Toy Models of Universe with an Effective Varying [Lambda]-Term in Lyra Manifold

Abstract

Research on the accelerated expansion of our Universe captures a lot of attention. The dark energy (DE) is a way to explain it. In this paper we will consider scalar field quintessence DE with [subscript]ωDE[/subscript] >-1 EoS, where the dynamics of the DE models related to the dynamics of the scalar field. We are interested in the study of the behavior of the Universe in the presence of interacting quintessence DE models in Lyra manifold with a varying Λ(t). In a considered framework we also would like to propose a new form for Λ(t). We found that the models correspond to the transit Universe, which will enter the accelerated expansion phase and will remain there with a constant deceleration parameter q. We found also that the Λ(t) is a decreasing function which takes a small positive value with [subscript]Ωm[/subscript] ≠0 and [subscript]ΩQ[/subscript] [arrow right]const dominating [subscript]Ωm[/subscript] in the old Universe. Observational constraints are applied and causality issue via [superscript]CS2[/superscript] is discussed as a possible way to either reject or accept the models.

Academic Editor:Sally Seidel

Max Planck Institute of Colloids and Interfaces, Potsdam-Golm Science Park, Am Muhlenberg 1 OT Golm, 14476 Potsdam, Germany

Received 6 October 2014; Accepted 21 December 2014; 12 January 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³ .

1. Introduction

Analysis of the observational data shows that our Universe for later stages of the evolution indicates accelerated expansion [1-3]. According to the observational data we know that in the Universe one of the main components is dark energy and its negative pressure (positive energy density) has enough power to work against gravity and provide accelerated expansion of the Universe. To have a balance in the Universe the second component known as dark matter (DM) is considered, which is responsible for the completely other phenomenon known as structure formation. According to different estimations the DE occupies about [figure omitted; refer to PDF] of the energy of our Universe, while DM is about [figure omitted; refer to PDF] , and usual baryonic matter is about [figure omitted; refer to PDF] . The surveys on clusters of galaxies showed that the density of matter is very much less than critical density [4]; observations of cosmic microwave background (CMB) anisotropies indicate that the Universe is at and the total energy density is very close to the critical [figure omitted; refer to PDF] [5]. However, an interesting question is how does our Universe approach the flatness observed today. It is interesting to mention theoretical investigations which show that our Universe can approach the isotropy monotonically even in the presence of an anisotropic fluid; moreover the anisotropy of the fluid also isotropizes at later times for the accelerating expansion of the Universe. For instance, such scenarios were obtained in [6]. Deeper analysis reveals that in cosmological models the Universe can achieve the slightly anisotropic geometry in spite of the inflation. Therefore, we can classify two groups of the models depending on whether this happens as it is discussed in [6]. Various interesting papers and cosmological models working with the anisotropy of the Universe, anisotropy of the DE, and vector DE models exist in literature and we will refer our readers to [6] and references therein, because discussion of such cosmology is out of the main goal in this work. The simple model for the DE is the cosmological constant with two problems called fine-tuning and coincidence [7]. These problems have opened ways for alternative models for the DE including dynamical forms of it, as a variable cosmological constant [8, 9], [figure omitted; refer to PDF] -essence model [10, 11], and Chaplygin gas models [12-28] to mention a few. In recent times it was shown that certain type of interactions between DE and DM also could solve the mentioned problems. To solve dark energy problem, on the other hand, we can modify the left-hand side of Einstein equations and obtain theories such as [figure omitted; refer to PDF] theory of the gravity [29-38]. Modifications of these types provide a natural way to explain the origin of the dark energy. But such theories with different forms of modifications still should pass experimental tests, because they contain ghosts, finite-time future singularities, and so on, which is the base of other theoretical problems. One of the well studied DE models is a quintessence model [39-47], which is a scalar field model described by a scalar field [figure omitted; refer to PDF] and [figure omitted; refer to PDF] potential and it is the simplest scalar field scenario without having theoretical problems such as the appearance of ghosts and Laplacian instabilities. Energy density and pressure of quintessence DE are given as [figure omitted; refer to PDF]

