Figure 1: The expansion of the Einstein-de Sitter cosmology.

Figure 2: The expansion of the dust-filled Robertson-Walker cosmology.

Figure 3: The hyperbolic analog of the dust-filled Robertson-Walker cosmology.

Figure 4: The expansion of the radiation-filled Robertson-Walker cosmology.

The standard cosmological models are models without cosmological constant ($\Lambda=0$). We first consider Friedmann models ($p=0$), for which \begin{equation} \dot{a}^2 = \frac{C}{a} - k \end{equation} If $k=0$, we have \begin{equation} \sqrt{a}\dot{a} = \sqrt{C} \end{equation} so that \begin{equation} a = \left(\frac{9Ct^2}{4}\right)^{1/3} \end{equation} which is plotted in Figure 1. This model is called the Einstein-de Sitter cosmology, and is flat, but nonetheless expanding.

If instead $k=1$, we have \begin{equation} \dot{a}^2 = \frac{C}{a} - 1 \end{equation} which can be solved parametrically, yielding \begin{align} \frac{t}{C} &= \frac12 \left( \eta - \sin\eta \right) \\ \frac{a}{C} &= \frac12 \left( 1 - \cos\eta \right) \end{align} which is plotted in Figure 2. This model is called the dust-filled Robertson-Walker cosmology, and is closed; this universe ends with a “Big Crunch”.

If $k=-1$, the solution is formally similar, but with circular trigonometric functions replaced by hyperbolic trigonometric functions. The parametric solution in this case is \begin{align} \frac{t}{C} &= \frac12 \left( \sinh\eta - \eta \right) \\ \frac{a}{C} &= \frac12 \left( \cosh\eta - 1 \right) \end{align} which is plotted in Figure 3. This model is open (expands forever).

Finally, if $k=1$ and \begin{equation} p = \frac{\rho}{3} \end{equation} we have the radiation-filled Robertson-Walker cosmology, for which it turns out that \begin{equation} \rho a^4 = \hbox{constant} \end{equation} and which admits a parametric solution of the form \begin{align} \frac{t}{B} &= 1 - \cos\eta \\ \frac{a}{B} &= \sin\eta \end{align} which is plotted in Figure 4. and which is again closed.