Past exam of the astrophysics course of the University of Oxford / 2022 / 3 / b / Solution

Insert the diagonal covariance from a into the two-point function and apply the spherical harmonic addition theorem:The Legendre polynomials encode the angular separation. Since $P_l(1)=1$, the zero-separation value $C(0)=⟨{\Theta }^2⟩$ is the total variance. For measured Cosmic microwave background anisotropy, remove the monopole and usually the kinematic dipole, so the corresponding sum starts at $l=2$; an instrument's angular response supplies an additional window function.