Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 3 / a / Solution

Separate conservation of a fluid with fixed $w=P/\rho$ gives $\dot{\rho }+3H(1+w)\rho =0$, hence $\rho \propto a^{-3(1+w)}$. The $w=-1/3$ component therefore has ${\rho }_Q\propto a^{-2}$. Flatness today fixes its normalization to ${\rho }_{Q0}=3H_0^2(1-{\Omega }_{M0})/(8\pi G)$, soThis is a coasting fluid: its contribution to $\rho +3P$ is zero, so it does not itself drive accelerated expansion. Its background Friedmann contribution resembles curvature, although the actual spatial geometry remains flat.