Our model of the Universe contains an effective two-component fluid with an effective energy density and a pressure given as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the energy density and the pressure of a barotropic fluid, respectively, which will model the DM in our Universe with a [figure omitted; refer to PDF] EoS equation. The last assumption concerning the energy density and the pressure of the effective fluid can work particularly for the old and large scale Universe, where quantum and nonequilibrium effects are not considered. Whether the last assumption works in the early Universe is an open question, because, for the early Universe with high energy, small scales quantum effects can have unexpected effects and how the situation should be modified is not clear yet. As long as we have other conceptual problems, for instance, we do not know how correctly we can model the content of the early Universe. The question of how an interaction between the fluid components arose is not answered yet as well. One of the assumptions concerning the interaction between components is probably the same origin of the DE and the DM; however, this hypothesis is not a satisfactory approach to the problem. Despite the fact that the question is not answered yet, we continue the consideration of the different interactions; moreover we continue also performing some modifications based mainly on phenomenology. In literature we can find different cosmological models admitting different forms of the interaction [figure omitted; refer to PDF] between the fluid components of the Universe. One of them with a general form is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive constants, [figure omitted; refer to PDF] is the deceleration parameter, and [figure omitted; refer to PDF] is a constant. [figure omitted; refer to PDF] will correspond to a sign-changeable interaction [48]. The typical value of the constants is about [figure omitted; refer to PDF] . A phenomenological modification of [figure omitted; refer to PDF] can include a possibility with [figure omitted; refer to PDF] or [figure omitted; refer to PDF] to be functions of a cosmological parameter, that is, to consider [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In this study we consider a quintessence DE with exponential self-interacting potential [figure omitted; refer to PDF] of the form [49, 50] [figure omitted; refer to PDF] interacting with a barotropic fluid via [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constants, obtained from (5) with [figure omitted; refer to PDF] . The dynamics defined in Lyra manifold with a varying [figure omitted; refer to PDF] will be considered instead of the dynamics provided by GR. This work will differ from the other similar works in literature by the models with a new form for the [figure omitted; refer to PDF] . We know that a [figure omitted; refer to PDF] should be a decreasing function over time and has a small positive value in recent Universe. It will be seen that the proposed [figure omitted; refer to PDF] , which is a function from the Hubble parameter [figure omitted; refer to PDF] , scalar field [figure omitted; refer to PDF] , and self-interacting potential [figure omitted; refer to PDF] like [51] [figure omitted; refer to PDF] can achieve the desirable behavior.

The paper is organized as follows. In Section 2 we review the modified field equations. In Section 3 we analyse the models. In Section 4 we discuss the causality issue and observational constraints on the models. The last section includes conclusion and discussion about the cosmological consequence provided by the suggested cosmological models.

2. The Field Equations

Lyra geometry is an example of scalar tensor theory and one of the modifications of GR suggested by Lyra as a modification of Riemannian geometry [52]. In this modification the Weyl's gauge is modified. Field equations that constructed an analogue of the Einstein field equations based on Lyra's geometry can be written as [53, 54] [figure omitted; refer to PDF]

It was pointed out that the constant displacement field [figure omitted; refer to PDF] of this theory can be interpreted as a cosmological constant [figure omitted; refer to PDF] in the normal relativistic treatment [55]. We are interested in the other modification of the field equations which contain varying cosmological constant [figure omitted; refer to PDF] and which can be written as [56] [figure omitted; refer to PDF]

Considering the content of the Universe to be a perfect fluid, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a 4-velocity of the comoving observer, satisfying [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a time-like vector field of displacement, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a function of time alone, and the factor [figure omitted; refer to PDF] is substituted in order to simplify the writing of all the following equations. By using FRW metric for a flat Universe, [figure omitted; refer to PDF] field equations can be reduced to the following Friedmann equations: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Hubble parameter, and an overdot stands for differentiation with respect to cosmic time [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] represents the scale factor. The [figure omitted; refer to PDF] and [figure omitted; refer to PDF] parameters are the usual azimuthal and polar angles of spherical coordinates, with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The coordinates ( [figure omitted; refer to PDF] ) are called comoving coordinates.

