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Galaxies 2013, 1(2), 107-113; doi:10.3390/galaxies1020107
Abstract: The conditions for the appearance of the Little Rip, Pseudo Rip and Quasi Rip universes in the terms of the parameters in the equation of state of some dark fluid are investigated. Several examples of the Rip cosmologies are investigated.
The appearance of new cosmological models is connected with the discovery of the accelerated expansion of the universe. Cosmic acceleration can be introduced via dark energy [1,2] or via modification of gravity [3,4]. Dark energy should have strong negative pressure and can be characterized by an equation of state (EoS) parameter w. The thermodynamic parameter w = p/ρ, where ρ is the dark energy and p is the dark pressure, is known to be negative. The theory predicts many interesting ways in which the universe could have evolved, including the Big Rip (BR) [5,6], the Little Rip (LR) [7,8,9,10,11,12,13,14], the Pseudo Rip (PR)  and Quasi Rip (QR)  cosmological models. The BR singularity phenomenon means that the physical quantities become infinite at the finite Rip time t. In the LR scenario, an infinite time is required to reach the singularity. In the PR cosmology, the Hubble parameter tends to the “cosmological constant” in the remote future. In the Rip phenomena, like LR or PR, the parameter w asymptotically tends to −1. These models are based on the assumption that the dark energy density ρ is a monotonically increasing function. In the cosmological model, the QR the dark energy density ρ monotonically increases when EoS parameter w < −1 in the first stage, and monotonically decreases (w > −1) in the second stage.
In this review article we study the influence of the time-dependent thermodynamic parameter w and the cosmological constant Λ from the EoS on the occurrence the Rip phenomena of some cosmological models. Section 2 is devoted the non-viscous models of the cosmic fluid. In Section 3 we consider the description of viscous LR cosmology for dark fluid in the late universe.
2. Dark Fluid Inhomogeneous Equation of State in the Some Cosmological Models
We suppose that our universe is filled with an ideal fluid (dark energy) obeying an inhomogeneous EoS :
The Friedmann equation for a spatially flat universe is:
Let us write down the energy conservation law:
We now consider examples of the dark energy models corresponding to the LR, QR and PR universes. For simplicity it will be assumed that the universe consists of the dark energy only.
2.1. The Little Rip Case
Let us a Hubble parameter has the following form :
We assume that the parameter w does not depend on the time w(t) = w0.
Taking into account Equations (1)–(4) and solve the Equation (3) with respect to Λ(t), we obtain :
Thus, if we assume an ideal fluid obeying the EoS Equations (1) and (5), then we obtain the LR scenario.
Let us consider another LR model :
Now writing the parameter w(t) in the form:
Let us choose the cosmological model with more complicated behavior of H :
Here we have defined Ln as:
Let us choose the parameter w(t) as:
As result, we have obtained a brane dark energy universe from the standpoint of 4d FRW cosmology without introducing the brane conception.
2.2. The Pseudo Rip Case
Let us investigate a PR model with the parameter Hubble :
If the Hubble ratio tends to a constant value H0 and the universe asymptotically approaches the de Sitter space. It may correspond to a PR model.
We will consider this cosmological model in analogy with the LR model.
Let us take the parameter w(t) in the view Equation (7), we obtain :
Now writing the parameter w(t) in the view:
2.3. The Quasi Rip Case
In this case we have modeled the QR universe induced by the dark fluid EoS. Let us take the energy density as a function of the scale factor a :
Choosing the parameter w(a) in the EoS as:
3. Examples of the Viscous Little Rip Cosmology
In this section we will consider the examples of the viscous LR cosmology in an isotropic cosmic fluid in the later stage of the evolution of the universe.
3.1. Dark Fluid with Bulk Viscosity
Let us write the expression for the time-dependent energy density for the viscous LR cosmology :
If the parameter w(t) has the form:
The validity of the Equations (24) and (25) means an equivalent description the viscous LR (23).
3.2. The Turbulent Description
In the later stages of the evolution of the universe near the future singularity it is necessary to take into account a transition into the turbulence motion. Let us consider the cosmic fluid as a two-component fluid and introduce the effective energy density in the view :
Let us consider the case wturb = w <−1, that is the turbulent matter behaves similar the non-turbulence matter in the phantom region. We will investigate the LR model and take the effective energy density as :
The viscous LR model for a perfect fluid can be realized via the choice in the EoS the parameter w(t) in the view:
Note, that there is another method of the solving this problem, which is connected with the transition of a one-component cosmic fluid from the viscous era into the turbulent era .
Several dark energy models have been analyzed in the present review article. We showed that these cosmological models can be caused via the corresponding choice of the cosmological constant or the thermodynamic parameter in the dark fluid inhomogeneous EoS within the framework of 4d FRW cosmology.
This work has been supported by project 2.1839.2011 of Ministry of Education and Science (Russia) and LRSS project 224.2012.2 (Russia). We are very grateful to Professor Sergei Odintsov for helpful discussions.
Conflict of Interest
The authors declare no conflict of interest.
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