Advances in High Energy Physics

Volume 2015, Article ID 796168, 10 pages

http://dx.doi.org/10.1155/2015/796168

## Toy Models of Universe with an Effective Varying -Term in Lyra Manifold

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

Academic Editor: Sally Seidel

Copyright © 2015 Martiros Khurshudyan. 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 SCOAP^{3}.

#### 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 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 . In a considered framework we also would like to propose a new form for . 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 . We found also that the is a decreasing function which takes a small positive value with and dominating in the old Universe. Observational constraints are applied and causality issue via is discussed as a possible way to either reject or accept the models.

#### 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 of the energy of our Universe, while DM is about , and usual baryonic matter is about . 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 [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], -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 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 and 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

Our model of the Universe contains an effective two-component fluid with an effective energy density and a pressure given as where and are the energy density and the pressure of a barotropic fluid, respectively, which will model the DM in our Universe with a 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 between the fluid components of the Universe. One of them with a general form is where and are positive constants, is the deceleration parameter, and is a constant. will correspond to a sign-changeable interaction [48]. The typical value of the constants is about . A phenomenological modification of can include a possibility with or to be functions of a cosmological parameter, that is, to consider and . In this study we consider a quintessence DE with exponential self-interacting potential of the form [49, 50] interacting with a barotropic fluid via where , , , and are constants, obtained from (5) with . The dynamics defined in Lyra manifold with a varying 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 . We know that a should be a decreasing function over time and has a small positive value in recent Universe. It will be seen that the proposed , which is a function from the Hubble parameter , scalar field , and self-interacting potential like [51] 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]

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

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

The continuity equation reads as

With an assumption that

Equation (13) will give a link between and of the following form:

To introduce an interaction between the DE and the DM, we should mathematically split (14) and consider the following two equations:

Cosmological parameters of our interest are EoS parameters of each fluid component , EoS parameter of composed fluid and deceleration parameter , which can be written as

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: to accept the cosmological models. However, the last opinion also can be challenged. Constant cosmological constant produces models of the Universe where , with , because we have the following form for the : as a solution of (15), where is the scale factor and 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 and the potential given by (6) and (4) will determine the behavior of the as

Having the interaction between the fluid components gives us a transit Universe (i.e., a transition to the Universe with to the Universe with , where the Hubble parameter is a decreasing function (Figure 1) which enters the ever accelerating expansion phase, where , with are small constants and constantly exist in the old Universe, while and dominates to the DM (Figure 2). The behavior of the 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 . In such Universe the dynamics of the energy densities of the fluid components will be governed according to the two following equations: