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

2.3. The Correlation Function

It is conventional in CMB work to expand the temperature fluctuations in spherical harmonics

(3)

and work in terms of the multipole moments a_m. One can define the sky correlation function

(4)

where the average is taken over the (observed) sky with the separation angle ₂₁ held fixed. Using the properties of Y_m the correlation function C_sky(₂₁) (hereafter we will drop the subscript "sky") is

(5)

where we have introduced the rotationally symmetric quantity a² _m |a_m|². The a² are not to be confused with (2 + 1) C (see later) which is often used to define the power spectrum of fluctuations in a Gaussian theory. At this point we have made no assumption about the underlying theory of fluctuations or the model of structure formation - the a² are purely measured quantities on the sky. [Note that other authors' definitions of various quantities can differ from ours by 4, (2 + 1), or similar factors.] The COBE team (C. Bennett et al 1992, Kogut et al 1993, Fixsen et al 1994, Bennett et al 1994) quote results for the first two moments, i.e. a T₀ / (4) for = 1, 2,

(6)

A common way of comparing theory and experiment is through the a_m. Of course an actual measurement of a temperature difference on the sky involves finite resolution and specific measurement strategies modifying Equation 5. These are usually included in the theoretically predicted correlation function through a window or filter function, W, described in Section 4.2.