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Transcript of Cosmic Variance from Superhorizon Mode Coupling -...
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):
2/8
⇣(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
⇣(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
Monday, January 13, 2014 Elliot Nelson, Penn State6/8
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