Unexplored regions in teleparallel $f(T)$ gravity: Sign-changing dark energy density

While $f(T)$ gravity has shown considerable potential in addressing cosmological tensions, we explore previously overlooked solution spaces that hold further promise. We examine the case where the customary assumption of a strictly positive effective DE density may not apply, offering new possibilities. Focusing on $f(T) = T e^{T_*/T}$, we investigate cosmological solutions parametrized by the parameter $\beta = T_*/T_0$. This parameter uniquely determines $\Omega_{\rm m0}$, and its sign plays a crucial role in characterizing deviations from the $\Lambda$CDM. We elaborate on the structural asymmetry between the positive- and negative-$\beta$ branches: while the $\beta_{+}$ leads to dynamics with modest departures from $\Lambda$CDM, the $\beta_{-}$ yields more pronounced and nontrivial deviations. Despite these deviations, the negative-$\beta$ branch can remain consistent with local gravity constraints through an effective chameleon-like mechanism. We also examine the model in the context of dynamical DE. Ensuring consistency with CMB data, the widely studied $\beta_{+}$ exhibits phantom behavior, while the previously overlooked $\beta_{-}$ features a sign-changing DE density that transitions smoothly from negative to positive values at $z_{\dagger} \sim 1.5$. Though the sign-changing DE leads to a larger-than-expected enhancement, we extend the analysis by incorporating $\Lambda$. This extension broadens the solution space consistent with the SH0ES measurement while maintaining consistency with CMB. Additionally, it introduces richer phenomenological possibilities, including the potential moderation or cessation of cosmic acceleration at very low redshifts, aligning with recent observational analyses, such as those from DESI BAO data. Our findings suggest that existing $f(T)$ models, as well as $f(Q)$ models, should be revisited in light of the novel theoretical insights presented here.