Slightly rewriting Einstein's equation for the Robertson-Walker metric, as given in §Robertson-Walker Metrics, we obtain first \begin{equation} \frac{\dot{a}^2+k}{a^2} = \frac{8\pi\rho+\Lambda}{3} \end{equation} and then \begin{equation} \frac{2\ddot{a}}{a} = \Lambda - 8\pi p - \frac{\dot{a}^2+k}{a^2} = \frac23 \Lambda - \frac{8\pi}{3} \left(\rho+3p\right) \end{equation} A physically realistic model that is not empty will have strictly positive energy density $(\rho>0)$ and nonnegative pressure density $(p\ge0)$.

Einstein initially assumed that $\Lambda=0$. In this case (and also if $\Lambda<0$), we must have \begin{equation} \ddot{a} < 0 \end{equation} In other words, in the absence of a cosmological constant, the universe cannot be static! Einstein was unhappy with this conclusion, since the then-current belief was that the universe was indeed static; he later added the cosmological constant term precisely so as to make static solutions possible.

However, later observations showed that the universe is currently expanding, that is, that \begin{equation} \dot{a} \big|_{\hbox{now}} > 0 \end{equation} In an expanding universe, we expect to see all other stellar objects receding from us. To see that this property does not violate our assumptions of homogeneity and isotropy, consider again an expanding balloon, shown at two different stages in Figure 1. Imagine that the surface of the balloon represents our universe, and the black dots represent galxies. As the balloon expands, the distance between every pair of galaxies increases.

Elementary calculus shows that, if $\dot{a}>0$ now, and if $\ddot{a}<0$ always, then it must be true that $a$ was zero at some time in the past. Thus, Robertson-Walker models with $\Lambda\le0$ (and with expansion now) must have a past singularity, called the Big Bang, when the universe had zero size.

This result is a special case of a much more general principle in relativity, which, loosely stated, says that “gravity attracts”. More precisely, reasonable matter (positive energy density) always leads to a singularity somewhere, either in the past (e.g. the Big Bang) or in the future (such as a “Big Crunch” for cosmological models, or a black hole formed by a collapsing star).

        
Figure 1: Distances between stellar objects increase as the universe expands, illustrated as the distance between dots on an expanding balloon.