$f(T)$ cosmology: From Pseudo-Bang to Pseudo-Rip
  We investigate the complete universe evolution in the framework of $f(T)$
cosmology. We first study the requirements at the kinematic level and we
introduce a simple scale factor with the necessary features. Performing a
detailed analysis of the phase portrait we show that the universe begins in the
infinite past from a phase where the scale factor goes to zero but the Hubble
parameter goes to a constant, and its derivative to zero. Since these features
resemble those of the Pseudo-Rip fate but in a reverted way, we call this
initial phase as Pseudo-Bang. Then the universe evolves in a first inflationary
phase, a cosmological turnaround and a bounce, after which we have a second
inflationary regime with a successful exit. Subsequently we obtain the standard
thermal history and the sequence of radiation, matter and late-time
acceleration epochs, showing that the universe will result in an everlasting
Pseudo-Rip phase. Finally, taking advantage of the fact that the field
equations of $f(T)$ gravity are of second order, and therefore the
corresponding autonomous dynamical system is one dimensional, we incorporate
the aforementioned kinematic features and we reconstruct the specific $f(T)$
form that can dynamically generate the Pseudo-Bang cosmological scenario.
Lastly, we examine the evolution of the primordial fluctuations showing that
they are initially sub-horizon, and we show that the total fluid does not
exhibit any singular behaviour at the phantom crossing points, while the
torsional fluid experiences them as Type II singular phases.
