A constant overdensity is created by taking a spherical region of a flat matterdominated universe with radius $\bar{R}$ and compressing it into a region with radius $R<\bar{R}$. The evolution is governed by the parametric equations

$$R=A R_{0}(1-\cos \theta), \quad t=B(\theta-\sin \theta)$$

where $R_{0}$ is a constant and

$$A=\frac{\Omega_{m, 0}}{2\left(\Omega_{m, 0}-1\right)}, \quad B=\frac{\Omega_{m, 0}}{2 H_{0}\left(\Omega_{m, 0}-1\right)^{3 / 2}}$$

where $H_{0}$ is the Hubble constant and $\Omega_{m, 0}$ is the fractional overdensity at time $t_{0}$.

Show that, as $t \rightarrow 0^{+}$,

$$R(t)=R_{0} \Omega_{m, 0}^{1 / 3} a(t)\left(1-\frac{1}{20}\left(\frac{6 t}{B}\right)^{2 / 3}+\ldots\right)$$

where the scale factor is given by $a(t)=\left(3 H_{0} t / 2\right)^{2 / 3}$.

that, when the spherical overdensity has collapsed to zero radius, the linear perturbation has value $\delta_{\text {linear }}=\frac{3}{20}(12 \pi)^{2 / 3}$.