Redshift and redshift-drift in $\Lambda = 0$ quasi-spherical Szekeres cosmological models and the effect of averaging

Since the advent of the accelerated expanding homogeneous universe model, some other explanations for the supernova Ia dimming have been explored, among which there are inhomogeneous models constructed with exact $\Lambda = 0$ solutions of Einstein's equations. They have been used either as one patch or to build Swiss-cheese models. The most studied ones have been the Lema\^itre-Tolman-Bondi (LTB) models. However, these models being spatially spherical, they are not well designed to reproduce the large scale structures which exhibit clusters, filaments and non spherical voids. This is the reason why Szekeres models, which are devoid of any symmetry, have recently come into play. In this paper, we give the equations and an algorithm to compute the redshift-drift for the most general quasi-spherical Szekeres (QSS) models with no dark energy. We apply it to a QSS model recently proposed by Bolejko and Sussman (BSQSS model) who averaged their model to reproduce the density distribution of the Alexander and collaborators' LTB model which is able to fit a large set of cosmological data without dark energy. They concluded that their model represents a significant improvement over the observed cosmic structure description by spherical LTB models. We show here that this QSS model is ruled out by a negative cosmological redshift, i.e. a blueshift, which is not observed in the Universe. We also compute a positive redshift and the redshift-drift for the Alexander et al.'s model and compare this redshift-drift to that of the $\Lambda$CDM model. We conclude that the process of averaging an unphysical QSS model can lead to obtain a physical model able to reproduce our observed local Universe with no dark energy need and that the redshift-drift can discriminate between this model and the $\Lambda$CDM model. For completeness, we also compute the blueshift-drift of the BSQSS model.