There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is: a ˙ ˙ 2 + k c 2 a 2 = 8 π π G ρ ρ + Λ Λ c 2 3 , which is derived from the 00 component of the Einstein field equations. The second is: a ¨ ¨ a = − − 4 π π G 3 ( ρ ρ + 3 p c 2 ) + Λ Λ c 2 3 which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time).