Cosmological Imprints of the SuperhorizonUniverse (from Numerical Relativity)
Jonathan Braden
University College London
PASCOS 2017, UAM, June 20, 2017
w/ Hiranya Peiris, Matthew Johnson, and Anthony Aguirre
based on arXiv:1604.04001 and in progress
The CMB and LSS
Inflation: Only a few parameters
Pζ(k) = Akns−1 r = 16ε fNL
I Nature of Inflaton?
I Initial Conditions?
What About Ultra-Large Scales
Evolve long wavelength modes dynamicallyCMB scales see locally homogeneous background
Ultra-Large Scale Structure
Local Remnants of Ultra-Large Scale Structure?
I Structure present at start of inflation
I Conversion of structure during or after inflation
Ultra-Large Scale Structure
Local Remnants of Ultra-Large Scale Structure?
I Structure present at start of inflation
I Conversion of structure during or after inflation
Ultra-Large Scale Structure
Local Remnants of Ultra-Large Scale Structure?
I Structure present at start of inflation
I Conversion of structure during or after inflation
Modelling Initial Conditions
Monte Carlo Sampling: Planar Symmetry
ds2 = −dτ2 + a2‖(x , τ)dx2 + a2⊥(x , τ)(dy2 + dz2
)
Inflaton on a‖(τ = 0) = 1 = a⊥(τ = 0)
φ(x) = φ+ δφ
φ gives N e-folds 3H2I ≡ V (φ)
Field Fluctuations
δφ(xi ) = Aφ∑
n=1
G e iknxi√P(kn) G =
√−2 ln βe2πiα
P(k) = Θ(kmax − k) H−1I kmax = 2π√
3
Modelling Initial Conditions
Monte Carlo Sampling: Planar Symmetry
ds2 = −dτ2 + a2‖(x , τ)dx2 + a2⊥(x , τ)(dy2 + dz2
)
Inflaton on a‖(τ = 0) = 1 = a⊥(τ = 0)
φ(x) = φ+ δφ
φ gives N e-folds 3H2I ≡ V (φ)
Field Fluctuations
δφ(xi ) = Aφ∑
n=1
G e iknxi√P(kn) G =
√−2 ln βe2πiα
P(k) = Θ(kmax − k) H−1I kmax = 2π√
3
Observational Constraints
Pr(Aφ,HILobs|C obs2 , . . . ) ∝ L(Aφ,HILobs)Pr(Aφ,HILobs| . . . )L = Pr(C obs
2 |Aφ,HILobs, . . . )
I Aφ : Fluctuation Amplitude P(k) ∝ A2φ
I HILobs : Uncertain post-inflation expansion historyI . . . : V (φ), spectrum shape, IC hypersurface, Chigh−`
2 , etc.
Planck measured C`
C obs2 = 253.6µK 2 Chigh−`
2 = 1124.1µK 2
Numerical GR Input
Pr(C2|Aφ,HILobs, . . .
)
Required Evolution
0 50 100 150 200
HIx/√
3
−0.06
−0.04
−0.02
0.00
0.02
0.04
0.06
lna
0 50 100 150 200
HIx/√
3
0.934
0.936
0.938
0.940
0.942
0.944
0.946
φ/M
P
0 50 100 150 200
HIx/√
3
0.000000
0.000002
0.000004
0.000006
0.000008
0.000010
0.000012
0.000014
H
+5.773×10−1
Initial Conditions (τ = 0)
Required Evolution
0 50 100 150 200
HIx
59.560.060.561.061.562.062.563.063.564.0
lna
0 50 100 150 200
HIx/√
3
0.112
0.114
0.116
0.118
0.120
0.122
0.124
0.126
φ/M
P
0 50 100 150 200
HIx/√
3
0.375
0.380
0.385
0.390
0.395
0.400
0.405
0.410
0.415
0.420
H
End of Inflation (εH = −d lnH/d ln a = 1)
Observer Weighting
Observer Weighting
Evaluation of CMB Quadrupole
C2 = 15
[(a(UL)20 + a
(Q)20
)2+∑m=2
m=−2,m 6=0
(a(Q)2m
)2]
a(Q)2m : Gaussian with 〈(a(Q)
2m )2〉 = 1124.1µK 2
−150 −100 −50 0 50 100 150
a(Q)2m [µK]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
P( a
(Q)
2m
)[(µK
)−1]
Evaluation of CMB Quadrupole
C2 = 15
[(a(UL)20 + a
(Q)20
)2+∑m=2
m=−2,m 6=0
(a(Q)2m
)2]
a(Q)2m : Gaussian with 〈(a(Q)
2m )2〉 = 1124.1µK 2
−150 −100 −50 0 50 100 150
a(Q)2m [µK]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
P( a
(Q)
2m
)[(µK
)−1]
Evaluation of CMB Quadrupole
C2 = 15
[(a(UL)20 + a
(Q)20
)2+∑m=2
m=−2,m 6=0
(a(Q)2m
)2]
a(UL)20 : NR simulations
−2 −1 0 1 2
a(UL)20 /σ
(UL)20
0
1
2
3
4
P( a
(UL
)20
/σ
(UL
)20
∣ ∣ ∣Aφ,.
