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 .
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