Friedmann's Equation can be derived from Einstein's 4×4 matrix equation of general relativity (see for example Landau & Lifshitz 1975, Kolb and Turner 1992 or Liddle & Lyth 2000):

(10) |

where
*R*_{µ} is
the Ricci tensor,
is the Ricci scalar,
*g*_{µ} is
the metric tensor describing the local curvature of space (intervals of
spacetime are described by *ds*^{2} = *g*_{µ}
*dx*^{µ}
*dx*^{}),
*T*_{µ} is
the stress-energy tensor and
is the
cosmological constant. Taking the
(*µ*, ) = (0, 0)
terms of Eq. 10
and making the identifications of the metric tensor with the terms in
the FRW metric of Eq. 1, yields the Friedmann Equation:

(11) |

where *R* is the scale factor of the Universe,
*H* = /
*R* is Hubble's constant,
is the density
of the Universe in relativistic or non-relativistic matter, *k* is
the constant from Eq. 1 and
is the
cosmological constant.
In words: the expansion (*H*) is controlled by the density
(), the
geometry (*k*) and the cosmological
constant ().
Dividing through by *H*^{2} yields

(12) |

where the critical density
_{c} =
3 *H*^{2} /
8 *G*.
Defining _{} =
/
_{c}
and _{} =
/
3*H*^{2} and using
=
_{} +
_{} we
get,

(13) |

or equivalently,

(14) |

If we are interested in only post-inflationary expansion in the
radiation- or matter-dominated epochs we can ignore the
term and multiply
Eq. 11 by 3 / 8 *G*
to get

(15) |

which can be rearranged to give

(16) |

A more heuristic Newtonian analysis can also be used to derive Eqs. 14
& 16 (e.g.
Wright 2003).
Consider a spherical shell of radius *R* expanding at a velocity
*v* = *HR*, in a universe of density
.
Energy conservation requires,

(17) |

By setting the total energy equal to zero we obtain a critical density
at which *v* = *HR* is the escape velocity,

(18) |

However, by requiring only energy conservation (2*E* =
*constant* not necessarily *E* = 0) in Eq. 17, we find,

(19) |

Dividing Eq. 19 by *H*^{2} *R*^{2} we get

(20) |

which is the same as Eq. 14. Multiplying Eq. 19 by
3 / (8 *G*
*R*^{2}) we get

(21) |

which is the same as Eq. 16.