Separating Cosmological BModes with FastICA

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Separating Cosmological B-Modes with FastICA

• Stivoli F.
• Baccigalupi C.
• Maino D.
• Stompor R.
• Orsay
  15/09/2005

2
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

3

• 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

4
Component separation
problemThe problemHere 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 when
• the a priori knowledge about the signals is poor.

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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

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• 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

7
Estimation of
gaussianityThe 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.
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• 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

9
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.

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• 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
• 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.

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Whats
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.

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• 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

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CMB simulated sky

• The CMB polarized emission is simulated
  accordingly to the
• cosmological concordance model (Spergel et al.
  2003)
• 
  
• 
  gt The primordial GW
  
• 
  contribution is set to a 10
  
• 
  of the scalar perturbation
  
• 
  amplitude (r0.1).
• 
  
• 
  gt Lensing effect.

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• 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

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Synchrotron simulated
templateWe adopted the Giardinos model (2002).
gt Over the degree scale total
intensity synchrotron survey at 408 MHz
(Haslam et al. 1982)
gt
Polarization fraction 75
Random Gaussian polarization angle with
power scaling as gt 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.
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Dust simulated templateDetected for
the first time by Archeops, indicating 5 of
polarization fraction (Benoit et al., 2004)gt
Total intensity emission is well known at 100
µmgt It can be extrapolated to microvawes
frequenciesgt Great uncertainties aboutthe
polarization angle!
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Polarized Skies (Q)
Simulations indicate that the foreground
contamination is challenging for the cosmological
B mode measurements
40 GHz
90 GHz
150 GHz
350 GHz
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• 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

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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, G?rski and Hivon 2002) .
• These pseudo power spectra can be written as
• Obviously the mixing gets reduced as the size of
  the cut increases.

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10 deg radius
20 deg radius

Pseudo-B power spectra - circular cuts
The E modes contaminate substantially the B power
spectrum.
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• 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

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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.

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• 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

24
We perform separation of 100 realizations of
CMB against thesereference templates of
foregrounds.As a starting point we chose the
following parameters

• Noise S/N2 (Gaussian distributed, homogenous
  noise)
• Area of the patch a circular cut of 10 deg
  radius
• Foreground amplitude as it is in the template

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Recovered pseudo power spectrum (E modes) at 40
GHz

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Recovered pseudo power spectrum (B modes) at 40
GHz

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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 didnt improve the
  separation significantly.
• Changing the noise rms
• The code proved to be stable from noiseless
  case up to S/N1.
• 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).

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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?
B modes

• Inversion of the power spectra mixing equations
  is not trivial (Lewis, 2003).
• The most important feature of the B modes would
  appear around multipole l100.
• Then we can still infer whether or not FastICA is
  able to reject a model without tensor
  contribution to the anisotropies (r0).

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Rejecting r0 model
10 deg cut
10 deg caseinside 1-s 4inside 2-s 16
20 deg cut
20 deg caseinside 1-s 0inside 2-s 0
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• 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

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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.