Cosmic microwave background power spectrum (CMB power spectrum)

The CMB angular power spectrum decomposes the variance of temperature or polarization anisotropy by angular multipole. Its large-angle plateau and acoustic peaks encode primordial perturbations and photon-baryon evolution.

For a scale-invariant potential spectrum with ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }/(2{\pi }^2)$ constant, the matter-era Sachs-Wolfe effect gives $C_{\ell }=(4\pi {\mathcal{A}}_{\Phi }/9)\int j_{\ell }(x{)}^{2}dx/x$. The logarithmic spherical Bessel square integral then gives $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{A}}_{\Phi }/9$. This plateau applies to large-angle temperature anisotropies, neglecting acoustic evolution and late evolving potentials; it is not a constant $C_{\ell }$ spectrum.

Cosmic variance is the sampling uncertainty caused by observing only one realization of a cosmological random field. For a Gaussian isotropic full-sky temperature field and an ideal noiseless estimate, $\mathrm{Var}({\hat{C}}_{\ell })=2C_{\ell }^2/(2\ell +1)$ because there are only $2\ell +1$ independent real angular degrees of freedom.