Dual relativity and its cosmological consequences

A new version of the modified theory of gravity is formulated in which two physical metrics are constructed out of two vierbeins connected with each other by the duality condition including the flat metric of the prior geometry. The duality condition plays a crucial role in this theoretical scheme, and thus gives the name to the whole approach: the theory of dual relativity (TDR). The energy-momentum tensor density of a closed system is defined, and the conservation law for this tensor density is deduced. In the TDR it is assumed that there exist two kinds of matter governed by two physical metrics connected to each other by the duality relations, and it is shown that the interaction between these kinds of matter has antigravitational character. A cosmological limit of the field equations of the TDR is considered. In this limit, there are two types of solutions; with positive and with negative energy density. The first type admits the existence of a stable Universe as a whole. The second type gives the oscillations of the cosmological scale factor within the finite limits excluding the zero point, that can be treated as a solution for a certain domain of the Universe. For this type of solutions, the model formula for the dependence of the Hubble parameter $H$ on the redshift $z$ has been obtained. The values of the parameters of this formula are found from the fit to the available $H(z)$ data. It is obtained that the TDR gives a better description of the $H(z)$ data as compared to the flat $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. The important consequences of the obtained results are, first, their incompatibility with the standard big bang model and, second, the existence of two critical values of the scale factor determining the points of its sharp change in the course of oscillations in the solutions of the second type.