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

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.

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.

The ordinary Sachs-Wolfe effect combines intrinsic last-scattering temperature with the gravitational redshift of photons escaping the last-scattering gravitational potential. For adiabatic large-scale perturbations in matter domination, with the potential convention in the question, the combination is $\Theta =\varphi /(3c^2)$, up to the observer monopole. It is distinct from the Integrated Sachs-Wolfe effect caused by a time-varying potential along the later line of sight.

Use the Fourier transform convention in which $P_{\delta }(k)=Ak$ is the matter power spectrum and $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_D(\mathbf{k}-{\mathbf{k}}^{\prime })P_{\delta }(k)$. The Poisson equation becomes $-k^2{\varphi }_{\mathbf{k}}=4\pi G{\bar{\rho }}_0{\delta }_{\mathbf{k}}$, givingWith the matter-dominated critical-density approximation ${\bar{\rho }}_0=3H_0^2/(8\pi G)$,independent of $k$. Thus the Harrison-Zeldovich spectrum gives scale-invariant potential fluctuations and the requested $H_0^4$ dependence. Keeping a general present matter fraction would insert ${\Omega }_m^2$; the answer here follows the stipulated critical matter-dominated normalization. In this Fourier transform convention, $A$ has dimensions of length to the fourth power.

The finite angular resolution of COBE DMR suppresses small angular scales, so only low spherical harmonic multipoles can be recovered usefully. The approximate relation $\theta \sim \pi /l$ associates $l\simeq 17$ with scales of order ten degrees; this is an effective sensitivity and beam-window limit, not a mathematical hard truncation.

Interpret the printed superscript uncertainty as $(1.1\pm 0.1)\times 10^{-5}$. Its square estimates the band-limited zero-separation variance, rather than the full $C(0)$:Use the coefficient supplied in the paper for this numerical inference. With ${\sigma }_{2:17}=1.1\times 10^{-5}$,so c givesThe stated amplitude uncertainty gives approximately $18\%$ relative uncertainty in $A$, or $A\simeq (1.38\pm 0.25)\times 10^6\mathrm{Mp}{\mathrm{c}}^4$, keeping $H_0$ fixed. The ideal matter-era Sachs-Wolfe plateau with precisely the c spectrum convention has coefficient $2\pi /9=0.698$, rather than $0.732$; the supplied numerical coefficient is about $5\%$ larger and has been treated as the paper's prescribed approximation. A real beam-convolved analysis retains its window function and additional cosmological transfer effects.