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

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 mea- sure 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% condence 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%) condence intervals be given when reporting experimental results.