Post on 25-Apr-2018
Cosmic Variance from Mode Coupling
Sarah ShanderaPenn State University
Shandera, CAP, 23 Sept 2013
L
Us
Nelson, Shandera, 1212.4550 (PRL);LoVerde, Nelson, Shandera, 1303.3549 (JCAP);Bramante, Kumar, Nelson, Shandera, 1307.5083;
Work in progress (S. Watson; Sohyun Park)
Linde, Muhkanov; Salopek, Bond; Mollerach et al; Boubekeur, Lyth; Byrnes et al;papers related to anomaly...
today: Peloso, Khoury, LoVerde
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
The Plan
• Cosmic Variance from a generalized local ansatz
• Big Picture implications for inflation theory (important if we want to push observers/data analysis)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
gNL = 105
Cosmic Variance and the local ansatz:
LoVerde, Nelson, Shandera, 1303.3549 (JCAP);Nelson, Shandera, 1212.4550 (PRL);
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
gNL = 105
Cosmic Variance and the local ansatz:
LoVerde, Nelson, Shandera, 1303.3549 (JCAP);Nelson, Shandera, 1212.4550 (PRL);
f(ψG)→ φG + fNL(B, f̃NL, g̃NL, . . . )φ2G + gNL(B, f̃NL, g̃NL, . . . )φ3
G + . . .
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Re-interpreting Planck results
No detection of local NG means cosmic variance may not be too large
(whew!)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Re-interpreting Planck results
No detection of local NG means cosmic variance may not be too large
(whew!)
Planck says NG is small? Or, universe is very big?
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Beyond the local ansatz:
Consider a simple scale dependence:
ζ(x) = φG(x) + σG(x) +35fNL � [σG(x)2 − �σG(x)2�]
+925
gNL � σG(x)3 + ...
Bramante, Kumar, Nelson, Shandera, 1307.5083;
fNL(k) = fNL(kp)�
k
kp
�nf
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Subsampling the generalized local ansatz
P obsζ (k) = Pζ(k)
�1 +
125
fNL(k)σGl + 35f2
NL(k)(σ2Gl − �σ2
Gl�)1 + 36
25f2NL(k)�σ2
G(k)�
�,
Shift to observed spectral index
Bramante, Kumar, Nelson, Shandera, 1307.5083;
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
More simply
Different short wavelength modes couple with different strengths to the (locally constant) background:
Locally observed spectral index is biased in subvolumes
SL
Us
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
The spectral index
Bramante, Kumar, Nelson, Shandera, 1307.5083;
(solid lines show 0.5 sigma fluctuations)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
The spectral index, cont’d
Bramante, Kumar, Nelson, Shandera, 1307.5083;
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Model Building Consequences
Bramante, Kumar, Nelson, Shandera, 1307.5083;
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
How to rule cosmic variance of spectral index out .....
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
How to rule cosmic variance of spectral index out .....
......of observational relevance:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
How to rule cosmic variance of spectral index out .....
......of observational relevance:
Observationally rule out any significant blue tilt in f local
NL
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
How to rule cosmic variance of spectral index out .....
Importance of smaller scale probes.
......of observational relevance:
Observationally rule out any significant blue tilt in f local
NL
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Generalizations
ζk = σG,k +�
L−1
d3p1
(2π)3d3p2
(2π)3(2π)3δ3(p1 + p2 − k)F (p1, p2, k)σG,p1σG,p2 + ...,
Effect of a general bispectrum on the power spectrum:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Generalizations
ζk = σG,k +�
L−1
d3p1
(2π)3d3p2
(2π)3(2π)3δ3(p1 + p2 − k)F (p1, p2, k)σG,p1σG,p2 + ...,
Effect of a general bispectrum on the power spectrum:
Equilateral bispectrum has no effectBispectrum can weight IR modes differently
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Generalizations
If tensor modes have an independent clock: same story (eg, non Bunch Davies)
ζk = σG,k +�
L−1
d3p1
(2π)3d3p2
(2π)3(2π)3δ3(p1 + p2 − k)F (p1, p2, k)σG,p1σG,p2 + ...,
Effect of a general bispectrum on the power spectrum:
Equilateral bispectrum has no effectBispectrum can weight IR modes differently
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scale-dependence in the bispectrum
ξs,m(k) = ξs,m(kp)�
k
kp
�n(s),(m)f
Ratio of contributions of each field
Self-interactions of one field
BΦ(k1,k2,k3) = ξs(k3)ξm(k1)ξm(k2)PΦ(k1)PΦ(k2) + 5 perm .