..)
α = 2300
Aφ
1 × 10−5
3 × 10−5
6 × 10−5
Dependence of C2 on Model Parameters
0 1500 3000 4500
C2 [µK2]
0.0
0.2
0.4
0.6
0.8
P( C
2|A
φ,
σ(UL)
σ(Q
),.
..)
×10−3
σ(UL)
σ(Q) = 4
Aφ
1 × 10−5
3 × 10−5
6 × 10−5
χ2dof=5
Final Posterior
−5 −4 −3 −2 −1 0
log10(HILobs)
−6
−5
−4
−3lo
g10(A
φ)
α = 2300
0
2
4
6
8
P(log
10A
φ,l
og10H
ILobs|C
obs
2)×10−2
Significant deviations from Gaussian approximation
Heuristic Explanation: Initial Conditions
Heuristic Explanation: End-of-Inflation
Distribution of a20 with δζ dependence
−3 −2 −1 0 1 2 3
a(UL)20 /σ
(UL)20
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
δζ
σζ = 0.51
0.0
0.5
1.0
1.5
2.0
P( ζ
,a(U
L)
20
/σ
(UL)
20
)
Analytic Approximation
ζ and comoving derivatives nearly Gaussian
−4 −3 −2 −1 0 1 2 3 4
ζ/σζ
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
P(ζ/σ
ζ)
−4 −3 −2 −1 0 1 2 3 4
ζ ′′/σζ′′
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
P(ζ′′ /σζ′′)
Treat as Gaussian random field
Large-Scale Approximation for a20
a20(x0) ∼ −Ae−2ζ(x0)(ζ ′′(x0)−O(ζ ′(x0)2)
)
Analytic Approximation
ζ and comoving derivatives nearly Gaussian
−4 −3 −2 −1 0 1 2 3 4
ζ/σζ
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
P(ζ/σ
ζ)
−4 −3 −2 −1 0 1 2 3 4
ζ ′′/σζ′′
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
P(ζ′′ /σζ′′)
Treat as Gaussian random field
Large-Scale Approximation for a20
a20(x0) ∼ −Ae−2ζ(x0)(ζ ′′(x0)−O(ζ ′(x0)2)
)
Anaytic a20 Distributions
−2 −1 0 1 2
a(UL)20 /σ
(UL)20
0
1
2
3
4
5P
( a(U
L)
20
/σ(U
L)
20
)
Vary σζ at fixed σζ(p)/σζ
Recall The Numerical Result
−2 −1 0 1 2
a(UL)20 /σ
(UL)20
0
1
2
3
4P
( a(U
L)
20
/σ(U
L)
20
∣ ∣ ∣Aφ,.
..)
α = 2300
Aφ
1 × 10−5
3 × 10−5
6 × 10−5
Conclusions
I Numerical relativity is a useful framework for makingcosmological predictions
I Sometimes it is a necessary tool (deviations from Gaussianity)
I Robust qualitative conclusions over a variety of inflationarymodels
I Constraining large amplitude superhorizon structure is hardeven in the most optimistic case
I Inflation is effective at hiding large amplitude initialfluctuations
I Gaussianity of ζ in comoving coordinates suggests analyticapproach in 3D
