= Scalar-field reconstruction of a coasting component
{title2=$V(\phi)=2M^2H_0^2\Omega_X/a(\phi)^2$}
For conserved $w=-1/3$ scalar energy, $\rho_X=3M^2H_0^2\Omega_X/a^2$ and $\phi'=\sqrt2MH_0\sqrt{\Omega_X}$ 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,
$$
a(\phi)=A\cosh\left[\frac{\phi-\phi_0}{\sqrt2M}+C\right]-b,\qquad C=\operatorname{arcosh}\frac{1+b}{A},
$$
and $V(\phi)=2M^2H_0^2\Omega_X/a(\phi)^2$. The formulas retain all density parameters without eliminating one by the flatness sum.
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