Constraining the Ultra-Large Scale Structure of the...
Transcript of Constraining the Ultra-Large Scale Structure of the...
![Page 1: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/1.jpg)
Constraining the Ultra-Large Scale Structure ofthe Universe
Jonathan Braden
University College London
ICG Portsmouth, May 25, 2016
w/ Hiranya Peiris, Matthew Johnson and Anthony Aguirre
based on arXiv:1604.04001 and in progress
![Page 2: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/2.jpg)
Inhomogeneous Nonlinear Ultra-Large Scale Cosmology
![Page 3: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/3.jpg)
... A Mosaic of Interesting Dynamics
I Strongly inhomogeneous and nonlinear cosmologicalinitial conditions (this talk) [JB, Peiris, Johnson, Aguirre]
I Isocurvature mode conversion into intermittent densityperturbations [JB, Bond, Frolov, Huang]
I Caustic formation in chaotic long wavelength dynamicsI Generalised form of local nonGaussianity
I First order phase transitions [JB, Bond, Mersini-Houghton]
I Entropy production in highly inhomogeneous nonlinear fieldtheories [JB, Bond]
Spatially intermittent dynamics, fundamental issues in QFT, noveland poorly constrained observables
![Page 4: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/4.jpg)
Numerical Approach is Essential [JB, in preparation]
Hybrid MPI/OpenMP Lattice Code
I Solve field equation (e.g.)
φi + 3a
aφi −
∇2φia2
+ V ′(~φ) = 0
I 10th order Gauss-Legendreintegration (general) or 8th orderYoshida (nonlinear sigma models)
I Finite-difference (fully parallel) orPseudospectral (OpenMP)
I Optional absorbing boundaries
I Quantum fluctuations →realization of random field
I Energy conservationO(10−9 − 10−14)
![Page 5: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/5.jpg)
Numerical Approach is Essential [JB, in preparation]
Hybrid MPI/OpenMP Lattice Code
I Solve field equation (e.g.)
φi + 3a
aφi −
∇2φia2
+ V ′(~φ) = 0
I 10th order Gauss-Legendreintegration (general) or 8th orderYoshida (nonlinear sigma models)
I Finite-difference (fully parallel) orPseudospectral (OpenMP)
I Optional absorbing boundaries
I Quantum fluctuations →realization of random field
I Energy conservationO(10−9 − 10−14)
![Page 6: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/6.jpg)
Dawn of Numerical Relativity in Cosmology
I Bentivegna, Bruni
I Mertens, Giblin, Starkman
I Kleban, Linde, Senatore, West
I Peiris, Johnson, Feeney, Aguirre, Wainwright (symmetryreduced bubbles)
I Adamek, Daverio, Durrer, Kunz (weak field subhorizon)
Connection of Initial Conditions with Observables
I Requires Monte Carlo sampling of Initial Conditions
I Highly accurate integrators to achieve machine precision
![Page 7: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/7.jpg)
Dawn of Numerical Relativity in Cosmology
I Bentivegna, Bruni
I Mertens, Giblin, Starkman
I Kleban, Linde, Senatore, West
I Peiris, Johnson, Feeney, Aguirre, Wainwright (symmetryreduced bubbles)
I Adamek, Daverio, Durrer, Kunz (weak field subhorizon)
Connection of Initial Conditions with Observables
I Requires Monte Carlo sampling of Initial Conditions
I Highly accurate integrators to achieve machine precision
![Page 8: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/8.jpg)
Outline
I Brief Review of Cosmology and Role of Inflation
I Inflationary Models and Initial Conditions
I Choice of Initial Conditions
I Planar Symmetric Dynamics
I Projection of Dynamics into Observations (CMB quadrupole)
I Semi-analytic Approximation (work in progress)
![Page 9: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/9.jpg)
History of the Universe
![Page 10: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/10.jpg)
Model Choices
V (φ) = V0
(1− e
−√
23α
φMP