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
But...
nsq. ≡d lnBζ(kL, kS , kS)
d ln kL− (ns − 1)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
But...
nsq. ≡d lnBζ(kL, kS , kS)
d ln kL− (ns − 1)
Just from shift to power spectrum:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
But...
nsq. ≡d lnBζ(kL, kS , kS)
d ln kL− (ns − 1)
Just from shift to power spectrum:
∆nsq.(k) ≡ nobssq. (k)− nLargeVol.
sq. (k)
≈ −65fNL(kL)σGl nf
1 + 65fNL(kL)σGl
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
But...
nsq. ≡d lnBζ(kL, kS , kS)
d ln kL− (ns − 1)
Just from shift to power spectrum:
Allowing gNL(k) adds another shift
∆nsq.(k) ≡ nobssq. (k)− nLargeVol.
sq. (k)
≈ −65fNL(kL)σGl nf
1 + 65fNL(kL)σGl
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scaling Patterns
Self-interactions; eg, local ansatz
Mn,R =�δn
R�c
�δ2R�n/2
c
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scaling Patterns
Self-interactions; eg, local ansatz
Hierarchical M(h)n = n! 2
n−3
�M(h)
36
�n−2
Mn,R =�δn
R�c
�δ2R�n/2
c
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scaling Patterns
Self-interactions; eg, local ansatz
Hierarchical M(h)n = n! 2
n−3
�M(h)
36
�n−2
Mn,R =�δn
R�c
�δ2R�n/2
c
∝ fNLP1/2ζ
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scaling Patterns
Extra source, eg, gauge field
(Barnaby, Shandera; 1109.2985)
Self-interactions; eg, local ansatz
δA
Hierarchical M(h)n = n! 2
n−3
�M(h)
36
�n−2
Mn,R =�δn
R�c
�δ2R�n/2
c
∝ fNLP1/2ζ
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Scaling Patterns
Extra source, eg, gauge field
(Barnaby, Shandera; 1109.2985)
Self-interactions; eg, local ansatz
δA
Feeder M(f)n = (n− 1)! 2n−1
�M(f)
38
�n/3
Hierarchical M(h)n = n! 2
n−3
�M(h)
36
�n−2
Mn,R =�δn
R�c
�δ2R�n/2
c
∝ fNLP1/2ζ
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Hybrid
(Quasi-single field)
M(h)n ∝ (A)n (B)n−2
Scaling Patterns, cont’d
(Chen, Wang)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Cosmic Variance and scaling patterns?
Φ(�x) = φ(�x) + ψG(�x) + f̃NL
�ψG(�x)2 − �ψG(�x)2�
�0
Gives `feeder’ scaling non-Gaussianity
But: in biased subvolumes, back to hierarchical
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Cosmic Variance and scaling patterns?
Φ(�x) = φ(�x) + ψG(�x) + f̃NL
�ψG(�x)2 − �ψG(�x)2�
�0
Gives `feeder’ scaling non-Gaussianity
But: in biased subvolumes, back to hierarchical
Biased subvolumes: Generate whole local ansatz series, with bias B controlling size of terms
Φ(�x) = φ(�x) + f̃NLψG(�x)p + . . .
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Power spectrum:
Inflation
Spectrum of scale-invariant, adiabatic, super horizon modes as “initial conditions”
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Power spectrum:
Inflation
Spectrum of scale-invariant, adiabatic, super horizon modes as “initial conditions”
Bispectrum, trispectrum, etc?
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Power spectrum:
Inflation
Spectrum of scale-invariant, adiabatic, super horizon modes as “initial conditions”
Bispectrum, trispectrum, etc? Mathematically independent....
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Power spectrum:
Inflation
Spectrum of scale-invariant, adiabatic, super horizon modes as “initial conditions”
Bispectrum, trispectrum, etc? Mathematically independent....
....up to relationships of the type τNL ≥ (6/5fNL)2
(Smith, LoVerde, Zaldarriaga)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Power spectrum:
Inflation
Spectrum of scale-invariant, adiabatic, super horizon modes as “initial conditions”
Bispectrum, trispectrum, etc?
Perturbation theory that obeys usual (effective) field theory notions: patterns
Non-zero NG, but no patterns in shape or size
Mathematically independent....
....up to relationships of the type τNL ≥ (6/5fNL)2
(Smith, LoVerde, Zaldarriaga)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Mode Coupling Cosmic Var:Derivations didn’t depend on where you got the fluctuations:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Mode Coupling Cosmic Var:Derivations didn’t depend on where you got the fluctuations:
Good: can we find a “natural” pattern inflation does not predict? (If not, is that bad for inflation?)
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Mode Coupling Cosmic Var:Derivations didn’t depend on where you got the fluctuations:
Good: can we find a “natural” pattern inflation does not predict? (If not, is that bad for inflation?)
Bad: a random pattern in a big universe may not look random to local observers
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
A familiar aspect of inflation, dressed up in different clothes: IC problem for our observable 60 e-folds region
Mode Coupling Cosmic Var:Derivations didn’t depend on where you got the fluctuations:
Good: can we find a “natural” pattern inflation does not predict? (If not, is that bad for inflation?)
Bad: a random pattern in a big universe may not look random to local observers
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
What about describing only our Hubble patch?
• Apply cosmological principle
• eg, 60 e-folds + non-Bunch Davies is fine tuned.
S =�
d4x√−gL(Xi,φi, A
µj , . . . )Local:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
What about describing only our Hubble patch?
• Apply cosmological principle
• eg, 60 e-folds + non-Bunch Davies is fine tuned.
With local type mode-coupling, this line is blurred: observer’s version of the landscape needed
S =�
d4x√−gL(Xi,φi, A
µj , . . . )Local:
Monday, September 23, 2013
Shandera, CAP, 23 Sept 2013
Easy to apply this to other models to check what level of confusion can occur (quasi-single field in progress)
More work:
Monday, September 23, 2013