Generalized teleparallel cosmology and initial singularity crossing

We present a class of cosmological solutions for a generalized teleparallel gravity with, $f(T)=T+\tilde{\alpha}(-T)^n$, where $\tilde{\alpha}$ is some parameter and $n$ is an integer or half-integer. Choosing $\tilde{\alpha} \sim G^{n-1}$, where $G$ is the gravitational constant, and working with an equation of state $p=w\, \rho$, one obtains a cosmological solution with multiple branches. The dynamics of the solution describes standard cosmology at late times, but the higher-torsion correction changes the nature of the initial singularity from big bang to a sudden singularity. The milder behavior of the sudden singularity enables us to extend timelike or lightlike curves, through joining two disconnected branches of solution at the singularity, leaving the singularity traversable. We show that this extension is consistent with the field equations through checking the known junction conditions for generalized teleparallel gravity. This suggests that these solutions describe a contracting phase a prior to the expanding phase of the universe.