# Dynamical System Analysis of Interacting Hessence Dark Energy inf(T) Gravity

Authors: Jyotirmay Das Mandal; Ujjal Debnath
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We have carried out dynamical system analysis of hessence field coupling with dark matter inf(T) 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., ΛCDM model is a subset of the solution set). Also, all the fixed points of our model are able to avoid Big Rip singularity.


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## Body

## 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 ΛCDM 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 p/ρ≡ω<-1/3. Accordingly, a priori choice for dark energy is a time-independent positive “cosmological constant” which relates to the equation of state (EoS) ω=-1. 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 ω close to the value of cosmological constant of ω does not indicate whether ω changes with time or not. So, theoretically, one could consider ω 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 ω of quintessence satisfies -1≤ω≤1. Some current data indicates that ω lies in small neighbourhood of ω=-1. Hence, it is technically feasible to relax ω to go down the line ω<-1 [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 ω<-1 (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 ω>-1 of recent past to ω<-1 at present time [41]. The line ω=-1 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, f(R) 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 f(T) gravity as in the same vein of f(R) gravity theory [54]. Although the EoS of “cosmological constant” (ΛCDM 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 f(T)) is really worth discussing. The work in [56] is a nice account in establishing matter stability of f(T) theory in weak field limit in contrast to f(R) theory. It is shown that any choice of f(T) 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 inf(T) gravity. Since the system is complex, we have preferred a dynamical analysis. As we have mentioned previously, hessence field and f(T) 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 f(T) 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 f(T) gravity has been discussed in [60]. But, to the best of our knowledge, hessence in f(T) gravity has not been considered before.We arrange the paper in the following manner. Short sketch off(T) theory is presented in Section 2. Hessence field in f(T) 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 8πG=ℏ=c=1 in this paper.


## 2. A Brief Outline off(T) Gravity: Some Basic Equations

In teleparallelism [54, 61, 62], eAμ are called the orthonormal tetrad components (A=0,1,2,3). The index A is used for each point xμ for a tangent space of the manifold; hence each eAμ represents a tangent vector to the manifold (i.e., the so-called vierbein). Also the inverse of the vierbein is obtained from the relation eAμeνA=δνμ. The metric tensor is given as gμν=ηABeμAeνA (μ,ν=0,1,2,3); μ,ν are coordinate indices on the manifold (here, ηAB=diag(1,-1,-1,-1)). Recently, to explain the acceleration, the teleparallel torsion (T) in Lagrangian density has been modified from linear torsion to some differentiable function of T [63, 64] (i.e., f(T)) like f(R) theory mentioned earlier. In this new setup of gravity, the field equation is of second order unlike f(R) (which is of fourth order). In f(T) theory of gravitation, corresponding action reads(1)A=12κ2∫d4x-gT+fT+Lm,where T is the torsion scalar, f(T) is some differentiable function of torsion T, Lm is the matter Lagrangian, -g=det(eμA), and κ2=8πG. The torsion scalar T mentioned above is defined as(2)T=SρμνTρμνwith the components of torsion tensor Tρμν of (2) given by(3)Tρμν=ΓWλνμ-ΓWλμν=eAλ∂μeνA-∂νeμA,where ΓWλνμ=eAλ∂μeνA is the Weitzenböck connection. Here, the superpotential Sρμν (2) is defined as follows:(4)Sρμν=12Kμνρ+δρμTθνθ-δρνTθμθ,Kμνρ=-12Tμνρ-Tνμρ-Tρμν.Kμνρ 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 f(T)=0. Actually this is the correspondence between teleparallel gravity and Einsteinian theory [53]. It is noticed that f(T) theory can address early acceleration and late evolution of universe depending on the choice of f(T). For example, power law or exponential form cannot overcome phantom divide [65], but some other choices of f(T) [66] can cross phantom divide. The reconstruction of f(T) model [67, 68], various cosmological [69, 70] and thermodynamical [71] analysis, has been reported. It is so interesting to note that linear f(T) model (i.e., when dF/dT = constant) behaves as cosmological constant. Anyway, a preferable choice of f(T) 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., f/T→0 for a≪1). Moreover, in future, it should give de-Sitter-like state. One such choice is given in power form as in [72]; namely,(5)fT=β-Tn,with β being a constant. In particular, n=1/2 gives same expanding model as the theory referred to in [72, 73]. Current data needs the bound “n≪1” to permit f(T) as an alternate gravity theory. The effective DE equation of state varies from ω=-1+n of past to ω=-1 in future.Throughout the work, we assume flat, homogeneous, isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric,(6)ds2=dt2-a2t∑i=13dxi2,which arises from the vierbein eμA=diag(1,a(t),a(t),a(t)). Here, a(t) is the scale factor as a function of cosmic time t. Using (3) and (4), one gets (7)T=SρμνTρμν=-6H2,where H=a˙(t)/a(t) is the Hubble factor (from here and in the rest of the paper “overdot” will mean the derivative operator d/dt).


