= Constant-equation-of-state luminosity distance
{title2=$D_L=(1+z)c\int_0^z dz'/H(z')$}
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> gives
$$
D_L=\frac{2c(1+z)}{H_0(1+3w)}[1-(1+z)^{-(1+3w)/2}]
$$
when $w\ne-1/3$. 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.
\Image[/past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2013-constant-w-luminosity-distance.png]
{title=Flat single-fluid luminosity distances and their quadratic low-redshift approximations}
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