Semi-Symmetric Metric Gravity: from the Friedmann-Schouten geometry with torsion to dynamical dark energy models

In the present paper we introduce a geometric generalization of standard general relativity, based on a geometry initially introduced by Friedmann and Schouten in 1924, through the notion of a semi-symmetric connection. The semi-symmetric connection is a particular connection that extends the Levi-Civita one, by allowing for the presence of torsion. While the mathematical landscape of the semi-symmetric metric connections is well-explored, their physical implications remain to be investigated. After presenting in detail the differential geometric aspects of the geometries with semi-symmetric metric connection, we formulate the Einstein field equations, which contain additional terms induced by the presence of the specific form of torsion we are studying. We consider the cosmological applications of the theory by deriving the generalized Friedmann equations, which also include some supplementary terms as compared to their general relativistic counterparts and can be interpreted as a geometric type dark energy. To evaluate the proposed theory, we consider three cosmological models - the first with constant effective density and pressure, the second with the dark energy satisfying a linear equation of state, and a third one one with a polytropic equation of state. We compare the predictions of the semi-symmetric metric gravitational theory with the observational data for the Hubble function, and with the predictions of the standard $\Lambda$CDM model. Our findings indicate that the semi-symmetric metric cosmological models give a good description of the observational data, and for certain values of the model parameters, they can reproduce almost exactly the predictions of the $\Lambda$CDM paradigm. Consequently, Friedmann's initially proposed geometry emerges as a credible alternative to standard general relativity, in which dark energy has a purely geometric origin.