Past exam of the astrophysics course of the University of Oxford / 2021 / 2 / b / Solution

For $r=A(1-\cos \theta )$ and $t+T=B(\theta -\sin \theta )$, differentiation givesComparison with $-GM/[A^2(1-\cos \theta {)}^2]$ proves $A^3=GM{B}^2$. Substitution into the conserved specific orbital energy gives $ϵ=-GM/(2A)$. ThereforeThese constants exactly reproduce both the initial radius and expanding velocity. The initial parameter obeysIn an Einstein-de Sitter universe whose background big bang is $t=0$, $t_i=2/(3H_i)$. Thus the exact required offset isExpanding ${\theta }_i-\sin {\theta }_i=\frac{4}{3}{\delta }_i^{3/2}-\frac{8}{5}{\delta }_i^{5/2}+O({\delta }_i^{7/2})$ givesOne cannot set $T=0$ exactly at finite ${\delta }_i$ while retaining the stated initial Hubble velocity and background time origin. For example ${\delta }_i=0.01$, $t_i=1$ gives $T=-0.00199148$. Shifting to $t^{\prime }=t+T$ removes the offset from the local orbit, but the background then scales as $a\propto (t^{\prime }-T{)}^{2/3}$, not $t^{\prime 2/3}$. Setting $T\simeq 0$ is justified in the early-initial-time limit for subsequent evolution; the following standard growing-mode formulae use that approximation.