CMB, lensing, and non-Gaussianities Antony Lewis http://cosmologist.info/ Lewis arXiv:1107.5431 Hanson & Lewis arXiv:0908.0963 Lewis, Challinor & Hanson arXiv:1101.2234 Pearson, Lewis & Regan arXiv:1201.1010 Howlett, Lewis, Challinor & Hall arXiv:1201.3654 + (mostly) work by other people Moriond March 2012

Evolution of the universe Opaque Transparent Hu & White, Sci. Am., 290 44 (2004)

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

z=0 θ 14 000 Mpc z~1000

Observed CMB temperature power spectrum WMAP 7 Larson et al, arXiv:1001.4635 Keisler et al, arXiv:1105.3182 Constrain theory of early universe + evolution parameters and geometry Observations

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

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

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

WMAP 7

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:1201.3654 Use other information to break remaining degeneracies, and provide complementary information

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

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

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 )

Millennium simulation

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

+ 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

Primordial local non-Gaussianity e.g. 𝜁= 𝜁 0 1+ 6 5 𝑓 𝑁𝐿 𝜁 0,𝑙 ⇒𝑇∼ 𝑇 𝑔 (1+ 6 5 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

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

Magnified Unlensed Demagnified + shear (shape) modulation

- 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

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..

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)

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)

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 0 1+ 𝜏 𝑁𝐿 1 2 𝜙 ⇒𝑇∼ 𝑇 𝑔 (1+ 𝜏 𝑁𝐿 1 2 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)

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/0108090 Engelen et al, arXiv:1202.0546 + Das et al, arXiv:1103.2124 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

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, 1202.0546 Perotto et al. 2006 WMAP+SPT Δ∑ 𝑚 𝜈 ~0.1𝑒𝑉 (1𝜎) Planck forecast

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

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….

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)

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