**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 *a*_{0} 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.

Type of Energy | (a) |
a(t) |

Dust | a^{-3} |
t^{2/3} |

Radiation | a^{-4} |
t^{1/2} |

Cosmological Constant | constant | e^{Ht} |

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
*H*_{0}^{-1} is known as the
*Hubble time*, and provides a useful estimate of the time scale
for which the universe has been around.