 Solution

First retain the exact initial normalization. The scale factor obeys $a/a_i=(t/t_i{)}^{2/3}$, so the cosmological redshift isAlso $\rho =3M/(4\pi r^3)$ and ${\rho }_m=1/(6\pi G{t}^2)$. Using $A^3=GM{B}^2$ givesThese equations exactly accommodate the finite-time initial conditions in b.

Now neglect $T$ and retain leading order in ${\delta }_i\ll 1$. Since $t_i/B=4{\delta }_i^{3/2}/[3(1+{\delta }_i)]$, the cosmological redshift relation reduces toDefine ${\delta }_0=\frac{3}{5}{\delta }_i(1+z_i)$. ThenThe second equality is exact for the ideal common-bang growing-mode spherical-collapse model; it is approximate for the paper's finite-time initial Hubble-velocity data.

The factor $3/5$ has a physical origin. Linear mass conservation gives ${\dot{\delta }}_i=0$ when the initial local expansion equals the background Hubble flow. The general linear dust solution is ${\delta }_L=C_g(t/t_i{)}^{2/3}+C_d(t/t_i{)}^{-1}$. Imposing ${\delta }_L(t_i)={\delta }_i$ and ${\dot{\delta }}_L(t_i)=0$ yieldsThus ${\delta }_0$ is the present extrapolation of the growing component, after neglecting the decaying transient. It is not the exact present value of the full linear solution for finite $t_i$.