The distance-redshift relation for the inhomogeneous and anisotropic universe

Recently, there have been a indication that the universe is in a state of accelerated expansion. This accelerated expansion is supposed to be caused by unidentified dark energy as a result of fitting the observations of type Ia supernovae to a distanceredshift relation of homogeneous and isotropic universe model. But the universe is not perfectly homogeneous and isotropic because of structures such as galaxies or groups of galaxies. In this work, by taking into account effects of such inhomogeneities and anisotropies of the matter distribution, we explain the distance-redshift relation suggested by the observations of type Ia supernovae without dark energy. First, we show that after averaging the optical scalar equation of an inhomogeneous and anisotropic spacetime over the sphere, we can obtain a simplified optical scalar equation, which coincides with that in the Lemâitre-Tolman-Bondi spacetime with a Dyer-Roeder-like extention. Then, we show that the distance-redshift relation obtained from our equation provides better fitting of the observational data than the ΛCDM model and the void model.