Rip Cosmology via Inhomogeneous Fluid

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.

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]:

(1)

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

The Friedmann equation for a spatially flat universe is:

(2)

where ρ is the energy density, 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:

(3)

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]:

(4)

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]:

(5)

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]:

(6)

Here, H₀, C and λ are positive constants.

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

(7)

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

(8)

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

(9)

where, C₀, C₁, …, 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]:

(10)

Here we have defined L_n as:

(11)

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

(12)

where λ is a positive tension (λ > 0).

Let us choose the parameter w(t) as:

(13)

then we find in analogy the “cosmological constant” [19]:

(14)

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]:

(15)

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

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 [18]:

(16)

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]:

(17)

Here is the dimensionless parameter, , where t₀ 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:

(18)

with δ > 0, we obtain [19]:

(19)

where .

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]:

(20)

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:

(21)

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

(22)

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]:

(23)

with the viscous condition , 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:

(24)

with δ > 0, then the “cosmological constant” is equal [22]:

(25)

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]:

(26)

The first term ρ denotes the ordinary laminar energy density and the second term ρ_turb denotes the turbulent part. Now we present analogously the effective pressure:

(27)

The dependence of p on ρ is given with simple relation:

(28)

Analogously the turbulent quantities p_turb and ρ_turb are connected by the similar form:

(29)

where w_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]:

(30)

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

(31)

and corresponding expression for Λ(t) [22]:

(32)

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].