The gravitational field with torsion is being constructed by using the parametrized absolute parallelism geometry. A generalized law of variation of Hubble’s parameter in evolutionary cosmological models is used. The cosmological models under the influence of the gravitational field with torsion are obtained and discussed. A new model of the Universe is presented using a special class of Riemann–Cartan geometry. This model is oscillating with expansion and contraction at different stages. It behaves normally as the conventional Big Bang model in the first half-age until it reaches the moment of a Big Rip, then reverses its behavior as a result of a changes in the pressure and torsion until it reaches a Big Crunch at the end of the second half-age. We suppose that the Big Rip singularity is replaced by a regular maximum of the scale factor at the Big Rip due to a possible physical mechanism of quantum nature. The positivity condition for the energy density of matter leads to exclusion of open and closed universes.

We used the absolute parallelism geometry to obtain a new formula for the Ricci scalar. We consider \(f(R,\Sigma,T)\) modified theories of gravity, where the gravitational Lagrangian is given by three arbitrary functions of the Ricci scalar \(R\), Ricci torsion scalar \(\Sigma\), and the trace of the stress-energy tensor \(T\). We obtain the gravitational field equations in the metric formalism. The evolution of the function \(f(R)\) withr time is studied, and we discuss the parameters that make up the function and impose constraints on these parameters. The solution of the \(f(R,\Sigma,T)\) gravity equations are obtained under a varying polynomial deceleration parameter. The effect of torsion on cosmological models is also discussed. Physical aspects of the energy density, pressure, and energy conditions of the cosmological models proposed in this article are studied, and the evolution of the physical parameters is shown in figures. Evolution of the fluid pressure and energy density parameter as a function of redshift has been obtained. The \(f(R)\) gravity and \(f(R,T)\) gravity theories as special cases could be inferred from \(f(R,\Sigma,T)\) gravity. Several special cases have been studied, with illustrations for each case.