)2
I α� 1→ small-field model
I α = 1→ Starobinski
I α� 1→ m2φ2/2
![Page 11: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/11.jpg)
Model Choices
V (φ) = V0
(1− e
−√
23α
φMP
)2
0 1 2 3 4 5 6
φ/MP
0.0
0.2
0.4
0.6
0.8
1.0
1.2
V(φ
)
×10−11
α = 0.1
I α� 1→ small-field model
I α = 1→ Starobinski
I α� 1→ m2φ2/2
![Page 12: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/12.jpg)
Model Choices
V (φ) = V0
(1− e
−√
23α
φMP
)2
0 2 4 6 8 10 12
φ/MP
0.0
0.2
0.4
0.6
0.8
1.0
V(φ
)
×10−10
α = 1.0
I α� 1→ small-field model
I α = 1→ Starobinski
I α� 1→ m2φ2/2
![Page 13: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/13.jpg)
Model Choices
V (φ) = V0
(1− e
−√
23α
φMP
)2
0 5 10 15 20 25 30
φ/MP
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
V(φ
)
×10−9
α = 100.0
I α� 1→ small-field model
I α = 1→ Starobinski
I α� 1→ m2φ2/2
![Page 14: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/14.jpg)
Model Choices
V (φ) = V0
(1− e
−√
23α
φMP
)2
0 5 10 15 20 25 30 35
φ/MP
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
V(φ
)
×10−8
α = 100000.0
I α� 1→ small-field model
I α = 1→ Starobinski
I α� 1→ m2φ2/2
![Page 15: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/15.jpg)
CMB Parameter Predictions
10−5 10−3 10−1 101 103 105 107 109
α
−0.050
−0.045
−0.040
−0.035
−0.030
−0.025
ns−
1
∆ ln a60
50
40
35 40 45 50 55 60
∆ ln a
−0.060
−0.055
−0.050
−0.045
−0.040
−0.035
−0.030
−0.025
ns−
1
α0.01
1.0
100.0
10000.0
1000000.0
10−510−410−310−210−1100 101 102 103 104 105 106 107
α
10−810−710−610−510−410−310−210−1100101102103104105
r
∆ ln a = 60
∆ ln a = 50
∆ ln a = 40
35 40 45 50 55 60
∆ ln a
10−5
10−4
10−3
10−2
10−1
100
r
α = 0.01
α = 0.1
α = 1.0
α = 100.0
α = 10000.0
![Page 16: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/16.jpg)
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
![Page 17: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/17.jpg)
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
![Page 18: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/18.jpg)
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
![Page 19: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/19.jpg)
Dimensional Reduction: Planar Symmetry
Synchronous Gauge
ds2 = −dτ2 + a2‖(x , τ)dx2 + a2
⊥(x , τ)(dy2 + dz2)
Residual gauge freedom : a‖(x , τ = 0), a⊥(x , τ = 0)
Isotropised Expansion
a ≡ det(γij)1/6 =
(a‖a
2⊥)1/3
H ≡ −1
3γ ijKij = −1
3(K x
x + 2K yy )
![Page 20: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/20.jpg)
Initial Conditions: Field
Work in Spatially Flat Gauge a‖(τ = 0) = 1 = a⊥(τ = 0)
φ(x) = φ+ δφ
φ gives desired N e-folds of inflation in homogeneous limit3H2
I ≡ 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
![Page 21: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/21.jpg)
Initial Conditions: Field
Work in Spatially Flat Gauge a‖(τ = 0) = 1 = a⊥(τ = 0)
φ(x) = φ+ δφ
φ gives desired N e-folds of inflation in homogeneous limit3H2
I ≡ 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
![Page 22: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/22.jpg)
Initial Conditions: Metric
Work in Spatially Flat Gauge a‖(τ = 0) = 1 = a⊥(τ = 0)
Momentum Constraint Sews Neighbouring Grid Sites Together
0 = P = K yy′ − a′⊥
a⊥(K x
x − K yy )− φ′Πφ
2a‖M2P
Hamiltonian Constraint Enforces Energy Conservation
0 = H =2a⊥a
′‖a′⊥ − a‖a
′2⊥ − 2a‖a⊥a
′′⊥
a3‖a
2⊥
+ 2K xxK
yy + K y
y2
−M−2P
(φ′2 + Π2
φ
2a2‖
+ V
)= 0
![Page 23: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/23.jpg)
Distribution of Initial Conditions
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
−6
−5
−4
−3
−2
Kxx
×10−5 − 5.7728×10−1
0 50 100 150 200
HIx/√
3
−0.61
−0.60
−0.59
−0.58
−0.57
−0.56
−0.55
−0.54
Kyy
What’s the response in the curvature perturbation?
