# What is the Friedmann Equation

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

The Friedmann equation is a dynamical equation that describes the expansion of the universe.

Equations

$$H^2 = \left( \frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2}$$

Extended explanation

The Friedmann equation is derived from the 0-0 component of the Einstein field equations of General Relativity, on invoking the Friedmann Robertson Walker metric as the correct metric for the spacetime of the universe.

Note that, coincidentally, the equation can be derived by using Newtonian mechanics. We present this derivation here, with the caveat that it should not be taken as a rigorous derivation.

Consider a particle of mass m, a radius r from a uniform expanding medium of density ρ. The total mass of the material within the radius r is given by:

$$M = \frac{4\pi \rho r^3}{3}$$

Therefore the force from Newton's universal law of gravitation is given by:

$$F = -\frac{4\pi G\rho r m}{3}$$

And the gravitational potential energy of the particle is:

$$V= -\frac{4\pi G r^2 m}{3}$$

If we consider the energy conservation of the particle,

$$U= E_k + V$$

where $E_k = \frac{1}{2} m \dot{r}^2$

U must remain a constant and with r being written as comoving coordinates $\mathbf{r} = a(t) \mathbf{x}$

we have:

$$U = \frac{1}{2}m\dot{a}^2x^2 - \frac{4}{3} \pi G \rho a^2 x^2 m$$

Which gives the familiar:

$$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2}$$

where we have defined $kc^2 = -\frac{2U}{mx^2}$

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