Equivalence of $f(Q)$ cosmology with quintom-like scenario: the phantom field as effective realization of the non-trivial connection

Abstract: We show that $f(Q)$ cosmology with a non-trivial connection, namely the Connection II of the literature, is dynamically equivalent with a quintom-like model. In particular, we show that the scalar field arising from the non-linear $f(Q)$ form, and the scalar field associated to the non-trivial connection, are combined to provide one canonical and one phantom field in the minisuperspace Lagrangian. Hence, the combination of two well-behaved fields appears effectively as a phantom field. This is fundamentally different from the usual approach in phantom and quintom cosmology, in which the phantom field is introduced ad hoc, and thus it may open a new way to handle the phantom fields, namely as effective realizations of the connection-related canonical fields.