there is a good quantity to know, a coefficient that you use
"ate pig over three see square" 8 pi G/(3 c^2)
The critical relation needed for a nice universe is:
square of expansion rate equals that coefficient times the density.
H^2 = 8 pi G/(3 c^2) * density
If you know H, like it's the observed 7.26 percent per 10^9 years, then you can solve for the critical or perfect density to go with that rate of expansion.
This involves turning over the coefficient and getting
"three see square over ate pig" 3 c^2/(8 pi G)
If we solve our densityexpansion equation for the density we get:
density = (3 c^2/(8 pi G))*H^2
In other words, using what we know about the present expansion rate:
[critical] density = (3 c^2/(8 pi G))*(7.26 percent per 10^9 years)^2
Now that is ready to go into the google calculator except we need to specify what UNITS we want the density expressed in. I'm assuming we convert everything into the common currency of energy and want it expressed in the standard metric form of joules per cubic meter. The matter is all converted to energy so we can add it together with the dark energy and radiation energy etc.
Since the actual density in the universe is very sparse it will turn out nanojoules per m^3, so we can specify that in what we paste into the google window:
(3 c^2/(8 pi G))*(7.26 percent per 10^9 years)^2 in nanojoules per m^3
Let's see how that works. Does it give us some reasonable density figure in nanojoules per cubic meter?
YES!!! Check it out. Google says the presentday critical density is 0.85 nanojoule per m^3.
