Advances in High Energy Physics

Volume 2017 (2017), Article ID 2864784, 14 pages

https://doi.org/10.1155/2017/2864784

## Dynamical System Analysis of Interacting Hessence Dark Energy in Gravity

Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711 103, India

Correspondence should be addressed to Ujjal Debnath

Received 24 January 2017; Revised 31 March 2017; Accepted 15 May 2017; Published 25 July 2017

Academic Editor: George Siopsis

Copyright © 2017 Jyotirmay Das Mandal and Ujjal Debnath. 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

We have carried out dynamical system analysis of hessence field coupling with dark matter in gravity. We have analysed the critical points due to autonomous system. The resulting autonomous system is nonlinear. So, we have applied the theory of nonlinear dynamical system. We have noticed that very few papers are devoted to this kind of study. Maximum works in literature are done treating the dynamical system as done in linear dynamical analysis, which are unable to predict correct evolution. Our work is totally different from those kinds of works. We have used nonlinear dynamical system theory, developed till date, in our analysis. This approach gives totally different stable solutions, in contrast to what the linear analysis would have predicted. We have discussed the stability analysis in detail due to exponential potential through computational method in tabular form and analysed the evolution of the universe. Some plots are drawn to investigate the behaviour of the system* (this plotting technique is different from usual phase plot and that devised by us)*. Interestingly, the analysis shows that the universe may resemble the “cosmological constant” like evolution (i.e., CDM model is a subset of the solution set). Also, all the fixed points of our model are able to avoid Big Rip singularity.

#### 1. Introduction

High end cosmological observations of the Supernova of type Ia (SN Ia), WMAP, and so forth [1–19] suggest the fact that the universe may be accelerating lately again after the early phase. Many theories are formulated to explain this late time acceleration. However, these theories can be divided mainly into two categories fulfilling the criteria of a homogeneous and isotropic universe. The first kind of theory (better to known as “standard model” or CDM model) assumes a fluid of negative pressure named as “dark energy” (DE). The name arises from the fact the exact origin of this energy is still unexplained in theoretical setup. Observations, anyway, indicate that nearly 70% of the universe may be occupied by this kind of energy. Dust matter (cold dark matter (CDM) and baryon matter) comprises the remaining 30% and there is negligible radiation. Cosmologists are inclined to suspect dark energy as the primal cause of the late acceleration of universe. Theory of dark energy has remained one of the foremost areas of research in cosmology till the discovery of acceleration of the universe at late times [20–25]. One could clearly notice from the second field equation that the expansion would be accelerated if the equation of state (EoS) parameter satisfies . Accordingly, a priori choice for dark energy is a time-independent positive “cosmological constant” which relates to the equation of state (EoS) . This gives a universe which is expanding forever at exponential rate. Anyway, cosmological constant has some severe shortcomings like fine tuning problem and so forth (see [20] for a review); some recent data [26, 27] in some sense agrees with this choice. By the way, observation which constrains close to the value of cosmological constant of does not indicate whether changes with time or not. So, theoretically, one could consider as a function of cosmic time, such as inflationary cosmology (see [28–32] for review). Scalar fields evolve in particle physics quite naturally. Till date, a large variety of scalar field inflationary models are discussed. This theory is active area in literature nowadays (see [20]). The scalar field which lightly interacts with gravity is called “quintessence.” Quintessence fields are first-hand choice because this field can lessen fine tuning problem of cosmological constant to some extent. Needless to say, some common drawbacks for quintessence also exist. Observations point that, at current epoch energy, density of scalar field and matter energy density are comparable. But we know that they evolve from different initial conditions. This discrepancy (known as “coincidence problem”) arises for any scalar field dark energy; quintessence too suffers from this problem [33]. Of course, there is resolution of this problem; it is called “tracking solution” [34]. In the tracking regime, field value should be of the order of Planck mass. Anyway, a general setback is that we always need to seek for such potentials (see [35] for related discussion). EoS parameter of quintessence satisfies . Some current data indicates that lies in small neighbourhood of . Hence, it is technically feasible to relax to go down the line [36]. There exists another scalar field with negative kinetic energy term, which can describe late acceleration. This is named as phantom field, which has EoS (see details in [20, 37]). Phantom field energy density increases with time. As a result, Hubble factor and curvature diverges in finite time causing “Big Rip” singularity (see [38–40]). By the way, some specific choice of potential can avoid this flaw. Present data perhaps favours a dark energy model with of recent past to at present time [41]. The line is known as “phantom” divide. Evidently, neither quintessence nor phantom field alone can cross the phantom divide. In this direction, a first-hand choice is to combine both quintessence and phantom field. This is known in the literature as “quintom” (i.e., hybrid of quintessence and phantom) [41]. This can serve the purpose but still has some fallacy. A single canonical complex field is quite natural and useful (like “spintessence” model [42, 43]). However, canonical complex scalar fields suffer a serious setback, namely, the formation of “Q-ball” (a kind of stable nontopological soliton) [42, 43].