## 3. Hessence Dark Energy inf(T) Gravity Theory: Formation of Dynamical Equations

Here, we consider a noncanonical complex scalar field:(8)Φ=ϕ1+iϕ2,where i=-1 with Lagrangian density:(9)LDE=14∂μΦ2+∂μΦ∗2-VΦ,Φ∗.Clearly the Lagrangian density is identical to the Lagrangian given by two real scalar fields, which looks like(10)LDE=12∂μϕ12-12∂μϕ22-Vϕ1,ϕ2,where ϕ1 and ϕ2 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 Lagrangian(11)LDE=12∂μΨ∗∂μΨ-VΨ,where |Ψ| denotes the absolute value of Ψ; that is, |Ψ|2=Ψ∗Ψ. 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 ρh and pressure ph of hessence field can be written as(12)ρh=12ϕ˙2-Q2a6ϕ2+Vϕ,(13)ph=12ϕ˙2-Q2a6ϕ2-Vϕ,where Q 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:(14)pm=wmρm≡γ-1ρm,where γ is the barotropic index satisfying 0<γ≤2. Also pm and ρm, respectively, denote the pressure and energy density of matter. In particular γ=1 and γ=4/3 indicate dust matter and radiation, respectively. We suppose that hessence and background fluid interact through a term C. This term C indicates energy transfer between dark energy and dark matter. Positive C is needed to solve coincidence problem, since positive magnitude of C indicates energy transfer from dark energy to dark matter. Also 2ND 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 C 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 C=δϕ˙ρm, where δ is a real constant of small magnitude, which may be chosen as positive or negative at will, such that C remains positive. Also, ϕ˙ may be positive or negative according to the hessence field ϕ. So we have(15)ρ˙h+3Hρh+ph=-C,(16)ρ˙m+3Hρm+pm=C,preserving the total energy conservation equation: (17)ρ˙total+3Hρtotal+ptotal=0. The modified field equations in f(T) gravity are(18)H2=12fT+113ρh+ρm-f6,(19)H˙=-12ρh+ph+ρm1+fT+2TfT.In view of (12) and (15), we have(20)ϕ¨+3Hϕ˙+Q2a6ϕ2+V′=-δρm.Here, “′” means “d/dϕ.” Similarly, (14) and (16) give(21)ρ˙m+3Hγρm=δϕ˙ρm.Now, we introduce five auxiliary variables:(22)x=ϕ˙6H,y=V3H,u=6ϕ,v=Q6Ha3ϕ,Ωm=ρm3H2.We form the following autonomous system after some manipulation:(23)dxdN=-3x-uv2-λ32y2-δ32Ωm+3x22x2-2v2+Ωm,dydN=λ32xy+32y2x2-2v2+Ωm,dudN=-xu2,dvdN=-xuv-3v+32v2x2-2v2+Ωm,dΩmdN=Ωm-3γ-δ6x+32x2-2v2+Ωm.In above calculations, N=∫a˙/adt=ln⁡a denotes the “e-folding” number. We have chosen N as independent variable. We have taken f(T)=β-T for above derivation of autonomous system. Also, we have chosen exponential form of potential, that is, V′/V=λ (where λ 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 f(T) theory, is also done with exponential potential. But, to our knowledge, hessence, matter in f(T) theory, has not been considered before. In view of (22), the Friedmann equation (18) reduces as(24)x2+y2-v2+Ωm=1.The Raychaudhuri equation becomes(25)-H˙H2=322x2-2v2+Ωm.The density parameters of hessence (Ωh) dark energy and background matter (Ωm) are obtained in the following forms:(26)Ωh=ρh3H2=x2+y2-v2,Ωm=ρm3H2=1-x2+y2-v2.The EoS of hessence ωh dark energy and total EoS of the system ωtotal are calculated in the following forms:(27)ωh=phρh=x2-y2-v2x2+y2-v2,ωtotal=ph+pmρh+ρm=x2-y2-v2+γ-1Ωm.Also, the deceleration parameter q can be expressed as(28)q=-1-H˙H2=-1+322x2-2v2+Ωm.


## 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,V′/V=λ (where λ is a real constant). The fixed points Pi, the coordinates of Pi, that is, (xc, yc, uc, vc, Ωmc), are given in Table 1 with relevant parameters and existence condition(s).Table 1
Fixed points of the autonomous system of equations (23) and various physical parameters with existence conditions. Here A=-3/2γ/δ+λ and B=6+λ6A-6A2/9.
Pi
x
c, yc, uc, vc, Ωmc
Ωm
ωh
ωtotal
Ωh
q
Existence condition(s)
P
1
1, 0, 0, 0, 0
0
1
1
1
2
Always
P
2
−1, 0, 0, 0, 0
0
1
1
1
2
Always
P
3
±
±
1 or 6-3γδ6, 0, 0, 0, 0
0
1
1
1
2
6
-
3
γ
δ
6
=
±
1
P
4
-
2
3
δ, 0, 0, 0, 6-3γ+2δ23
6-3γ+2δ23
1
γ1-2δ23+4δ23-1
2
δ
2
3
12+δ2
2
δ
2
3
+
6
-
3
γ
+
2
δ
2
3
=
1
P
5
x
=
6
-
3
γ
δ
6, 0, 0, x2-1, 0
0
1
1
1
2
6
δ
2
≤
(
6
-
3
γ
)
2
P
6
x
=
6
-
3
γ
δ
6, 0, 0, -x2-1, 0
0
1
1
1
2
6
δ
2
≤
(
6
-
3
γ
)
2
P
7
0, 1, any value, 0, 0
0
−1
−1
1
−1
γ
=
0
P
8
0, 1, any value, 0, 0
0
−1
−1
1
−1
γ
=
0
P
9
-
λ
6, 1-λ26, 0, 0, 0
0
-1+λ23
-1+λ23
1
-1+λ22
λ
2
≤
6
P
10
-
λ
6, -1-λ26, 0, 0, 0
0
-1+λ23
-1+λ23
1
-1+λ22
λ
2
≤
6
P
11
A, 1-A2-B2, 0, 0, B
B
-1+2A2+B21-B
-1+A2+B2+(γ-2)B
1-B
-1+32B+3B2
δ
+
λ
≠
0
P
12
A, -1-A2-B2, 0, 0, B
B
-1+2A2+B21-B
-1+A2+B2+(γ-2)B
1-B
-1+32B+3B2
δ
+
λ
≠
0
P
13
-
6
λ, 0, 0, 6λ2-1, 0
0
1
1
1
2
λ
2
≤
6
P
14
-
6
λ, 0, 0, -6λ2-1, 0
0
1
1
1
2
λ
2
≤
6
P
15
x
=
-
6
λ
=
6
-
3
γ
δ
6, 0, 0, x2-1, 0
0
1
1
1
2
-
6
λ
=
6
-
3
γ
δ
6 and λ≠2δ
P16
x
=
-
6
λ
=
6
-
3
γ
δ
6, 0, 0, -x2-1, 0
0
1
1
1
2
-
6
λ
=
6
-
3
γ
δ
6 and λ≠2δFrom Table1, we note the following.Case  1. Fixed points P1,P2=(±1,0,0,0,0) always exist with the physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  2. Fixed point P3±=(±1or6-3γ/δ6,0,0,0,0) exists under the condition that 6-3γ/δ6=±1 with the physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2, that is, same as P1 and P2.Case  3. Fixed point P4=(-2/3δ,0,0,0,6-3γ+2δ2/3) exists under the condition that 2δ2/3+6-3γ+2δ2/3=1 with physical parameters Ωm=6-3γ+2δ2/3, ωh=1, ωtotal=-1+γ(1-2δ2/3)+4δ2/3, Ωh=2δ2/3, and q=1/2+δ2.Case  4. Fixed points P5,P6=(x=6-3γ/δ6,0,0,±x2-1,0) exist under the condition that 6δ2≤(6-3γ)2 with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  5. Fixed points P7,P8=(0,1,anyvalue,0,0) exist under the condition that γ=0 with physical parameters Ωm=0, ωh=-1, ωtotal=-1, Ωh=1, and q=-1.Case  6. Fixed points P9,P10=(-λ/6, ±1-λ2/6,0,0,0) exist under the condition that λ2≤6 with physical parameters Ωm=0, ωh=-1+λ2/3, ωtotal=-1+λ2/3, Ωh=1, and q=-1+λ2/2.Case  7. Fixed points P11,P12=(A,±1-A2-B2,0,0,B) exist under the condition that δ+λ≠0 with physical parameters Ωm=B, ωh=-1+2A2+B2/1-B, ωtotal=-1+A2+B2+(γ-2)B, Ωh=1-B, and q=-1+3/2B+3B2.Case  8. Fixed points P13,P14=(-6/λ,0,0,±6/λ2-1,0) exist under the condition that λ2≤6 with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  9. Fixed points P15,P16=(x=-6/λ=6-3γ/δ6,0,0,±x2-1,0) exist under the condition that -6/λ=6-3γ/δ6 and λ≠2δ with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.


