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

Inhomogeneous Nonlinear Ultra-Large Scale Cosmology

... 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

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)

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)

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

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

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)

History of the Universe

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

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

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

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

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

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

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

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 )

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

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

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

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?

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‖

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

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‖

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‖

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

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

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

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

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‖

]

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)

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

Weighting of Observers

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

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

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

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‖

]

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

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

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

Heuristic Picture of Large Amplitude Effects

Heuristic Picture of Large Amplitude Effects

Modelling With Toy Distributions

Model behaviour with Johnson distribution

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〉)

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

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⟩

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⟩

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

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

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

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