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### Will the expansion of the universe continue?

In the '20s Alexander Friedman proposed solutions to the problem of the evolution (dynamics) of the universe.
Three basic models, resting on the theory of general relativity, were proposed, and are distinguished by the value of the density (mass per unit volume) r (rho) of matter. The motion of stars and galaxies is governed by their mutual gravitational interaction. The gravitational force is always attractive and is proportional to the masses of the objects and inversely proportional to the distance squared, or

Thus, we expect that all the galaxies and stars would collapse to a point. In reality, if we give galaxies enough initial velocity and the density is low enough, gravity will not win out and galaxies will move further apart forever. This is analogous to throwing an object in the air. Momentarily, the object moves against gravity. If the velocity of the throw is large enough, then the object will escape the gravitational pull of the Earth, otherwise it will fall back down. The same type of argument applies for the dynamics (evolution) of the universe.
Let's introduce W= (density of the universe)/ rcrit, where rcrit is the density where the universe is tethering between collapsing into itself and expanding forever.

W < 1 gives a universe that will expand forever.
W > 1 is a universe that, at a certain point, will contract.
W = 1 is the case in between.

Schematic plot for the three different models of the universe

The Einstein - de Sitter model

This is a static model where the universe is closed. This can be visualized by thinking that we live on the surface of a sphere. When we travel, we think we go in a straight line, but in reality we go around the sphere. Thus, in this model of the universe, there are no boundaries. In this model, W=1 and the force of gravity is balanced by the "cosmological constant", which is a term in Einstein's equations that balances the gravitational force. This model has been superseded by models that incorporate the fact that the universe is expanding (see previous section), and, therefore, is not static.

For more details see Einstein theory of general relativity