Cosmological parameters from CMB maps without likelihood approximation

Abstract

We propose an efficient Bayesian Markov chain Monte Carlo (MCMC) algorithm for estimating cosmological parameters from cosmic microwave background (CMB) data without the use of likelihood approximations. It builds on a previously developed Gibbs sampling framework that allows for exploration of the joint CMB sky signal and power spectrum posterior, P(s,C{sub ℓ}|d), and addresses a long-standing problem of efficient parameter estimation simultaneously in regimes of high and low signal-to-noise ratio. To achieve this, our new algorithm introduces a joint Markov chain move in which both the signal map and power spectrum are synchronously modified, by rescaling the map according to the proposed power spectrum before evaluating the Metropolis–Hastings accept probability. Such a move was already introduced by Jewell et al., who used it to explore low signal-to-noise posteriors. However, they also found that the same algorithm is inefficient in the high signal-to-noise regime, since a brute-force rescaling operation does not account for phase information. This problem is mitigated in the new algorithm by subtracting the Wiener filter mean field from the proposed map prior to rescaling, leaving high signal-to-noise information invariant in the joint step, and effectively only rescaling the low signal-to-noise component. To explore the full posterior, the new joint move is then interleaved with a standard conditional Gibbs move for the sky map. We apply our new algorithm to simplified simulations for which we can evaluate the exact posterior to study both its accuracy and its performance, and find good agreement with the exact posterior; marginal means agree to ≲0.006σ and standard deviations to better than ∼3%. The Markov chain correlation length is of the same order of magnitude as those obtained by other standard samplers in the field.