How was the critical density of the universe calculated?

Asked by: Michel Parent

Answer

The critical density of the universe is actually not 'calculated' in the normal sense of
the word. It actually comes out as a parameter in the Friedmann equation for the expansion
of the universe. When we compare the true density of the universe to the 'critical
density', we determine whether we live in a 'closed,' 'flat,' or 'open' universe. What do
these words mean? If the density of our universe is greater than the critical density, our
universe is 'closed,' which means our universe will eventually stop expanding and start
contracting back on itself. If the density of the universe is equal to the critical
density, then we live in a 'flat' universe. In a flat universe, the universal expansion
slows, but it never reverses into a contraction. Lastly, if the density of the universe is
less than the critical density, then the universe is 'open.' In an open universe, the
expansion continues forever.

As for how to see what the critical density is, we should turn to the Friedmann equation:

This equation determines how the scale factor of the universe (essentially the size of the
universe), R, changes with time in terms of the energy density, , and k, a measure of
the curvature of the universe. The first term in this equation is just the square of the
Hubble parameter, H. We now just rearrange this equation:

This is how the critical density is defined. To calculate it, you just need to measure the
Hubble parameter H and Newton's constant G. Currently, the best known values for H and G
give a value for the critical density of about 1x10^{-29} grams per cubic centimeter. This
seems very small, but we have to remember that most of the universe is practically empty.
Recent measurements indicate that the actual density of our universe is very close to the
critical density.
Answered by: Andreas Birkedal-Hansen, M.A., Physics Grad Student, UC Berkeley

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