The continuity equation reads as [figure omitted; refer to PDF]

With an assumption that [figure omitted; refer to PDF]

Equation (13) will give a link between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of the following form: [figure omitted; refer to PDF]

To introduce an interaction between the DE and the DM, we should mathematically split (14) and consider the following two equations: [figure omitted; refer to PDF]

Cosmological parameters of our interest are EoS parameters of each fluid component [figure omitted; refer to PDF] , EoS parameter of composed fluid [figure omitted; refer to PDF] and deceleration parameter [figure omitted; refer to PDF] , which can be written as [figure omitted; refer to PDF]

To study the causality issue we need also the behavior of the square of the sound speed with the widespread accepted opinion with the following constraint on it: [figure omitted; refer to PDF] to accept the cosmological models. However, the last opinion also can be challenged. Constant cosmological constant [figure omitted; refer to PDF] produces models of the Universe where [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] , because we have the following form for the [figure omitted; refer to PDF] : [figure omitted; refer to PDF] as a solution of (15), where [figure omitted; refer to PDF] is the scale factor and [figure omitted; refer to PDF] is the integration constant, which means that for the very large scale Universe the dynamics again will correspond to the dynamics given by GR. In the next section we start the analysis of the models.

3. The Model and the Cosmological Parameters

The cosmological model with the varying cosmological constant [figure omitted; refer to PDF] and the potential given by (6) and (4) will determine the behavior of the [figure omitted; refer to PDF] as [figure omitted; refer to PDF]

Having the interaction [figure omitted; refer to PDF] between the fluid components gives us a transit Universe (i.e., a transition to the Universe with [figure omitted; refer to PDF] to the Universe with [figure omitted; refer to PDF] , where the Hubble parameter [figure omitted; refer to PDF] is a decreasing function (Figure 1) which enters the ever accelerating expansion phase, where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] are small constants and constantly exist in the old Universe, while [figure omitted; refer to PDF] and dominates to the DM (Figure 2). The behavior of the [figure omitted; refer to PDF] clearly shows that the dynamics of the old Universe will be described by the theory differing from GR, but the proof of this fact could not appear in a simple way from the observations due to the very small value of [figure omitted; refer to PDF] . In such Universe the dynamics of the energy densities of the fluid components will be governed according to the two following equations: [figure omitted; refer to PDF]

Behavior of the Hubble parameter [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the deceleration parameter [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] . The cosmological constant [figure omitted; refer to PDF] is given by (6) and the potential [figure omitted; refer to PDF] by (4).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Behavior of the critical densities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of the DE and DM against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] for the model described via cosmological constant [figure omitted; refer to PDF] given by (6) and the potential given by (4).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Including the interaction under consideration into the model and having a strong interaction described by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we see that we can increase values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the later stages of the evolution. The behavior of the [figure omitted; refer to PDF] indicates that our Universe will start its evolution with the fluid with a positive [figure omitted; refer to PDF] ; then we will have a transition to the Universe where [figure omitted; refer to PDF] . The effective fluid and DE are quintessence fluids with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . We see that increasing [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (and [figure omitted; refer to PDF] ) will decrease [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the accelerated expansion phase of the old Universe (Figure 3). The last two cosmological parameters are not strongly dependent on the EoS parameter of the barotropic fluid, and with big accuracy the fact that changes in [figure omitted; refer to PDF] will not change [figure omitted; refer to PDF] and [figure omitted; refer to PDF] could be accepted. It is also interesting to understand the behavior of the model for different forms of the interactions. We investigated the behavior of the Universe in the presence of the two other phenomenological forms of the [figure omitted; refer to PDF] given as [figure omitted; refer to PDF]