### 4.2. Stability of the Fixed Points

Dynamical analysis is a powerful technique to study cosmological evolution, where exact solution could not be found due to complicated system. This can be done without any information of specific initial conditions. The dynamical systems mostly encountered in cosmological system are nonlinear systems of differential equations (DE). Here the dynamical system is also nonlinear. Very few works in literature are devoted to analysing nonlinear dynamical systems. But we used the methods developed till now [76]. Also we devised some method (as in the plotting of the dynamical evolution and use of normally hyperbolic fixed points). We now analyse stability of the fixed points. In this regard, we find the eigenvalues of the linear perturbation matrix of the dynamical system (23). Due to the Friedmann equation (24), we have four independent perturbed equations. The eigenvalues of the 4×4 linear perturbation matrix corresponding to each fixed point Pi are given in Table 2. Before further discussion, we state some basics from nonlinear system of differential equation (DE) [76]. If the real part of each eigenvalue is nonzero, then the fixed point is called hyperbolic fixed point (otherwise, it is called nonhyperbolic). Let us write a nonlinear system of DE in Rn(then-dimensional Euclidean plane) as(29)x˙=fx,where f:E→Rn is derivable and E is an open set in R. For nonlinear system, the DE cannot be written in matrix form as done in linear system. Near hyperbolic fixed point, a nonlinear dynamical system could be linearized and stability of the fixed point is found by Hartman-Grobman theorem. As we can see from the following, let xc be a fixed point and let ζ(t) be the perturbation from xc; that is, ζ(t)=x-xc; that is, x=xc+ζ(t). We find the time evolution of ζ(t) for (29) as(30)ζ˙=ddtx-xc=x˙=fx=fxc+ζ.Table 2
Eigenvalues of the fixed points of the autonomous system of equations (23) and the nature of stability (if any), where D=-12+24a2+12b2+26aδ+6aλ and Δ=-144a2+144a4+144a2b2+36b4+486aδ-486a3δ-726ab2δ+24a2δ2-726aλ+726a3λ+846ab2λ+48δλ-72a2δλ-48b2δλ-48λ2+54a2λ2+48b2λ2.
P
i
Eigenvalues
Nature of stability (ifexists†)
P
1
0,0
,
3
+
δ
6
,
3
+
3
2
λ
2D stable manifold
P
2
0,0
,
3
-
δ
6
,
3
-
3
2
λ
2D stable manifold
P
3
±
0,0
,
3
±
δ
6
,
3
±
3
2
λ
2D stable manifold
P
4
0
,
-
3
2
+
δ
2
,
-
3
2
+
δ
2
,
3
2
+
δ
2
-
δ
λ
3D stable manifold
P
5
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P
6
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P
7
-
3,0
,
-
3
-
3
(
δ
λ
-
λ
2
)
,
-
3
+
3
(
δ
λ
-
λ
2
)
stable
P
8
-
3,0
,
-
3
-
3
(
δ
λ
-
λ
2
), -3+3(δλ-λ2)
stable
P
9
0
,
-
3
+
λ
2
2
,
-
3
+
λ
2
2
,
-
3
-
δ
λ
+
λ
2
3D stable manifold
P10
0
,
-
3
+
λ
2
2
,
-
3
+
λ
2
2
,
-
3
-
δ
λ
+
λ
2
3D stable manifold
P
11
0
,
-
3
+
3
a
2
+
3
b
2
2
,
1
4
(
D
-
Δ
), 14(D+Δ)
3D stable manifold
P
12
0
,
-
3
+
3
a
2
+
3
b
2
2
,
1
4
(
D
-
Δ
), 14(D+Δ)
3D stable manifold
P
13
0,0
,
0
,
-
3
(
2
δ
λ
-
λ
2
)
λ
2
1D stable manifold
P14
0,0
,
0
,
-
3
(
2
δ
λ
-
λ
2
)
λ
2
1D stable manifold
P
15
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P16
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
†Nature of stability is discussed in detail.Sincef is assumed to be derivable, we use the Taylor expansion of f to get(31)fxc+ζ=fxc+ζDfxc+⋯.Df(x)=∂fi/∂xj,i,j=1,2,…,n; as ζ is very small, higher order terms are neglected above. As f(xc)=0, (30) reduces to(32)ζ˙=ζDfxc.This is called the linearization of the DE near a fixed point. Stability of the fixed point xc is inferred from the sign of eigenvalues of Jacobian matrix Df(xc). If the fixed point is hyperbolic, then stability is concluded from Hartman-Grobman theorem, which states the following.Theorem (Hartman-Grobman).Given the nonlinear DE (29) in Rn, where f is derivable with flow ϕt, if xc is a hyperbolic fixed point, then there exists a neighbourhood N of xc, on which ϕt is homeomorphic to the flow of linearization of the DE near xc.But for nonhyperbolic fixed point this cannot be done and the study of stability becomes hard due to lack of theoretical setup. If at least one eigenvalue corresponding to the fixed point is zero, then it is termed as nonhyperbolic. For this case, we cannot find out stability near the fixed point. Consequently, we have to resort to other techniques like numerical solution of the system near fixed point and to study asymptotic behaviour with the help of plot of the solution, as is done in this work (details are described later). However, we can find the dimension of stable manifold (if exists) with the help of centre manifold theorem. There is a separate class of important nonhyperbolic fixed points known as normally hyperbolic fixed points, which are rarely considered in literature (see [77]). As some fixed points encountered in our work are of this kind, we state the basics here. We are also interested in nonisolated normally hyperbolic fixed points of a given DE (e.g., a curve of fixed points; such a set is called equilibrium set). If an equilibrium set has only one zero eigenvalue at each point and all other eigenvalues have nonzero real part, then the equilibrium set is called normally hyperbolic. The stability of normally hyperbolic fixed point is deduced from invariant manifold theorem, which states the following.Theorem (Invariant Manifold). Let x=xc be a fixed point of the DE x˙=f(x) on Rn and let Es, Eu, and Ec denote the stable, unstable, and centre subspaces of the linearization of the DE at xc. Then there exist a stable manifold Ws tangent to Es, an unstable manifold Ws tangent to Eu, and a centre manifold Wc tangent to Ec at xc. In other words, the stability depends on the sign of remaining eigenvalues. If the sign of remaining eigenvalues is negative, then the fixed point is stable; otherwise it is unstable. Table 2 shows the eigenvalues corresponding to the fixed points given in Table 1 and existence for hyperbolic, nonhyperbolic, or normally hyperbolic fixed points with the nature of stability (if any).We see from Table2 that each fixed point Pi is nonhyperbolic, except P7 and P8 (which are normally hyperbolic). So we cannot use linear stability analysis. Hence, we have utilised the following