An approximation to the likelihood function for band–power estimates of CMB anisotropies

¹ Observatoire Midi-Pyrénées, Unité associée au CNRS, 14 avenue E. Belin, 31400 Toulouse, France http://www.omp.obs-mip.fr/omp
² Observatoire de Strasbourg, Université Louis Pasteur, Unité associée au CNRS, 11 rue de l'Université, 67000 http://astro.u-strasbg.fr/Obs.html

Send offprint request to: J.G. Bartlett, e-mail: barlett@ast.obs-mip.fr

Received: 5 March 1999
Accepted: 18 August 2000

Abstract

Band–power estimates of cosmic microwave background fluctuations are now routinely used to place constraints on cosmological parameters. For this to be done in a rigorous fashion, the full likelihood function of band–power estimates must be employed. Even for Gaussian theories, this likelihood function is not itself Gaussian, for the simple reason that band–powers measure the variance of the random sky fluctuations. In the context of Gaussian sky fluctuations, we use an ideal situation to motivate a general form for the full likelihood function from a given experiment. This form contains only two free parameters, which can be determined if the 68% and 95% confidence intervals of the true likelihood function are known. The ansatz works remarkably well when compared to the complete likelihood function for a number of experiments. For application of this kind of approach, we suggest that in the future both 68% and 95% (and perhaps also the 99.7%) confidence intervals be given when reporting experimental results.