The Friedmann equation and the fluid conservation equation for a closed isotropic and homogeneous cosmology are given by

$$\begin{aligned} &\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{1}{a^{2}} \\ &\dot{\rho}+3 \frac{\dot{a}}{a}(\rho+P)=0 \end{aligned}$$

where the speed of light is set equal to unity, $G$ is the gravitational constant, $a(t)$ is the expansion scale factor, $\rho$ is the fluid mass density and $P$ is the fluid pressure, and overdots denote differentiation with respect to the time coordinate $t$.

If the universe contains only blackbody radiation and $a=0$ defines the zero of time $t$, show that

$$a^{2}(t)=t\left(t_{*}-t\right)$$

where $t_{*}$ is a constant. What is the physical significance of the time $t_{*}$ ? What is the value of the ratio $a(t) / t$ at the time when the scale factor is largest? Sketch the curve of $a(t)$ and identify its geometric shape.

Briefly comment on whether this cosmological model is a good description of the observed universe at any time in its history.