Finite-time Hubble-velocity spherical-collapse offset ($T/t_i=-{\delta }_i/5+O({\delta }_i^2)$)

For a pressureless overdensity initialized with the background Hubble flow in an Einstein-de Sitter universe, the cycloid $r=A(1-\cos \theta )$, $t+T=B(\theta -\sin \theta )$ has $A=r_i(1+{\delta }_i)/(2{\delta }_i)$, $B=(1+{\delta }_i)/(2H_i{\delta }_i^{3/2})$, and ${\theta }_i=2\arctan \sqrt{{\delta }_i}$. These follow from conservation of energy and matching the initial radius and velocity. With $t_i=2/(3H_i)$, $T=B({\theta }_i-\sin {\theta }_i)-t_i=-t_i{\delta }_i/5+O(t_i{\delta }_i^2)$. Thus an exactly common background and perturbation bang time is incompatible with these finite-time data. Linear mass conservation instead gives a growing amplitude $3{\delta }_i/5$ and a decaying amplitude $2{\delta }_i/5$, whose initial time derivatives cancel.