Efficient cosmological parameter estimation from microwave background anisotropies

We reexamine the issue of cosmological parameter estimation in light of current and upcoming high-precision measurements of the cosmic microwave background power spectrum. Physical quantities which determine the power spectrum are reviewed, and their connection to familiar cosmological parameters is explicated. We present a set of physical parameters, analytic functions of the usual cosmological parameters, upon which the microwave background power spectrum depends linearly (or with some other simple dependence) over a wide range of parameter values. With such a set of parameters, microwave background power spectra can be estimated with a high accuracy and negligible computational effort, vastly increasing the efficiency of the cosmological parameter error determination. The techniques presented here allow calculation of microwave background power spectra ${10}^5$ times faster than comparably accurate direct codes (after precomputing a handful of power spectra). We discuss various issues of parameter estimation, including parameter degeneracies, numerical precision, mapping between physical and cosmological parameters, systematic errors, and illustrate these considerations with an idealized model of the Microwave Anisotropy Probe experiment.