The CMB angular power spectrum

As the CMB appears to come from spherical surface known as the last-scattering surface, which corresponds to the epoch of re-combination at $Z\approx 1000$, the CMB anisotropy field $\Delta T/T$ can naturally be described in terms of spherical harmonics as in equation .

$$\Delta T(\theta,\phi) = \sum_{m,\ell} a_{m\ell} Y_{m\ell}(\theta,\phi)$$ (1)

The lack of any preferred direction requires that there is no dependence on m and so the angular power spectrum can be expressed purely in terms of $\ell$ with $(2\ell+1)$ statistically independent sub-components contributing to each power spectrum estimator $C_\ell = \sum_m{\vert a_{m\ell}\vert^2/(2\ell+1)}$. The angular power spectrum is usually plotted out in a log-linear form in terms of $C_\ell\ell(\ell+1)$, as this shows the relative contributions to anisotropy from each angular frequency range and so a flat response here will correspond to a scale-invariant spectrum of the kind expected in inflation theories. This is the format used in figure , which has been normalized to the quadruple term (C₂).