Reconstruct the Distance Duality Relation by Gaussian Process

In this letter, the distance-duality (DD) relation is reconstructed by Gaussian process (GP) which is cosmological model-independent. Generally, the GP plays two important roles. One is to shape the $\eta$ tendency which denotes the deviation from the DD relation, the other one is to produce the luminosity-distance (LD, $D_{L}$) and the angular-diameter-distance (ADD, $D_{A}$) data at the same redshift. The shapes of $\eta$ are given out based on SNe Ia (Type Ia supernovae) data with different light-curve fitters (including MLCS2K2 and SALT2) and ADD data with different galaxy cluster morphologies (including the elliptical $\beta$ and spherical $\beta$ models). The data related to MLCS2K2 light-curve fitter have higher values of $\eta$ compared to that related to the SALT2 light-curve fitter. As for the morphology of galaxy cluster, the DD relation is favored by the elliptical one.