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^{}.

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

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
-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 -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 -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).

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