Attractor Solutions in Cosmology
Abstract
Abstract: In this paper, we explore the cosmological implications of interacting dark energy model in a torsion based gravity namely . Assuming dark energy interacts with dark matter and radiation components, we examine the stability of this model by choosing different forms of interaction terms. We consider three different forms of dark energy: cosmological constant, quintessence and phantom energy. We then obtain several attractor solutions for each dark energy model interacting with other components. This model successfully explains the coincidence problem via the interacting dark energy scenario.
pacs:
04.20.Fy; 04.50.+h; 98.80.kI Introduction
Its now wellaccepted in astrophysics community that the observable universe is in a phase of rapid expansion whose rate of expansion is increasing, so called ‘accelerated expansion’. This conclusion has been drawn by numerous recent cosmological and astrophysical data findings of supernovae SNe Ia c1 (), cosmic microwave background radiations via WMAP c2 (), galaxy redshift surveys via SDSS c3 () and galactic Xray c4 (). This phenomenon is commonly termed ‘dark energy’ (DE) in the literature, and suggests that a cosmic dark fluid possessing negative pressure and positive energy density. Although the phenomenon of dark energy in cosmic history is very recent , it has opened new areas in cosmology research. Two important problems like ‘fine tuning’ and ‘cosmic coincidence’ are related to dark energy. It is thought that the most elegant solution to DE paradigm is the Einstein’s cosmological constant c7 () but it cannot resolve the two problems mentioned above. Hence cosmologists looked for other theoretical models by considering the dynamic nature of dark energy like quintessence scalar field quint (), a phantom energy field phant () and fessence f (). Another interesting set of proposals to DE puzzle is the ‘modified gravity’ which was proposed after the failure of general relativity (GR). This new set of gravity theories passes several solar system and astrophysical tests successfully sergei2 ().
For the past few years, models based on DE interacting with dark matter or any other exotic component have gained great impetus. Such interacting DE models can successfully explain numerous cosmological puzzles including dynamic DE, phantom crossing, cosmic coincidence and cosmic age jamil () and also in good compatibility with the astrophysical observations of cosmic microwave background, supernova type Ia, baryonic acoustic oscillations and galaxy redshift surveys obs (). There are some criticisms on interacting DE models for not being favored from observations and that the usual CDM model is favorable cr (). However the thermal properties of this model in various gravities have been discussed in literature jamil2 (). The model in which dark energy interacts with two different fluids has been investigated in literature. In cruz (), the two fluids were dark matter and another was unspecified. However, in another investigation jamil78 (), the third component was taken as radiation to address the cosmictriplecoincidence problem and study the generalized second law of thermodynamics. In a recent investigation, the authors investigated the coincidence problem in loop quantum gravity with triple interacting fluids including DE, dark matter and unparticle jamil8 ().
There is no need to construct a gravitational theory on a Riemannian manifold. A manifold can be divided into two separate but connected parts; one with Riemannian structure with a definite metric and another part, with a nonRiemannian structure and with torsion or nonmetricity. That part which has the zero Riemannian tensor but has non zero torsion is based on a tetrad basis, and defines a Weitzenbock spacetime. gravity is an alternative theory for GR, defined on the Weitzenbock nonRiemannian manifold, working only with torsion. This model firstly proposed by Einstein for unifying the electromagnetism and the gravity. If , this theory is called teleparallel gravity hayashi (); hehl (). It has been shown that with linear , this model has many common features like GR and is in good agreement with some standard tests of the GR in solar system hayashi (). But introducing a general model backs to few years f(T) (). This model has many features, for example Birkhoff’s theorem has been studied in this gravity birkhoff (). Earlier the authors in zheng () investigated perturbation in and found that the perturbation in gravity grows slower than that in Einstein general relativity. Bamba et al bamba () studied the evolution of equation of state parameter and phantom crossing in model. Emergent universes in chameleon model is investigated in ujjal (). As a thermodynamical view, there are many strange features. It has been shown in gravity, the famous formula of entropyarea of the black hole thermodynamics is not valid miaoli (). The reason goes back to the violation of the local lorentz invariance of this theory sotirio (); miaoli (). It shows that it is not possible to use the Wald conjecture wald () for calculating the entropy as a Noether charge in . The Hamiltonian formulation of gravity has been studied in miao () and shown that there are five degrees of freedom. In this paper we study the triple coincidence problem: why we happen to live during this special epoch when ? nima (). We investigate this problem in the framework of gravity.
