Critical Density

 

The energy density Friedmann Equation with  has the following form

 

(32.1)

 

Using the definition of the Hubble parameter H given by

 

(32.2)

 

 

the above Friedmann equation can be rewritten as

 

(32.3)

 

 

 

Rearranging the terms we find the following.

 

(32.4)

 

 

 

This equation implies that there is a critical density associated with the spatially flat universe given by:

 

 

 

(32.5)

Using the values for the present era one obtains the following estimate for :

 

 

(32.6)

 

The density of luminous matter in the universe has been calculated by astronomers to be

 

 

(32.7)

 

Hence, if this represents all of the matter that exists there is not enough matter to close the universe since a density greater than the critical density would be required for that.

Since astronomers think that the luminous matter estimate does not represent the amount of matter in the universe very well, a search is on to find the ‘dark’ matter that would get the universe at the critical density as the Inflationary Universe Model predicts. This is called The Dark Matter Problem.

 

It is standard in cosmology to introduce the density parameter  defined as follows

 

(32.8)

 

 

 

Hence we see that

 

(32.9)