# Luminosity distance from angular diameter distance

Consider that I know the cosmological angular diameter distance at a given redshift :

$$D_{A}\left(z\right)=\frac{x_{object}}{\theta_{observer}}$$

Is there a general formula to compute the luminosity distance $D_{L}$ from $D_{A}$ without assuming an homogeneous cosmology ?

• So are you interested in some sort of general integration over the different epochs of the universe with metric perturbations accounted for and not the simple $D_L=(1+z)^2D_A$? – Jim May 15 '14 at 15:21
• @Jim I wonder whether the relation you point out is valid in a perturbed Universe. – Vincent May 15 '14 at 15:23
• Roughly speaking, it's about as valid as the unperturbed equations for $D_L$ and $D_A$ are individually. For $z<1$ these are great approximations. For $z>1$ they still hold and the data does follow the predictions, but perturbations have increasingly larger effects – Jim May 15 '14 at 15:29
• @Jim Ok, I will rephrase the problem the other way. Consider a completely perturbed Universe in which I know, for a particular object: $x_{object}$ and $\theta_{observer}$ (I know them numerically, I do not have any expressions of them). Is there a way to compute $D_{L}$ from that ? – Vincent May 15 '14 at 15:40
• I don't know it off-hand. There is definitely a way; you should look around at some observational cosmology papers. Failing that method, the next step would be to re-derive the equations for $D_L$and $D_A$ only start with a perturbed metric. A majority of the math would be easy, but it would get really hairy right at the end. – Jim May 15 '14 at 15:47

The Etherington theorem states that the relation $$D_L=(1+z)^2D_A$$ remains valid for any space-time. The theorem shows the relation without using Einstein equations nor the matter content of the Universe. It only depends on photon conservation, and in the fact that photons travel through null geodesics in a Riemannian geometry.