Talk given at Queen Mary, University of London in March 2010. Cosmological perturbation theory is well established as a tool for probing the inhomogeneities of the early universe. In this talk I will motivate the use of perturbation theory and outline the mathematical formalism. Perturbations beyond linear order are especially interesting as non-Gaussian effects can be used to constrain inflationary models. I will show how the Klein-Gordon equation at second order, written in terms of scalar field variations only, can be numerically solved. The slow roll version of the second order source term is used and the method is shown to be extendable to the full equation. This procedure allows the evolution of second order perturbations in general and the calculation of the non-Gaussianity parameter in cases where there is no analytical solution available.

1. Cosmological Perturbations and
Numerical Simulations
Ian Huston
Astronomy Unit
24th March 2010
arXiv:0907.2917, JCAP 0909:019

2. perturbations
Long review: Malik & Wands 0809.4944
Short technical review: Malik & Matravers 0804.3276

4. T(η, xi) = T0(η) + δT(η, xi)
∞ n
i
δT(η, x ) = δTn(η, xi)
n=1
n!
1
ϕ = ϕ0 + δϕ1 + δϕ2 + . . .
2

5. T(η, xi) = T0(η) + δT(η, xi)
∞ n
i
δT(η, x ) = δTn(η, xi)
n=1
n!
1
ϕ = ϕ0 + δϕ1 + δϕ2 + . . .
2

6. Gauges
Background split not
covariant
Many possible descriptions
Should give same physical
answers!

9. First order transformation
µ
ξ1 = (α1, β1,i + γ1)
i
⇓
δϕ1 = δϕ1 + ϕ0α1

10. Perturbed FRW metric
g00 = −a2(1 + 2φ1) ,
g0i = a2B1i ,
gij = a2 [δij + 2C1ij ] .

11. Choosing a gauge
Longitudinal: zero shear
Comoving: zero 3-velocity
Flat: zero curvature
Uniform density: zero energy
density
...

12. δGµν = 8πGδTµν
⇓
Eqs of Motion

13. non-Gaussianity
Some reviews: Chen 1002.1416, Senatore et al. 0905.3746

14. Sim 1
Simulations from Ligouri et al, PRD (2007)

15. Sim 2
Simulations from Ligouri et al, PRD (2007)

16. Gaussian fields:
All information in
ζ(k1 )ζ(k2 ) = (2π)3 δ 3 (k1 + k2 )Pζ (k1 ) ,
where ζ is curvature perturbation on uniform
density hypersurfaces.
ζ(k1 )ζ(k2 )ζ(k3 ) = 0 ,
ζ 4 (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 ) .

17. Bispectrum:
ζ(k1 )ζ(k2 )ζ(k3 ) = (2π)3 δ 3 (k1 +k2 +k3 )B(k1 , k2 , k3 )
Local (squeezed) Equilateral

18. B(k1, k2, k3) fNLF (x2, x3) ,
xi = ki/k1 , 1 − x2 ≤ x3 ≤ x2 .
Local 1 Higher Deriv. 1
0.9 x2 0.9 x2
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
8 1
6 0.75
4 F x2 , x3 0.5 F x2 , x3
2 0.25
0 0
0.2 0.4 0.2
0.6 0.4 0.8 0.6
1 0.8 1 x3
x3
Babich et al. astro-ph/0405356

19. WMAP7 bounds (95% CL)
loc
−10 < fNL < 74
loc
fNL > 1
rules out ALL single field
inflationary models.

20. WMAP7 bounds (95% CL)
loc
−10 < fNL < 74
loc
fNL > 1
rules out ALL single field
inflationary models.

21. One way of getting local fNL
ζ(x) = ζL (x) + 3 fNL ζL (x)
5
loc 2
∆T 1 loc
− ζ, fNL > 0
T 5
⇓
∆T < ∆TL

22. Sim 1: fNL = 1000
Simulations from Ligouri et al, PRD (2007)

23. Sim 2: fNL = 0
Simulations from Ligouri et al, PRD (2007)

24. code():
Paper: Huston & Malik 0907.2917, JCAP
2nd order equations: Malik astro-ph/0610864, JCAP

