1. CMB, lensing, and non-Gaussianities
Antony Lewis
Lewis arXiv:
Hanson & Lewis arXiv: Lewis, Challinor & Hanson arXiv:
Pearson, Lewis & Regan arXiv: Howlett, Lewis, Challinor & Hall arXiv:
+ (mostly) work by other people
Moriond March 2012

2. Evolution of the universe
Opaque
Transparent
Hu & White, Sci. Am., (2004)

3. CMB temperature Last scattering surface End of inflation
gravity+ pressure+ diffusion
10 −5 perturbations ⇒ Linear theory predictions very accurate
⇒ Gaussian fluctuations from inflation remain Gaussian

4. z=0 θ
Mpc
z~1000

5. Observed CMB temperature power spectrum
WMAP 7
Larson et al, arXiv:
Keisler et al, arXiv:
Constrain theory of early universe + evolution parameters and geometry
Observations

6. e.g. Geometry: curvature
closed
flat θ θ
We see:

7. or is it just closer?? flat flat θ θ We see:
Degeneracies between parameters

8. Sample from 𝑃 {𝜃} 𝑑𝑎𝑡𝑎) (sample all possible model universes given the data)
Samples in 6D parameter space
(e.g. using MCMC, adaptive importance sampling, etc. CosmoMC, PMC, MultiNest…).
Density of samples ⇒ relative probability of that region of parameter space

9. WMAP 7

10. Constrain some combinations of parameters accurately
Acoustic-scale degeneracy: Ω 𝑚 ℎ 3 ∼ constant Assume flat, w=-1
Planck forecast
Howlett, Lewis, Challinor & Hall arXiv:
Use other information to break remaining degeneracies, and provide complementary information

11. Beyond the power spectrum
Flat sky approximation: Θ 𝑥 = 1 2𝜋 ∫ 𝑑 2 𝑙 Θ(𝑙)𝑒 𝑖𝑥⋅ 𝑙
(Θ=𝑇)
Gaussian + statistical isotropy:
Θ 𝑙 1 Θ 𝑙 2 =𝛿 𝑙 1 + 𝑙 2 𝐶 𝑙 - power spectrum encodes all the information
- modes with different wavenumber are independent

12. Bispectrum Trispectrum N-spectra…
Non-Gaussianity – general possibilities
Bispectrum
𝑙 3
𝑙 1
𝒍 𝟏 + 𝒍 𝟐 + 𝒍 𝟑 =𝟎
𝑙 2
1 2𝜋 𝛿 𝑙 1 + 𝑙 2 + 𝑙 3 𝑏 𝑙 1 𝑙 2 𝑙 3
Flat sky approximation:
〈Θ 𝑙 1 Θ 𝑙 2 Θ( 𝑙 3 )〉=
If you know Θ 𝑙 1 , Θ 𝑙 2 , sign of 𝑏 𝑙 1 𝑙 2 𝑙 3 tells you which sign of Θ 𝑙 3 is more likely
Trispectrum
𝑙 4
𝑙 1 𝐿
𝑙 3
N-spectra…
𝑙 2

13. Equilateral
𝑘 1 + 𝑘 2 + 𝑘 3 =0, 𝑘 1 = 𝑘 2 =| 𝑘 3 |
𝑘 3
𝑘 1
𝑘 2 +
b>0
𝑇( 𝑘 3 ) = + +
b<0
𝑇( 𝑘 1 )
𝑇( 𝑘 2 )
−𝑇( 𝑘 3 )

14. Millennium simulation

15. Near-equilateral to flattened:
b>0
b<0
𝑘 1
𝑘 3
𝑘 2

16. + Local (squeezed) 𝑘 1 + 𝑘 2 + 𝑘 3 =0, 𝑘 1 ≪ 𝑘 2 , 𝑘 3 𝑘 2 ∼− 𝑘 3 𝑘 2
𝑘 1 + 𝑘 2 + 𝑘 3 =0, 𝑘 1 ≪ 𝑘 2 , 𝑘 3
𝑘 2 ∼− 𝑘 3
𝑘 2
𝑘 3
𝑘 1
b>0 + +
𝑇(𝑘 1 ) = +
𝑇(𝑘 2 )
T( 𝑘 3 )
b<0
−𝑇(𝑘 1 )
Squeezed bispectrum is a correlation of small-scale power with large-scale modes

17. Primordial local non-Gaussianity
e.g. 𝜁= 𝜁 𝑓 𝑁𝐿 𝜁 0,𝑙
⇒𝑇∼ 𝑇 𝑔 ( 𝑓 𝑁𝐿 𝜁 ∗,𝑙 )
Liguori et al 2007
Single-field slow-roll inflation 𝑓 𝑁𝐿 ∼ 𝑛 𝑠 −1 ≪1
⇒ Any significant detection would rule out large classes of inflation models
New information that is not present in the power spectrum

18. e.g. CMB Lensing: modulation due to large-scale gravitational lenses
Last scattering surface
Inhomogeneous universe
- photons deflected
Observer

19. Magnified
Unlensed
Demagnified
+ shear (shape) modulation

20. - Non-Gaussian statistically isotropic temperature distribution
Marginalized over modulation field X (unobservable directly) : 𝑇∼∫𝑃(𝑇,𝑋)𝑑𝑋
- Non-Gaussian statistically isotropic temperature distribution
Power spectra of lensed CMB
(no longer contains all the information) TT EE BB

