Empirical measurement of cosmic luminosity-angular distance relation

One of the keys to understanding the universe is the proper measurement of the relation between the luminosity distance $d_L$ and the angular diameter distance $d_A$. In 1933 Etherington deduced from general relativity the reciprocity equation $d_L=d_A(1+z{)}^{\gamma }$, with $\gamma =1$ for a local (non-expanding) universe. This relation has been adapted to an expanding universe with the value $\gamma =2$ by the introduction of the concept of comoving distance ($d_M$). In this work, we developed the method \textit{Cosmic Redshift Inference} to measure experimentally the value of $\gamma$ defined above independently of any possible galaxy evolution. The method has been applied to 1.2 million galaxy of SDSS DR15 obtaining the value $\gamma\simeq 1$.