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

Primordial density contrasts perturb the intrinsic photon temperature, the gravitational potential, and the velocity of the last-scattering material. Intrinsic photon-density fluctuations obey $\delta T/T={\delta }_{\gamma }/4$ because ${\rho }_{\gamma }\propto T^4$; adiabatic initial conditions relate these to matter density contrasts. Subsequent acoustic evolution, Doppler effect and gravitational frequency shifts determine the observed Cosmic microwave background anisotropy. There is no scale-independent identification of the final sky map with matter density contrast alone.

The spherical harmonic coefficients are random variables inherited from these perturbations. After subtracting the mean sky temperature, $⟨a_{lm}⟩=0$ for the anisotropy modes. Gaussian primordial perturbations give approximately Gaussian random variables; this extra Gaussian assumption is not needed for diagonal covariance. Reality of the sky map implies $a_{l,-m}=(-1{)}^m{a}_{lm}^{\ast }$.

For a statistically isotropic sky, its two-point function depends only on $\hat{n}\cdot \hat{m}$. Rotations mix the $m$ components at fixed $l$ and do not mix different irreducible $l$ spaces. A rotation-invariant covariance matrix is therefore a multiple of the identity in each $l$ block:On our one sky, average the $2l+1$ angular components at fixed $l$, or equivalently suitable rotated sky pairs, rather than treating different universes as observed samples. The CMB angular power spectrum is estimated byUnder the Gaussian, full-sky, noiseless assumptions, there are $2l+1$ independent real degrees of freedom and $\mathrm{Var}({\hat{C}}_l)=2C_l^2/(2l+1)$. This cosmic variance remains even with perfect detectors; a finite sky mask and detector noise add further uncertainty. The estimator is unbiased, $⟨{\hat{C}}_l⟩=C_l$, but does not turn one realization into the exact ensemble mean.