Mapping the Cosmological Confidence Ball Surface

Abstract

We present a new technique to compute simultaneously valid confidence intervals for a set of model parameters. We apply our method to the Wilkinson Microwave Anisotropy Probe's (WMAP) Cosmic Microwave Background (CMB) data, exploring a seven dimensional space (tau, Omega_DE, Omega_M, omega_DM, omega_B, f_nu, n_s). We find two distinct regions-of-interest: the standard Concordance Model, and a region with large values of omega_DM, omega_B and H_0. This second peak in parameter space can be rejected by applying a constraint (or a prior) on the allowable values of the Hubble constant. Our new technique uses a non-parametric fit to the data, along with a frequentist approach and a smart search algorithm to map out a statistical confidence surface. The result is a confidence ``ball'': a set of parameter values that contains the true value with probability at least 1-alpha. Our algorithm performs a role similar to the often used Markov Chain Monte Carlo (MCMC), which samples from the posterior probability function in order to provide Bayesian credible intervals on the parameters. While the MCMC approach samples densely around a peak in the posterior, our new technique allows cosmologists to perform efficient analyses around any regions of interest: e.g., the peak itself, or, possibly more importantly, the 1-alpha confidence surface.