scheme to infer the stability of nonhyperbolic fixed points. We find the numerical solutions of the system of differential equations (23). Then, we have investigated the variation of the dynamical variables x,y,u,v,Ωm against e-folding N, which in turn gives the variation against time t through graphs in the neighbourhood of each fixed point, and notice if the dynamical variables asymptotically converge to any of the fixed points. In that case, we can say that the fixed point is stable (otherwise, it is unstable). This method is used nowadays in absence of proper mathematical analysis of nonlinear dynamical system. But we must remember that the method is not full proof, since we have to consider the neighbourhood of N as large as possible (i.e., |N|→∞), because a small perturbation can lead to instability. The graphs corresponding to each fixed point are given and analysed below. We consider the fixed points one by one.We note from Figure1 that P1 is not a stable fixed point. Similar is the case of P2, as is evident from Figure 2. We note that if λ≤-6 and δ≤-3/2 (or λ≥6 and δ≥3/2) (equality should occur in one of them), then P1 (or P2) may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence and total EoS being 1, and universe decelerates.Figure 1
Plot of (1) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P1, for γ=1, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 2
Plot of (2) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P2, for γ=1, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note thatP3± bears same feature as P1 and P2. So, none of P1, P2, and P3 describes the current phase of universe. The points bear no physical significance.Ifδ2≤3/2 and δ2-δλ≤-3/2 (equality should occur in one of them), P4 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence being 1 and total EoS is -1+γ(1-2δ2/3)+4δ2/3 and universe decelerates. Here, the plot in Figure 3 indicates that with a small increase of N the solution moves away from P4. This is an unstable fixed point.Figure 3
Plot of (3) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P4, for γ=4/3, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that, forP5 and P6, if xδ≤-3/2 and xλ≤-6 (equality should occur in one of them), P5 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues and P6 too may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence being 1 and total EoS is 1 and universe decelerates. Figure 4 indicates that the three of the variables (namely, x, v, and Ωm) are moving away from P5 and intruding in a neighbourhood of N=10. This may denote the stable manifold corresponding to the negative eigenvalues. However, this point gives the decelerated phase of the universe. Similar phenomena can be noted from Figure 5.Figure 4
Plot of (4) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P5, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 5
Plot of (5) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P6, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifδλ-λ2≤3, both P7 and P8 are normally hyperbolic set of fixed points and as the rest three nonzero eigenvalues are negative, they are stable. The set of fixed points has EoS of hessence of -1 and total EoS is also -1 and universe accelerates like “cosmological constant.” We note clearly from Figures 6 and 7 that all lines from negative and positive values of N (i.e., from past and future) are converging towards N=0 (i.e., the set of fixed points).Figure 6
Plot of (6) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P7, for γ=0, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 7
Plot of (7) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P8, for γ=0, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifλ2≤6 and λ2-δλ≤3 (equality should occur in one of them), P9 and P10 may admit 3-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence being -1+λ2/3 and total EoS also being -1+λ2/3 (i.e., both EoS are “quintessence-like” if λ2<3 or “dust-like” if λ2=3). The graphs in Figures 8 and 9 also support the fact corresponding to the stable manifolds. For λ=-0.5, the EoS of hessence and total EoS both behave like “quintessence.”Figure 8
Plot of (8) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P9, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 9
Plot of (9) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P10, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifa2+b2/2≤1,D≤-Δ (equality should occur in one of them), P11 and P12 may admit 3-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence being -1+2A2+B2/1-B and total EoS being -1+A2+B2+(γ-2)B. We see from Figure 10 that the system is moving away from the fixed point P11. Similar phenomena happen for fixed point P12 as seen from Figure 11.Figure 10
Plot of (10) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P11, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 11
Plot of (11) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P12, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that if2δ<λ<0 or 0<λ<2δ, then P13 and P14 may admit 1-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence and total EoS being 1 and universe decelerates. The graphs in Figures 12 and 13 show that the system is diverging from the fixed points P13 and P14. So, both the points are unstable in nature.Figure 12
Plot of (12) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P13, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 13
Plot of(13) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P14, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifxδ≤-3/2 and xλ≤-6, then P15 and P16 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence and total EoS being 1 and universe decelerates. Here, we note that the solution set of the dynamical system moves rapidly from the fixed points P15 and P16 as clear from Figures 14 and 15. The fixed points are unstable.Figure 14
Plot of (14) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P15, for γ=1, δ=1/(46), and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 15
Plot of (15) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P16, for γ=1, δ=1/(46), and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.


