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 cosmic microwave background data, exploring a seven-dimensional space (τ,ΩDE,ΩM,ωDM,ωB,fν,ns). We find two distinct regions of interest: the standard concordance model and a region with large values of ωDM, ωB, and H0. 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 nonparametric 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-α. 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-α confidence surface.