The Dynamics of the Expanding Universe

We'll illustrate expansion dynamics using Newtonian gravitational dynamics. Happily, the same dynamical equations 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 is

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, also remember:

So that we have

Or,

Replacing rho₀ with rho, and dividing by R²,

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 changes sign. Expansion turns into contraction. 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.

Note: we are ignoring any cosmological constant here!