Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 1

For a component with the barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation isUsing $H=\dot{a}/a$ makes this a separable differential equation:Taking $a_0=1$ gives the constant-equation-of-state density scalingThe spatially flat Friedmann equation and the cosmological density parameters therefore giveAlong a radial light ray, $d\chi =dt/a$. Since the redshift-time relation is $dt=-dz/[(1+z)H]$ and $a=(1+z{)}^{-1}$,Hence the comoving radial distance is

In the flat Lambda-CDM model,At low redshift, $H(z)=H_0+O(z)$, so the angular diameter distance satisfies $d_A(z)=z/H_0+O(z^2)$ and initially increases from zero. At high redshift, pressureless matter dominates and the integral converges:Thus the positive continuous function $d_A$ rises from zero and returns toward zero, so it attains an interior maximum. This is the angular diameter distance turnover; beyond it, a fixed physical size appears larger at higher redshift.

If the variable dark-energy equation of state differs slightly from $-1$, matter still controls the large-$z$ expansion, while the same low-$z$ limit holds. The turnover therefore persists under such a small change. More generally, it persists whenever the high-redshift comoving distance grows more slowly than $1+z$.

For the transverse baryon acoustic oscillation ruler, $d_A=a\chi$ and the physical ruler length is $a{\chi }_d$. Its observed angle is thereforewhereThe unknown cosmological standard ruler length and $H_0$ occur only through the common amplitude $H_0{\chi }_d$. Measurements at two distinct redshifts givewhich generically determines the one shape parameter ${\Omega }_{m,0}$. Thus two redshifts suffice; additional redshifts overdetermine the model and improve precision.

To first order, the radial and transverse comoving separations arewhereThe local spatial geometry is that of Euclidean space, so the Pythagorean theorem givesConsequently the requested function is

Yes. A radial baryon acoustic oscillation measurement and a transverse baryon acoustic oscillation measurement of the same ruler at one redshift obeyTheir ratio cancels both the unknown ruler length and the overall Hubble scale:This is the Alcock-Paczyński parameter. At a specified redshift its dependence on ${\Omega }_{m,0}$ permits a one-redshift determination within the assumed flat Lambda-CDM model.