Cosmic Variance from Superhorizon Mode Coupling

Elliot Nelson Institute for Gravitation and the Cosmos

The Pennsylvania State University !EN & S. Shandera, 1212.4550 (PRL) M. LoVerde, EN, & S. Shandera, 1303.3549 (JCAP) J. Bramante, J. Kumar, EN, & S. Shandera, 1307.3549 (JCAP) EN, S. Park, S. Shandera, work in progress. !!Related work: Nurmi, Byrnes, & Tasinato, 1301.3128 Nurmi, Byrnes, Tasinato, & Wands, 1306.2370 LoVerde, 1310.5739

Monday, January 13, 2014 Elliot Nelson, Penn State

Observable universe = finite subvolume of the larger universe

Finite sample: volume, range of scales, number of galaxies, etc.

Is Our Universe a Biased Sample?

Monday, January 13, 2014 Elliot Nelson, Penn State

Statistics Question: Is our Hubble volume a representative or biased sample? How limited is our access to global statistics?

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Is Our Universe a Biased Sample?

average over observable universe global statisticssingle roll of the dice

c

H0

correlated modes

L

⇣

⇣obs.

h⇣k1 ⇥ · · · ⇣kN i|⇣L 6= h⇣k1 ⇥ · · · ⇣kN i|0

M ⇠

...Local statistics biased:

Observable modes coupled to superhorizon modes...

= haiM (1 + ⇣

obs

(x)) , x 2 VolM

haiL(1 + ⇣l)=

a(x) = haiL(1 + ⇣(x)) , x 2 VolL

Monday, January 13, 2014 Elliot Nelson, Penn State

⇣Consider the primordial curvature perturbation on a fixed spatial slice (at end of inflation):

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⇣(x) = ⇣G(x) +3

5fNL(⇣

2G(x)� h⇣2Gi) +

9

25gNL⇣

3G(x) + · · ·

fobs

NL

= fNL

� 12

5f2

NL

⇣Gl +9

5gNL

⇣Gl + . . .

P obs

⇣ (k) =

1 +

12

5fNL⇣Gl + . . .

�PG(k)

Case Study: Local Non-Gaussianity

Perturbations in subvolume:

Local statistics modulated by the local background:

⇣obs(x) = ⇣obsG (x) +3

5fobs

NL

[⇣obsG (x)2 � h⇣obsG (x)2i] + 9

25gobsNL

⇣obsG (x)3 + . . .

(1 +6

5fNL⇣Gl)⇣Gs + · · ·

=

Z M�1

L�1

d3k

(2⇡)3⇣G(k)e

ik·x +

Z kmax

M�1

d3k

(2⇡)3⇣G(k)e

ik·x

k < M�1

Long-short wavelength split:

⇣G ⌘ ⇣Gs + ⇣Gl

⇣Gl > 0

⇣Gl < 0

Nearly Gaussian:

[M. LoVerde, EN, & S. Shandera, 1303.3549] [Nurmi, Byrnes, & Tasinato, 1301.3128]

L

k < M�1

k < M�1

Monday, January 13, 2014 Elliot Nelson, Penn State3/8

⇣(x) = ⇣G(x) +3

5fNL(⇣

2G(x)� h⇣2Gi) +

9

25gNL⇣

3G(x) + · · ·

fobs

NL

= fNL

� 12

5f2

NL

⇣Gl +9

5gNL

⇣Gl + . . .

P obs

⇣ (k) =

1 +

12

5fNL⇣Gl + . . .

�PG(k)

Case Study: Local Non-Gaussianity

Perturbations in subvolume:

Local statistics modulated by the local background:

⇣obs(x) = ⇣obsG (x) +3

5fobs

NL

[⇣obsG (x)2 � h⇣obsG (x)2i] + 9

25gobsNL

⇣obsG (x)3 + . . .

(1 +6

5fNL⇣Gl)⇣Gs + · · ·

=

Z M�1

L�1

d3k

(2⇡)3⇣G(k)e

ik·x +

Z kmax

M�1

d3k

(2⇡)3⇣G(k)e

ik·x

k < M�1

Long-short wavelength split:

⇣G ⌘ ⇣Gs + ⇣Gl

⇣Gl > 0

⇣Gl < 0

Nearly Gaussian:

[M. LoVerde, EN, & S. Shandera, 1303.3549] [Nurmi, Byrnes, & Tasinato, 1301.3128]

L

k < M�1

k < M�1

Monday, January 13, 2014 Elliot Nelson, Penn State3/8

Planck satellite measurements:

Planck CMB Measurements

f local

NL

= 2.7± 5.8[Red tilt]ns = 0.9603± 0.0073

[Nearly Gaussian perturbations]

Monday, January 13, 2014 Elliot Nelson, Penn State4/8

Gaussian Patches in an NG Universe

Strongly NG field can appear nearly Gaussian:

⇣(x) = ⇣pG(x)� h⇣pGi

⇣obs(x) = p⇣p�1

Gl ⇣Gs(x) +p!

2!(p� 2)!⇣p�2

Gl (⇣2Gs(x)� h⇣2Gsi) + ...

3

5fobs

NL

⇡ p� 1

2p⇣pGl

⇠ 1

⇣l

Constrain superhorizon cosmic variance

by tightening boundsfobs

NL

Recover statistically natural, nearly Gaussian series, in typical subvolumes; series controlled by

[EN, S. Shandera, 1212.4550; LoVerde et. al., 1303.3549]

!

Minimum level of NG in subvolumes:

!

h⇣2Gsi1/2/⇣Gl

⇣Gl �q

h⇣2Gsi $L

M� k

max

M $ Nsuperhorizon

� Nsubhorizon

Monday, January 13, 2014 Elliot Nelson, Penn State5/8

⇣ = ⇣pG � h⇣pGi

Numerical Realizations: Suppression of : :

Gaussian Patches in an NG Universe

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Biased Scale Dependence:

fNL(k) = fNL(kp)

✓k

kp

◆nf

Weak scale-dependence:

Biased power spectrum:

Generalized Local Ansatz:

P obs

⇣ (k) = P⇣(k)

1 +

12

5⇠m(k)

fNL

(k)�Gl +3

5

f2

NL

(k)(�2

Gl � h�2

Gli)1 + 36

25

f2

NL

(k)h�2

Gli

�

⇣(k) = �G(k) + �G(k) +3

5fNL(k)[�

2G](k) +

9

25gNL(k)[�

3G](k) + ...

nobs.

s 6= ns

�ns(k) ⇡12

5⇠m(k)fNL(k)�Gl(nf + n(m)

f ),6

5fNL(k)�Gl ⌧ 1

Running in global bispectrum leaks into observed power spectrum

Monday, January 13, 2014 Elliot Nelson, Penn State7/8

Summary

Local statistics can be biased by background modes Constraints on global statistics ( ) are probabilistic

Parameters from models need only render our observations typical among Hubble volumes

Bias grows with size of universe ~ length of inflation

Local and global statistics can be qualitatively different

Nearly Gaussian Hubble volumes in a NG universe Statistical naturalness of weakly NG local ansatz

Observations could tell us if this uncertainty is relevant to our interpretation of data

Ongoing work: bias from models of inflation, subsampling with nonlocal NG

P⇣ , fNL, ns, nt, nsq.

Monday, January 13, 2014 Elliot Nelson, Penn State8/8