![Page 24: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/24.jpg)
Time Evolution
a‖ = −a‖K xx , ˙a⊥ = −a⊥K y
y ,
˙K xx =
a′2⊥a2⊥a
2‖
+ K xx
2 − K yy
2 +
(Π2φ − φ′2
)
2a2‖M
2P
,
˙K yy = − a′2⊥
2a2⊥a
2‖
+3
2K y
y2 − V (φ)
2M2P
+
(Π2φ + φ′2
)
4a2‖M
2P
,
Πφ = 2K yyΠφ +
1
a‖φ′′ +
(2a′⊥a‖a⊥
−a′‖
a2⊥
)φ′ − a⊥∂φV (φ),
φ =Πφ
a‖
![Page 25: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/25.jpg)
Numerical Approach
Machine precision accuracy
I Gauss-Legendre time-integrator (O(dt10), symplectic)
I Fourier pseudospectral discretisation (exponentialconvergence)
Fast to allow sampling
I Adaptive time-stepping
I Adaptive grid spacing
O(1− 10)s to evolve through 60 e-folds of inflation
![Page 26: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/26.jpg)
Convergence Testing : Dynamical Variables
Vary grid spacing dx at fixed dx/dt
0 50 100 150 200 250
HIτ/√
3
10−16
10−15
10−14
10−13
10−12
10−11
‖Hdx−Hdx/2‖
0 50 100 150 200 250
HIτ/√
3
10−16
10−15
10−14
10−13
10−12
10−11
‖ln
(a) dx−
ln(a
) dx/2‖
Vary time-step dt at fixed dx
0 50 100 150 200 250
HIτ/√
3
10−1610−1510−1410−1310−1210−1110−1010−910−810−710−610−510−4
‖Hdx−Hdx/2‖
0 50 100 150 200 250
HIτ/√
3
10−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−2
‖ln
(a) dx−
ln(a
) dx/2‖
![Page 27: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/27.jpg)
Convergence Testing : Constraint Preservation
Vary grid spacing dx at fixed dx/dt
0 50 100 150 200 250
HIτ/√
3
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
‖H‖
0 50 100 150 200 250
HIτ/√
3
10−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1
100
‖P‖
Vary time-step dt at fixed dx
0 50 100 150 200 250
HIτ/√
3
10−1610−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−3
‖H‖
0 50 100 150 200 250
HIτ/√
3
10−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1
100‖P‖
![Page 28: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/28.jpg)
Self-Gravitating Dynamics: Isotropisation of Expansion
0 10 20 30 40 50 60 70
ln a
−0.5
0.0
0.5
1.0
1.5
2.0Kxx/Kyy−
1×10−5
Expansion quickly isotropises itself
![Page 29: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/29.jpg)
Self-Gravitating Dynamics: Attractor Behaviour
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
φ
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
φ
![Page 30: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/30.jpg)
Self-Gravitating Dynamics: Geometry Attractor
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
H
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5ε H
Individual Lattice Sites Evolve Along AttractorFixed H, εH are equivalent
![Page 31: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/31.jpg)
Self-Gravitating Dynamics: Spatially Dependent ExpansionHistory
0 10 20 30 40 50 60 7016 ln det(g)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
−1 3K
i i
![Page 32: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/32.jpg)
Observations: Relation to Local Multipoles
ζ(H) ≡ δ ln a|Ha‖=1=a⊥
Large-scale perturbations →evaluate ζ on last-scattering surface around a point x0
Expand for Large-Scale Fluctuations
ζ(x0+rls) ≈ ζ(x0)+(HI rls)∂ζ
∂(HI xp)(x0)+
(HI rls)2
2
∂2ζ
∂(HI xp)2(x0)+. . .
Matches onto Multipoles
a(UL)20 ≈ F (LobsHI)
2∂2xpζ ' F (LobsHI)
2 1
a‖∂x
[∂xζ
a‖
]
![Page 33: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/33.jpg)
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)
![Page 34: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/34.jpg)
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)
![Page 35: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/35.jpg)
Freeze-in of Superhorizon Perturbations
0 100 200 300 400 500 600
H0x/√
3
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0δζ
H/HI
0. 98
0. 95
0. 75
0. 50
0. 05
Initial transient modifies result from separate universeapproximation
![Page 36: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/36.jpg)
Weighting of Observers
![Page 37: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/37.jpg)
Distribution of Superhorizon a20
−2 −1 0 1 2
a(UL)20 /σ(UL)
0
1
2
3
4P( a
(UL
)20
/σ
(UL
)|Aφ,...)
