ball5.gif Conventions

The cosmic microwave background radiation (CMBR) fractional temperature perturbation, $\delta T/T$, is expressed as a function of angular position, $(\theta, \phi)$, on the sky via the spherical harmonic decomposition,

{\delta T \over T}(\theta , \phi) = \sum_{\ell=2}^\infty
...m=-\ell}^\ell a_{\ell m} Y_{\ell m}(\theta , \phi) .

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

\langle a_{\ell m} a_{\ell^\prime m^\prime}{}^* \rangle =
C_\ell \delta_{\ell\ell^\prime} \delta_{mm^\prime} .

Given a gaussian model $C_\ell$, the predicted root mean square (rms) temperature that would be measured by an experiment, $\delta T_{\rm rms}$, is defined by

(\delta T_{\rm rms})^2 = T_0{}^2 \sum_{\ell=2}^{\infty}
{(2\ell + 1) \over 4\pi} C_\ell W_\ell ,

where $T_0$ is the present value of the CMBR temperature and $W_\ell$ is the zero-lag window function of the experiment. $W_\ell$ 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 $\ell$ sensitivity due to the finite size of the telescope beam and the suppression of low $\ell$ 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 $\ell$ where $W_\ell$ is greatest, $\ell_{\rm m}$, the two values of $\ell$, $\ell_{e^{-0.5}}$, where

W_{\ell_{e^{-0.5}}} = e^{-0.5} W_{\ell_{\rm m}},

and the effective multipole $\ell_{\rm e} = I(\ell W_\ell)/I(W_\ell)$, where

I(W_\ell) = \sum_{\ell=2}^\infty {(\ell+0.5) W_\ell \over \ell(\ell+1)} .

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

C_\ell = {6C_2 \over \ell(\ell+1)} = {24 \pi \over 5}
{(Q_{\rm rms-PS}/T_0)^2 \over \ell(\ell+1)},

where $Q_{\rm rms-PS}$ is the corresponding quadrupole-moment amplitude of the model CMBR anisotropy and $T_0$ 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,

\delta T_\ell = {\delta T_{\rm rms} \over \sqrt{I(W_\ell)} }

(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 $COBE$ DMR-Normalized Open and Flat-$\Lambda$ Cold Dark Matter Cosmogonies", Astrophys. J. 481, 22 (1997).

Bharat Ratra and Tarun Souradeep
Department of Physics, Kansas State University
Last updated: 2000-08-31