CMB angular power spectrum

Remarks

The two-point correlation function is $$C(\vartheta) = \sum_{\ell} \frac{2 \ell + 1}{4 \pi} C_\ell \, P_\ell(\cos \vartheta),$$ where $\vartheta$ is the angle between the two directions and $P_\ell$ are the Legendre polynomials. This follows from the addition theorem for spherical harmonics. From one sky, $C_\ell$ is estimated by $\hat C_\ell = \frac{1}{2 \ell + 1} \sum_m \abs{a_{\ell m}}^2$. For a Gaussian field, this estimator has relative standard deviation $\sqrt{2 / (2 \ell + 1)}$, the cosmic variance. Spectra are usually plotted as $D_\ell = \ell (\ell + 1) C_\ell / (2 \pi)$. The dipole $\ell = 1$ is dominated by the motion of the observer and is usually removed.

Example: if $C_\ell = c / (\ell(\ell + 1))$ for a constant $c$, then $D_\ell = c / (2 \pi)$ is flat; this is roughly the shape at low $\ell$ for a scale-invariant spectrum of primordial fluctuations.

Non-example: a fixed map with a preferred direction, such as $\delta T / \bar T = \varepsilon \cos \vartheta_0$, is not statistically isotropic. Its coefficients are not described by a variance depending only on $\ell$.