## 4.1. Fixed Points with Exponential Potential

We have made the choice of exponential form of potential, that is,V′/V=λ (where λ is a real constant). The fixed points Pi, the coordinates of Pi, that is, (xc, yc, uc, vc, Ωmc), are given in Table 1 with relevant parameters and existence condition(s).Table 1
Fixed points of the autonomous system of equations (23) and various physical parameters with existence conditions. Here A=-3/2γ/δ+λ and B=6+λ6A-6A2/9.
Pi
x
c, yc, uc, vc, Ωmc
Ωm
ωh
ωtotal
Ωh
q
Existence condition(s)
P
1
1, 0, 0, 0, 0
0
1
1
1
2
Always
P
2
−1, 0, 0, 0, 0
0
1
1
1
2
Always
P
3
±
±
1 or 6-3γδ6, 0, 0, 0, 0
0
1
1
1
2
6
-
3
γ
δ
6
=
±
1
P
4
-
2
3
δ, 0, 0, 0, 6-3γ+2δ23
6-3γ+2δ23
1
γ1-2δ23+4δ23-1
2
δ
2
3
12+δ2
2
δ
2
3
+
6
-
3
γ
+
2
δ
2
3
=
1
P
5
x
=
6
-
3
γ
δ
6, 0, 0, x2-1, 0
0
1
1
1
2
6
δ
2
≤
(
6
-
3
γ
)
2
P
6
x
=
6
-
3
γ
δ
6, 0, 0, -x2-1, 0
0
1
1
1
2
6
δ
2
≤
(
6
-
3
γ
)
2
P
7
0, 1, any value, 0, 0
0
−1
−1
1
−1
γ
=
0
P
8
0, 1, any value, 0, 0
0
−1
−1
1
−1
γ
=
0
P
9
-
λ
6, 1-λ26, 0, 0, 0
0
-1+λ23
-1+λ23
1
-1+λ22
λ
2
≤
6
P
10
-
λ
6, -1-λ26, 0, 0, 0
0
-1+λ23
-1+λ23
1
-1+λ22
λ
2
≤
6
P
11
A, 1-A2-B2, 0, 0, B
B
-1+2A2+B21-B
-1+A2+B2+(γ-2)B
1-B
-1+32B+3B2
δ
+
λ
≠
0
P
12
A, -1-A2-B2, 0, 0, B
B
-1+2A2+B21-B
-1+A2+B2+(γ-2)B
1-B
-1+32B+3B2
δ
+
λ
≠
0
P
13
-
6
λ, 0, 0, 6λ2-1, 0
0
1
1
1
2
λ
2
≤
6
P
14
-
6
λ, 0, 0, -6λ2-1, 0
0
1
1
1
2
λ
2
≤
6
P
15
x
=
-
6
λ
=
6
-
3
γ
δ
6, 0, 0, x2-1, 0
0
1
1
1
2
-
6
λ
=
6
-
3
γ
δ
6 and λ≠2δ
P16
x
=
-
6
λ
=
6
-
3
γ
δ
6, 0, 0, -x2-1, 0
0
1
1
1
2
-
6
λ
=
6
-
3
γ
δ
6 and λ≠2δFrom Table1, we note the following.Case  1. Fixed points P1,P2=(±1,0,0,0,0) always exist with the physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  2. Fixed point P3±=(±1or6-3γ/δ6,0,0,0,0) exists under the condition that 6-3γ/δ6=±1 with the physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2, that is, same as P1 and P2.Case  3. Fixed point P4=(-2/3δ,0,0,0,6-3γ+2δ2/3) exists under the condition that 2δ2/3+6-3γ+2δ2/3=1 with physical parameters Ωm=6-3γ+2δ2/3, ωh=1, ωtotal=-1+γ(1-2δ2/3)+4δ2/3, Ωh=2δ2/3, and q=1/2+δ2.Case  4. Fixed points P5,P6=(x=6-3γ/δ6,0,0,±x2-1,0) exist under the condition that 6δ2≤(6-3γ)2 with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  5. Fixed points P7,P8=(0,1,anyvalue,0,0) exist under the condition that γ=0 with physical parameters Ωm=0, ωh=-1, ωtotal=-1, Ωh=1, and q=-1.Case  6. Fixed points P9,P10=(-λ/6, ±1-λ2/6,0,0,0) exist under the condition that λ2≤6 with physical parameters Ωm=0, ωh=-1+λ2/3, ωtotal=-1+λ2/3, Ωh=1, and q=-1+λ2/2.Case  7. Fixed points P11,P12=(A,±1-A2-B2,0,0,B) exist under the condition that δ+λ≠0 with physical parameters Ωm=B, ωh=-1+2A2+B2/1-B, ωtotal=-1+A2+B2+(γ-2)B, Ωh=1-B, and q=-1+3/2B+3B2.Case  8. Fixed points P13,P14=(-6/λ,0,0,±6/λ2-1,0) exist under the condition that λ2≤6 with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.Case  9. Fixed points P15,P16=(x=-6/λ=6-3γ/δ6,0,0,±x2-1,0) exist under the condition that -6/λ=6-3γ/δ6 and λ≠2δ with physical parameters Ωm=0, ωh=1, ωtotal=1, Ωh=1, and q=2.


