Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 74 / 1

On the sphere, both cosmic time and the comoving coordinate $r$ are constant. The positive spatial induced metric obtained from the FLRW metric is therefore $d{\ell }^2=a^2{r}^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$. Its area element is the square root of its determinant:Integrating over the spherical coordinates givesThus $ar$ is the areal radius. In a curved spatial slice it need not equal the proper radial distance $a{\int }_0^r(1-k{r}^{\prime 2}{)}^{-1/2}d{r}^{\prime }$; the angular part of the FLRW metric, rather than a Euclidean radial-distance assumption, establishes the surface area.

For a radial null geodesic, the FLRW metric givesTwo successive wave crests traverse the same comoving path between comoving observers. Subtracting their path integrals, to first order in their short emission and reception periods, gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Since cosmic time is each observer's proper time and the frequency is the reciprocal period,This neglects peculiar velocities and local gravitational redshift contributions.

For B's light passing through A on its way to Earth, with the comparison event at A being that passage, cosmological redshift factors multiply:This is the intended numerical answer with that reception-event specification. The reception-event dependence of intergalactic redshift is essential: Earth's two redshifts by themselves neither specify the A–B separation nor identify the emission event at B whose light A receives at a chosen time. For arbitrary directions or a different reception time at A, the general answer is $a(t_{A,\mathrm{rec}})/a(t_{B,\mathrm{em}})-1$, with the emission time fixed by the A–B null geodesic; it cannot be deduced from the two supplied numbers alone.

Introduce the radial comoving distance $\chi =\int dr/\sqrt{1-k{r}^2}$. A photon travels $d\chi =cdt/a$, so a finite comoving particle horizon since a big-bang origin at $t=0$ requiresTo relate this geometric criterion to a mass density, use the Friedmann equation as well as the FLRW metric. For an early epoch dominated by $\rho ={\rho }_{\ast }{a}^{-\alpha }$, with negligible curvature and cosmological constant in the leading balance,Equivalently $dt/a=da/(a^{2}H)\propto a^{\alpha /2-2}da$. Its integral at $a=0$ converges precisely when $\alpha /2-2>-1$, henceAt $\alpha =2$ the divergence is logarithmic, and at $0<\alpha <2$ it is a power divergence. If spatial curvature dominates a bang instead, $a\propto t$ and the integral again diverges; $\alpha >2$ also ensures that this curvature term, which scales as $a^{-2}$, is subdominant near the bang. This argument assumes the ordinary expanding big-bang branch of the Friedmann equation; a metric alone imposes no density law, and a universe with a different past boundary needs its own lower integration limit.

During radiation domination, conservation of energy gives $\rho \propto a^{-4}$: dilution contributes $a^{-3}$ and the cosmological redshift of each photon contributes another $a^{-1}$. Thus $\alpha =4>2$ and $a=A\sqrt{t}$ in the flat radiation-dominated approximation. The particle horizon measured as proper radial distance on a constant-time slice isEvery local comoving observer measures the speed of light to be $c$. However, the spatial separations traversed earlier have expanded by the time the final distance is measured; the present separation from the earliest accessible comoving worldline is not the integral of local path lengths measured at their different traversal times. Indeed ${\dot{s}}_{\mathrm{hor}}=H{s}_{\mathrm{hor}}+c=2c$ in this era, explicitly separating cosmic expansion from local propagation. Curvature or a later transition to matter domination changes the exact formula; $2ct$ is the radiation-dominated result.