We follow the plan: In section II we introduce the basic equations as an autonomous dynamical system. In section III we choose a model for gravity. In section IV we working on numerical analysis of the stability and the evolution of the functions of the model in details. We conclude and summarize in section V.
Ii Basic equations
One suitable form of action for gravity in Weitzenbock spacetime is f(T) ()
Here , and is the tetrad (vierbein) basis. The dynamical quantity of the model is the scalar torsion and is the matter Lagrangian. We start with the Friedmann equation in this form of the model f(T) ()
(1) 
where , while , and represent the energy densities of matter, dark energy and the radiation respectively.
Another FRW equation is
(2) 
For a spatially flat universe (), the total energy conservation equation is
(3) 
where is the Hubble parameter, is the total energy density and is the total pressure of the background fluid.
We assume a three component fluid containing matter, dark energy and radiation having an interaction. The corresponding continuity equations are jamil8 ()
(4)  
which satisfy collectively (3) such that .
We define dimensionless density parameters via
(5) 
The continuity equations (II) in dimensionless variables reduce to
where , is called the efolding parameter. The coupling functions , are in general functions of the energy densities and the Hubble parameter i.e. . The system of equations in (II) is analyzed by first equating them to zero to obtain the critical points. Next we perturb equations up to first order about the critical points and check their stability. Below for computation, we shall assume , and to be a general nonzero but negative parameter. We are interested in stable critical points (i.e. those points for which all eigenvalues of Jacobian matrix are negative) as these are attractor solutions of the dynamical system.
Iii model
To avoid analytic and computation problems, we choose a suitable expression which contains a constant, linear and a nonlinear form of torsion, specifically
(7) 
where , and are arbitrary constants. The first and the third terms (excluding the middle term) has correspondence with the cosmological constant EoS in gravity mirza (). There are many kinds of such these models, reconstructed from different kinds of the dark energy models. For example this form (7) may be inspired from a model for dark energy from proposed form of the Veneziano ghost kk (). But the linear term is needed to show the differences between our results in gravity from the Einstein gravity. Here we choose this model to simplify our numerical computations and for easier discussion on the difference of our results with the same results in GR. It is the minimum model, but our equations have been written for a general action. It is possible by repeating the numerical steps as we done in this paper, discuss the stability of other models. Recently Capozziello et al capo123 () inviestigated the cosmography of cosmology by using data of BAO, Supernovae Ia and WMAP. Following their interesting results, we notice that if we choose , and , than we can estimate the parameters of our proposed model as a function of Hubble parameter and the cosmographic parameters and the value of matter density parameter.
Iv Analysis of stability in phase space
In this section, we will construct four models by choosing different coupling forms and analyze the stability of the corresponding dynamical systems about the critical points. We shall plot the phase and evolutionary diagrams accordingly. For this reason, we must find the critical points of the (II), and then we linearize the system near the critical points up to first order.
iv.1 Interacting model  I
We consider the model with the following interaction terms
(8) 
where is a coupling parameter and we assume it to be a positive real number of order unity. Thus (8) says that both matter and radiation have increase in energy density with time while dark energy loses its energy density. Therefore it is a decay of dark energy into matter and radiation.
Using (8), the system (II) takes the form
(9)  
The critical points for this model are obtained by equating the left hand sides of (9) to zero. We obtain four critical points:

