Scalar-field reconstruction for a constant equation of state ($\varphi -{\varphi }_0=\sqrt{3(1+w)/(8\pi G)}\log (a/a_0)$)

A flat universe dominated by a canonical homogeneous scalar field with constant equation-of-state parameter $w>-1$ has $\rho ={\rho }_0(a/a_0{)}^{-3(1+w)}$ and ${\dot{\varphi }}^2=(1+w)\rho$. The positive rolling branch gives the displayed relation and $V(\varphi )=(1-w){\rho }_0\exp [-\sqrt{24\pi G(1+w)}(\varphi -{\varphi }_0)]/2$. The opposite rolling branch reverses the exponential slope. For $w=-1$ the field is constant; $w<-1$ cannot be realized by a real scalar with this positive kinetic term. An exponential slope $\lambda \sqrt{8\pi G}$ yields $w={\lambda }^2/3-1$, with accelerating nonphantom expansion when $0<{\lambda }^2<2$.