Cosmological Imprints of the Superhorizon Universe (from Numerical...
Transcript of Cosmological Imprints of the Superhorizon Universe (from Numerical...
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