= Scalar-field reconstruction for a constant equation of state
{title2=$\phi-\phi_0=\sqrt{3(1+w)/(8\pi G)}\log(a/a_0)$}
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 $\dot\phi^2=(1+w)\rho$. The positive rolling branch gives the displayed relation and $V(\phi)=(1-w)\rho_0\exp[-\sqrt{24\pi G(1+w)}(\phi-\phi_0)]/2$. 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 $0<\lambda^2<2$.
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