25. Approaches:
δN formalism
Moment transport equations
Field Equations

26. 1
ϕ = ϕ0 + δϕ1 + δϕ2
2

27. i i 2 i 2 8πG 2 8πG i
δϕ2 (k ) + 2Hδϕ2 (k ) + k δϕ2 (k ) + a V,ϕϕ + 2ϕ0 V,ϕ + (ϕ0 ) V0 δϕ2 (k )
H H
1 3 3 3 i i i 16πG i i 2 i i
+ d pd qδ (k − p − q ) Xδϕ1 (p )δϕ1 (q ) + ϕ0 a V,ϕϕ δϕ1 (p )δϕ1 (q )
(2π)3 H
8πG 2
2 i i i i
+ ϕ0 2a V,ϕ ϕ0 δϕ1 (p )δϕ1 (q ) + ϕ0 Xδϕ1 (p )δϕ1 (q )
H
4πG 2 ϕ X
0 i i i i i
−2 Xδϕ1 (k − q )δϕ1 (q ) + ϕ0 δϕ1 (p )δϕ1 (q )
H H
4πG i i 2 8πG i i
+ ϕ0 δϕ1 (p )δϕ1 (q ) + a V,ϕϕϕ + ϕ0 V,ϕϕ δϕ1 (p )δϕ1 (q )
H H
1 3 3 3 i i i 8πG pk q k i i i
+ d pd qδ (k − p − q ) 2 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
(2π)3 H q2
 
2 16πG i i 4πG 2 ϕ
0 pi qj kj ki
+p δϕ1 (p )ϕ0 δϕ1 (q ) + p q l −  ϕ δϕ (ki − q i )ϕ δϕ (q i )
l 0 1 0 1
H H H k2
X 4πG 2 p q l p q m + p2 q 2
l m i i i
+2 ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
H H k2 q 2
4πG q 2 + pl q l i i l i i
+ 4X δϕ1 (p )δϕ1 (q ) − ϕ0 pl q δϕ1 (p )δϕ1 (q )
H k2
4πG pl q l pm q m
2 ϕ
0 i i i i
+ Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
HH p2 q 2
 
ϕ0 pl q l + p2 2 i i q 2 + pl q l i i
+ 8πG  q δϕ1 (p )δϕ1 (q ) − δϕ1 (p )δϕ1 (q )
H k2 k2
4πG 2 kj k pi pj
i i i i
+ 2 Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) = 0
H k2 p2

28. Single field slow roll
Single field full equation
Multi-field calculation

29. δϕ1(q i)δϕ1(k i − q i)d3q

30. code():
1000+ k modes
python & numpy
parallel

31. Four potentials
×10−9
3.0
V (ϕ) = 1 m2 ϕ2
2
2.8 V (ϕ) = 1 λϕ4
4
2
V (ϕ) =σϕ 3
2.6 V (ϕ) = U0 + 1 m2 ϕ2
2 0
PR1
2.4
2
2.2
2.0
1.8 −61
10 10−60 10−59 10−58
k/MPL

32. Source term
10−1
V (ϕ) = 1 m2 ϕ2
2
V (ϕ) = 1 λϕ4
4
2
10−5 V (ϕ) =σϕ 3
V (ϕ) = U0 + 1 m2 ϕ2
2 0
|S|
10−9
10−13
10−17
0 10 20 30 40 50 60
N − N init

33. Second order perturbation
×10−95
4
3
2
1
√1 k 2 δϕ2
0
3
2π
−1
−2
−3
−4
64 63 62 61
Nend − N

34. Future Plans:
Full single field equation
Multi field equation
Vector & Vorticity similarities
Rework code for efficiency

35. Summary:
Perturbations seed structure
2nd order needed for fNL
Numerically intensive calculation

36. kmax
IA (k) = dq 3 δϕ1 (q i )δϕ1 (k i − q i ) = 2π dq q 2 δϕ1 (q i )A(k i , q i ) ,
kmin
√ √ √ √
πα2 kmax − k + kmax k + kmax + kmax
IA (k) = − 3k 3 log √ + log √ √
18k k kmin + k + kmin
√
π kmin
+ − arctan √
2 k − kmin
− kmax 3k 2 + 8kmax
2
k + kmax − kmax − k
+ 14kkmax k + kmax + kmax − k
+ kmin 3k 2 + 8kmin
2
k + kmin + k − kmin
+ 14kkmin k + kmin − k − kmin .

37. 10−6
10−7
rel
10−8
10−9
k ∈ K1
−10
k ∈ K2
10
k ∈ K3
−61
10 10−60 10−59 10−58 10−57
k/MPL
K1 = 1.9 × 10−5 , 0.039 Mpc−1 , ∆k = 3.8 × 10−5 Mpc−1 ,
K2 = 5.71 × 10−5 , 0.12 Mpc−1 , ∆k = 1.2 × 10−4 Mpc−1 ,
K3 = 9.52 × 10−5 , 0.39 Mpc−1 , ∆k = 3.8 × 10−4 Mpc−1 .