Best Unbiased Estimates for the Microwave Background Anisotropies

TL;DR

This paper addresses the challenge of estimating parameters of the cosmic microwave background anisotropies from a single sky map, providing optimal unbiased estimators and analyzing their theoretical uncertainties.

Contribution

It presents a detailed solution for deriving the best unbiased estimators of CMB anisotropy parameters and discusses their theoretical error bounds.

Findings

Derived theoretical error bars for estimators

Identified limitations due to non-ergodicity on a 2-sphere

Provided a methodology for parameter estimation from single realizations

Abstract

It is likely that the observed distribution of the microwave background temperature over the sky is only one realization of the underlying random process associated with cosmological perturbations of quantum-mechanical origin. If so, one needs to derive the parameters of the random process, as accurately as possible, from the data of a single map. These parameters are of the utmost importance, since our knowledge of them would help us to reconstruct the dynamical evolution of the very early Universe. It appears that the lack of ergodicity of a random process on a 2-sphere does not allow us to do this with arbitrarily high accuracy. We are left with the problem of finding the best unbiased estimators of the participating parameters. A detailed solution to this problem is presented in this article. The theoretical error bars for the best unbiased estimates are derived and discussed.