Luminosity distance and redshift in the Szekeres inhomogeneous cosmological models

The Szekeres inhomogeneous models can be used to model the true lumpy universe that we observe. This family of exact solutions to Einstein's equations was originally derived with a general metricthat has no symmetries. In this work, we develop and use a framework to integrate the angular diameterand luminosity distancesin the general Szekeres models. We use the affine null geodesicequations in order to derive a set of first-order ordinary differential equations that can be integrated numerically to calculate the partial derivativesof the null vectorcomponents. These equations allow the integration in all generality of the distances in the Szekeres models and some examples are given. The redshift is determined from simultaneous integration of the null geodesicequations. This work does not assume spherical or axial symmetry, and the results will be useful for comparisons of the general Szekeres inhomogeneous models to current and future cosmological data.