Galaxies 2013, 1(2), 107-113; doi:10.3390/galaxies1020107

Rip Cosmology via Inhomogeneous Fluid
Valerii V. Obukhov , Alexander V. Timoshkin * and Evgenii V. Savushkin
Department of Physics and Mathematics, Tomsk State Pedagogical University, Tomsk 634061, Russia; E-Mails: (V.V.O.); (E.V.S.)
Author to whom correspondence should be addressed; E-Mail:; Tel.: +3-822-996-728; Fax: +3-822-446-826.
Received: 31 July 2013; in revised form: 10 August 2013 / Accepted: 12 August 2013 /
Published: 15 August 2013


: 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.
cosmological models; dark fluid; equation of state

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]:

Galaxies 01 00107 i001
where p is the pressure and w(t), Λ(t) are time-dependent parameters.

The Friedmann equation for a spatially flat universe is:

Galaxies 01 00107 i002
where ρ is the energy density, Galaxies 01 00107 i041 is the Hubble parameter, a(t) is the scale factor, k2 = 8πG with Newton’s gravitational constant G.

Let us write down the energy conservation law:

Galaxies 01 00107 i003

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]:

Galaxies 01 00107 i004
where H0 > 0, λ > 0, and H0 is the present-time Hubble parameter.

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 [18]:

Galaxies 01 00107 i005

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]:

Galaxies 01 00107 i006
Here, H0, C and λ are positive constants.

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

Galaxies 01 00107 i007
and express Λ(t) from Equation (3) we obtain the solution, realizing the LR (6) [18]:
Galaxies 01 00107 i008

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

Galaxies 01 00107 i009
where, C0, C1, …, Cn, 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]:
Galaxies 01 00107 i010

Here we have defined Ln as:

Galaxies 01 00107 i011

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

Galaxies 01 00107 i012
where λ is a positive tension (λ > 0).

Let us choose the parameter w(t) as:

Galaxies 01 00107 i013
then we find in analogy the “cosmological constant” [19]:
Galaxies 01 00107 i014

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]:

Galaxies 01 00107 i015
where H0, H1 and λ are the positive constants. We assume that H0 > H1 when t > 0.

If Galaxies 01 00107 i034 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 [18]:

Galaxies 01 00107 i016
showing that the PR behavior is caused by the parameter w.

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]:

Galaxies 01 00107 i017
Here Galaxies 01 00107 i035 is the dimensionless parameter, Galaxies 01 00107 i036, where t0 is the present time and λ is a negative tension (λ < 0). If Galaxies 01 00107 i037, then the Hubble parameter. Galaxies 01 00107 i038 This situation corresponds to the universe expands in a quasi-de Sitter regime.

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

Galaxies 01 00107 i018
with δ > 0, we obtain [19]:
Galaxies 01 00107 i019
where Galaxies 01 00107 i039.

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]:

Galaxies 01 00107 i020
where α and β are a constants, ρ0 is the energy density at a present time t0. 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:

Galaxies 01 00107 i021
where δ is a constant, we obtain the solution realizing QR Equation (20) caused by the parameter w(a) [20]:
Galaxies 01 00107 i022

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]:

Galaxies 01 00107 i023
with the viscous condition Galaxies 01 00107 i040, where ζ is the bulk viscosity and A is a positive constant. Now we consider the viscous LR cosmology from the point of view of 4d FRW non-viscous cosmology, analogous to Section 2.

If the parameter w(t) has the form:

Galaxies 01 00107 i024
with δ > 0, then the “cosmological constant” is equal [22]:
Galaxies 01 00107 i025

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]:

Galaxies 01 00107 i026
The first term ρ denotes the ordinary laminar energy density and the second term ρturb denotes the turbulent part. Now we present analogously the effective pressure:
Galaxies 01 00107 i027
The dependence of p on ρ is given with simple relation:
Galaxies 01 00107 i028
Analogously the turbulent quantities pturb and ρturb are connected by the similar form:
Galaxies 01 00107 i029
where wturb is a constant.

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 [21]:

Galaxies 01 00107 i030

The viscous LR model for a perfect fluid can be realized via the choice in the EoS the parameter w(t) in the view:

Galaxies 01 00107 i031
and corresponding expression for Λ(t) [22]:
Galaxies 01 00107 i032

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.


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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