Details

The Friedmann equation in its simplest form (flat universe, cosmological constant ) is . It describes the Einstein–de Sitter universe, in which the expansion depends sensitively on the density . The so-called critical density is given by . Cosmologists prefer the dimensionless parameter . Using , the expansion rate is and the relation implies that the amount of matter in the universe remains constant. Friedmann's equation can be transformed into , with the solution , since here. In the present epoch, , so . The Hubble time would give the age of the universe if were constant, so the graph must be shifted until the line through the point and the origin is tangent to the graph. Einstein–de Sitter space was the favorite model for an expanding universe until the 1980s. After the inflation theory was proposed, vacuum energy or dark energy had to be considered. This can be represented by a cosmological constant , which Einstein inserted into his theory of general relativity to achieve a static universe but later rejected. With the vacuum density , the Friedmann equation becomes . The solution believed to be most accurate today takes and . For , becomes negative, which would imply a contracting universe.

Reference

[1] B. A. Jordaan. "Cosmology and the Engineer." (Jan 9, 2015). www.einsteins-theory-of-relativity-4engineers.com/cosmology.html.