We present a reconstruction algorithm for cosmological models based on $f(\mathcal{Q})$ gravity. We specifically focus on Bianchi type I and Friedmann-Lemaître-Robertson-Walker spacetimes, obtaining exact solutions that might have application in a variety of scenarios such as spontaneous isotropization of Bianchi type I models, dark energy, and inflation, as well as pre-big bang cosmologies.

We study Dirac spinors in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity. We find four types of results: the resulting dynamic behavior of the universe depends on the particular choice of function f(R); some f(R) models do not isotropize and have no Einstein limit, so that they have no physical significance, whereas for other f(R) models isotropization and Einsteinization occur, and so they are physically acceptable, suggesting that phenomenological arguments may select f(R) models that are physically meaningful; the singularity problem can be avoided, due to the presence of torsion; the general conservation laws holding for f(R)-gravity with torsion ensure the preservation of the Hamiltonian constraint, so proving that the initial value problem is well-formulated for these models.

F(R)-gravity with geometric torsion (not related to any spin fluid) is considered in a cosmological context. We derive the field equations in vacuum and in presence of perfect-fluid matter and discuss the related cosmological models. Torsion vanishes in vacuum for almost all arbitrary functions f(R) leading to standard General Relativity. Only for f(R)=R^{2}, torsion gives contribution in the vacuum leading to an accelerated behavior . When material sources are considered, we find that the torsion tensor is different from zero even with spinless material sources. This tensor is related to the logarithmic derivative of f'(R), which can be expressed also as a nonlinear function of the trace of the matter energy-momentum tensor. We show that the resulting equations for the metric can always be arranged to yield effective Einstein equations. When the homogeneous and isotropic cosmological models are considered, terms originated by torsion can lead to accelerated expansion. This means that, in f(R) gravity, torsion can be a geometric source for acceleration.