Generic phase portrait analysis of finite-time singularities and generalized Teleparallel gravity

We analyze the four common types of finite-time singularity using a generic framework of the phase portrait geometric approach. This technique requires the Friedmann system to be written as a one-dimensional autonomous system. We employ a scale factor that has been used widely in the literature to realize the four finite time singularity types, then we give a detailed discussion for each case showing possible novel models. Moreover,we show how different singularity types can play essential roles in different cosmological scenarios. Among several modified gravity theories, we show that the f(T) cosmology is compatible with the phase portrait analysis, since the field equations include Hubble derivatives only up to first order. Therefore, we reconstruct the f(T) theory which generates these phase portraits. We also perform a complementary analysis using the effective equation of state. Furthermore, we investigate the role of the torsion fluid in realizing the cosmic singularities. (authors)