Annu. Rev. Astron. Astrophys. 1994. 32:
319-70
Copyright © 1994 by . All rights reserved |

**4.1. The Gaussian Auto-Correlation Function**

It has become common in analyzing data from small-scale experiments to
employ a Gaussian Auto-Correlation Function (GACF) as the assumed
underlying "theory". The GACF is parameterized by two numbers, its
amplitude *C*_{0} and correlation angle
_{c}:

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This "theory" is then convolved with the observing strategy and the
predictions compared to the data. Usually limits or best-fit values
are quoted on *C*_{0} for a range of
_{c}. In the
language of the multipole
moment expansion, the assumption of a GACF is equivalent to assuming

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Note that this is very different from CDM, where
(
+ 1)*C*_{} has a
Sachs-Wolfe plateau followed by Doppler/adiabatic peaks.

Commonly a plot of *C*_{0} vs
_{c} is used to
describe the sensitivity of a
particular experiment to fluctuations on various scales, under the
assumption that the sky correlation function is really a GACF. The
GACF approximation is a simple way to understand the sensitivity of an
experiment. By varying
_{c}, one can
match the "power spectrum" to the
window function of the experiment (especially multi-beam experiments
where the GACF power spectrum and the window function have similar
shapes, which we will call "Gaussian"). The experiment is most
sensitive to a GACF whose peak
(
+ 1)*C*_{}
occurs at essentially the
same place as the peak of its window function
*W*_{}. The
amplitude of
fluctuations to which one is sensitive (or the "area" under the window
function) is then parameterized by the minimum *C*_{0}.

For experiments in which the window function is similar to the GACF power spectrum, the approximation made in fitting with a GACf is numerically quite good. This is because, unless the underlying power spectrum varies rapidly on scales probed by the experiment, once the power spectrum is convolved with the window function, it has a "Gaussian" shape. The GACf retains its "Gaussian" shape when convolved with a Gaussian window function. Hence the two convolved spectra will look very similar (Bunn et al 1994b).

An analysis using a GACF will (roughly) take into account the peak
and are of the window function of the experiment. The minimal values
of *C*_{0}^{1/2} will therefore be more comparable
between experiments than
the measured rms temperature fluctuations (which could be defined in
an observer-dependent way). The correlations between nearby points
will also be approximately correct for experiments with "Gaussian"
window functions (and in which the window function approach is
applicable; see Section 4.2). Although for
some experiments, the GACf
approximation may be used to give a fit to the data, the GACF
assumption should be viewed with caution. One should bear in mind that
the quoted *C*_{0}^{1/2} is the best fit amplitude
of fluctuations for a power
spectrum that is a GACF with some fixed correlation angle
_{c}. There is
also little meaning in the values of *C*_{0}^{1/2}
at any point other than the minimum of the likelihood curve.