21. Reconstructing the modulation field
For a given (fixed) modulation field: 𝑇∼𝑃(𝑇|𝑋)
- Anisotropic Gaussian temperature distribution
- Modes correlated for 𝐤 2 ≠ 𝐤 3
Model-dependent function you can calculate for 𝑋(𝐊)=0
Can reconstruct the modulation field 𝑋!
- For small modulations can construct “optimal” QML estimator 𝑋 𝐾 by summing filtered fields appropriately over 𝑘 2 , 𝑘 3
𝑋 𝐾 ∼𝑁( 𝐤 2 , 𝐤 3 𝐴 𝐾, 𝑘 2 , 𝑘 3 𝑇 𝐤 2 𝑇 𝐤 3 −mean field)
Zaldarriaga, Hu, Hanson, etc..

22. Lensing: (nearly) optimal reconstruction of CMB lensing potential, 𝜓 𝑙𝑚
𝐴 𝐿, 𝑙 1 , 𝑙 2 ∼
True (simulated)
Reconstructed (Planck noise, Wiener filtered)
(Credit: Duncan Hanson)

23. How to detect squeezed non-Gaussianity?
1. Reconstruct the modulation field 𝑋 from quadratic combinations of small-sale modes
2. Correlate it with large-scale 𝑇 to measure bispectrum
Correlation of modulation with large-scale T
e.g. primordial local non-Gaussianity 𝑓 𝑁𝐿 ≠0: 𝑋= 6 5 𝑓 𝑁𝐿 𝜁 ∗
(this is a bit oversimplified for 𝑙 1 >40… but right idea)

24. 2b. Calculate modulation power spectrum: measures trispectrum
Power spectrum of modulation field
e.g. primordial local non-Gaussianity 𝜏 𝑁𝐿 ≠0
e.g. 𝜁= 𝜁 𝜏 𝑁𝐿 𝜙
⇒𝑇∼ 𝑇 𝑔 (1+ 𝜏 𝑁𝐿 𝜙 ∗ )
𝜏 𝑁𝐿 𝐶 𝐿 𝜙 ∗ ( 𝐶 𝑙 1 + 𝐶 𝑙 2 )( 𝐶 𝑙 3 + 𝐶 𝑙 4 )
Reduced trispectrum ∼
Power spectrum of the modulation field at recombination
(this is an accurate practical method for 𝜏 𝑁𝐿 because almost all the signal is in very squeezed shapes, 𝐿<10)

25. From reconstructed 𝜓 𝒍𝒎 can estimate lensing power spectrum 𝐶 𝑙 𝜓𝜓 (∝ trispectrum)
Reduced trispectrum ∼
Real data! South Pole Telescope CMB lensing reconstruction
Forecasts
Hu: astro-ph/
Engelen et al, arXiv:
+ Das et al, arXiv:
Full non-Gaussian analysis ≈ Power spectrum analysis with 𝐶 𝑙 𝑇𝑇 𝐶 𝑙 𝐸𝐸 𝐶 𝑙 𝑇𝐸 𝐶 𝑙 𝐵𝐵 𝐶 𝑙 𝜓𝜓 𝐶 𝑙 𝑇𝜓 𝐶 𝑙 𝐸𝜓

26. What does an estimate of 𝐶 𝑙 𝜓𝜓 do for us?
Probe 0.5≤𝑧≤6: depends on geometry and matter power spectrum
break degeneracies in the linear CMB power spectrum
- Better constraints on neutrino mass, dark energy, Ω 𝐾 , …
Neutrino mass fraction with and without lensing (Planck only)
Engelen et al,
Perotto et al. 2006
WMAP+SPT
Δ∑ 𝑚 𝜈 ~0.1𝑒𝑉 (1𝜎)
Planck forecast

27. Conclusions
CMB is still by far the cleanest probe of early-universe physics. Accurately constrain some combinations of cosmological parameters.
Linear power spectrum constraints subject to degeneracies, and miss information if the fluctuations not Gaussian, homogeneous and isotropic
Non-Gaussianities contain more information. Squeezed shapes can be studied by reconstructing modulations with quadratic estimators.
- Primordial non-Gaussianity
- Potentially powerful way to refute classes of simple inflation models - No detection as yet, may be very small
- CMB Lensing – definitely there
- Large trispectrum; equivalently reconstruct the lensing potential
Break primary CMB degeneracies, improve parameters
- Well-detected by ACT & SPT, already becoming useful
- Planck and polarization will improve things significantly
Future: digging out small B-mode signal may require removal of lensing B-modes

28. Why lensing is important
Known effect, small but significant amplitude (∼ 10 −3 )
Modifies the power spectra on small-scales (∼ 10 −2 )
Lensing of E gives B-mode polarization (confusion for tensors/strings)
Anisotropy/non-Gaussianities….

29. e.g. CMB lensing
CMB temperature
Lensed CMB
Lensing deflection angles
𝐴 𝐿, 𝑙 1 , 𝑙 2 ∼
Anisotropic
(anisotropic ⇒ easily distinguished from scalar primordial non-Gaussianity)

30. Deflection angle power spectrum
Non-linear
Linear
Deflections O(10-3), but coherent on degree scales  important!