Towards a de Sitter quintom-like cosmology from a reduction of a 7-dimensional $BF$ theory

In this article, we analyze a 7-dimensional $BF$ theory that, upon dimensional reduction, transforms into an effective 4-dimensional action in which a generic metric is involved. A constraint is imposed in such a way that this is the Friedmann-Lemaître-Robertson-Walker metric characterized by a scale factor $a(t)=\mathrm{e}\mathrm{x}\mathrm{p}(H_{0}t)$ with a Hubble constant $H_0=67.70[\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}]$. We also compute the barotropic parameter $\omega \equiv p\mathrm{/}\rho$ from our solutions. Three different scenarios are explored, in which the model transitions from quintessence to phantom regimes, crossing the theoretical phantom divide line. Consequently, the effective action represents a quintom-like scenario. Moreover, we derive some analytic solutions for the scale factor by treating it as a dynamical variable in the effective action. In this case, different outcomes are obtained, where the model can transit from a cosmological constant to a quintessence regime; however, it cannot cross the theoretical phantom divide line. However, this simplistic model successfully captures distinct phases of cosmic expansion.