Behavior of the EoS parameter of the effective fluid [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] for the model described via cosmological constant [figure omitted; refer to PDF] given by (6) and the potential given by (4).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The interaction given by (23) was considered by us in GR with varying [figure omitted; refer to PDF] and [figure omitted; refer to PDF] recently in [51]. The second form of the interaction given via (24) was considered in [57]. Analysis reveals an interesting fact that all models having different forms of the interaction term [figure omitted; refer to PDF] considered in this work reproduce the same behavior for the Universe and the cosmological parameters. The last part of this section is devoted to the cosmological model with [figure omitted; refer to PDF] . Consideration of the cosmological model with the constant [figure omitted; refer to PDF] allowed us to find that we have a transit Universe and with increasing value of the [figure omitted; refer to PDF] we will decrease the Hubble parameter [figure omitted; refer to PDF] and the deceleration parameter [figure omitted; refer to PDF] (Figure 4). In this case we are able to have the Universe which will enter the phase with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , to be very close to [figure omitted; refer to PDF] but not exactly [figure omitted; refer to PDF] (Figure 5). It is easy to see that the increasing [figure omitted; refer to PDF] will decrease cosmological parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In this case, we will have the old Universe which indicates accelerated expansion, where a very tiny amount of the barotropic fluid exists ( [figure omitted; refer to PDF] ); the dynamics of the Universe will be dominated by [figure omitted; refer to PDF] and DE only; because [figure omitted; refer to PDF] , increasing [figure omitted; refer to PDF] will decrease [figure omitted; refer to PDF] which can be seen from Figure 6. Consideration of the other forms for the interaction term [figure omitted; refer to PDF] given by (23)-(24) does not reveal significant changes in the model; therefore we decided to save a place and time and not to discuss the graphical behavior of the cosmological parameters for the other models. In the next section we would like to illuminate some physical aspects of the phenomenological model via the square of sound speed [figure omitted; refer to PDF] and observational constraints.

Behavior of the Hubble parameter [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the deceleration parameter [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] . The potential [figure omitted; refer to PDF] is given by (4) and [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Behavior of the EoS parameter of the effective fluid [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] . The potential [figure omitted; refer to PDF] is given by (4) and [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Behavior of the critical densities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of the DE and DM against time [figure omitted; refer to PDF] represents (a). (b) represents the behavior of the [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] . The potential [figure omitted; refer to PDF] is given by (4) and [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

4. [figure omitted; refer to PDF] and the Observational Constraints on the Model

The simplest way to reject the cosmological model is to consider the behavior of the square of sound speed [figure omitted; refer to PDF] . Today, the widespread opinion is to have the following constraint on it: [figure omitted; refer to PDF]

The criteria expressed in (25) indicate the bounds at which small perturbations of the background energy density propagate. It is possible to find argumentation challenging the case of [figure omitted; refer to PDF] [58]. A negative [figure omitted; refer to PDF] will indicate that given perturbations are unstable. One should not discard the possibility of [figure omitted; refer to PDF] without careful investigation [59]. For our model in both cases with constant cosmological constant [figure omitted; refer to PDF] and varying cosmological constant [figure omitted; refer to PDF] , we found that, for the reasonable values of the parameters of the model obtained from the observational constraints, the [figure omitted; refer to PDF] for the DE and for the effective fluid are positive for the early stages of the evolution, while they are negative for the old Universe (Figure 7). The observational constraints on the models are obtained based on the SNIa test, which is based on the distance modulus [figure omitted; refer to PDF] (Figure 8) related to the luminosity distance [figure omitted; refer to PDF] by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] The quantities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the apparent and the absolute magnitudes, respectively. There are many different [figure omitted; refer to PDF] data sets, obtained with different techniques. In some cases, these different samples may give very different results. Our observational analysis of the background dynamics uses the following three tests: the differential age of old objects based on the [figure omitted; refer to PDF] dependence as well as the data from [figure omitted; refer to PDF] and from [figure omitted; refer to PDF] . A fourth test could potentially be added: the position of the first peak of the anisotropy spectrum of the cosmic microwave background radiation ( [figure omitted; refer to PDF] ). However, the [figure omitted; refer to PDF] test implies integration of the background equations until [figure omitted; refer to PDF] which requires the introduction of the radiative component.

Behavior of the [figure omitted; refer to PDF] for the DE against time [figure omitted; refer to PDF] represents (a) with the potential [figure omitted; refer to PDF] given by (4) for the varying [figure omitted; refer to PDF] given by (6). (b) represents the behavior of the [figure omitted; refer to PDF] against time [figure omitted; refer to PDF] for the potential given by (4) and [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 8: Observational data [figure omitted; refer to PDF] for distance modulus versus our theoretical results.

[figure omitted; refer to PDF]

But the inclusion of such radiative component considerably changes the structure of the equations and no analytic expression for [figure omitted; refer to PDF] is available. Hence, we will limit ourselves to the mentioned three tests for which a reliable estimation is possible. We presented Table 1 containing information on the values of the parameters of the model with the varying cosmological constant [figure omitted; refer to PDF] obtained from the observational constraints.

