Separating Cosmological B-Modes with FastICA Stivoli F. Baccigalupi C. Maino D. Stompor R. Orsay – 15/09/2005

Outline Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

Component separation problem The problem: Here s j are the n unknown signals mixed up by the unknown mixing matrix A. Vectors x i are the m observed mixtures (time stream signals, pixel maps, etc). Surprisingly, it is possible to solve this ‘blind’ problem under a few assumptions on the signal s j. They must: Be statistically independent Have a different frequency scaling Have non-Gaussian distributions (save at most one) Of course, this component separation method is very appealing when the a priori knowledge about the signals is poor.

How ICA solves the problem The key point is non-gaussianity. Central Limit Theorem: “The distribution of a linear combination of independent variables is more Gaussian than the original variables.” It is possible to select one of the original signals by reaching the minimum of the gaussianity of y:

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

Estimation of gaussianity The entropy H of a random variable y is defined as: Entropy is the degree of information the variable gives. A Gaussian variable has the largest entropy among all the variables with the same mean and variance. The negentropy: This is the optimal estimator of gaussianity.

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

FastICA It is an application of the ICA principle. The actual version works on the map, searching for the minimum of the negentropy on the pixel domain (Maino et al. 2002).  It separates both total intensity and polarization.  Convergence is fast (less than 20 seconds on a common desktop).  It allows to deal with instrumental noise (with some limitation).  It recovers the image of the CMB.

Until now, FastICA was successfully tested on: All sky Planck simulated data. FastICA was able to recover the CMB power spectrum up to multipoles 2000 in total intensity (Maino et al. 2002). The real data of the COsmic Background Explorer, recovering the amplitude and spectral index of primordial cosmological perturbations (Maino et al. 2003). Planck simulated data in polarization. FastICA recovered E modes an all relevant scales (Baccigalupi et al. 2004). Reasons for these promising results are essentially two: I.We deal with very large statistical samples because of the high precision that CMB experiments are able to achieve. II.CMB largely satisfies one of the ICA requirements, since it is really uncorrelated with galactic emission.

What’s new in this work In the forthcoming years, the detection of the CMB B modes will be attempted by many experiments targeting limited region of the sky  to focus where the foreground emission in total intensity is known to be low.  to increase the S/N of this tiny polarized signal In this work the FastICA performance in the reconstruction of the CMB polarized emission on a portion of the sky is tested, focusing on the recovering of B modes.

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

CMB simulated sky The CMB polarized emission is simulated accordingly to the cosmological concordance model (Spergel et al. 2003) > The primordial GW contribution is set to a 10% of the scalar perturbation amplitude (r=0.1). > Lensing effect.

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

Synchrotron simulated template We adopted the Giardino’s model (2002). > Over the degree scale: total intensity synchrotron survey at 408 MHz (Haslam et al. 1982) > Polarization fraction 75% Random Gaussian polarization angle with power scaling as > On smaller scales: observations on the radio band (Uyaniker et al. 1999, Duncan et al. 1999) Frequency scaling exhibits a steep power law ( ) according to WMAP observations.

Dust simulated template Detected for the first time by Archeops, indicating ~5% of polarization fraction (Benoit et al., 2004) > Total intensity emission is well known at 100 μm > It can be extrapolated to microvawes frequencies > Great uncertainties about the polarization angle!

Polarized Skies (Q) 40 GHz 150 GHz350 GHz 90 GHz Simulations indicate that the foreground contamination is challenging for the cosmological B mode measurements

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

Pseudo power spectra On a finite portion of the sky a transfer of power from the E to the B modes and vice versa occurs (Hansen, Gόrski and Hivon 2002). These pseudo power spectra can be written as: Obviously the mixing gets reduced as the size of the cut increases.

10° deg radius 20° deg radius Pseudo-B power spectra - circular cuts The E modes contaminate substantially the B power spectrum.

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

FastICA goes on parallel To evaluate errors on the reconstruction and stability of its quality, a statistically significant number of simulation is needed. Moreover, the high speed of the code on a single separation, makes FastICA particularly suited for parallelization. For these reasons a parallel version of FastICA has been implemented, able to perform hundreds of separation in a very short time.

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

We perform separation of 100 realizations of CMB against these reference templates of foregrounds. As a starting point we chose the following parameters:  Noise: S/N=2 (Gaussian distributed, homogenous noise)  Area of the patch: a circular cut of 10 deg radius  Foreground amplitude: as it is in the template

Recovered pseudo power spectrum (E modes) at 40 GHz

Recovered pseudo power spectrum (B modes) at 40 GHz

After that we run many MonteCarlo simulations after changing the parameters one by one:  Doubling the radius of the cut Since the sample of the data (i.e. the number of pixels) was large enough for FastICA to converge in the 10 deg case, this didn’t improve the separation significantly.  Changing the noise rms The code proved to be stable from noiseless case up to S/N~1.  Increasing the foreground amplitude Convergence is achieved against a foreground signal a few times bigger than the original one (6x for the synchrotron, 10x for the dust).

Detection of B modes would reveal the imprinting of primordial GW. So, what can we say about the recovering of the true B modes, in our analysis?  Inversion of the power spectra mixing equations is not trivial (Lewis, 2003).  The most important feature of the B modes would appear around multipole ℓ=100.  Then we can still infer whether or not FastICA is able to reject a model without tensor contribution to the anisotropies (r=0). B modes

10 deg case inside 1-σ ~ 4% inside 2-σ ~ 16% 20 deg case inside 1-σ = 0% inside 2-σ = 0% Rejecting r=0 model 10 deg cut 20 deg cut

Independent Component Analysis (ICA)  Generalities  Negentropy  FastICA Polarization Reference Sky  CMB  Foregrounds  Power spectra on circular sky cut MonteCarlo Simulation  Parallelization of FastICA  Results Comments

 FastICA proved to be able to recover the CMB polarized signal properly against the foreground templates of diffuse galactic emission  Pseudo power spectra are very well reconstructed too Limitations of this analysis: Absence of systematics (1/f noise, elliptic and not symmetric beam cases are being tested) No inversion to get the whole true B power spectra It is clear that our knowledge about the properties of the foreground polarization is still very poor. A blind separation method can play a crucial role in the forthcoming experiments. Comments