= 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 $\rho_{Q0}=3H_0^2(1-\Omega_{M0})/(8\pi G)$, so
$$
\boxed{\rho_Q(a)=\frac{3H_0^2(1-\Omega_{M0})}{8\pi Ga^2}}.
$$
This 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.
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