Phase space analysis of Tsallis agegraphic dark energy

Abstract

Based on the generalized Tsallis entropy and holographic hypothesis, the Tsallis agegraphic dark energy (TADE) was proposed by introducing the timescale as infrared cutoff. In this paper, we analyze the evolution of the universe in the TADE model and the new Tsallis agegraphic dark energy (NTADE) model by considering an interaction between dark matter and dark energy as $Q=H(\alpha \rho _{m}+\beta \rho _{D})$. Through the phase space and stability analysis, we find an attractor which represents a late-time accelerated expansion phase can exist only in NTADE model. When $0\le \alpha <1$ and $\beta =0$, this attractor becomes a dark energy dominated de Sitter solution and the universe can eventually evolve into an accelerated expansion era which is depicted by the $\Lambda$ cold dark matter model. Thus, the expansion history of the universe can be depicted by the NTADE model.

Appendices

Appendix A: Centre manifold theory

In this Appendix, we briefly review the centre manifold theory and the detailed presentation is give in Ref. [69, 72, 78, 79, 82, 83]. If one of the eigenvalues of a critical point vanishes, linear stability theory fails to provide information on the stability of this point. Then, centre manifold theory can be applied to analyze the stability of this critical point. The main aim of the centre manifold theory is to analyze the dynamics on the dimensionally reduced space. This reduced space, which is identified by the eigenvectors of the zero eigenvalues, is called centre manifold and its existence is always guaranteed.

In general, we assume the critical point is located at the origin of this phase space. Then, an arbitrary dynamical system with negative and zero eigenvalues can be written in the form

$$\begin{aligned} u'= & {} Au+f(u,v),\nonumber \\ v'= & {} Bv+g(u,v), \end{aligned}$$

(22)

where $(u,v)\in R^{c}\times R^{s}$ and

$$\begin{aligned} f(0,0)= & {} 0, \quad Df(0,0)=0,\nonumber \\ g(0,0)= & {} 0, \quad Dg(0,0)=0. \end{aligned}$$

(23)

Here, Df denotes the matrix of first derivative of function f, A is a $c\times c$ matrix with eigenvalues have zero real parts and B is a $s\times s$ matrix with eigenvalues have negative real parts. The centre manifold is characterized by a function $h: R^{c}\rightarrow R^{s}$ and is defined as

$$\begin{aligned} W^{c}(0)=\{(u,v)\in R^{c}\times R^{s}:v=h(u),\mid u\mid <\delta ,h(0)=0,Dh(0)=0\}, \end{aligned}$$

(24)

for $\delta$ sufficiently small. The dynamics of system  (22) restricted to the centre manifold $W^{c}(0)$ is determined by

$$\begin{aligned} u'=Au+f(u,h(u)), \end{aligned}$$

(25)

for a sufficiently small $u\in R^{c}$. The stability properties of the reduced system  (25) implies the stability properties of the full system  (22). Now, the problem becomes how to determine h. In Ref. [82, 83], h was proven to satisfy

$$\begin{aligned} Nh(u)=Dh(u)[Au+f(u,h(u))]-Bh(u)-g(u,h(u))=0, \end{aligned}$$

(26)

from which h can be found and inserted into  (25) to analyze the reduced system. However, it is impossible to solve Eq. (26) analytically. Fortunately, the analytical solution of h is not needed since we are only interested in the reduced system near this critical point, and it can be approximated by a power series expansion $h(u)=a u^{2}+b u^{3}+O(u^{4})$.

Appendix B: Centre manifold dynamics for $A_{5}$

In this section, we apply centre manifold theory to analyze the stability of point $P_{5}$. Following [79], we move point $A_{5}(\frac{\beta }{3-\alpha +\beta },\frac{3-\alpha }{3-\alpha +\beta },0)$ to the origin of this phase space by using the transformation $x\rightarrow x+\frac{\beta }{3-\alpha +\beta }$, $y\rightarrow y+\frac{3-\alpha }{3-\alpha +\beta }$, $z\rightarrow z$. Then, Eq. (15) becomes

$$\begin{aligned} x'= & {} -3\Big (x+\frac{\beta }{3-\alpha +\beta }\Big )+\alpha \Big (x+\frac{\beta }{3-\alpha +\beta }\Big )\nonumber \\&\quad +\beta \Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big )+2\Big (x+\frac{\beta }{3-\alpha +\beta }\Big )\zeta ,\nonumber \\ y'= & {} (2\delta -4)\Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big ) z+2\Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big )\zeta ,\nonumber \\ z'= & {} -z^{2}+z\zeta , \end{aligned}$$

(27)

with

$$\begin{aligned}&\zeta =2-2\Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big )-\frac{1}{2}\Big (x+\frac{\beta }{3-\alpha +\beta }\Big )\\&\quad -\frac{1}{2}\bigg [\alpha \Big (x+\frac{\beta }{3-\alpha +\beta }\Big )+\beta \Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big )\bigg ]-(\delta -2)\Big (y+\frac{3-\alpha }{3-\alpha +\beta }\Big ) z. \end{aligned}$$

After using the eigenvectors of the stability matrix of this system, we can introduce another set of new coordinates

$$\begin{aligned} \left( \begin{array}{c} X\\ Y\\ Z\\ \end{array} \right) = \left( \begin{array}{ccc} 0 &{} 0 &{} 1\\ \frac{3-\alpha }{3-\alpha +\beta } &{} \frac{3-\alpha }{3-\alpha +\beta } &{} 0\\ \frac{-3+\alpha }{3-\alpha +\beta } &{} \frac{\alpha }{3-\alpha +\beta } &{} \frac{2(2-\delta )\beta }{(3-\alpha +\beta )^{2}}\\ \end{array} \right) \left( \begin{array}{c} x\\ y\\ z\\ \end{array} \right) . \end{aligned}$$