## 4.2. Stability of the Fixed Points

Dynamical analysis is a powerful technique to study cosmological evolution, where exact solution could not be found due to complicated system. This can be done without any information of specific initial conditions. The dynamical systems mostly encountered in cosmological system are nonlinear systems of differential equations (DE). Here the dynamical system is also nonlinear. Very few works in literature are devoted to analysing nonlinear dynamical systems. But we used the methods developed till now [76]. Also we devised some method (as in the plotting of the dynamical evolution and use of normally hyperbolic fixed points). We now analyse stability of the fixed points. In this regard, we find the eigenvalues of the linear perturbation matrix of the dynamical system (23). Due to the Friedmann equation (24), we have four independent perturbed equations. The eigenvalues of the 4×4 linear perturbation matrix corresponding to each fixed point Pi are given in Table 2. Before further discussion, we state some basics from nonlinear system of differential equation (DE) [76]. If the real part of each eigenvalue is nonzero, then the fixed point is called hyperbolic fixed point (otherwise, it is called nonhyperbolic). Let us write a nonlinear system of DE in Rn(then-dimensional Euclidean plane) as(29)x˙=fx,where f:E→Rn is derivable and E is an open set in R. For nonlinear system, the DE cannot be written in matrix form as done in linear system. Near hyperbolic fixed point, a nonlinear dynamical system could be linearized and stability of the fixed point is found by Hartman-Grobman theorem. As we can see from the following, let xc be a fixed point and let ζ(t) be the perturbation from xc; that is, ζ(t)=x-xc; that is, x=xc+ζ(t). We find the time evolution of ζ(t) for (29) as(30)ζ˙=ddtx-xc=x˙=fx=fxc+ζ.Table 2
Eigenvalues of the fixed points of the autonomous system of equations (23) and the nature of stability (if any), where D=-12+24a2+12b2+26aδ+6aλ and Δ=-144a2+144a4+144a2b2+36b4+486aδ-486a3δ-726ab2δ+24a2δ2-726aλ+726a3λ+846ab2λ+48δλ-72a2δλ-48b2δλ-48λ2+54a2λ2+48b2λ2.
P
i
Eigenvalues
Nature of stability (ifexists†)
P
1
0,0
,
3
+
δ
6
,
3
+
3
2
λ
2D stable manifold
P
2
0,0
,
3
-
δ
6
,
3
-
3
2
λ
2D stable manifold
P
3
±
0,0
,
3
±
δ
6
,
3
±
3
2
λ
2D stable manifold
P
4
0
,
-
3
2
+
δ
2
,
-
3
2
+
δ
2
,
3
2
+
δ
2
-
δ
λ
3D stable manifold
P
5
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P
6
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P
7
-
3,0
,
-
3
-
3
(
δ
λ
-
λ
2
)
,
-
3
+
3
(
δ
λ
-
λ
2
)
stable
P
8
-
3,0
,
-
3
-
3
(
δ
λ
-
λ
2
), -3+3(δλ-λ2)
stable
P
9
0
,
-
3
+
λ
2
2
,
-
3
+
λ
2
2
,
-
3
-
δ
λ
+
λ
2
3D stable manifold
P10
0
,
-
3
+
λ
2
2
,
-
3
+
λ
2
2
,
-
3
-
δ
λ
+
λ
2
3D stable manifold
P
11
0
,
-
3
+
3
a
2
+
3
b
2
2
,
1
4
(
D
-
Δ
), 14(D+Δ)
3D stable manifold
P
12
0
,
-
3
+
3
a
2
+
3
b
2
2
,
1
4
(
D
-
Δ
), 14(D+Δ)
3D stable manifold
P
13
0,0
,
0
,
-
3
(
2
δ
λ
-
λ
2
)
λ
2
1D stable manifold
P14
0,0
,
0
,
-
3
(
2
δ
λ
-
λ
2
)
λ
2
1D stable manifold
P
15
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
P16
0,0
,
3
+
6
x
δ
,
3
+
3
2
x
λ
2D stable manifold
†Nature of stability is discussed in detail.Sincef is assumed to be derivable, we use the Taylor expansion of f to get(31)fxc+ζ=fxc+ζDfxc+⋯.Df(x)=∂fi/∂xj,i,j=1,2,…,n; as ζ is very small, higher order terms are neglected above. As f(xc)=0, (30) reduces to(32)ζ˙=ζDfxc.This is called the linearization of the DE near a fixed point. Stability of the fixed point xc is inferred from the sign of eigenvalues of Jacobian matrix Df(xc). If the fixed point is hyperbolic, then stability is concluded from Hartman-Grobman theorem, which states the following.Theorem (Hartman-Grobman).Given the nonlinear DE (29) in Rn, where f is derivable with flow ϕt, if xc is a hyperbolic fixed point, then there exists a neighbourhood N of xc, on which ϕt is homeomorphic to the flow of linearization of the DE near xc.But for nonhyperbolic fixed point this cannot be done and the study of stability becomes hard due to lack of theoretical setup. If at least one eigenvalue corresponding to the fixed point is zero, then it is termed as nonhyperbolic. For this case, we cannot find out stability near the fixed point. Consequently, we have to resort to other techniques like numerical solution of the system near fixed point and to study asymptotic behaviour with the help of plot of the solution, as is done in this work (details are described later). However, we can find the dimension of stable manifold (if exists) with the help of centre manifold theorem. There is a separate class of important nonhyperbolic fixed points known as normally hyperbolic fixed points, which are rarely considered in literature (see [77]). As some fixed points encountered in our work are of this kind, we state the basics here. We are also interested in nonisolated normally hyperbolic fixed points of a given DE (e.g., a curve of fixed points; such a set is called equilibrium set). If an equilibrium set has only one zero eigenvalue at each point and all other eigenvalues have nonzero real part, then the equilibrium set is called normally hyperbolic. The stability of normally hyperbolic fixed point is deduced from invariant manifold theorem, which states the following.Theorem (Invariant Manifold). Let x=xc be a fixed point of the DE x˙=f(x) on Rn and let Es, Eu, and Ec denote the stable, unstable, and centre subspaces of the linearization of the DE at xc. Then there exist a stable manifold Ws tangent to Es, an unstable manifold Ws tangent to Eu, and a centre manifold Wc tangent to Ec at xc. In other words, the stability depends on the sign of remaining eigenvalues. If the sign of remaining eigenvalues is negative, then the fixed point is stable; otherwise it is unstable. Table 2 shows the eigenvalues corresponding to the fixed points given in Table 1 and existence for hyperbolic, nonhyperbolic, or normally hyperbolic fixed points with the nature of stability (if any).We see from Table2 that each fixed point Pi is nonhyperbolic, except P7 and P8 (which are normally hyperbolic). So we cannot use linear stability analysis. Hence, we have utilised the following scheme to infer the stability of nonhyperbolic fixed points. We find the numerical solutions