Point ,

Point ,

Point

Point
The eigenvalues of the Jacobian matrix for these critical points are:

Point

Point ,

Point ,

Point
Point is stable when one of these conditions is satisfied:
(10)  
(11)  
(12) 
is an unstable critical point since if than but . is stable if . Similarly is unstable since .
In figures (16), we plot the parameters of modelI. In figure 1, we observe that the dimensionless density parameters evolve from their currently observed values to vanishing densities. In figure2, we observe similar behavior when plotted against efolding parameter: the density parameters evolve from their current values to zero. In figure1, we deal with phantom energy, in figure3, the quintessence case while in figure5, the cosmological constant case.
iv.2 Interacting model  II
We study another model with the choice of the interaction terms
(13) 
This model effectively describes the situation when dark energy loses energy density to matter while the radiation density increases due to interaction with the matter.
There are four critical points:

Point ,

Point ,

Point ,

Point
The eigenvalues of the Jacobian matrix for these critical points are:

Point

Point ,

Point ,

Point
, are conditionally stable if (for ) and and (for ) . But and are unstable since .
In figures (712), we show the dynamics of ModelII. Attractor solutions are shown in figures 7,9 and 11. In figure 8, we see that the energy density of dark energy decays like quintessence. Also the energy density of radiation first increases till and then starts decreasing. The matter density always decreases and approaches zero nearly . In figure 10, the energy density of dark energy increases rapidly behaving like phantom energy, energy density of matter rises almost exponentially at later times, while radiation density increases slower compared to both matter and dark energy. These novel behaviors appear on account of interaction between three components. In figure 12, the dark energy density behaves like quintessence, while matter and radiation density falls with expansion.
iv.3 Interacting Model  III
Let us take the interaction terms jamil8 ()
(17) 
The system in (II) takes the form
(18)  
There are six critical points:

Point ,

Point ,

Point ,

Point ,

Point ,

Point
The eigenvalues of the Jacobian matrix for these critical points are:

Point

Point ,

Point ,

Point
is stable for . , are unstable. is conditionally stable when , .
In figures (1318), we have plotted the cosmological parameters of modelIII. In figures 13, 15 and 17, we show the attractor solutions of the differential equations equation. In figure 14, we observe the oscillatory behavior of dark energy and other cosmic components. It shows that when DE energy density decays then corresponding densities of dark matter and radiation increases and vice versa. Figure 16 shows that all forms of energy densities vanish by From figure 18, the radiation density stays zero while matter energy density decreases and vanish by . The dark energy density increases by while it decreases and stays constant at later epochs.
iv.4 Interacting Model  IV
Consider another model with the interaction terms jamil8 ()
(19) 
The system in (II) takes the form
(20)  
There are seven critical points:

Point ,

Point ,

Point ,

Point ,

Point ,

Point ,

Point .
For points , the eigenvalues of the Jacobian matrix are

Point ,

Point ,

Point ,

Point ,

Point
It is observed that in this model, is stable for , is stable for . is unstable. is stable for . It’s not possible to determine the stability of the point . But is unstable.
In figures (1923), we give description about modelIV. Figures 19,21,23 represent the phase space diagrams for different forms of dark energy. Figures 22 and 23 shows that energy density of dark energy decreases while in figure 20, the density of DE increases. It’s the typical behavior of the phantom fields. Thus, our model predicts the correct evolutionary schem for the DE density in regime of phantom.
V Conclusion
gravity is a powerful and novel theory for explanation of the acceleration expansion of the universe. In this paper we discussed the stability of the interactive models of the dark energy, matter and radiation in a FRW model, for a general theory. We derived the equations and show that why we have some attractor solutions for some specific forms of the interactions. We numerically integrate the equations and show that the evolution of the dark energy density mimics three diffract behaviors phantom, quintessence and cosmological constant in some interactive forms. We like to comment that this interaction is purely phenomenological required to meet some observational consequences. Since the phase space of the system of the evolutionary equations has a definitive end point, trackers are exist and depending on the initial conditions, there are different kinds of the trackers.
Acknowledgment
We would like to thank anonymous referees for giving useful comments to improve this paper. M. Jamil and D. Momeni would like to thank the warm hospitality of Eurasian National University where this work was started and completed.
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