Properties and use of CMB power spectrum likelihoods

Abstract

Fast robust methods for calculating likelihoods from CMB observations on small scales generally rely on approximations based on a set of power spectrum estimators and their covariances. We investigate the optimality of these approximation, how accurate the covariance needs to be, and how to estimate the covariance from simulations. For a simple case with azimuthal symmetry we compare optimality of hybrid pseudo-C_l CMB power spectrum estimators with the exact result, indicating that the loss of information is not negligible, but neither is it enough to have a large effect on standard parameter constraints. We then discuss the number of samples required to estimate the covariance from simulations, with and without a good analytic approximation, and assess the use of shrinkage estimators. Finally we discuss how to combine an approximate high-ell likelihood with a more exact low-ell harmonic-space likelihood as a practical method for accurate likelihood calculation on all scales.

15 pages, 11 figures; updated to match version accepted by PRD

References in corpus (33)

• HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results
• Cosmological parameters from CMB and other data: a Monte-Carlo approach
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data
• Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis
• Why your model parameter confidences might be too optimistic -- unbiased estimation of the inverse covariance matrix
• MASTER of the CMB Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex CMB Data Sets
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Parameter Estimation Methodology
• How to measure CMB power spectra without losing information
• High resolution CMB power spectrum from the complete ACBAR data set
• Radical Compression of Cosmic Microwave Background Data
• Likelihood Analysis of CMB Temperature and Polarization Power Spectra
• A Measurement of the Angular Power Spectrum of the CMB Temperature Anisotropy from the 2003 Flight of Boomerang
• Myths and Truths Concerning Estimation of Power Spectra
• Global, Exact Cosmic Microwave Background Data Analysis Using Gibbs Sampling
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission
• The Pseudo-$C_l$ method: Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology
• Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology
• An Efficient Technique to Determine the Power Spectrum from Cosmic Microwave Background Sky Maps
• Signal-to-Noise Eigenmode Analysis of the Two-Year COBE Maps
• CMB temperature and polarisation pseudo-Cl estimators and covariances
• Cosmological parameters from WMAP 5-year temperature maps
• A general solution to the E-B mixing problem
• Exact likelihood evaluations and foreground marginalization in low resolution WMAP data
• Shrinkage Estimation of the Power Spectrum Covariance Matrix
• Hybrid Estimation of CMB Polarization Power Spectra
• Harmonic E/B decomposition for CMB polarization maps
• Gabor Transforms on the Sphere with Applications to CMB Power Spectrum Estimation
• What can be learned from the lensed cosmic microwave background B-mode polarization power spectrum?
• Analysis of cosmic microwave background data on an incomplete sky
• Making Maps from Planck LFI 30GHz Data
• CMB likelihood approximation by a Gaussianized Blackwell-Rao estimator
• PReBeaM for Planck: A Polarized Regularized Beam Deconvolution Map-Making Method

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• Putting the Precision in Precision Cosmology: How accurate should your data covariance matrix be?
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• Estimating the large-scale angular power spectrum in the presence of systematics: a case study of Sloan Digital Sky Survey quasars
• Combining power spectrum and bispectrum measurements to detect oscillatory features
• Polyspectra searches for sharp oscillatory features in cosmic microwave sky data
• The skewed weak lensing likelihood: why biases arise, despite data and theory being sound
• Cosmological parameter estimation using Particle Swarm Optimization (PSO)
• Using inpainting to construct accurate cut-sky CMB estimators
• Euclid-era cosmology for everyone: Neural net assisted MCMC sampling for the joint 3x2 likelihood
• Investigating scalar-tensor-gravity with statistics of the cosmic large-scale structure
• A Bayesian method for combining theoretical and simulated covariance matrices for large-scale structure surveys
• Extreme data compression for the CMB
• Exact joint likelihood of pseudo-$C_{\ell }$ estimates from correlated Gaussian cosmological fields
• The Atacama Cosmology Telescope: Semi-Analytic Covariance Matrices for the DR6 CMB Power Spectra
• The impact of signal-to-noise, redshift, and angular range on the bias of weak lensing 2-point functions
• Constraining Neutrinos and Dark Energy with Galaxy Clustering in the Dark Energy Survey
• Improved Gibbs samplers for Cosmic Microwave Background power spectrum estimation
• Robust Likelihoods for Inflationary Gravitational Waves from Maps of Cosmic Microwave Background Polarization
• Estimation of Constrained Mean-Covariance of Normal Distributions
• The Parameter-Level Performance of Covariance Matrix Conditioning in Cosmic Microwave Background Data Analyses