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
*a*_{}^{2}
_{m}
|*a*_{m}|^{2}. The
*a*_{}^{2}
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
*a*_{}^{2}
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*_{}
*T*_{0} /
(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.