of the system of differential equations (23). Then, we have investigated the variation of the dynamical variables x,y,u,v,Ωm against e-folding N, which in turn gives the variation against time t through graphs in the neighbourhood of each fixed point, and notice if the dynamical variables asymptotically converge to any of the fixed points. In that case, we can say that the fixed point is stable (otherwise, it is unstable). This method is used nowadays in absence of proper mathematical analysis of nonlinear dynamical system. But we must remember that the method is not full proof, since we have to consider the neighbourhood of N as large as possible (i.e., |N|→∞), because a small perturbation can lead to instability. The graphs corresponding to each fixed point are given and analysed below. We consider the fixed points one by one.We note from Figure1 that P1 is not a stable fixed point. Similar is the case of P2, as is evident from Figure 2. We note that if λ≤-6 and δ≤-3/2 (or λ≥6 and δ≥3/2) (equality should occur in one of them), then P1 (or P2) may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence and total EoS being 1, and universe decelerates.Figure 1
Plot of (1) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P1, for γ=1, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 2
Plot of (2) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P2, for γ=1, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note thatP3± bears same feature as P1 and P2. So, none of P1, P2, and P3 describes the current phase of universe. The points bear no physical significance.Ifδ2≤3/2 and δ2-δλ≤-3/2 (equality should occur in one of them), P4 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence being 1 and total EoS is -1+γ(1-2δ2/3)+4δ2/3 and universe decelerates. Here, the plot in Figure 3 indicates that with a small increase of N the solution moves away from P4. This is an unstable fixed point.Figure 3
Plot of (3) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P4, for γ=4/3, δ=0.5, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that, forP5 and P6, if xδ≤-3/2 and xλ≤-6 (equality should occur in one of them), P5 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues and P6 too may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence being 1 and total EoS is 1 and universe decelerates. Figure 4 indicates that the three of the variables (namely, x, v, and Ωm) are moving away from P5 and intruding in a neighbourhood of N=10. This may denote the stable manifold corresponding to the negative eigenvalues. However, this point gives the decelerated phase of the universe. Similar phenomena can be noted from Figure 5.Figure 4
Plot of (4) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P5, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 5
Plot of (5) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P6, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifδλ-λ2≤3, both P7 and P8 are normally hyperbolic set of fixed points and as the rest three nonzero eigenvalues are negative, they are stable. The set of fixed points has EoS of hessence of -1 and total EoS is also -1 and universe accelerates like “cosmological constant.” We note clearly from Figures 6 and 7 that all lines from negative and positive values of N (i.e., from past and future) are converging towards N=0 (i.e., the set of fixed points).Figure 6
Plot of (6) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P7, for γ=0, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 7
Plot of (7) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P8, for γ=0, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifλ2≤6 and λ2-δλ≤3 (equality should occur in one of them), P9 and P10 may admit 3-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence being -1+λ2/3 and total EoS also being -1+λ2/3 (i.e., both EoS are “quintessence-like” if λ2<3 or “dust-like” if λ2=3). The graphs in Figures 8 and 9 also support the fact corresponding to the stable manifolds. For λ=-0.5, the EoS of hessence and total EoS both behave like “quintessence.”Figure 8
Plot of (8) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P9, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 9
Plot of (9) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P10, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifa2+b2/2≤1,D≤-Δ (equality should occur in one of them), P11 and P12 may admit 3-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence being -1+2A2+B2/1-B and total EoS being -1+A2+B2+(γ-2)B. We see from Figure 10 that the system is moving away from the fixed point P11. Similar phenomena happen for fixed point P12 as seen from Figure 11.Figure 10
Plot of (10) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P11, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 11
Plot of (11) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P12, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that if2δ<λ<0 or 0<λ<2δ, then P13 and P14 may admit 1-dimensional stable manifold corresponding to the negative eigenvalues with EoS of hessence and total EoS being 1 and universe decelerates. The graphs in Figures 12 and 13 show that the system is diverging from the fixed points P13 and P14. So, both the points are unstable in nature.Figure 12
Plot of (12) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P13, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 13
Plot of(13) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P14, for γ=1, δ=1, and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.We note that ifxδ≤-3/2 and xλ≤-6, then P15 and P16 may admit 2-dimensional stable manifold corresponding to the two negative eigenvalues with EoS of hessence and total EoS being 1 and universe decelerates. Here, we note that the solution set of the dynamical system moves rapidly from the fixed points P15 and P16 as clear from Figures 14 and 15. The fixed points are unstable.Figure 14
Plot of (14) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P15, for γ=1, δ=1/(46), and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.Figure 15
Plot of (15) variations of x (blue), y (green), u (orange), v (red), and Ωm (yellow) versus N near P16, for γ=1, δ=1/(46), and λ=-0.5. The position corresponding to N=0 is the fixed point under consideration.