To overcome various difficulties with above-mentioned models, Wei et al. in their paper [44, 45] introduced a noncanonical complex scalar field which plays the role of quintom [45–47]. They name this unique model as “hessence.” However, hessence is unlike other canonical complex scalar fields which suffer from the formation of Q-ball. Second kind of theory modifies the classical general relativity (GR) by higher degree curvature terms (namely, theory) [48–50] or by replacing symmetric Levi-Civita connection in GR theory by antisymmetric Weitzenböck connection. In other words, torsion is taken for gravitational interaction instead of curvature. The resulting theory [51–53] (called “teleparallel” gravity) was considered initially by Einstein to unify gravity with electromagnetism in non-Riemannian Weitzenböck manifold. Later, further modification was done to obtain gravity as in the same vein of gravity theory [54]. Although the EoS of “cosmological constant” (CDM model) is well within the various dataset, till now not a single observation can detect DE or DM, and search for possible alternative is on the way [55]. In this regard, alternate gravity theory (like ) is really worth discussing. The work in [56] is a nice account in establishing matter stability of theory in weak field limit in contrast to theory. It is shown that any choice of can be used. Other reasons for the theoretical advantage for their choice are discussed in the next section.

We, in this work, have chosen hessence in gravity. Since the system is complex, we have preferred a dynamical analysis. As we have mentioned previously, hessence field and theory both are promising candidates to explain present accelerated phase. So, we merged them to find if they can highlight present acceleration more accurately with current dataset. A mixed dynamical system with tachyon, quintessence, and phantom in theory is considered in [57]. Dynamical systems with quintom also exist in literature (see [58, 59] for review). The dynamical system analysis for normal scalar field model in gravity has been discussed in [60]. But, to the best of our knowledge, hessence in gravity has not been considered before.

We arrange the paper in the following manner. Short sketch of theory is presented in Section 2. Hessence field in gravity is introduced to form dynamical system in Section 3. Section 4 is devoted to dynamical system analysing and the stability of the system for hessence dark energy model. The significance of our result is discussed in Section 5 in light of recent data. We conclude the paper with relevant remarks in Section 6. We use normalized units as in this paper.

#### 2. A Brief Outline of Gravity: Some Basic Equations

In teleparallelism [54, 61, 62], are called the orthonormal tetrad components . The index is used for each point for a tangent space of the manifold; hence each represents a tangent vector to the manifold (i.e., the so-called vierbein). Also the inverse of the vierbein is obtained from the relation . The metric tensor is given as (); are coordinate indices on the manifold (here, ). Recently, to explain the acceleration, the teleparallel torsion in Lagrangian density has been modified from linear torsion to some differentiable function of [63, 64] (i.e., ) like theory mentioned earlier. In this new setup of gravity, the field equation is of second order unlike (which is of fourth order). In theory of gravitation, corresponding action readswhere is the torsion scalar, is some differentiable function of torsion , is the matter Lagrangian, , and . The torsion scalar mentioned above is defined aswith the components of torsion tensor of (2) given bywhere is the Weitzenböck connection. Here, the superpotential (2) is defined as follows: is called contortion tensor. The contortion tensor measures the difference between symmetric Levi-Civita connection and antisymmetric Weitzenböck connection. It is easy to check that the equation of motion reduces to Einstein gravity if . Actually this is the correspondence between teleparallel gravity and Einsteinian theory [53]. It is noticed that theory can address early acceleration and late evolution of universe depending on the choice of . For example, power law or exponential form cannot overcome phantom divide [65], but some other choices of [66] can cross phantom divide. The reconstruction of model [67, 68], various cosmological [69, 70] and thermodynamical [71] analysis, has been reported. It is so interesting to note that linear model (i.e., when = constant) behaves as cosmological constant. Anyway, a preferable choice of is such that it reduces to general relativity (GR) when redshift is large in tune with primordial nucleosynthesis and cosmic microwave data at early times (i.e., for ). Moreover, in future, it should give de-Sitter-like state. One such choice is given in power form as in [72]; namely,with being a constant. In particular, gives same expanding model as the theory referred to in [72, 73]. Current data needs the bound “” to permit as an alternate gravity theory. The effective DE equation of state varies from of past to in future.

