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


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

What does this mean?