## 5. Cosmological Significance of the Fixed Points

In this section, we discuss the possible singularities that any dark energy model could have and compare the fixed points against recent dataset’s, Planck 2015, data [27]. If the EoS ω≤-1 (i.e., the null energy condition p+ρ≥0 is violated) and Big Rip singularity happen within a finite time [20], this singularity happens when, at finite time t→ts, a→∞, ρ→∞, and |p|→∞.We now analyse the stable fixed points to see if they can avoid (or suffer) Big Rip singularity. For the stable fixed pointsP7 and P8, we have H˙/H2=0 which gives H=k (the integral constant); we get a∝ekt. Also, in these cases, ωtotal=-1 which with energy conservation equation gives ρ=constant. Hence universe suffers no Big Rip singularity here. Fixed points P7 and P8 exist with physical parameters Ωm=0, ωh=-1, and ωtotal=-1. The values of the parameters are well within the best fit of Planck 2015 data; that is, Ωm=0.3089±0.0062 from TT,TE,EE + lowP + lensing + ext data, and EoS of dark energy ω=-1.019-0.080+0.075.Now, we consider the unstable fixed points. An unstable fixed point may describe the initial phase of universe, whereas a stable fixed point may be the end phase of the universe. For fixed pointsP1, P2, and P3 existing with the physical parameters Ωm=0, ωh=1, and ωtotal=1, clearly, no Big Rip singularity occurs here. Here, the parameter Ωm lies within the best fit of Planck 2015 data; that is, Ωm=0.3089±0.0062 from TT,TE,EE + lowP + lensing + ext data. But ωh and ωtotal defy the EoS of dark energy ω=-1.019-0.080+0.075.Fixed pointP4 has values of physical parameters Ωm=6-3γ+2δ2/3, ωh=1, and ωtotal=-1+γ(1-2δ2/3)+4δ2/3. Here, ωh and ωtotal both are greater than −1; no Big Rip singularity occurs here too. Wide choices of γ and δ can fit Ωm and ωtotal within Planck 2015 data; that is, Ωm=0.3089±0.0062, but ωh disobey the EoS of dark energy ω=-1.019-0.080+0.075.Fixed pointsP5 and P6 exist with physical parameters Ωm=0, ωh=1, and ωtotal=1. We observe that this solution is devoid of Big Rip singularity. Here, Ωm lies within the best fit of Planck 2015 data. But ωh and ωtotal defy the EoS of dark energy ω=-1.019-0.080+0.075.Fixed pointsP9 and P10 admit physical parameters as Ωm=0, ωh=-1+λ2/3, and ωtotal=-1+λ2/3 and so avoid Big Rip singularity. Also, Ωm is within Planck 2015 data. Also, suitable choice of λ fits ωh and ωtotal within dataset.Fixed pointsP11 and P12 have physical parameters Ωm=B, ωh=-1+2A2+B2/1-B, and ωtotal=-1+A2+B2+(γ-2)B, where A=-3/2γ/δ+λ and B=6+λ6A-6A2/9. Here, we can adjust A and B to make ωh and ωtotal≥-1 miss Big Rip singularity. Since only 0<γ≤2, δ can take arbitrary small value and λ can have any real value, A, and hence B can be adjusted well within Planck Ωm=0.3089±0.0062 from TT,TE,EE + lowP + lensing + ext and EoS of dark energy ω=-1.019-0.080+0.075 data.Fixed pointsP13, P14, P15, and P16 can avoid Big Rip singularity, as they bear physical parameters Ωm=0, ωh=1, and ωtotal=1. Here, the parameter Ωm lies within the best fit of Planck 2015 data; that is, Ωm=0.3089±0.0062 from TT,TE,EE + lowP + lensing + ext data. But ωh and ωtotal totally defy the EoS of dark energy ω=-1.019-0.080+0.075.


## 6. Concluding Remarks

In this paper, we have performed a dynamical system study of a unique scalar field hessence coupling with dark matter in an alternate theory of gravity, namely,f(T) gravity. The system is unconventional, complex, but quite interesting. The model is chosen to explore one of the various possibilities about the fate of the universe. The sole purpose is to explain the current acceleration of universe. An unstable fixed point may describe the initial phase of universe, whereas a stable fixed point may be the end of the universe. We have chosen exponential form of potential of the form V=V0eλϕ (where V0 and λ are real constants and ϕ is the hessence field) for simplicity. The interaction term C is chosen to solve the so-called cosmological constant problem in tune with second law of thermodynamics and is quite arbitrary (only C should remain positive), since C=δϕ˙ρm, where δ is a real constant of small magnitude, which may be chosen as positive or negative, such that C remains positive. Also, ϕ˙ may be positive or negative according to the hessence field ϕ. The resulting nonlinear dynamical system gives sixteen possible fixed points. Among them, P7 and P8 are stable set of normally hyperbolic fixed points, which resembles “cosmological constant,” so it explains the current phase of acceleration of universe. But, interestingly, it does not show “hessence-like” nature. Among the other fixed points, the initial phases of evolution may begin. However, the complexity of the system is the main obstacle for a precise explanation. Anyway, in future work, we may try some other possible alternatives.



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*Source: 2864784-2017-07-25.xml*