1 Introduction

Six years after their first detection by the COBE satellite (Smoot et al. 1992), it is now well appreciated that cosmic microwave background (CMB) temperature fluctuations contain rich information concerning virtually all the fundamental cosmological parameters of the Big Bang model (Bond et al. 1994; Knox 1995; Jungman et al. 1996). New observations from a variety of experiments, ground-based and balloon-borne, as well as the two planned satellite missions, MAP and Planck Surveyor, are and will be supplying a constant stream of ever more precise data over the next decade.

It is in fact already possible to extract interesting information from the existing data set, consisting of almost 20 different experimental results (Lineweaver et al. 1997; Bartlett et al. 1998a,b; Bond & Jaffe 1999; Efstathiou et al. 1999; Hancock et al. 1998; Lahav & Bridle 1998; Lineweaver & Barbosa 1998a,b; Lineweaver 1998; Webster et al. 1998; Lasenby et al. 1999). These experimental results are most often given in the literature as power estimates within a band defined over a restricted range of spherical harmonic orders. Our compilation, similar to those of Lineweaver et al. (1997) and Hancock et al. (1998), is shown in Fig. 1 and may be accessed at our web site.

Figure 1: Present CMB power spectrum estimates. Flat band-powers are shown as a function of multipole order l. The data and references can be found at http://astro.u-strasbg.fr/Obs/cosmo/CMB/tabcmb.html. The solid curve is a flat CDM model with $\Omega =0.4$, $\Lambda =0.6$, H0=40 km s-1/Mpc, $Q=19\;\mu$K, $\Omega _{\rm b} h^2 = 0.006$ and n=0.94, while the dotted line represents an open model with $\Omega =0.2$, $\Lambda =0$, H0=60 km s-1/Mpc, $Q=20\;\mu$K, $\Omega _{\rm b} h^2 = 0.015$ and n=1 (no gravitational waves and no reionization)

The band is defined either directly by the observing strategy, or during the data analysis, e.g., the electronic differencing scheme introduced by Netterfield et al. (1997). This permits a concise representation of a set of observations, reducing a large number of pixel values to only a few band-power estimates, and for this reason the procedure has been referred to as "radical compression'' (Bond et al. 2000). If the sky fluctuations are Gaussian, as predicted by inflationary models, then little or nothing has been lost by the reduction to band-powers (Tegmark 1997). This is extremely important, because the limiting factor in statistical analysis of the next generation of experiments, such as, e.g., BOOMERanG, MAXIMA, and Archeops, is calculation time. Working with a much smaller number of band-powers, instead of the original pixel values, will be essential for such large data sets. The question then becomes how to correctly treat the statistical problem of parameter constraints starting directly with band-power estimates.

Standard approaches to parameter determination, whether they be frequentist or Bayesian, begin with the construction of a likelihood function. For Gaussian fluctuations, the only kind we consider here, this is a multivariant Gaussian in the pixel temperature values, where the covariance matrix is a function of the model parameters (see below). The likelihood is then used as a function of the parameters, but as just mentioned, the large number of pixels makes this object very computationally cumbersome. It would be extremely useful to be able to define a likelihood function starting directly with the power estimates in Fig. 1. This is the concern of this paper, where we develop an approximation to the the full likelihood function which requires only band-power estimates and very limited experimental details. As always in such procedures, it is worth emphasizing that the likelihood function, and therefore all derived constraints, only applies within the context of the particular model adopted. In our discussion, we shall focus primarily on inflationary scenarios, whose theoretical predictions have become easily calculable thanks to the development of fast Boltzmann codes, such as CMBFAST (Seljak & Zaldarriaga 1996; Zaldarriaga et al. 1998).

Much of the recent work on parameter determination has relied on the traditional $\chi^2$-fitting technique. As is well known, this amounts to a likelihood approach for observables with a Gaussian probability distribution. Band-power estimates do not fall into this category (Knox 1995; Bartlett et al. 1998c; Bond et al. 2000; Wandelt et al. 1998) - they are not Gaussian distributed variables, not even in the case of underlying Gaussian temperature fluctuations. The reason is clear: power estimates represent the variance of Gaussian distributed pixel values (the sky temperature fluctuations), and they therefore have a distribution more closely related to the $\chi^2$-distribution.

We begin, in the following section, by a general discussion of the likelihood approach applied to CMB observations. In the context of an ideally simple situation, we find the exact analytic form for the likelihood function of a band-power estimate. Reflections concerning the likelihood function in the context defined by actual experiments motivates us to propose this analytic form as an approximation, or ansatz, in the more general case. It is extremely easy to use, requiring little information in order to be applied to an experimental setup, because it contains only two adjustable parameters. These can be completely determined if one is given two confidence intervals, say the 68% and 95% confidence intervals, of the true, underlying likelihood distribution (notice that here we see the non-Gaussian nature of the likelihood - a Gaussian function would only require one confidence interval, besides the best power estimate, to be completely determined). We ask that in the future at least two confidence intervals be given when reporting experimental band-power estimates (more would be better, say for adjusting more complicated functional forms). An important limitation of the approach is the inability at present to account for more than one, correlated band-powers, as will be discussed further below.

We quantitatively test the accuracy of the approximation in Sect. 3 by comparison to several experiments for which we have calculated the full likelihood function. The approximation works remarkably well, and it can represent a substantial improvement over both single and "2-winged'' Gaussian forms commonly used in standard $\chi^2$-analyses; and it is as easy to use as the latter. The proposed likelihood approximation, the main result of this paper, is given in Eqs. (19-20). We plan to maintain a web page with a table of the best fit parameters required for its use. Detailed application of the approximate likelihood function to parameter constraints and to tests of the Gaussianity of the observed fluctuations is left to future papers. Other, similar work has been performed by Bond et al. (2000) and Wandelt et al. (1998).