separating cosmological b-modes with fastica stivoli f. baccigalupi c. maino d. stompor r. orsay –...

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



Comments

Component separation problem

The problem:

Here sj are the n unknown signals mixed up by the unknown mixing matrix A. Vectors xi 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 sj. 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 whenthe a priori knowledge about the signals is poor.

jjii sAx

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:

jj

jji

ii

i sksAwxwy




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.

dyyfyfyH )(log)()(

)()()( yHyHyJ gauss

The negentropy:

This is the optimal estimator of gaussianity.




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.




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.




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.

2

3

2

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 aboutthe polarization angle!

Polarized Skies (Q)

40 GHz

150 GHz 350 GHz

90 GHz

Simulations indicate that the foreground contamination is

challenging for the cosmological B mode

measurements




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.

BEE CKCKC '2'2'

)',()',(

EBB CKCKC '2'2

')',()',(

10° deg radius 20° deg radius

Pseudo-B power spectra - circular cuts

The E modes contaminate substantially the B power spectrum.




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.




Comments

We perform separation of 100 realizations of CMB against thesereference 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 caseinside 1-σ ~ 4%inside 2-σ ~ 16%

20 deg caseinside 1-σ = 0%inside 2-σ = 0%

Rejecting r=0 model

10 deg cut

20 deg cut




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 theforeground polarization is still very poor.A blind separation method can play a crucial role in the forthcoming experiments.

Comments

separating cosmological b-modes with fastica stivoli f. baccigalupi c. maino d. stompor r. orsay –...

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