The inconvenience of a single Universe. J.-F. Cardoso, C.N.R.S. - PowerPoint PPT Presentation

The inconvenience of a single Universe. J.-F. Cardoso, C.N.R.S. Institut d’Astrophysique de Paris - The photons who come in cold - A single Universe Mathematics of Imaging Workshop #2 Institut Henri Poincaré, March 2019

. The inconvenience of a single planet .

The inconvenience of a single planet There is no planet B.

. The inconvenience of a single Universe .

. The photons who come in cold: Big Bang theory .

The Planck mission from the European Spatial Agency

Cosmic background (and pesky foregrounds)

Extracting the CMB from Planck frequency channels Color scale: hundreds of micro-Kelvins. Credits: ESA, FRB.

Multipole decomposition and angular frequencies • A spherical field X ( θ, φ ) can be decomposed into ‘harmonic’ components: � X ( ℓ ) ( θ, φ ) X ( θ, φ ) = [ θ, φ ] = [ (co)latitude, longitude ] ℓ ≥ 0 called monopole, dipole, quadrupole, octopole, . . . , multipole, indexed by the (discrete) angular frequency, ℓ = 0 , 1 , 2 , . . . , thusly: = + + + + + · · · X ( θ, φ ) = X (0) ( θ, φ ) + X (1) ( θ, φ ) + X (2) ( θ, φ ) + X (3) ( θ, φ ) + · · · • Projection onto the (2 ℓ + 1) -dimensional spherically invariant subspaces. • Distribution of energy across (angular) scales quantified by �� 1 def S 2 X ( ℓ ) ( θ, φ ) 2 . � the (empirical) angular spectrum: = C ℓ 2 ℓ + 1

Fourier on the sphere: Spherical harmonic decomposition • An ortho-basis for spherical fields: the spherical harmonics Y ℓm ( θ, φ ) : � � � � X ( θ, φ ) = a ℓm Y ℓm ( θ, φ ) ← → a ℓm = φ Y ℓm ( θ, φ ) X ( θ, φ ) θ ℓ ≥ 0 − ℓ ≤ m ≤ ℓ • Multipole decomposition and angular spectrum: m = − ℓ m = − ℓ � � 1 X ( ℓ ) ( θ, φ ) = a 2 � a ℓm Y ℓm ( θ, φ ) C ℓ = ℓm 2 ℓ + 1 m = ℓ m = ℓ

Angular spectrum of the CMB (as measured/fitted by W-MAP) • Large scales dominate. We plot: D ( ℓ ) = C ( ℓ ) × ℓ ( ℓ + 1) / 2 π • Acoustic peaks !!! • One Universe: cosmic variance. � 1 a 2 � If C ℓ = ℓm , 2 ℓ + 1 − ℓ ≤ m ≤ ℓ � � � 2 C ℓ / E � then Var C ℓ = 2 ℓ + 1 .

Theoretical angular spectrum of the CMB A cosmological model predicts the angular spectrum of the CMB as a function of “cosmo- logical parameters”. Examples of the dependence of the spectrum on some parame- ters of the Λ − CDM model.

Angular spectrum and likelihood (ideally) • The spherical harmonic coefficients a ℓm of a stationary random field are uncorrelated with variance C ℓ , defining the angular power spectrum: � � E a ℓm a ℓ ′ m ′ = C ℓ δ ℓℓ ′ δ mm ′ • Thus, for a stationary Gaussian field, the empirical spectrum m = − ℓ � 1 a 2 � C ℓ = ℓm 2 ℓ + 1 m = ℓ is a sufficient statistic since the likelihood then reads: � � � � C ℓ − 2 log P ( X |{ C ℓ } ) = (2 ℓ + 1) + log C ℓ + cst C ℓ ℓ ≥ 0 • Also reads like a spectral mismatch: � � � � � � C ℓ C ℓ k ( u ) def + log C ℓ = k + cst’ = u − log u − 1 ≥ 0 C ℓ C ℓ

. Extracting the CMB .

Extracting the CMB from Planck frequency channels How to do it?

Some requirements for producing a CMB map • The method should be robust, accurate and high SNR (obviously). Special features: data set is expensive and there is ground truth. • The method should be linear in the data: 1. It is critical not to introduce non Gaussianity 2. Propagation of simulated individual inputs should be straightforward • The method should be able to support wide dynamical ranges, over the sky, over angular frequencies, across channel frequencies. • (Wo.manpower behind the method should be aplenty).

Wide dynamics over the sky Left: The W-MAP K band. Natural color scale [-200, 130000] µK . Middle: Same map with an equalized color scale. Right: Same map with a color scale adapted to CMB: [-300, 300] µK . Average power as a function of latitude on a log scale for the same map.

Wide spectral dynamics, SNR variations 10 4 30 GHz 44 GHz 70 GHz 100 GHz 10 2 143 GHz 217 GHz 353 GHz 545 GHz 857 GHz 10 0 10 -2 10 -4 10 -6 0 1000 2000 S & N angular spectra in Planck channels (unbeamed) for f sky = 0 . 40 .

