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CMB Temperature and Polarization Anisotropies on an incomplete sky

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Title: CMB Temperature and Polarization Anisotropies on an incomplete sky


1
CMB Temperature and PolarizationAnisotropies on
an incomplete sky
CMB
  • Moumita Aich
  • The Inter-University Centre for Astronomy and
    Astrophysics
  • Pune, India

2
Evolution of the universe
Evolution of the universe
Opaque
Transparent
Hu White, Sci. Am., 290 44 (2004)
3
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

4
(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
5
(No Transcript)
6
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
7
Polarization from Thomson Scattering
Differential cross section depends on
polarization and angle
8
Polarization from Thomson Scattering
Isotropic radiation scatters into unpolarized
radiation
9
Polarization from Thomson Scattering
Quadrupole anisotropies scatter into linear
polarization
10
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
11
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.

12
CMB Cross-Power Spectrum
(WMAP 3-year Paper Figures)
13
Systematic Effects
  • Incomplete sky coverage
  • Foreground contamination
  • Noise covariance
  • Non-circular beam correction
  • Lensing Effects

14
Systematic Effects
  • Incomplete sky coverage
  • Foreground contamination
  • Noise covariance
  • Non-circular beam correction
  • Lensing Effects

15
Incomplete sky coverage A RealityGround
based(e.g. MAT/TOCO, DASI, ACBAR, Clover) -
Small portion of sky obtained. (MAT, Clover
website)
16
Balloon based(e.g.BOOMERanG, MAXIMA, Archeops,
SPIDER) - Small portion of sky
obtained. (Archeops, SPIDER website)
17
Satellite based(e.g. COBE, WMAP, Planck) - Data
from galactic region is contaminated by residual
foreground emission. (WMAP 3 Year Papers)
18
The CMB Temperature and Polarization anisotropies
19
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)
20
Incomplete sky coverage Using Slepian Functions
Cut sky
Choose weight function
Maximization of the ratio of norms
where
21
Axisymmetric Cases A Single Polar Cap
block-diagonal matrix
The integral eigen-value equation
22
A Commuting differential Operator
Tridiagonal Matrix
they have identical eigenvectors
Since
Gilbert Slepian, 1977 Grunbaum, 1982
23
E-B Mixing of CMB Polarization
(Smith, 2006)
24
Pseudo Cl Construction for weighted polarization
sky
where
Construct W(n) such that terms due to leakage
0 and variance becomes small.
25
  • 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.
26
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.

27
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.

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