow roll
2� Single field full equation
2 Multi-field calculation
∫kjqjδϕ1(q
i)δϕ1(ki− qi)d3q
Bump potential
Vb(ϕ) =1
2m2ϕ2
[1 + c sech
(ϕ− ϕb
d
)]
0102030405060Nend −N
10−5
10−4
10−3
10−2
k3/2|δϕ
1|/M
−1/2
PL
Full Bump Potential
Half Bump Potential
Zero Bump Potential
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
δϕ′′2(k
i) + 2Hδϕ′2(k
i) +Mδϕ2(ki) = 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
0102030405060Nend −N
10−9
10−7
10−5
|δϕ2(k
)|/M−2 PL
Full Bump Potential
Half Bump Potential
Zero Bump Potential
Second order perturbation
5354555657Nend −N
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
|δϕ2(k
)|/M−2 PL
×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
|δϕ2(k
)|/|δϕ
2qu
ad|
Sub-Horizon Bump
Super-Horizon Bump
Standard Quadratic Potential
Future Plans
Three-point function of δϕ
Multi-field equation
Tensor & Vorticity similarities
Summary
Perturbations seed structure
Non-linear regime observationally
interesting
Numerically intensive calculation
Code available now
(http://pyflation.ianhuston.net)
δϕ′′2 (ki) + 2Hδϕ
′2(ki) + k
2δϕ2(ki) + a
2[V,ϕϕ +
8πG
H
(2ϕ′0V,ϕ + (ϕ′0)2
8πG
HV0
)]δϕ2(ki)
+1
(2π)3
∫d
3pd
3qδ
3(ki − pi − q
i)
{16πG
H
[Xδϕ
′1(pi)δϕ1(qi) + ϕ
′0a
2V,ϕϕδϕ1(pi)δϕ1(qi)
]
+
(8πG
H
)2ϕ′0
[2a
2V,ϕϕ
′0δϕ1(pi)δϕ1(qi) + ϕ
′0Xδϕ1(pi)δϕ1(qi)
]
−2
(4πG
H
)2 ϕ′0X
H
[Xδϕ1(ki − q
i)δϕ1(qi) + ϕ′0δϕ1(pi)δϕ
′1(qi)
]
+4πG
Hϕ′0δϕ
′1(pi)δϕ
′1(qi) + a
2[V,ϕϕϕ +
8πG
Hϕ′0V,ϕϕ
]δϕ1(pi)δϕ1(qi)
}
+1
(2π)3
∫d
3pd
3qδ
3(ki − pi − q
i)
{2
(8πG
H
)pkqk
q2δϕ′1(pi)
(Xδϕ1(qi) + ϕ
′0δϕ
′1(qi)
)
+p2 16πG
Hδϕ1(pi)ϕ′0δϕ1(qi) +
(4πG
H
)2 ϕ′0H
[ plql −
piqjkjki
k2
ϕ′0δϕ1(ki − q
i)ϕ′0δϕ1(qi)
]
+2X
H
(4πG
H
)2 plqlpmqm + p2q2
k2q2
[ϕ′0δϕ1(pi)
(Xδϕ1(qi) + ϕ
′0δϕ
′1(qi)
) ]
+4πG
H
[4X
q2 + plql
k2
(δϕ′1(pi)δϕ1(qi)
)− ϕ
′0plq
lδϕ1(pi)δϕ1(qi)
]
+
(4πG
H
)2 ϕ′0H
[plqlpmqm
p2q2
(Xδϕ1(pi) + ϕ
′0δϕ
′1(pi)
) (Xδϕ1(qi) + ϕ
′0δϕ
′1(qi)
) ]
+ϕ′0H
[8πG
plql + p2
k2q2δϕ1(pi)δϕ1(qi) −
q2 + plql
k2δϕ′1(pi)δϕ
′1(qi)
+
(4πG
H
)2 kjki
k2
(2
pipj
p2
(Xδϕ1(pi) + ϕ
′0δϕ
′1(pi)
)Xδϕ1(qi)
)]}= 0
