Constant-equation-of-state luminosity distance ($D_L=(1+z)c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$)

For one conserved constant-$w$ perfect fluid in a flat expanding FLRW metric, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Integrating the radial null geodesic giveswhen $w\ne -1/3$. Its continuous coasting limit is $D_L=c(1+z)\log (1+z)/H_0$. Expansion gives $D_L=(c/H_0)[z+(1-3w)z^2/4+O(z^3)]$, agreeing with the deceleration parameter $q_0=(1+3w)/2$. Curvature, additional independently evolving fluids or a varying $w$ change this single-fluid expression.