Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 3 / ii

In an Einstein-de Sitter universe, $H(z)=100h(1+z{)}^{3/2}\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. At the cloud's approximate epoch $z_{\ast }=3$, $H=800h\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, and $\delta z=0.1$ corresponds to $\delta v\simeq 7500\mathrm{km}{\mathrm{s}}^{-1}$. ThereforeChoosing the midpoint $z_{\ast }=3.05$ instead gives $9.1h^{-1}\mathrm{Mpc}$ proper. Thus the intended approximate proper distance is about $9h^{-1}\mathrm{Mpc}$; the small variation reflects the epoch choice in the local expansion, not a factor-of-four physical discrepancy. For example the exact comoving separation is $2c{H}_0^{-1}[(1+z_c{)}^{-1/2}-(1+z_q{)}^{-1/2}]$, whose conversion at the midpoint agrees with that estimate.