- freely available
- re-usable

*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.

## 1. Introduction

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) [15] and Quasi Rip (QR) [16] 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 [17]:

The Friedmann equation for a spatially flat universe is:

^{2}= 8πG with Newton’s gravitational constant G.

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 [9]:

_{0}> 0, λ > 0, and H

_{0}is the present-time Hubble parameter.

We assume that the parameter w does not depend on the time w(t) = w_{0}.

Taking into account Equations (1)–(4) and solve the Equation (3) with respect to Λ(t), we obtain [18]:

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 [9]:

_{0}, C and λ are positive constants.

Now writing the parameter w(t) in the form:

Let us choose the cosmological model with more complicated behavior of H [9]:

_{0}, C

_{1}, …, C

_{n}, are the positive constants, and use Equations (6) and (7) for the solution Equation (3) with respect to Λ(t). By generalizing Equation (8), we obtain [18]:

Here we have defined L_{n} as:

We consider also the example of the brane LR cosmology [12]. The Hubble parameter is equal [19]:

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 [9]:

_{0}, H

_{1}and λ are the positive constants. We assume that H

_{0}> H

_{1}when t > 0.

If the Hubble ratio tends to a constant value H_{0} 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 [18]:

In the next example we have investigated the appearance of the asymptotic de Sitter regime on the brane from 4d cosmology [12]. The Hubble parameter is equal [19]:

_{0}is the present time and λ is a negative tension (λ < 0). If , then the Hubble parameter. This situation corresponds to the universe expands in a quasi-de Sitter regime.

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 [14]:

_{0}is the energy density at a present time t

_{0}. Now we write the EoS parameters w(a) and Λ(a) depending on 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 [21]:

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 [21]:

_{turb}denotes the turbulent part. Now we present analogously the effective pressure:

_{turb}and ρ

_{turb}are connected by the similar form:

_{turb}is a constant.

Let us consider the case w_{turb} = 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 [21]:

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 [21].

## 4. Conclusions

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.

## Acknowledgements

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.

## References

- Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J.
**1998**, 116, 1009–1038. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Bamba, K.; Capozzielo, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci.
**2012**, 342, 155–228. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: From F(R) theory to lorentz non-invariant models. Phys. Rept.
**2011**, 505, 59–144. [Google Scholar] [CrossRef] - Caldwell, R.R.; Kamionkowski, M.; Weinberg, N.N. Phantom energy and cosmic doomsday. Phys. Rev. Lett.
**2003**, 91, 071301:1–071301:4. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Methods Mod. Phys.
**2007**, 4, 115–145. [Google Scholar] [CrossRef] - Frampton, H.F.; Ludwick, K.J.; Scherrer, R.J. The little rip. Phys. Rev. D
**2011**, 84, 063003:1–063003:5. [Google Scholar] - Brevik, I.; Elizalde, E.; Nojiri, S.; Odintsov, S.D. Viscous little rip cosmology. Phys. Rev. D
**2011**, 84, 103508:1–103508:6. [Google Scholar] - Frampton, P.H.; Ludwick, K.J.; Nojiri, S.; Odintsov, S.D.; Scherrer, R.J. Models for little rip dark energy. Phys. Lett. B
**2012**, 708, 204–211. [Google Scholar] [CrossRef] - Astashenok, A.V.; Nojiri, S.; Odintsov, S.D.; Yurov, A.V. Phantom cosmology without big rip singularity. Phys. Lett. B
**2012**, 709, 396–403. [Google Scholar] [CrossRef] - Astashenok, A.V.; Nojiri, S.; Odintsov, S.D.; Scherrer, R.J. Scalar dark energy models mimicking ΛCDM with arbitrary future evolution. Phys. Lett. B
**2012**, 713, 145–153. [Google Scholar] [CrossRef] - stashenok, A.V.; Elizalde, E.; Odintsov, S.D.; Yurov, A.V. Equation-of-State formalism for dark energy models on the brane and the future of brane universes. Eur. Phys. J. C
**2012**, 72, 2260:1–2260:10. [Google Scholar] - Nojiri, S.; Odintsov, S.D.; Saez-Gomez, D. Cyclic, ekpyrotic and little rip universe in modified gravity. ArXiv E-Prints
**2012**. arXiv:1108.0767. [Google Scholar] - Makarenko, A.N.; Obukhov, V.V.; Kirnos, I.V. From big to little rip in modified F(R, G) gravity.
**2013**, 343, 481–488. [Google Scholar] - Frampton, P.H.; Ludwick, K.J.; Scherrer, R.J. Pseudo-rip: Cosmological models intermediate between the cosmological constant and the little rip. Phys. Rev. D
**2012**, 85, 083001:1–083001:5. [Google Scholar] - Wei, H.; Wang, L.F.; Guo, X.J. Quasi-rip: A new type of rip model without cosmic doomsday. Phys. Rev. D
**2012**, 86, 083003:1–083003:7. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Inhomogeneous equation of state of the universe: Phantom era, future singularity and crossing the phantom barrier. Phys. Rev. D
**2005**, 72, 023003:1–023003:12. [Google Scholar] - Brevik, I.; Obukhov, V.V.; Osetrin, K.E.; Timoshkin, A.V. Little rip cosmological models with time-dependent equation of state. Mod. Phys. Lett. A
**2012**, 27, 1250210:1–1250210:8. [Google Scholar] - Brevik, I.; Obukhov, V.V.; Timoshkin, A.V.; Rabochaya, Y. Rip brane cosmology from 4d inhomogeneous dark fluid universe. Astrophys. Space Sci.
**2013**, 346, 267–271. [Google Scholar] [CrossRef] - Brevik, I.; Obukhov, V.V.; Timoshkin, A.V. Quasi-Rip and Pseudo-Rip universes induced by the fluid inhomogeneous equation of state. Astrophys. Space Sci.
**2013**, 344, 275–279. [Google Scholar] - Brevik, I.; Myrzakulov, R.; Nojiri, S.; Odintsov, S.D. Turbulence and little rip cosmology. Phys. Rev. D
**2012**, 86, 063007:1–063007:8. [Google Scholar] - Brevik, I.; Timoshkin, A.V.; Rabochaya, Y.; Zerbini, S. Turbulence accelerating cosmology from an inhomogeneous dark fluid. Astrophys. Space Sci.
**2013**, 347, 203–208. [Google Scholar] [CrossRef]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).