Useful Solutions
The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of state
where is the pressure, is the mass density of the fluid in the comoving frame and is some constant.
In spatially flat case (k = 0), the solution for the scale factor is
where is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by is extremely important for cosmology. E.g. describes a matter-dominated universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as
- matter-dominated
Another important example is the case of a radiation-dominated universe, i.e., when . This leads to
- radiation dominated
Note that this solution is not valid for domination of the cosmological constant, which corresponds to an . In this case the energy density is constant and the scale factor grows exponentially.
Solutions for other values of k can be found at Tersic, Balsa. "Lecture Notes on Astrophysics". http://nicadd.niu.edu/~bterzic/PHYS652/PHYS652_notes.pdf. Retrieved 20 July 2011..
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