CMB Temperature and Polarization Anisotropies on an incomplete sky

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CMB Temperature and PolarizationAnisotropies on
an incomplete sky
CMB

• Moumita Aich
• The Inter-University Centre for Astronomy and
  Astrophysics
• Pune, India

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

• Photon temperature scales as 1z
• Before z 1000 (3000K), CMB photons ionized
  hydrogen in the universe
• Equilibrium via Compton scattering
• After recombination, CMB photons free stream
  through universe to today T 2.725 K
• CMB spectrum today reflects the temperature
  differences on the surface of last scattering
• Photon-baryon fluid moved through potential wells
  before last scattering, giving rise to structure
  in the CMB

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(almost) uniform 2.726K blackbody
Dipole (3.4 mK)
O(10-5) perturbations (20
µK)
Observations the microwave sky today
Source NASA/WMAP Science Team
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(No Transcript)
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Acoustic Peaks
Power Spectrum vs. Multipole Moment

• Before recombination, free electrons act like
  glue to CMB photons and baryons
• Photon-baryon fluid has pressure coming from the
  radiation
• Pressure counteracts gravitational collapse,
  causing oscillating sequence of compression and
  rarefaction
• Acoustic Oscillations - whose frequency scales
  with size
• Fluctuations imprinted on CMB photons on last
  scattering surface

Source NASA/WMAP Science Team
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Polarization from Thomson Scattering
Differential cross section depends on
polarization and angle
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Polarization from Thomson Scattering
Isotropic radiation scatters into unpolarized
radiation
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Polarization from Thomson Scattering
Quadrupole anisotropies scatter into linear
polarization
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CMB Polarization

• Stokes parameters - Q and U which are coordinate
  dependent
• Spatially varying mixtures of Q and U gives the
  E B modes which
• retain their character on rotation of the local
  coordinate system

Grad (or E) modes
Curl (or B) modes
Kamionkowski et al. 1997 Seljak Zaldarriaga
1997
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CMB Anisotropies

• Temperature - Fluctuation of photon density at
  recombination.
• Linear Polarization - Thomson scattering of CMB
  photons and electrons at last scattering.
• Measure the angular coherence of the temperature
  and polarization fluctuations.
• Since CMB fluctuations are on the 2-D sky,
  angular decomposition of the fluctuations in
  multipole space.
• Gaussian density fluctuations, as predicted by
  inflation.
• Multipole moments have zero mean
• and variance
• Unbiased, minimum variance angular power spectrum
  estimators.

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CMB Cross-Power Spectrum
(WMAP 3-year Paper Figures)
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Systematic Effects

• Incomplete sky coverage
• Foreground contamination
• Noise covariance
• Non-circular beam correction
• Lensing Effects

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Systematic Effects

• Incomplete sky coverage
• Foreground contamination
• Noise covariance
• Non-circular beam correction
• Lensing Effects

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Incomplete sky coverage A RealityGround
based(e.g. MAT/TOCO, DASI, ACBAR, Clover) -
Small portion of sky obtained. (MAT, Clover
website)
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Balloon based(e.g.BOOMERanG, MAXIMA, Archeops,
SPIDER) - Small portion of sky
obtained. (Archeops, SPIDER website)
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Satellite based(e.g. COBE, WMAP, Planck) - Data
from galactic region is contaminated by residual
foreground emission. (WMAP 3 Year Papers)
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The CMB Temperature and Polarization anisotropies
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Finite Area Fields
Spin - weighted harmonic space window function
where s 0, 2 for T P (E,B) respectively
The Pseudo-Cl estimators are
Covariance Matrix
where
the subscripts (T,E,B)
(Brown et al. 2005 Wandelt et al. 2006)
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Incomplete sky coverage Using Slepian Functions
Cut sky
Choose weight function
Maximization of the ratio of norms
where
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Axisymmetric Cases A Single Polar Cap
block-diagonal matrix
The integral eigen-value equation
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A Commuting differential Operator
Tridiagonal Matrix
they have identical eigenvectors
Since
Gilbert Slepian, 1977 Grunbaum, 1982
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E-B Mixing of CMB Polarization
(Smith, 2006)
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Pseudo Cl Construction for weighted polarization
sky
where
Construct W(n) such that terms due to leakage
0 and variance becomes small.
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• Pure B-mode ?? the weight function W(n) and
  its gradient
• ?W(n) vanish on the boundary of the region.
• Pure pseudo Cl estimators

Pure pseudo-Cl estimators do not mix E B in
the mean, as well as the variance is minimized.
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Future Prospects

• Testing the covariance matrix analysis for
  different simulated CMB
• temperature and polarization data sets.
• Compute optimal weight functions for a survey
  algorithmically.
• Study pure pseudo-Cl estimators for realistic
  noise models, including models with correlated
  noise.
• The methodology of Slepian functions could be
  generalized for higher spin (i.e. CMB
  Polarization) both for single and double polar
  cap.
• Include higher m in harmonic transform wlm of the
  window function WT(n), thus deviating from the
  axisymmetric case.
• Extend the method of Slepian functions for such
  cases. More
• interest lies in obtaining a commuting
  differential operator which would reduce the
  computational cost.

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References

• M. L. Brown, P. G. Castro, and A. N. Taylor. CMB
  temperature and polarisation Pseudo-Cl estimators
  and covariances. astro-ph/0410394.
• Benjamin D. Wandelt, Eric Hivon, and Krzysztof M.
  Gorski. The Pseudo-Cl
• method CMB anisotropy power spectrum
  statistics for high precision cosmology.
  astro-ph/0008111.
• Kendrick M. Smith. Pseudo-Cl estimators which do
  not mix E and B modes.
• astro-ph/0511629.
• Kendrick M. Smith. Pure Pseudo-Cl estimators for
  CMB B-modes.
• astro-ph/0608662.
• Frederik J. Simons, F. A. Dahlen, and Mark A.
  Wieczorek. Spatiospectral
• concentration on a sphere.
  arXivmath/0408424v1.
• E. N. Gilbert and D. Slepian. Doubly orthogonal
  concentrated polynomials.

Thank You !