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


Replacing rho0 with rho, and dividing by R2,

What does this mean?

Note: we are ignoring any cosmological constant here!