Information gains from cosmic microwave background experiments

Abstract

To shed light on the fundamental problems posed by dark energy and dark matter, a large number of experiments have been performed and combined to constrain cosmological models. We propose a novel way of quantifying the information gained by updates on the parameter constraints from a series of experiments which can either complement earlier measurements or replace them. For this purpose, we use the Kullback-Leibler divergence or relative entropy from information theory to measure differences in the posterior distributions in model parameter space from a pair of experiments. We apply this formalism to a historical series of cosmic microwave background experiments ranging from Boomerang to WMAP, SPT, and Planck. Considering different combinations of these experiments, we thus estimate the information gain in units of bits and distinguish contributions from the reduction of statistical errors and the "surprise" corresponding to a significant shift of the parameters' central values. For this experiment series, we find individual relative entropy gains ranging from about 1 to 30 bits. In some cases, e.g. when comparing WMAP and Planck results, we find that the gains are dominated by the surprise rather than by improvements in statistical precision. We discuss how this technique provides a useful tool for both quantifying the constraining power of data from cosmological probes and detecting the tensions between experiments.

Quantifying concordance in cosmology

“…As the relative entropy measures both changes in precision and shifts in parameter space when updating or replacing data, it can be interpreted as the information gain when going from p(Θ|D 1 ) to p(Θ|D 2 ). It has been used in this sense to study the information gains from a historical sequence of CMB experiments [10] and from combining observations from a wide range of cosmological probes [16].…”

“…A measure of consistency, the Surprise of the constraints derived from D 2 , was consequently defined in [10] as:…”

“…The first application of the Surprise in [10] was a study of the agreement between constraints on a flat ΛCDM cosmology from a historical sequence of CMB experiments. One of the main results of this study was the detection of a significant Surprise when comparing constraints from WMAP 9 data [12,13] and the constraints from the 2013 release of Planck data [14,15].…”

“…In [10], we proposed a new measure that we called Surprise. It is derived from the relative entropy, or KullbackLeibler divergence [11], between two posteriors and is a global measure of consistency over the entire cosmological parameter space.…”

“…It is derived from the relative entropy, or KullbackLeibler divergence [11], between two posteriors and is a global measure of consistency over the entire cosmological parameter space. In [10], we also applied the Surprise to a historical sequence of CMB experiments. One of the key results was a large Surprise when comparing the constraints from observations of the Wilkinson Microwave Anisotropy Probe (WMAP) [12,13] to the 2013 release of Planck data [14,15].…”