Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section V

The official notes, PDF p. 7, handwritten p. 1 derive the affine choice of coordinate time and the circular-orbit angular velocity from the geodesic equations.

The official notes, PDF pp. 7–8, handwritten pp. 1–2 calculate both elapsed proper times and their weak-field ratio for satellite A and the polar station.

The official notes, PDF pp. 8–9, handwritten pp. 2–3 calculate the geostationary satellite/station proper time ratio, including the equatorial station's rotation. They explicitly correct the formula printed in the paper; the printed target is not the first-order result of the stated Schwarzschild metric.

The official notes, PDF p. 9, handwritten p. 3 compare the two clock ratios and identify the station's rotational contribution. PDF p. 10, handwritten p. 3a, separately clarifies the normalization when coordinate time is the affine parameter.

The official notes, PDF p. 11, handwritten p. 4 give the Lagrangian for the geodesic equations.

The official notes, PDF pp. 11–13, handwritten pp. 4–6 derive all four geodesic equations and list the twelve nonzero Christoffel symbols up to symmetry of the lower indices, using time coordinate $t$.

The official notes, PDF pp. 13–14, handwritten pp. 6–7 integrate the radial geodesic equations and establish the vanishing integration constant for a null geodesic.

The official notes, PDF pp. 14–16, handwritten pp. 7–9 integrate the exponential scale factor, discuss the sign required for an incoming photon, and obtain the distance–redshift relation. They use $a(t_e)$ for the paper's otherwise undefined $R(t_e)$.

The official notes, PDF pp. 17–18, handwritten pp. 10–11 give the matter density, curvature and cosmological constant scalings and the requested logarithmic sketch. For the closed model the curvature contribution to the Friedmann equations is negative; the sketched curvature magnitude must be read with that sign restored.

The official notes, PDF p. 18, handwritten p. 11 explain the imminent maximum expansion, recollapse, remaining time of order the present age, and total lifetime of approximately twice that age.

The official notes, PDF p. 19, handwritten p. 12 integrate the radial null geodesic, express the radial coordinate in terms of comoving distance, and give its maximum value in that coordinate chart. The maximum radial coordinate is not the total path length around a closed spatial slice.

The official notes, PDF pp. 20–21, handwritten pp. 13–14 combine the Friedmann equations and Raychaudhuri equation to derive the density and scale factor of the Einstein static universe.

The official notes, PDF p. 21, handwritten p. 14 give the acceleration signs and the instability conclusion for the Einstein static universe. One wording correction is necessary: their word “oscillatory” for the overdense perturbation is a slip. Starting near rest, an overdense model contracts and its matter density rises, whereas an underdense model expands and its matter density falls; both move away from the static balance.

The official notes, PDF pp. 22–23, handwritten pp. 15–16 give both density scalings, integrate the radiation- and matter-dominated Friedmann equations, normalize the late solution today, and match the two approximations at equality.

The official notes, PDF p. 24, handwritten p. 17 integrate the physical particle horizon and derive its early-time redshift dependence, explicitly stating the present matter-dominated approximation used in the printed expression.

The official notes, PDF pp. 24–27, handwritten pp. 17–20 obtain the photon number density by integrating the Planck distribution, explain the temperature and dilution scalings, and compare the interaction time with the expansion time for thermal equilibrium. They also correct the missing cube and factor-of-two error in the paper's printed photon number density.

The official notes, PDF p. 27, handwritten p. 20 derive the early-time photon count, its vanishing limit, and the failure of thermal equilibrium when the count is of order one. Two details need supplementing.

First, their constant $N_{\gamma 0}$ is the normalization obtained by extrapolating the early-time particle horizon formula to $z=0$. It is not the count within today's actual particle horizon. Let $N_0=N_{\gamma }(t_0)$ and let $r_{h0}$ be that horizon's physical radius. Using the paper's early-time approximation givesFor a present matter-dominated approximation, $r_{h0}\simeq 2c/H_0$. Alternatively, retaining both components in the flat model exactly, ${\Omega }_R=1/(2+z_{\mathrm{eq}})$, and integrating the particle horizon givesThe second boxed formula uses the exact two-component normalization and the radiation-dominated early-time limit. Its large-$z_{\mathrm{eq}}$ leading term agrees with the first expression and $r_{h0}\simeq 2c/H_0$.

Second, a particle horizon containing only order-one photons cannot support a smooth, locally representative radiation density: discrete-particle fluctuations are of the same order as the mean, and a single photon's direction is strongly anisotropic. Thus the fluid description assuming spatial homogeneity and cosmological isotropy is not self-consistent on that causal scale. This does not by itself disprove an ensemble-averaged cosmological symmetry; it limits the extrapolation of the smooth thermal equilibrium description.