(28)

In these coordinates, the system of equations are transformed into

$$\begin{aligned} \left( \begin{array}{c} X'\\ Y'\\ Z'\\ \end{array} \right) = \left( \begin{array}{ccc} 0 &{} 0 &{} 0\\ 0 &{} -4 &{} 0\\ 0 &{} 0 &{} \alpha -3\\ \end{array} \right) \left( \begin{array}{c} X\\ Y\\ Z\\ \end{array} \right) + \left( \begin{array}{c} f\\ g_{1}\\ g_{2}\\ \end{array} \right) \end{aligned}$$

(29)

with

$$\begin{aligned} f= & {} -\frac{[4Y+(3-\alpha )Z](3-\alpha +\beta )}{2(3-\alpha )}X+[1+2Y+2Z-(1+Y+Z)\delta ]X^{2}\nonumber \\&\quad -\frac{2\beta (\delta -2)^{2}}{(3-\alpha +\beta )^{2}}X^{3}, \end{aligned}$$

(30)

$$\begin{aligned} g_{1}= & {} \Big [(-3+\alpha -\beta )Z+2(2-\delta )(1+Z)X-\frac{4\beta (\delta -2)^{2}}{(3-\alpha +\beta )^{2}}X^{2}\Big ]Y\nonumber \\&\quad +\Big [-4+\frac{4\beta }{\alpha -3}-2(\delta -2)X\Big ]Y^{2}, \end{aligned}$$

(31)

$$\begin{aligned} g_{2}= & {} \Big [\frac{(-6+2\alpha -\beta )(\delta -2)}{3-\alpha +\beta }X+\frac{4(3-\alpha +\beta )}{\alpha -3}Y+(4-2\delta )XY-\frac{6\beta (\delta -2)^{2}}{(3-\alpha +\beta )}X^{2}\Big ]Z\nonumber \\&\quad +[-3+\alpha -\beta -2(\delta -2)X]Z^{2}-\frac{2\beta (\alpha -1)(\delta -2)}{(\alpha -3)(-3+\alpha -\beta )}XY-\frac{2\beta (\delta -2)^{2}}{(3-\alpha +\beta )^{2}}YX^{2}\nonumber \\&\quad -\frac{2\beta (\delta -2)[9-3\alpha -\beta +(-3+\alpha +\beta )\delta ]}{(-3+\alpha -\beta )^{3}}X^{2}-\frac{4\beta ^{2}(\delta -2)^{3}}{(3-\alpha +\beta )}X^{3}. \end{aligned}$$

(32)

Compare this dynamical system with Eq. (22), we obtain

$$\begin{aligned} A=0, \quad B= \left( \begin{array}{cc} -4 &{} 0\\ 0 &{} \alpha -3\\ \end{array} \right) , \quad g= \left( \begin{array}{c} g_{1}\\ g_{2}\\ \end{array} \right) . \end{aligned}$$

(33)

Now, the centre manifold can be assumed to the form

$$\begin{aligned} h= \left( \begin{array}{c} a_{2}X^{2}+a_{3}X^{3}+O(X^{4})\\ b_{2}X^{2}+b_{3}X^{3}+O(X^{4})\\ \end{array} \right) , \end{aligned}$$

(34)

and Eq. (26) becomes

$$\begin{aligned} N= \left( \begin{array}{c} 4a_{2}X^{2}+(-2a_{2}+4a_{3})X^{3}+O(X^{4})\\ \sigma _{1}X^{2}+\sigma _{2}X^{3}+O(X^{4})\\ \end{array} \right) =0, \end{aligned}$$

(35)

with

$$\begin{aligned}&\sigma _{1}=\frac{2\beta (\delta -2)[9-3\alpha -\beta +(-3+\alpha +\beta )\delta ]}{(-3+\alpha -\beta )^{3}}+(3-\alpha )b_{2},\\&\sigma _{2}=\frac{4\beta ^{2}(\delta -2)^{3}}{(3-\alpha +\beta )^{4}}+\frac{2\beta (\alpha -1)(\delta -2)a_{2}}{(\alpha -3)(-3+\alpha -\beta )}+\frac{(6-2\alpha +\beta \delta )b_{2}}{-3+\alpha -\beta }+(3-\alpha )b_{3}. \end{aligned}$$

where $a_{2}$, $a_{3}$, $b_{2}$ and $b_{3}$ are computed as

$$\begin{aligned} a_{2}= & {} 0, \quad a_{3}=0,\\ b_{2}= & {} \frac{2\beta (\delta -2)[9-3\alpha -\beta +(-3+\alpha +\beta )\delta ]}{(\alpha -3)(-3+\alpha -\beta )^{3}},\\ b_{3}= & {} 2\beta (2-\delta )\Big [\frac{2(\alpha -3)(9-3\alpha -5\beta )+[2(\alpha -3)^{2}+13(\alpha -3)\beta +\beta ^{2}]\delta +\beta (9-3\alpha -\beta )\delta ^{2}}{(\alpha -3)^{2}(3-\alpha +\beta )^{4}}\Big ]. \end{aligned}$$

Then, the dynamics of the system restricted to the centre manifold is determined by

$$\begin{aligned} X'=AX+f(X,h(X))=(1-\delta )X^{2}-\frac{\beta (\delta -1)(\delta -2)}{(3-\alpha )(3-\alpha +\beta )}X^{3}+O(X^{4}). \end{aligned}$$

(36)

It is obvious from this result that $A_{5}$ is unstable since the first term is even power. For any equation of the type $X'=\gamma X^{n}$, where $\gamma$ is a constant and n is a positive integer number, the critical point can be stable only for $\gamma <0$ and n is odd-parity. Otherwise, it is unstable.