Dark energy cosmology: the equivalent description via different
  theoretical models and cosmography tests
  We review different dark energy cosmologies. In particular, we present the
$\Lambda$CDM 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 $\Lambda$CDM 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 $\Lambda$CDM 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 $F(R)$ Ho\v{r}ava-Lifshitz gravity and the
teleparallel $f(T)$ gravity. The cosmology of such models representing the
$\Lambda$CDM-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.
