We review different dark energy cosmologies. In particular, we present the ΛCDM cosmology, Little Rip and Pseudo-Rip universes, the phantom and quintessence cosmologies with Type I, II, III and IV finite-time future singularities and non-singular dark energy universes. In the first part, we explain the ΛCDM model and well-established observational tests which constrain the current cosmic acceleration. After that, we investigate the dark fluid universe where a fluid has quite general equation of state (EoS) [including inhomogeneous or imperfect EoS]. All the above dark energy cosmologies for different fluids are explicitly realized, and their properties are also explored. It is shown that all the above dark energy universes may mimic the ΛCDM model currently, consistent with the recent observational data. Furthermore, special attention is paid to the equivalence of different dark energy models. We consider single and multiple scalar field theories, tachyon scalar theory and holographic dark energy as models for current acceleration with the features of quintessence/phantom cosmology, and demonstrate their equivalence to the corresponding fluid descriptions. In the second part, we study another equivalent class of dark energy models which includes F (R) gravity as well as F (R) Hořava-Lifshitz gravity and the teleparallel f (T ) gravity.The cosmology of such models representing the ΛCDM-like universe or the accelerating expansion with the quintessence/phantom nature is described. Finally, we approach the problem of testing dark energy and alternative gravity models to general relativity by cosmography. We show that degeneration among parameters can be removed by accurate data analysis of large data samples and also present the examples.PACS numbers: 04.50. Kd, 95.36.+x, 2 Contents

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...