The Density Parameter

So we have seen that the shape of space and fate of the universe depend on k:

k=0: flat space, universe coasts to a certain size
k=1: positively curved space, universe eventually recollapses
k=-1: negatively curved space, ever expanding universe

What does k depend on?

Go back to our equation for the expansion:

Realize that

So that

(This, incidently, is called the Friedmann equation)

Let's look at a flat (k=0) universe:

This is true if

This value of the density is called the critical density. It is not (necessarily) the density of the universe, it is the density the universe would have if it were flat. Putting in H₀=72 km/s/Mpc, rho_c=1.4x10^-7 Msun/pc³.

We write the density parameter as

And then we can characterize different models using different values for Omega₀:

What is Omega₀? Good question - it's very difficult to measure this!

Current observations suggest

Omega₀ ~ 0.2 - 0.4 (this is all matter, including dark matter)
Omega_baryon ~ 0.04 (this is only baryonic [ie, normal] matter)