Coasting fluid ($w=-\frac{1}{3},\rho \propto a^{-2}$)

A coasting fluid has pressure $p=-\rho /3$. The cosmological perfect-fluid continuity equation makes its density scale as $a^{-2}$. If it alone fills a flat expanding universe, the scale factor grows linearly in physical time. Its background density has the same scale-factor dependence as a spatial-curvature term, although the fluid does not itself create spatial curvature.

For conserved $w=-1/3$ scalar energy, ${\rho }_X=3M^2{H}_0^2{\Omega }_X/a^2$ and ${\varphi }^{\prime }=\sqrt{2}M{H}_0\sqrt{{\Omega }_X}$ on the increasing branch. With dust and radiation, define $b={\Omega }_m/(2{\Omega }_X)$ and $A=\sqrt{{\Omega }_m^2/4-{\Omega }_r{\Omega }_X}/{\Omega }_X$. When the discriminant is positive,and $V(\varphi )=2M^2{H}_0^2{\Omega }_X/a(\varphi {)}^2$. The formulas retain all density parameters without eliminating one by the flatness sum.