The cosmic microwave background radiation (CMBR) fractional temperature
perturbation, , is expressed as a function of angular position,
, on the sky via the spherical harmonic decomposition,

The CMBR spatial anisotropy in a gaussian model can then be characterized by the angular perturbation power spectrum , defined in terms of the ensemble average,

Given a gaussian model , the predicted root mean square (rms)
temperature that would be measured by an experiment,
,
is defined by

where is the present value of the CMBR temperature and is the zero-lag window function of the experiment. is a mathematical description of the sensitivity of the model of the experiment to CMBR anisotropy perturbations on different angular scales. It accounts for a number of experimental and observational effects, including the suppression of high sensitivity due to the finite size of the telescope beam and the suppression of low sensitivity due to differential (as opposed to absolute) temperature measurements (for almost all sub-orbital experiments).

It has become conventional to summarize zero-lag window functions in terms
of a few parameters (e.g., Bond; Ratra et al.). These are the value of
where is greatest, , the two values of ,
, where

and the effective multipole , where

It is of interest to use anisotropy observations to measure the power
spectrum over the entire range of to which an experiment is
sensitive. However, the broad width of the window functions of
current experiments prevents determination of each individual
. To analyze experimental data it is therefore necessary to
assume a functional form for the 's over the range of the
experiment is sensitive to, with the overall normalization allowed to
be a free parameter and with the shape of the allowed to depend
on a small number of other parameters. Comparing these 's to the
data then allows for a determination of the normalization and of the shape
parameters which best agree with the data.

Most analyses of data make use of the flat angular power spectrum,

where is the corresponding quadrupole-moment amplitude of the model CMBR anisotropy and is the CMBR temperature now. This spectrum is an approximation to the CMBR anisotropy spectrum in the fiducial cold dark matter model on large angular scales (Peebles 1982). And it has become conventional to quote the results of analyses in terms of the bandtemperature,

(e.g., Bond; Ratra et al.).

J.R. Bond, ``Theory and Observations of the Cosmic Background Radiation", in
Cosmology and Large Scale Structure, ed. R. Schaeffer, J. Silk, M. Spiro,
and J. Zinn-Justin (Amsterdam: Elsevier Science), 469 (1996).

P.J.E. Peebles, ``Large-Scale Background Temperature and Mass Fluctuations
due to Scale-Invariant Primeval Perturbations", *Astrophys. J. Lett.*
**263**, L1 (1982).

B. Ratra, N. Sugiyama, A.J. Banday, and K.M. Górski, ``Cosmic Microwave
Background Anisotropy in DMR-Normalized Open and Flat- Cold
Dark Matter Cosmogonies", *Astrophys. J.* **481**, 22 (1997).

Department of Physics, Kansas State University

Last updated: 2000-08-31