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Talk given at Queen Mary, University of London in March 2010. Cosmological perturbation theory is well established as a tool forprobing the inhomogeneities of the early universe.In this talk I will motivate the use of perturbation theory andoutline the mathematical formalism. Perturbations beyond linear orderare especially interesting as non-Gaussian effects can be used toconstrain inflationary models.I will show how the Klein-Gordon equation at second order, written interms of scalar field variations only, can be numerically solved.The slow roll version of the second order source term is used and themethod is shown to be extendable to the full equation. This procedureallows the evolution of second order perturbations in general and thecalculation of the non-Gaussianity parameter in cases where there isno analytical solution available.
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Cosmological Perturbations andNumerical Simulations
Ian Huston
Astronomy Unit
24th March 2010
arXiv:0907.2917, JCAP 0909:019
perturbations
Long review: Malik & Wands 0809.4944
Short technical review: Malik & Matravers 0804.3276
T(η, xi) = T0(η) + δT(η, xi)
δT(η, xi) =∞∑
n=1
εn
n!δTn(η, xi)
ϕ = ϕ0 + δϕ1 +1
2δϕ2 + . . .
T(η, xi) = T0(η) + δT(η, xi)
δT(η, xi) =∞∑
n=1
εn
n!δTn(η, xi)
ϕ = ϕ0 + δϕ1 +1
2δϕ2 + . . .
Gauges
Background split notcovariant
Many possible descriptions
Should give same physicalanswers!
First order transformation
ξµ1 = (α1, β
i1, + γi
1)
⇓δ̃ϕ1 = δϕ1 + ϕ′0α1
Perturbed FRW metric
g00 = −a2(1 + 2φ1) ,
g0i = a2B1i ,
gij = a2 [δij + 2C1ij] .
Choosing a gauge
Longitudinal: zero shear
Comoving: zero 3-velocity
Flat: zero curvature
Uniform density: zero energydensity
. . .
δGµν = 8πGδTµν
⇓Eqs of Motion
non-Gaussianity
Some reviews: Chen 1002.1416, Senatore et al. 0905.3746
Sim 1
Simulations from Ligouri et al, PRD (2007)
Sim 2
Simulations from Ligouri et al, PRD (2007)
Gaussian fields:All information in
〈ζ(k1)ζ(k2)〉 = (2π)3δ3(k1 + k2)Pζ(k1) ,
where ζ is curvature perturbation on uniformdensity hypersurfaces.
〈ζ(k1)ζ(k2)ζ(k3)〉 = 0 ,
〈ζ4(ki)〉 = 〈ζ(k1)ζ(k2)〉〈ζ(k3)ζ(k4)〉+ 〈ζ(k2)ζ(k3)〉〈ζ(k4)ζ(k1)〉+ 〈ζ(k1)ζ(k3)〉〈ζ(k2)ζ(k4)〉 .
Bispectrum:
〈ζ(k1)ζ(k2)ζ(k3)〉 = (2π)3δ3(k1+k2+k3)B(k1, k2, k3)
Local (squeezed) Equilateral
B(k1, k2, k3) ' fNLF (x2, x3) ,
xi = ki/k1 , 1− x2 ≤ x3 ≤ x2 .
Local
0.50.6
0.70.8
0.91
x2
0.20.40.60.81 x3
0
2
4
6
8
FHx2 , x3L
0
2
4
Higher Deriv.
0.50.6
0.70.8
0.91
x2
0.20.40.60.81 x3
00.25
0.5
0.75
1
FHx2 , x3L
00.25
0.5
Babich et al. astro-ph/0405356
WMAP7 bounds (95% CL)
−10 < f locNL < 74
f locNL > 1
rules out ALL single fieldinflationary models.
WMAP7 bounds (95% CL)
−10 < f locNL < 74
f locNL > 1
rules out ALL single fieldinflationary models.
One way of getting local fNL
ζ(x) = ζL(x) + 35f
locNLζ2
L(x)
∆T
T' −1
5ζ , f loc
NL > 0
⇓∆T < ∆TL
Sim 1: fNL = 1000
Simulations from Ligouri et al, PRD (2007)
Sim 2: fNL = 0
Simulations from Ligouri et al, PRD (2007)
code():
Paper: Huston & Malik 0907.2917, JCAP
2nd order equations: Malik astro-ph/0610864, JCAP
Approaches:
δN formalism
Moment transport equations
Field Equations
ϕ = ϕ0 + δϕ1 +1
2δϕ2
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
2� Single field slow roll
2 Single field full equation
2 Multi-field calculation
∫δϕ1(q
i)δϕ1(ki − qi)d3q
code():
1000+ k modes
python & numpy
parallel
Four potentials
10−61 10−60 10−59 10−58
k/MPL
1.8
2.0
2.2
2.4
2.6
2.8
3.0P2 R
1×10−9
V (ϕ) = 12m2ϕ2
V (ϕ) = 14λϕ4
V (ϕ) =σϕ23
V (ϕ) =U0 + 12m2
0ϕ2
Source term
0 10 20 30 40 50 60N − Ninit
10−17
10−13
10−9
10−5
10−1|S|
V (ϕ) = 12m2ϕ2
V (ϕ) = 14λϕ4
V (ϕ) =σϕ23
V (ϕ) =U0 + 12m2
0ϕ2
Second order perturbation
61626364Nend −N
−4
−3
−2
−1
0
1
2
3
4
1 √2πk
3 2δϕ
2
×10−95
Future Plans:
Full single field equation
Multi field equation
Vector & Vorticity similarities
Rework code for efficiency
Summary:
Perturbations seed structure
2nd order needed for fNL
Numerically intensive calculation
IA(k) =
∫dq3δϕ1(q
i)δϕ1(ki − qi) = 2π
∫ kmax
kmin
dq q2δϕ1(qi)A(ki, qi) ,
IA(k) = −πα2
18k
{3k3
[log
(√kmax − k +
√kmax√
k
)+ log
(√k + kmax +
√kmax√
kmin + k +√
kmin
)
+π
2− arctan
( √kmin√
k − kmin
)]
−√
kmax
[ (3k2 + 8k2
max
) (√k + kmax −
√kmax − k
)+ 14kkmax
(√k + kmax +
√kmax − k
)]
+√
kmin
[ (3k2 + 8k2
min
) (√k + kmin +
√k − kmin
)+ 14kkmin
(√k + kmin −
√k − kmin
)]}.
10−61 10−60 10−59 10−58 10−57
k/MPL
10−10
10−9
10−8
10−7
10−6
ε rel
k ∈ K1
k ∈ K2
k ∈ K3
K1 =[1.9× 10−5, 0.039
]Mpc−1 , ∆k = 3.8× 10−5Mpc−1 ,
K2 =[5.71× 10−5, 0.12
]Mpc−1 , ∆k = 1.2× 10−4Mpc−1 ,
K3 =[9.52× 10−5, 0.39
]Mpc−1 , ∆k = 3.8× 10−4Mpc−1 .
