COSMIC MICROWAVE BACKGROUND LIKELIHOOD APPROXIMATION BY A GAUSSIANIZED BLACKWELL-RAO ESTIMATOR

We introduce a new cosmic microwave background (CMB) temperature likelihood approximation called the Gaussianized Blackwell-Rao estimator. This estimator is derived by transforming the observed marginal power spectrum distributions obtained by the CMB Gibbs sampler into standard univariate Gaussians, and then approximating their joint transformed distribution by a multivariate Gaussian. The method is exact for full-sky coverage and uniform noise and an excellent approximation for sky cuts and scanning patterns relevant for modern satellite experiments such as the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck. The result is a stable, accurate, and computationally very efficient CMB temperature likelihood representation that allows the user to exploit the unique error propagation capabilities of the Gibbs sampler to high ls. A single evaluation of this estimator between l = 2 and 200 takes similar to 0.2 CPU milliseconds, while for comparison, a singe pixel space likelihood evaluation between l = 2 and 30 for a map with similar to 2500 pixels requires similar to 20 s. We apply this tool to the five-year WMAP temperature data, and re-estimate the angular temperature power spectrum, C-l, and likelihood, L(C-l), for l <= 200, and derive new cosmological parameters for the standard six-parameter Lambda CDM model. Our spectrum is in excellent agreement with the official WMAP spectrum, but we find slight differences in the derived cosmological parameters. Most importantly, the spectral index of scalar perturbations is n(s) = 0.973 +/- 0.014, 1.9 sigma away from unity and 0.6 sigma higher than the official WMAP result, n(s) = 0.965 +/- 0.014. This suggests that an exact likelihood treatment is required to higher ls than previously believed, reinforcing and extending our conclusions from the three-year WMAP analysis. In that case, we found that the suboptimal likelihood approximation adopted between l = 12 and 30 by the WMAP team biased n(s) low by 0.4 sigma, while here we find that the same approximation between l = 30 and 200 introduces a bias of 0.6 sigma in n(s).