constraining the ultra-large scale structure of the...

Constraining the Ultra-Large Scale Structure ofthe Universe Using Numerial Relativity

Jonathan Braden

University College London

GRG21, Columbia University, New York, July 12, 2016

w/ Hiranya Peiris, Matthew Johnson, and Anthony Aguirre

based on arXiv:1604.04001 and in progress

Ultra-Large Scale Structure

Local Remnants of Ultra-Large Scale Structure?

IStructure present at start of inflation

IConversion of structure during or after inflation

Modelling Initial Conditions

Monte Carlo Sampling: Planar Symmetry

ds2 = �d⌧2

+ a2k(x , ⌧)dx2 + a2?(x , ⌧)�dy2 + dz2

�

Inflaton on ak(⌧ = 0) = 1 = a?(⌧ = 0)

�(x) = ¯� + � ˆ�

¯� gives N e-folds 3H2

I

⌘ V (

¯�)

Field Fluctuations

� ˆ�(xi ) = A�

X

n=1

ˆGe iknxipP(kn) ˆG =

q�2 ln

ˆ�e2⇡i↵

P(k) = ⇥(kmax

� k) H�1

I

kmax

= 2⇡p3

Model Choices

V (�) = V0

✓1 � e

�q

2

3↵�

MP

◆2

I ↵ ⌧ 1 ! small-field model

I ↵ = 1 ! Starobinski

I ↵ � 1 ! m2�2/2

Model Choices

V (�) = V0

✓1 � e

�q

2

3↵�

MP

◆2

0 1 2 3 4 5 6

�/MP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

V(�)

⇥10�11

↵ = 0.1



I ↵ � 1 ! m2�2/2

Model Choices

V (�) = V0

✓1 � e

�q

2

3↵�

MP

◆2

0 2 4 6 8 10 12

�/MP

0.0

0.2

0.4

0.6

0.8

1.0

V(�)

⇥10�10

↵ = 1.0



I ↵ � 1 ! m2�2/2

Model Choices

V (�) = V0

✓1 � e

�q

2

3↵�

MP

◆2

0 5 10 15 20 25 30

�/MP

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

V(�)

⇥10�9

↵ = 100.0



I ↵ � 1 ! m2�2/2

Model Choices

V (�) = V0

✓1 � e

�q

2

3↵�

MP

◆2

0 5 10 15 20 25 30 35

�/MP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

V(�)

⇥10�8

↵ = 100000.0



I ↵ � 1 ! m2�2/2

Cosmological Observables Encoded in ⇣ = � ln a

Cosmological Observables Encoded in ⇣ = � ln a

a ⌘ det(�ij)1/6

=

�aka

2

?�1/3

H ⌘ �1

3

� ijKij = �1

3

(K xx + 2K y

y )

Required Evolution

0 50 100 150 200

HI

x/p

3

�0.06

�0.04

�0.02

0.00

0.02

0.04

0.06

ln

a

0 50 100 150 200

HI

x/p

3

0.934

0.936

0.938

0.940

0.942

0.944

0.946

�/M

P

0 50 100 150 200

HI

x/p

3

0.000000

0.000002

0.000004

0.000006

0.000008

0.000010

0.000012

0.000014

H

+5.773⇥10�1

Initial Conditions (⌧ = 0)

Required Evolution

0 50 100 150 200

HI

x/p

3

0.112

0.114

0.116

0.118

0.120

0.122

0.124

0.126

�/M

P

0 50 100 150 200

HI

x/p

3

0.3750.3800.3850.3900.3950.4000.4050.4100.4150.420

H

End of Inflation (✏H = �d lnH/d ln a = 1)

Solving Einstein’s Equations for a Self-Gravitating Inflaton

Machine precision accuracy

IGauss-Legendre time-integrator (O(dt10), symplectic)

IFourier pseudospectral discretisation (exponential

convergence)

Fast to allow sampling

IAdaptive time-stepping

IAdaptive grid spacing

O(1s � 10s) to evolve through 60 e-folds of inflationMachine precision convergence and constraint preservations

Observational Constraints

Pr(A�,HILobs|C obs

2

, . . . ) / L(A�,HILobs)Pr(A�,HILobs| . . . )L = Pr(C obs

2

|A�,HILobs, . . . )

I A� : Fluctuation Amplitude P(k) / A2

�I HILobs : Uncertain post-inflation expansion history

I . . . : V (�), spectrum shape, IC hypersurface, Chigh�`2

, etc.

