2.3. Flat Universes

It is much easier to find exact solutions to cosmological equations of motion when k = 0. Fortunately for us, nowadays we are able to appeal to more than mathematical simplicity to make this choice. Indeed, as we shall see in later lectures, modern cosmological observations, in particular precision measurements of the cosmic microwave background, show the universe today to be extremely spatially flat.

In the case of flat spatial sections and a constant equation of state parameter w, we may exactly solve the Friedmann equation (27) to obtain

(35)

where a₀ is the scale factor today, unless w = - 1, in which case one obtains a(t) e^Ht. Applying this result to some of our favorite energy density sources yields table 1.

Table 1. A summary of the behaviors of the most important sources of energy density in cosmology. The behavior of the scale factor applies to the case of a flat universe; the behavior of the energy densities is perfectly general.

Type of Energy	(a)	a(t)
Dust	a-3	t2/3
Radiation	a-4	t1/2
Cosmological Constant	constant	eHt

Note that the matter- and radiation-dominated flat universes begin with a = 0; this is a singularity, known as the Big Bang. We can easily calculate the age of such a universe:

(36)

Unless w is close to -1, it is often useful to approximate this answer by

(37)

It is for this reason that the quantity H₀^-1 is known as the Hubble time, and provides a useful estimate of the time scale for which the universe has been around.