The Friedmann equation for the scale factor $a(t)$ of a homogeneous and isotropic universe of mass density $\rho$ is

$$H^{2}=\frac{8 \pi G \rho}{3}-\frac{k c^{2}}{a^{2}}, \quad\left(H=\frac{\dot{a}}{a}\right)$$

where $\dot{a}=d a / d t$ and $k$ is a constant. The mass conservation equation for a fluid of mass density $\rho$ and pressure $P$ is

$$\dot{\rho}=-3\left(\rho+P / c^{2}\right) H$$

Conformal time $\tau$ is defined by $d \tau=a^{-1} d t$. Show that

$$\mathcal{H}=a H, \quad\left(\mathcal{H}=\frac{a^{\prime}}{a}\right)$$

where $a^{\prime}=d a / d \tau$. Hence show that the acceleration equation can be written as

$$\mathcal{H}^{\prime}=-\frac{4 \pi}{3} G\left(\rho+3 P / c^{2}\right) a^{2}$$

Define the density parameter $\Omega_{m}$ and show that in a matter-dominated era, in which $P=0$, it satisfies the equation

$$\Omega_{m}^{\prime}=\mathcal{H} \Omega_{m}\left(\Omega_{m}-1\right)$$

Use this result to briefly explain the "flatness problem" of cosmology.