= Finite-time Hubble-velocity spherical-collapse offset
{title2=$T/t_i=-\delta_i/5+O(\delta_i^2)$}
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.
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