Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3

The curvature constant specifies the intrinsic spatial curvature of homogeneous constant-time slices. Positive curvature has spherical geometry, zero curvature Euclidean geometry, and negative curvature hyperbolic geometry. A conventional coordinate normalization uses $k=+1,0,-1$, with the scale factor carrying the curvature radius; rescaling that factor changes the numerical magnitude assigned to a dimensional curvature coefficient.

The sign alone does not determine the future expansion once a cosmological constant is admitted. Nor does negative or zero local curvature by itself determine the global topology: quotient spaces can share the same local curvature.

For conserved nonrelativistic matter, ${\rho }_M={\rho }_{M0}{a}^{-3}$, where $a=R/R_0$ and $a_0=1$. Substitution into the Friedmann equation giveswithThe source writes the dimensionless normalized factor as $R$ in its final expression; this is equivalent to setting $R_0=1$ consistently in the curvature coefficient. Evaluating today gives ${\Omega }_M+{\Omega }_k+{\Omega }_{\Lambda }=1$. Here the printed $\Lambda$ already has dimensions of inverse time squared. If the cosmological constant is instead defined geometrically with inverse-length-squared units, replace this $\Lambda$ by ${\Lambda }_{\mathrm{geom}}{c}^2$.

With zero curvature and no cosmological constant, the present normalization requires ${\Omega }_M=1$. Thus $\dot{a}=H_0{a}^{-1/2}$, and choosing the big bang as $t=0$ givesFor an empty expanding model, ${\Omega }_k=1$ and the curvature must be negative. The equation becomes $\dot{a}=H_0$, givingThe latter is the Milne slicing of flat spacetime; its zero-scale-factor origin is not a matter-density curvature singularity. In the matter model, gravity decelerates expansion, so the earlier slope was greater than the present slope and the same present size was reached in less time. With the stipulated $H_0$, the two model ages are approximately $8.69$ and $13.0$ Gyr.

Adiabatic photon redshifting gives $T\propto a^{-1}$. Using the stipulated temperatures,In the flat dust approximation of part c, $t=t_0{a}^{3/2}$, so $t_{\mathrm{rec}}\simeq (2/3H_0)1000^{-3/2}\simeq 2.75\times 10^5$ years. This approximation omits the significant radiation contribution near the actual recombination epoch.

The elementary travel-distance estimate is $c{t}_{\mathrm{rec}}\simeq 8.4\times 10^4$ pc, whose corresponding comoving scale today is about $84$ Mpc. In an expanding universe, however, that is not the exact proper particle-horizon radius. Integrating the null geodesic givesfor dust. Thus the proper horizon at this epoch is about $0.25$ Mpc and its present comoving size about $250$ Mpc. The factor three accounts for expansion during light propagation. These scales describe the limited causal domain in the simple decelerating model. The near-uniform CMB temperature across much more widely separated last-scattering regions motivates the horizon problem; the question's rough $ct$ estimate and the exact dust horizon should not be conflated.