Planck measured C`

C obs

2

= 253.6µK 2 Chigh�`2

= 1124.1µK 2

Numerical GR Input

Pr

⇣ˆC2

|A�,HILobs, . . .⌘

Observer Weighting

Evaluation of CMB Quadrupole

ˆC2

=

1

5

⇣a(UL)

20

+ a(Q)

20

⌘2

+

Pm=2

m=�2,m 6=0

⇣a(Q)

2m

⌘2

�

a(Q)

2m : Gaussian with h(a(Q)

2m )

2i = 1124.1µK 2


ˆC2

=

1

5

⇣a(UL)

20

+ a(Q)

20

⌘2

+

Pm=2

m=�2,m 6=0

⇣a(Q)

2m

⌘2

�

a(Q)

2m : Gaussian with h(a(Q)

2m )

2i = 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

]


ˆC2

=

1

5

⇣a(UL)

20

+ a(Q)

20

⌘2

+

Pm=2

m=�2,m 6=0

⇣a(Q)

2m

⌘2

�

a(UL)

20

: NR simulations

�2 �1 0 1 2

a(UL)

20

/�(UL)

0

1

2

3

4

P⇣ a

(U

L)

20

/�(U

L)

|A�,.

..⌘

↵ =

2

300

A�

1 ⇥ 10

�5

3 ⇥ 10

�5

6 ⇥ 10

�5

Dependence of C2

on Model Parameters

0 1500 3000 4500

ˆC2

[µK

2

]

0.0

0.2

0.4

0.6

0.8

P⇣

ˆ C2

|A�,

�(UL)

�(Q

),.

..⌘

⇥10�3

�(UL)

�(Q) = 4

A�

1 ⇥ 10

�5

3 ⇥ 10

�5

6 ⇥ 10

�5

�2

dof=5

a(UL)

20

⇡ F (HILobs)2 @xpxp⇣ ⇡ F (HILobs)

2

1

ak@@x

⇣1

ak@x⇣

⌘

Final Posterior

�5 �4 �3 �2 �1 0

log

10

(HI

Lobs

)

�6

�5

�4

�3log

10

(A

�)

↵ =

2

300

0

2

4

6

8

P(log

10

A�,l

og

10

HI

Lobs

|ˆ Cobs

2

)⇥10�4

Significant deviations from Gaussian approximation

Marginalised Constraints on Model Parameters: Amplitude

�6.0 �5.5 �5.0 �4.5 �4.0 �3.5 �3.0log

10

(A�)

0.00

0.02

0.04

0.06

0.08

0.10

P

⇣ log

10

(A

�)|ˆ C

obs

2

⌘

↵ =

2

300

Numerical GR

Gaussian

Heuristic Explanation: Initial Conditions

Heuristic Explanation: End-of-Inflation

Analytic Approximation

⇣ and comoving derivatives nearly Gaussian

�4 �3 �2 �1 0 1 2 3 4

⇣/��

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

P(⇣/�

�)

�4 �3 �2 �1 0 1 2 3 4

⇣ ��/��

0.000.050.100.150.200.250.300.350.400.45

P(⇣

��/�

��)

Treat as Gaussian random field

Large-Scale Approximation for a20

a20

(x0

) ⇡ Ae�2⇣(x0

)

�⇣ 00(x

0

) � ⇣ 0(x0

)

2

�

Anaytic a20

Distributions

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.5

a20

/��

0.0

0.5

1.0

1.5

2.0

2.5

3.0P

(a20

/�

�)

Vary �⇣ at fixed �⇣(p)/�⇣

Conclusions

INumerical relativity is a useful framework for making

cosmological predictions

ISometimes it is a necessary tool (deviations from Gaussianity)

IRobust qualitative conclusions over a variety of inflationary

models

IInflation is e↵ective at hiding large amplitude initial

fluctuations

IGaussianity of ⇣ in comoving coordinates suggests analytic

approach in 3D

constraining the ultra-large scale structure of the...

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