Next Contents Previous


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):

 R_{\mu \nu} - \frac{1}{2}g_{\mu \nu} {\mathcal R} = 8\pi G\; T_{\mu
 \nu} + \Lambda g_{\mu \nu}
\end{equation} (10)

where Rµnu is the Ricci tensor, R is the Ricci scalar, gµnu is the metric tensor describing the local curvature of space (intervals of spacetime are described by ds2 = gµnu dxµ dxnu), Tµnu is the stress-energy tensor and Lambda is the cosmological constant. Taking the (µ, nu) = (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:

H^{2} = \frac{8\pi G \rho}{3} - \frac{k}{R^{2}} + \frac{\Lambda}{3}\\
\end{equation} (11)

where R is the scale factor of the Universe, H = dot{R} / R is Hubble's constant, rho is the density of the Universe in relativistic or non-relativistic matter, k is the constant from Eq. 1 and Lambda is the cosmological constant. In words: the expansion (H) is controlled by the density (rho), the geometry (k) and the cosmological constant (Lambda). Dividing through by H2 yields

1 = \frac{\rho}{\rho_{c}} - \frac{k}{H^{2}R^{2}} + \frac{\Lambda}{3H^{2}}
\end{equation} (12)

where the critical density rhoc = 3 H2 / 8 pi G. Defining Omegarho = rho / rhoc and OmegaLambda = Lambda / 3H2 and using Omega = Omegarho + OmegaLambda we get,

1 - \Omega = \frac{-k}{H^{2}R^{2}}
\end{equation} (13)

or equivalently,

(1 - \Omega)H^{2}R^{2} = constant.
\end{equation} (14)

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

\frac{3H^{2}}{8\pi G \rho} = 1 - \frac{3k}{8\pi G \rho R^{2}}
\end{equation} (15)

which can be rearranged to give

\left(\Omega^{-1} - 1\right)\rho R^{2} = constant
\end{equation} (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 rho. Energy conservation requires,

2E = v^{2} - \frac{2GM}{R}= H^{2}R^{2} - \frac{8\pi G R^{2}\rho}{3}.
\end{equation} (17)

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

\rho_{c} = \frac{3 H^{2}}{8 \pi G} = 1.879 \;h^{2} \times 10^{-29} g\;
cm^{-3} \sim 20\; protons\; m^{-3}.
\end{equation} (18)

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

constant = H^{2}R^{2} - \frac{8\pi G R^{2}\rho}{3}.
cm^{-3} \sim 20\; protons\; m^{-3}.
\end{equation} (19)

Dividing Eq. 19 by H2 R2 we get

(1 - \Omega)H^{2}R^{2} = constant,
\end{equation} (20)

which is the same as Eq. 14. Multiplying Eq. 19 by 3 / (8pi G rho R2) we get

(\Omega^{-1} - 1)\rho R^{2} = constant
\end{equation} (21)

which is the same as Eq. 16.

Next Contents Previous