α = 2300
Aφ
1× 10−5
3× 10−5
6× 10−5
![Page 38: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/38.jpg)
Statistical Framework
Pr(Aφ, Lobs|C obs2 ) ∝ L(Aφ, Lobs, . . . )Pr(Aφ, Lobs, . . . )
L(Aφ, Lobs) = Pr(C obs2 |Aφ, Lobs, . . . )
Planck high-` “theory” and low-` “observed”
C obs2 = 253.6µK2 C theory
2 = 1124.1µK2
![Page 39: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/39.jpg)
Distribution of CMB quadrupole
C2 =1
5
(a
(UL)20 + a
(Q)20
)2+
2∑
m=−2, m 6=0
(a
(Q)2m
)2
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
![Page 40: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/40.jpg)
Relative RMS of superhorizon contribution depends of distance tolast-scattering surface
a(UL)20 ≈ F (LobsHI)
2∂2xpζ ' F (LobsHI)
2 1
a‖∂x
[∂xζ
a‖
]
![Page 41: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/41.jpg)
Constraints from the CMB Quadrupole
−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)×10−4
Strong Modification from Gaussian Ansatz at Large Aφ
![Page 42: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/42.jpg)
Marginalised Constraints
−6.0 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0
log10(Aφ)
0.00
0.02
0.04
0.06
0.08
0.10P( lo
g10(A
φ)|C
obs
2
)
α = 2300
Numerical GR
Gaussian
Constraint of Aφ as we marginalise over Lobs
![Page 43: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/43.jpg)
Marginalised Constraints
−5 −4 −3 −2 −1 0
log10(HILobs)
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018P( lo
g10(H
ILobs)|C
obs
2
)
α = 2300
Numerical GR
Gaussian
Constraint of Lobs as we marginalise over Aφ
![Page 44: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/44.jpg)
Heuristic Picture of Large Amplitude Effects
![Page 45: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/45.jpg)
Heuristic Picture of Large Amplitude Effects
![Page 46: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/46.jpg)
Modelling With Toy Distributions
Model behaviour with Johnson distribution
![Page 47: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/47.jpg)
Removing Inhomogeneous Expansion
−4 −3 −2 −1 0 1 2 3 4
a2‖a20/
√〈a4‖a
220〉
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8P( (a
2 ‖a20/√〈a
4 ‖a2 20〉)
![Page 48: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/48.jpg)
Statistics of ζ
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/σζ)
−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′′)
40 50 60 70 80 90
ζ
40
50
60
70
80
90
ζ ‖
−0.006−0.005−0.004−0.003−0.002−0.0010.000 0.001
ζ − ζ‖
0
500
1000
1500
2000
2500
3000
3500
4000
P(ζ−ζ ‖
)Aφ = 10−3
![Page 49: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/49.jpg)
Analytical Modelling
ζ(H, xcom) ∼ GRF
Large-Scale Approximation for a20
a20(x0) ≈ Ae−2ζ‖(x0)(ζ ′′(x0)− ζ ′‖(x0)ζ ′(x0)
)
ζ, ζ ′, ζ ′′ are correlated Gaussian random deviates, with covariance
Cζ =
σ20 0 −σ2
1
0 σ21 0
−σ21 0 σ2
2
σi =∫dkk2i
⟨∣∣∣ζk∣∣∣2⟩
![Page 50: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/50.jpg)
Analytical Modelling
ζ(H, xcom) ∼ GRF
Large-Scale Approximation for a20
a20(x0) ≈ Ae−2ζ(x0)(ζ ′′(x0)− ζ ′(x0)2
)
ζ, ζ ′, ζ ′′ are correlated Gaussian random deviates, with covariance
Cζ =
σ20 0 −σ2
1
0 σ21 0
−σ21 0 σ2
2
σi =∫dkk2i
⟨∣∣∣ζk∣∣∣2⟩
![Page 51: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/51.jpg)
Anaytic a20 Distributions
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
a20/σζ
0.0
0.5
1.0
1.5
2.0
2.5
3.0P
(a20/σζ)
Vary σζ at fixed σζ(p)/σζ
![Page 52: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/52.jpg)
Anaytic a20 Distributions
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
a20/σζ
0.0
0.5
1.0
1.5
2.0P
(a20/σζ)
Vary σζ′/σζ at fixed σζ and σζ′′/σζ
![Page 53: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/53.jpg)
Future Steps
I Three-dimensional simulations (in progress)
I Multi-field models (additional isocurvature modes,non-attractor, etc.)
I Inclusion of stochastic effects from subhorizon fluctuations(technical challenge)
I Correlated anomalies
![Page 54: Constraining the Ultra-Large Scale Structure of the Universejbraden/Data/Talks/ulss-portsmouth.pdf · A Mosaic of Interesting Dynamics I Strongly inhomogeneous and nonlinear cosmological](https://reader035.fdocuments.us/reader035/viewer/2022071115/5ff2a59faa2f0750b55ad095/html5/thumbnails/54.jpg)
Conclusions
I Fully relativistic treatment of highly inhomogeneous initialconditions for inflation
I Constraints obtained from CMB quadrupole based on fullBayesian analysis
I Gravitational nonlinearities extremely important indetermining final distributions of observables
I Very large amplitude initial inhomogeneities more poorlyconstrained than intermediate amplitudes
I Comoving curvature perturbation ζ approximately GRF inappropriate coordinate system