Power Spectrum Estimation from High-Resolution Maps by Gibbs Sampling

We revisit a recently introduced power spectrum estimation technique based on Gibbs sampling, with the goal of applying it to the high-resolution WMAP data. In order to facilitate this analysis, a number of sophistications have to be introduced, each of which is discussed in detail. We have implemented two independent versions of the algorithm to cross-check the computer codes and to verify that a particular solution to any given problem does not affect the scientific results. We then apply these programs to simulated data with known properties at intermediate (Nside = 128) and high (Nside = 512) resolutions, to study effects such as incomplete sky coverage and white versus correlated noise. From these simulations we also establish the Markov chain correlation length as a function of signal-to-noise ratio and give a few comments on the properties of the correlation matrices involved. Parallelization issues are also discussed, with emphasis on real-world limitations imposed by current supercomputer facilities. The scientific results from the analysis of the first-year WMAP data are presented in a companion letter.

[1]  Edward J. Wollack,et al.  First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters , 2003, astro-ph/0302209.

[2]  Krzysztof M. Gorski,et al.  Gabor transforms on the sphere with applications to CMB power spectrum estimation , 2002 .

[3]  Edward J. Wollack,et al.  First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results , 2003, astro-ph/0302207.

[4]  Max Tegmark.  How to measure CMB power spectra without losing information , 1996, astro-ph/9611174.

[5]  Cmb power spectrum estimation via hierarchical decomposition , 2001, astro-ph/0104443.

[6]  C. B. Netterfield,et al.  MASTER of the Cosmic Microwave Background Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex Cosmic Microwave Background Data Sets , 2001, astro-ph/0105302.

[7]  Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology , 1998, astro-ph/9808292.

[8]  Fast, exact CMB power spectrum estimation for a certain class of observational strategies , 2001, astro-ph/0106515.

[9]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

[10]  David N. Spergel,et al.  An Efficient Technique to Determine the Power Spectrum from Cosmic Microwave Background Sky Maps , 1998, astro-ph/9805339.

[11]  Matias Zaldarriaga,et al.  Significance of the largest scale CMB fluctuations in WMAP , 2003, astro-ph/0307282.

[12]  Fast cosmic microwave background power spectrum estimation of temperature and polarization with Gabor transforms , 2002, astro-ph/0207526.

[13]  U. Toronto,et al.  Estimating the power spectrum of the cosmic microwave background , 1997, astro-ph/9708203.

[14]  Harmonic analysis of cosmic microwave background data – II. From ring‐sets to the sky , 2001, astro-ph/0112277.

[15]  M. Halpern,et al.  First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Profiles and Window Functions , 2003, astro-ph/0302214.

[16]  A. Banday,et al.  Evolution of large scale structure : from recombination to Garching : proceedings of the MPA-ESO Cosmology Conference : Garching, Germany, 2-7 August 1998 , 1999 .

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  Harmonic analysis of cosmic microwave background data — I. Ring reductions and point-source catalogue , 2001, astro-ph/0112276.

[19]  Krzysztof M. Gorski.  On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: Method , 1994 .

[20]  Edward J. Wollack,et al.  First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission , 2003, astro-ph/0302208.

[21]  C. H. Anderson,et al.  Application of Monte Carlo Algorithms to the Bayesian Analysis of the Cosmic Microwave Background , 2002, astro-ph/0209560.

[22]  Is the low cosmic microwave background quadrupole a signature of spatial curvature , 2003, astro-ph/0303127.

[23]  H. K. Eriksen,et al.  On Foreground Removal from the Wilkinson Microwave Anisotropy Probe Data by an Internal Linear Combination Method: Limitations and Implications , 2004, astro-ph/0403098.