Using the "teleparallel" equivalent of General Relativity as the gravitational sector, which is based on torsion instead of curvature, we add a canonical scalar field, allowing for a nonminimal coupling with gravity. Although the minimal case is completely equivalent to standard quintessence, the nonminimal scenario has a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing. The richer structure is manifested in the absence of a conformal transformation to an equivalent minimally-coupled model. 95.36.+x

We study the cosmological evolutions of the equation of state for dark energy w DE in the exponential and logarithmic as well as their combination f (T ) theories. We show that the crossing of the phantom divide line of w DE = −1 can be realized in the combined f (T ) theory even though it cannot be in the exponential or logarithmic f (T ) theory. In particular, the crossing is from w DE > −1 to w DE < −1, in the opposite manner from f (R) gravity models. We also demonstrate that this feature is favored by the recent observational data.Cosmic observations from Supernovae Ia (SNe Ia) [1], cosmic microwave background (CMB) radiation [2][3][4], large scale structure (LSS) [5], baryon acoustic oscillations (BAO) [6], and weak lensing [7] have implied that the expansion of the universe is currently accelerating. This is one of the most important issues in modern physics. Approaches to account for the late time cosmic acceleration fall into two representative categories: One is to introduce "dark energy" in the right-hand side of the Einstein equation in the framework of general relativity (for a review on dark energy, see [8]). The other is to modify the left-hand side of the Einstein equation, called as a modified gravitational theory, e.g., f (R) gravity [9][10][11].As another possible way to examine gravity beyond general relativity, one could use the Weitzenböck connection, which has no curvature but torsion, rather than the curvature defined by the Levi-Civita connection. Such an approach is referred to "teleparallelism" (see, e.g., [12][13][14][15]), which was also taken by Einstein [16]. To explain the late time acceleration of the universe, the teleparallel Lagrangian density described by the torsion scalar T has been extended to a function of T [17, 18] 1 . This idea is equivalent to the concept of f (R) gravity, in which the Ricci scalar R in the Einstein-Hilbert action is promoted to a function of R.Recently, f (T ) gravity has been extensively studied in the literature [20][21][22][23][24][25].In this paper, we explicitly examine the cosmological evolution in the exponential f (T ) theory [18,23] in more detail with the analysis method in Ref. [26]. In particular, we study the equation of state (w DE ) and energy density (ρ DE ) for dark energy. The recent cosmological observational data [27] seems to imply a dynamical dark energy of equation of state with the crossing of the phantom divide line w DE = −1 from the non-phantom phase to phantom phase as the redshift z decreases in the near past. However, we illustrate that the universe with the exponential f (T ) theory always stays in the non-phantom (quintessence) phase or the phantom one, and hence the crossing of the phantom divide cannot be realized [23]. It is interesting to mention that such an exponential type as f (R) gravity models has been investigated in Refs. [28][29][30]. We also present a logarithmic f (T ) theory and show that it has a similar feature as the exponential one. Our motivation in this paper is to build up a realistic f (T ) theor...

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We explore the cosmological evolution in the exponential gravity f (R) = R + c 1 1 − e −c 2 R (c 1,2 = constant). We summarize various viability conditions and explicitly demonstrate that the late-time cosmic acceleration following the matter-dominated stage can be realized. We also study the equation of state for dark energy and confirm that the crossing of the phantom divide from the phantom phase to the non-phantom (quintessence) one can occur. Furthermore, we illustrate that the cosmological horizon entropy globally increases with time.