Astron. Astrophys. Suppl. Ser. 146, 507-517

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

J.G. Bartlett¹ - M. Douspis¹ - A. Blanchard^1,2 - M. Le Dour¹

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

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 Strasbourg, France
http://astro.u-strasbg.fr/Obs.html

Received March 5, 1999; accepted August 18, 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.

Key words: cosmic microwave background -- Cosmology: observations -- maths: data analysis -- methods: statistical