Study of CMB temperature fluctuations have over the short interval of time since their discovery become the cosmological tool with the greatest potential for determining the values of the fundamental cosmological constants. The present data set is already capable of eliminating some regions of parameter space, and this is only a fore-taste of what is to come. Experimental results are often quoted as band-power estimates, and for Gaussian sky fluctuations, these represent a complete description of an observation. Because there are far fewer band-powers than pixel values for any given experiment, the reduction to band-powers has been called "radical compression'' (Bond et al. 2000); and as the number of pixels explodes with the next instrument generations, this kind of compression will become increasingly important in any systematic analysis of parameter constraints.
For these reasons, it is extremely useful to develop statistical methods which take as their input power estimates. Since most standard methods use as a starting point the likelihood function, one would like to have a simple expression for this quantity given a power estimate - one that does not require manipulation of the entire observational pixel set. One difficulty is that even for Gaussian sky fluctuations, the band-power likelihood function is not Gaussian, most fundamentally because the power represents an estimate of the variance of the pixel values. For any fiducial model, the data covariance matrix can be diagonalized and the likelihood function near this point in parameter space expressed as a product of individual Gaussians in the data elements (this is strictly speaking only possible for the model in question). This consideration lead us to examine the ideal situation where the eigenvalues of were all identical, for which we can analytically find the exact form of the likelihood function in terms of the best power estimate. Using this as motivation, we have proposed the same functional form for band-power likelihood functions, Eq. (19), as an ansatz in more general cases. It contains two free parameters, and , which may be uniquely determined if two confidence intervals of the full likelihood function (the thing one is trying to fit) are known; for example, the 68% and 95% confidence intervals (Eqs. 19, 20). We have seen that the resulting approximate distributions match remarkably well the complete likelihood functions for a number of experiments - those discussed here as well as 11 others (calculated by K. Ganga and B. Ratra). All of these comparisons may be viewed at our web site, where we also plan to provide and continually up-date the appropriate parameter values and for each published experiment.
Although at least one confidence interval is normally given in the literature (usually at 68%), a second confidence interval is rarely quoted. To aid the kind of approach proposed here, we would ask that in the future experimental band-power estimates be given with at least two likelihood-based confidence intervals (additional intervals, such as 99.8%, would allow one to fit other functional forms with 3 free parameters). This remains the surest way of finding the effective number of degrees-of-freedom of the likelihood, . An otherwise a priori choice for this number appears difficult, among other things because it depends on the nature of the scan strategy. We have noted in this light that , precisely because of correlations between pixels, which depend on the scan geometry.
One important aspect of the approximate nature of the proposed method is its inability to account for correlations between several band-powers. When analyzing a set of band-powers, one is obliged to simply multiply together their respective approximate likelihood functions. The accuracy of the approximation is thus limited by the extent to which inter-band correlations are important. Although one's desire is to give experimental results as independent power estimates, this is not always possible. Furthermore, and as discussed in Douspis et al. (2000), the very use of flat-band powers may lead to a loss of relevant experimental information otherwise contained in the original pixel data. The accuracy of any method based on their use is thus additionally limited by the importance of this lost information. These limitations define in practice the approximate nature of the proposed method.
Another important point to make is that the approximation is extremely easy to use, as easy as the (inappropriate) method; and for experiments with a small number of significant degrees-of-freedom, it represents a substantial improvement over the latter. This is the case, for example, with the MAX ID likelihood function, and it will always be the case when estimating power on the largest scales of a survey. When the effective number of degrees-of-freedom becomes large, a Gaussian becomes an acceptable approximation, and the gain in using the proposed ansatz is less significant. Nevertheless, the approximation's facile applicability promotes its use even in these cases. In the future, we will apply the proposed approximation in a systematic study of parameter constraints and for a test of the Gaussianity of the CMB fluctuations.
We are very grateful to K. Ganga and B. Ratra for so kindly providing us with an additional 11 likelihood functions with which to test the approximation; and we also thank D. Barbosa for supplying much information concerning current experimental results. We are happy to thank the referee for helpful comments.
Copyright The European Southern Observatory (ESO)