Past exam of the astrophysics course of the University of Oxford / 2021 / 4

The Einstein-de Sitter universe has $a\propto t^{2/3}$ and $H=\dot{a}/a\propto a^{-3/2}$. Normalizing $a_0=1$ therefore gives $H(z)=H_0(1+z{)}^{3/2}$. Differentiating $1+z=1/a$ givesIn the local comoving rest frame, a photon crosses proper path length $d\ell =c|dt|$. Galaxies with proper number density $n(z)$ and geometric cross section ${\sigma }_G=\pi r_G^2$ contribute the mean number of intersections $d\tau =n(z){\sigma }_{G}d\ell$. The infinitesimal probability of an intersection is the same to first order, givingThe radius is a physical radius, and $n(z)$ is a proper number density evaluated at epoch $z$, rather than a density already multiplied by $dz$. For a finite path with overlaps, the integrated mean count $\tau$ need not itself be the probability of at least one intersection.

Conservation of the comoving galaxy population gives $n(z)=n_0{a}^{-3}=n_0(1+z{)}^3$. Holding the proper galaxy radius fixed, integrate a:With $r_G=0.015\mathrm{Mpc}$, $n_0=0.01\mathrm{Mp}{\mathrm{c}}^{-3}$, and $c/H_0=4282.75\mathrm{Mpc}$, set $C=c\pi r_G^2{n}_0/H_0=0.030273$. ConsequentlyFor random uncorrelated intersections, the zero-intersection probability is $e^{-\tau }$ from the Poisson distribution, so the fraction of the sky covered by foreground galaxies out to $z=1$ isIgnoring overlaps gives $f\simeq \tau \simeq 3.7\%$. A literal shell of zero thickness exactly at $z=1$ has zero coverage; the integrated foreground interpretation is needed. Clustering changes the exact overlap probability, although the mean optical depth above still follows from the mean number density.

Finally,At this optical depth, the Poisson sky coverage is $1-e^{-1}\simeq 63\%$, not $100\%$. The large inferred cosmological redshift extrapolates conserved galaxy counts and fixed radii far beyond their plausible physical range; it is the answer within the prescribed toy model.

Use the dimensionally correct flat Friedmann-Lemaître-Robertson-Walker metricThe source's $-cd{t}^2$ is missing a factor $c$, and its angular term must mean the unit-sphere line element $d{\Omega }^2$, not the solid angle element $d\Omega =\sin \vartheta d\vartheta d\phi$. For example, $cd{t}^2$ has dimensions length times time, while $d{s}^2$ has dimensions length squared. The corrected convention above removes both ambiguities.

A radial light ray has $d{s}^2=0$ and $|d\chi |=c|dt|/a$. ThusA proper shell volume at emission is $d{V}_{\mathrm{proper}}=a^3{\chi }^{2}d\chi d\Omega$. Multiplying by $n(z)=n_0{a}^{-3}$ cancels the expansion factors, equivalently using the comoving volume element directly. HenceThis counts every galaxy in the shell per solid angle, rather than intersections weighted by individual galaxy areas. Near $z=0$ it reduces to $n_0(c/H_0{)}^3{z}^2$, as expected for a Euclidean volume shell.