 Solution

For a single constant-$w$ perfect fluid in a flat expanding FLRW metric model, cosmological energy conservation gives $\rho ={\rho }_0{a}^{-3(1+w)}$ and $H(z)=H_0(1+z{)}^{3(1+w)/2}$. With $a_0=1$, the luminosity distance follows by integrating the radial null geodesic:The constant-equation-of-state luminosity distance is consequentlyThis includes $w=-1$ regularly, giving $D_L=z(1+z)/H_0$. Restoring conventional units multiplies the distance formulas by $c$. Expanding about $z=0$ givesThe second result also follows directly from the Friedmann acceleration equation. The expansion agrees with the general coefficient $(1-q_0)/2$ of $z^2$.

For phantom energy, $w<-1$, put $n=3(1+w)/2<0$. Integration of $\dot{a}=H_0{a}^{1-n}$ with $a(t_{\mathrm{now}})=1$ givesBecause $1/n<0$, the scale factor diverges at the Big Rip timeThis is a property of the stipulated constant-$w$ fluid model; it is not realizable by the positive-density canonical scalar of the preceding parts.