Throughout the work, we assume flat, homogeneous, isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which arises from the vierbein . Here, is the scale factor as a function of cosmic time . Using (3) and (4), one gets where is the Hubble factor (from here and in the rest of the paper “overdot” will mean the derivative operator ).

#### 3. Hessence Dark Energy in Gravity Theory: Formation of Dynamical Equations

Here, we consider a noncanonical complex scalar field:where with Lagrangian density:Clearly the Lagrangian density is identical to the Lagrangian given by two real scalar fields, which looks likewhere and are quintessence and phantom fields, respectively. It is noteworthy that the Lagrangian in (9) consists of one field instead of two independent fields as in (10) of [41]. It also differs from canonical complex scalar field (like “spintessence” in [42, 43]) which has the Lagrangianwhere denotes the absolute value of ; that is, . However, hessence is unlike canonical complex scalar fields which suffer from the formation of “Q-ball” (a kind of stable nontopological soliton). Following Wei et al. as in [44, 45], the energy density and pressure of hessence field can be written aswhere is a constant and denotes the total induced charge in the physical volume (refer to [44, 45]). In this paper, we will consider interaction of hessence field and matter. The matter is perfect fluid with barotropic equation of state:where is the barotropic index satisfying . Also and , respectively, denote the pressure and energy density of matter. In particular and indicate dust matter and radiation, respectively. We suppose that hessence and background fluid interact through a term . This term indicates energy transfer between dark energy and dark matter. Positive is needed to solve coincidence problem, since positive magnitude of indicates energy transfer from dark energy to dark matter. Also law of thermodynamics is also valid with this choice. An interesting work to settle this problem is reviewed in [74]. A rigorous dynamical analysis is done there. Similar approach exists for quintom model, too. Various choices of this interaction term are used in the literature. Here, in view of dimensional requirement of energy conservation equation and to make the dynamical system simple, we have taken , where is a real constant of small magnitude, which may be chosen as positive or negative at will, such that remains positive. Also, may be positive or negative according to the hessence field . So we havepreserving the total energy conservation equation: The modified field equations in gravity areIn view of (12) and (15), we haveHere, “′” means “.” Similarly, (14) and (16) giveNow, we introduce five auxiliary variables:We form the following autonomous system after some manipulation:In above calculations, denotes the “-folding” number. We have chosen as independent variable. We have taken for above derivation of autonomous system. Also, we have chosen exponential form of potential, that is, (where is a real constant), for simplicity of the autonomous system. This kind of choice is standard in literature with coupled real scalar field [75] and complex field (like hessence in loop quantum cosmology) in [59]. The work in [60] dealing with quintessence, matter in theory, is also done with exponential potential. But, to our knowledge, hessence, matter in theory, has not been considered before. In view of (22), the Friedmann equation (18) reduces asThe Raychaudhuri equation becomesThe density parameters of hessence () dark energy and background matter () are obtained in the following forms:The EoS of hessence dark energy and total EoS of the system are calculated in the following forms:Also, the deceleration parameter can be expressed as

#### 4. Fixed Points and Stability Analysis of the Autonomous System

##### 4.1. Fixed Points with Exponential Potential

We have made the choice of exponential form of potential, that is, (where is a real constant). The fixed points , the coordinates of , that is, (, , , , ), are given in Table 1 with relevant parameters and existence condition(s).