Testing the anisotropy of the Universe with the distance duality relation

The distance duality relation (DDR) is a deduction of the standard cosmological model, which is based on the assumption that the universe is homogeneous and isotropic on large scale. However, the astronomical data hint that the universe may have certain preferred direction. If indeed the case, the anisotropy would be likely to impact the DDR. In this paper, we test the anisotropy of the universe by performing two anisotropic DDR parametrizations with the dipole and quadrupole structures. The DDR is tested by comparing the luminosity distance from type-Ia supernovae (Union 2.1 and JLA compilations) and the angular diameter distance from strong gravitational lensing (SL) systems at the same redshift. It is shown that, with the Union2.1 compilation, the DDR is valid in dipole parametrization, but it is violated in quadrupole parametrization at 1$\sigma$ confidence level. If JLA compilation is applied, the result is completely reversed, i.e. the DDR is valid in quadrupole parametrization, but it is violated in dipole parametrization at 2$\sigma$ confidence level. Additionally, we try to incorporate galaxy clusters into SL systems to strengthen the results, but find that it is inconsistent with the results of pure SL systems. Due to the large uncertainty of available data, no strong evidence for the anisotropy of the universe is found.