Table 1

γ	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

In the next section we will provide discussion of the obtained results to conclude our work and finalize the paper.

5. Discussion

In this work we proposed and considered cosmological models involving interacting quintessence DE on the Lyra manifold, which gives an interesting modification of the field equations. Moreover, the cosmology with a varying cosmological constant has been assumed. In our model we tried to make a connection between the Hubble parameter [figure omitted; refer to PDF] , the scalar field [figure omitted; refer to PDF] , and the potential [figure omitted; refer to PDF] to obtain the varying cosmological constant. Given form of the interaction term [figure omitted; refer to PDF] between the DE and a barotropic fluid results in a transit Universe, where the deceleration parameter [figure omitted; refer to PDF] changes the sign from the positive to negative, independently whether we consider constant cosmological constant or a varying cosmological constant [figure omitted; refer to PDF] under consideration. Analysis of the critical densities of the components reveals that in the initial Universe the barotropic fluid was dominant over the DE (for some values of the model parameters this domination can be neglected), which quickly changes during the evolution. The cosmological model with varying [figure omitted; refer to PDF] indicates that for the later stages of the evolution our Universe will enter the phase where DE will dominate in the dynamics of the Universe where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] will accept tiny small positive values. This means that with this case the evolution of the Universe from the early times to the future will be described by the theory differing from GR. We should note that the two quantities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are decreasing functions over time, which also corporates with the physical theories perfectly. With the constant cosmological constant we have rather simplified scenario, because we got [figure omitted; refer to PDF] immediately, when [figure omitted; refer to PDF] , which means that the latter Universe will recover GR to describe its dynamics. We should note that a transit Universe is obtained and as was expected increasing [figure omitted; refer to PDF] will suppress the domination of the DE in the old Universe, where [figure omitted; refer to PDF] . In both cases the EoS parameters of the effective fluid and the DE remain strictly above [figure omitted; refer to PDF] indicating the quintessence Universe with the true quintessence DE. Causality, based on the "good" theories, implies restriction on the square of speed sound to be [figure omitted; refer to PDF] , which is a simple and a good way to reject or accept cosmological models. However, as we mentioned in the main text, this restriction can be challenged with different arguments. Unfortunately, if we adhere to widespread opinion, then our models with [figure omitted; refer to PDF] for the later stages of the evolution, which indicates instability of the theory, should be rejected. Related to this, we assume that our model can represent a possible behavior of an early Universe only, because only there we are able to obtain [figure omitted; refer to PDF] . With the observational constraints we found constraints on the parameters of the models. We have some analysis in the literature according to which with [figure omitted; refer to PDF] level from [figure omitted; refer to PDF] data we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (Km/sMpc) [60]. On the other hand from data of [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (Km/sMpc) [60]. Also joint test using [figure omitted; refer to PDF] and [figure omitted; refer to PDF] gives [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (Km/sMpc) [60]. Recent astronomical data based on a new infrared camera on the [figure omitted; refer to PDF] gives [figure omitted; refer to PDF] (Km/sMpc) [61]. The other probe using galactic clusters data suggests [figure omitted; refer to PDF] (Km/sMpc) [62]. Finally, [figure omitted; refer to PDF] model suggests [figure omitted; refer to PDF] . Conclusion of the presented facts is that generally [figure omitted; refer to PDF] . We see that our models are able to reproduce the above-mentioned behavior for the Hubble parameter [figure omitted; refer to PDF] (in appropriate units) and the deceleration parameter [figure omitted; refer to PDF] . Obtained behavior of the models encouraged us to extend models and as a first step we considered other forms of the interaction (mentioned in the main text), but we do not see any significant changes in the behavior of the Universe, which deserves to be included in this work. Our future interest in this direction is to consider cosmological models with different forms of the varying [figure omitted; refer to PDF] on one hand, and on the other hand instead of the barotropic fluid consider different nonlinear EoS fluids, particularly, ones which include bulk viscosity. The last aspect is motivated by the fact that in an isotropic and homogeneous Universe bulk viscosity will model irreversible processes. Obtained results hopefully will be reported in forthcoming articles.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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