in an earlier post on this thread we had a little bit about the Friedmann eqns. but this is better and also here is a link to a Sean Carroll piece in LivingReviews. the people at Albert Einstein Institute-Potsdam MPI asked Carroll to do the piece on "Cosmological Constant" for LivingReviews
http://relativity.livingreviews.org/Articles/lrr-2001-1/node3.html
Sean Carroll is a blogger as well as one of the worlds foremost cosmologists. he's at chicago. check out his blog sometime--it can be entertaining---the name is "preposterousuniverse"
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In what follows I am using the same notation Sean Carroll uses in
LivingReviews which is pretty standard.
First here is a version of the Friedmann equations which conceals the cosmological constant as "dark energy" added into the rho term as another kind of energy density. So you don't see the Lambda explicitly in this version. This is how a lot of people do it nowadays, and the dark energy fraction is given as 73 percent of total energy density rho.
[tex](\frac{a'}{a})^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}[/tex]
[tex]\frac{a''}{a}= -\frac{4\pi G}{3}(\rho + 3p)[/tex]
Now I'm going to separate the cosmological constant part out as Lamda, an inverse distance squared term. Now rho is all the other stuff, not counting dark energy, and the equations are:
[tex](\frac{a'}{a})^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}[/tex]
[tex]\frac{a''}{a}= -\frac{4\pi G}{3}(\rho + 3p)+\frac{\Lambda}{3}[/tex]
EXPLAINING THE NOTATION
this is with c = 1 units, which simplifies things some.
the scale factor of the metric (whose increase is the expansion of the universe) is denoted by the letter a.
k is a spatial curvature parameter used to distinguish three cases
k = -1, 0, +1 for negative curvature, spatially flat, positive curvature
rho is an energy density, and easy to confuse with p pressure
the universe appears to be spatially flat, the critical density rhocrit is that needed for it to be perfectly flat with k = 0
HOW THE HUBBLE PARAMETER COMES IN
the Hubble parameter H is defined to be the time derivative a' of the scale parameter a, divided by a.
[tex]H^2 = (\frac{a'}{a})^2[/tex]
for the time being assume we've included the Lambda term in rho as "dark energy, because this is a convenient way to set things up for calculating stuff, like the critical density. In the case of a spatially flat universe the first Friedmann equation boils down to
[tex]H^2 = \frac{8\pi G}{3}\rho_{crit}[/tex]
algebraically that turns into the formula for the critical density
[tex]\rho_{crit} = \frac{3}{8\pi G}H^2[/tex]
the Hubble parameter has been measured really accurately at 71 km/s per Mpc
and this let's us calculate the critical density at 0.83 joule per cubic km.since the U tests out flat or very nearly so, this is taken to be the
density of all the stuff, stars galaxies, light, dark matter, dust, dark energy etc. It all amounts to 0.83 joule per cubic km.
And the dark energy being 73 percent (from supernova data) means that its share is 0.6 joule per cubic km.
