Graceful Exit Inflation in $f(T)$ Gravity
  We apply a quadratic teleparallel torsion scalar of the $f(T)=T+\alpha T^{2}$ field equations to the spatially flat Friedmann-Robertson-Walker (FRW) model. We assume two perfect fluid components, the matter component has a fixed equation of state (EoS) parameter $\omega$, while the torsion component has a dynamical EoS. We obtain an effective scale factor allowing a graceful exit inflation model with no need to slow roll technique. We perform a standard cosmological study to examine the cosmic evolution. In addition, the effective EoS shows consistent results confirming a smooth phase transition from inflation to radiation dominant universe. We consider the case when the torsion is made of a scalar field. This treatment enables us to induce a scalar field sensitive to the spacetime symmetry with an effective potential constructed from the quadratic $f(T)$ gravity. The model is parameterized by two parameters ($\alpha,\omega$) both derive the universe to exit out of de Sitter expansion. The first is purely gravitational and works effectively at large Hubble regime of the early stage allowing a slow roll potential. The second parameter $\omega$ is a thermal-like correction coupled to the kinetic term and works effectively at low Hubble regime of late stages. The slow roll analysis of the obtained potential can perform tensor-to-scalar ratio and spectral index parameters consistent with the recent Planck and BICEP2 data. Both cosmological and scalar field analyses show consistent results. 
G.G.L. Nashed, W. El Hanafy and Sh.Kh. Ibrahim
gr-qc