And what about the noise ? Local RMS ( µK ) of the noise in a reconstructed CMB map.

Foregrounds Various foreground emissions (both galactic and extra-galactic) pile up in front of the CMB. But they do so additively ! Noise Even better, most scale rigidly with frequency: each Dust frequency channel sees a different mixture of each astrophysical emission: Synchrotron   d 30 Free-free  .  . d = .  = As + n  Galaxies d 857 Such a linear mixture can be inverted . . . if the mixing Gal clusters matrix A is known. How to find it or do without it ? Gal clusters 1 Trust astrophysics and use parametric models, or CMB 2 Trust your data and the power of statistics.

Three contamination models based on matrices (or lack thereof) 1 � Nine Planck channels modeled as noisy linear mixtures of CMB and 6 (say) “foregrounds”         d 1 a 1 F 11 . . . F 16 n 1 s       � � d 2 a 2 F 21 . . . F 26 n 2         f 1 s . . . . .  .   . . .     .  . . . . . = . . . ×  + or d = [ a | F ] + n .      .   . f . . . . .  .   . . .   .  . . . . . . . . f 6 d 9 a 9 F 91 . . . F 96 n 9 2 � Interesting limiting case: maximal foreground, no noise, that is, Planck channels modeled as linear mixtures of CMB and 9 − 1 = 8 “foregrounds”       d 1 a 1 F 11 . . . . . . F 18 s     � � d 2 a 2 F 21 . . . . . . F 28 f 1         s . . . .  .   . . .  . . . . f 2 = . . . . . . ×   or d = [ a | F ]     f  .  . . . . . . . . .     . . . . . . . . . . . f 8 d 9 a 9 F 91 . . . . . . F 98 3 � No foreground/noise model at all, but an unstructured contaminant g : d = a s + g . s = � i w i d i = w † d of the input channels gives − A linear combination � unbiased CMB if w † a = 1 and perfect foreground rejection if w † F = 0 . Actually, we are after Span( F ) , the foreground subspace.

Four CMB maps in Planck releases NILC SEVEM SMICA Commander Wavelet space Pixel+Harmonic Harmonic space Pixel space � � � � s s d = a s + g d = a s + g d = [ a | F ] + n d = [ a | F ( θ )] + n f f • Various filtering schemes (space-dependent, multipole-dependent, or both): − NILC : Needlet (spherical wavelet) domain ILC. − SEVEM : Pixel domain, internal template fitting. − SMICA : Harmonic domain, ML approach, foreground subspace. − Commander : Pixel domain, Bayesian method with physical foreground models.

The SMICA method The CMB is statistically independent of the other components. SMICA = SM + ICA = Spectral Matching Independent Component Analysis � � s Do ICA : Blind estimation of the linear model d = [ a | F ] + n , f via spectral matching : if all fields are modeled as Gaussian stationary , then the likelihood is a mismatch between empirical and theoretical spectra. Q: How dare you do that on sky maps so blatantly non Gaussian/non stationary?

Combining all 9 Planck channels, non parametrically: the ILC 1/ Independent contamination model: Stack the 9 Planck channels into a 9 × 1 data vector d = [ d 30 , d 44 , . . . , d 545 , d 857 ] † d ( p ) = a s ( p ) + g ( p ) p = 1 , . . . , N pix 2/ Linear combination: Estimate the CMB signal s ( p ) in pixel p by weighting the inputs: s ( p ) = w † d ( p ) � Idea: The variance of independent variables add up. Hence. . . Minimize the pixel average � ( w † d ) 2 � p subject to w † a = 1 , yielding C − 1 a � C = � dd † � p , � w = with the sample covariance matrix . C − 1 a a † � Coined ILC (Internal Linear Combination) by astrophysicists, a.k.a. MVBF (Min. Variance Beam Former) in array processing, BLUE elsewhere. .

Is the ILC good enough for Planck data ? ILC looks good: linear, unbiased, min. MSE, very blind, very few assumptions: knowing a (calibration) and the CMB uncorrelated from the rest (very true). However, a simulation result shows poor quality: ILC map on a ← − ± 300 µK color scale Error on a ± 50 µK color scale − → Two things, at least, need fixing: • harmonic dependence and • chance correlations.

Linear filtering in harmonic space Since resolution, noise and foregrounds vary (wildly) in power over channels and angular frequency, the combining weights should depend on ℓ . Harmonic ILC (Tegmark): 3 CMB map synthesized from spherical 217 harmonic coefficients ˆ s ℓm , obtained as 2 143 linear combinations: 353 1 � µ K /µ K RJ s ℓm = w † i ˆ with, again, ℓ d ℓm 0 C − 1 w i ( ℓ ) � 100 a ℓ w ℓ = C ℓ = Cov( d ℓm ) 1 545 a † C − 1 030 a 857 044 ℓ 070 100 • At high ℓ , (spectral) matrices C ℓ well 143 2 217 C ℓ = � 353 m d ℓm d † estimated by � ℓm / 2 ℓ + 1 . 545 857 3 500 1000 1500 2000 2500 3000 3500 • At lower ℓ , we need to get smarter to ℓ fight chance correlation.