The Dynamics of the Expanding Universe

We'll illustrate expansion dynamics using Newtonian dynamics. Happily, we will derive the same dynamical equations that come out of general relativity for a relativistic cosmology, with a few terms redefined.

Start with a test particle on the surface of an expanding sphere of radius R. Its equation of motion starts with F=ma and works out to be:

Since density is proportional to R^-3, and we define "now" with a 0 subscript, and R₀=1, we have

Which we can insert into the equation of motion to get

Note that if rho₀ is nonzero, the Universe must be expanding or contracting. It cannot be static.

How do we integrate this? Multiply both side by Rdot to get

And remember that

So that

Now, another rememberance:

So that we have

Or,

where -k is a constant of integration. Replacing rho₀ with rho*R³, and then dividing by R², we finally get

What does this mean?

If k=0, then Rdot is always positive, and the expansion continues at an ever slowing pace (since rho is dropping). This is called a critical or flat universe.

If k>0, Rdot is initially positive, but will reach a point where it goes to zero. Expansion stops, gravity wins, and the universe then starts to collapse. This is a closed universe.

If k<0, Rdot is always positive, and never goes to zero -- expansion always continues. This is an open universe.