Effects of Anisotropy on the Sign-Changeable Interacting Tsallis Holographic Dark Energy

TL;DR

This paper investigates how anisotropy influences the behavior and stability of Tsallis holographic dark energy models with sign-changeable interactions in a Bianchi type I universe, revealing conditions for stability, phantom crossing, and parameter effects.

Contribution

It introduces a detailed analysis of anisotropic effects on Tsallis holographic dark energy with sign-changeable interactions, including stability, phantom crossing, and diagnostic plane behaviors.

Findings

Current universe can be classically stable in some models

All models show classical instability at future ($z ightarrow-1$)

Some models can cross the phantom line

Abstract

A spatially homogeneous and anisotropic Bianchi type I universe is considered while it is filled by pressureless dark matter (DM) and Tsallis holographic dark energy (DE) interacting with each other throughout a sign-changeable mutual interaction. Various infra-red (IR) cutoffs are studied, and it has been obtained that while the current universe can classically be stable for some cases, all models display classical instability by themselves at the future ($z\rightarrow-1$). Moreover, we find out that some models can cross the phantom line. In order to have a more comprehensive study, the statefinder diagnostic and the $\omega_D-{\omega}^{\prime}_{D}$ plane are also investigated showing that the model parameters significantly affect the evolution trajectories in the $r-s$ and $\omega_D-{\omega}^{\prime}_{D}$ planes.

Equations

$$d{s}^2=d{t}^2-A^2(t)d{x}^2-B^2(t)d{y}^2-C^2(t)d{z}^2,$$

$$3H^2-{\sigma }^2$$

$$3H^2+2\dot{H}+{\sigma }^2$$

$${\Omega }_m$$

$${\Omega }_m+{\Omega }_{\Lambda }=1-{\Omega }_{\sigma }.$$

$${\dot{\rho }}_m+3H{\rho }_m=Q,$$

$${\dot{\rho }}_D+3H(1+{\omega }_D){\rho }_D=-Q.$$

$$q=-\frac{\ddot{a}}{a{H}^2}=-1-\frac{\dot{H}}{H^2}.$$

$$v_s^2=\frac{d{P}_D}{d{\rho }_D}=\frac{{\dot{P}}_D}{{\dot{\rho }}_D}=\frac{{\rho }_D}{{\dot{\rho }}_D}{\dot{\omega }}_D+{\omega }_D.$$

$$s=\frac{r-1}{3(q-1/2)},r=\frac{\stackrel{...}{a}}{a{H}^3},$$

$$r=2q^2+q-\frac{\dot{q}}{H}.$$

$${\rho }_D=B{L}^{2\delta -4},$$

$${\rho }_D=B{H}^{-2\delta +4},$$

$${\dot{\rho }}_D={\rho }_D(-2\delta +4)\frac{\dot{H}}{H},$$

$$\frac{\dot{H}}{H^2}=\frac{-3{\Omega }_D(b^2(1+u)+u)}{2-2{\Omega }_{\sigma }+{\Omega }_D(3b^2(1+u)-(-2\delta +4))},$$

$$u=\frac{{\rho }_m}{{\rho }_D}=\frac{{\Omega }_m}{{\Omega }_D}=-1+\frac{1}{{\Omega }_D}(1-{\Omega }_{\sigma }).$$

$$q=-1+\frac{3{\Omega }_D(b^2(1+u)+u)}{2-2{\Omega }_{\sigma }+{\Omega }_D(3b^2(1+u)-(-2\delta +4))}.$$

$${\omega }_D=-1-b^{2}q(1+u)+(1+q)(\frac{-2\delta +4}{3}).$$

$${\Omega }_D^{\prime }=-2{\Omega }_D(-\delta +1)(1+q),$$

$$R_h=a(t){\int }_t^{\infty }\frac{dt}{a(t)},$$

$${\dot{R}}_h=H{R}_h-1,$$

$${\rho }_D=B{R}_h^{2\delta -4}.$$

$${\dot{\rho }}_D={\rho }_D(2\delta -4)H(1-F),$$

$${\omega }_D=-1-b^{2}q(1+u)-(\frac{2\delta -4}{3})(1-F).$$

$$\frac{\dot{H}}{H^2}=\frac{{\Omega }_D(-3b^2(1+u)-3u+(2\delta -4)(1-F)}{2-2{\Omega }_{\sigma }+3b^2{\Omega }_D(1+u)},$$

$$q=-1-\frac{{\Omega }_D(-3b^2(1+u)-3u+(2\delta -4)(1-F))}{2-2{\Omega }_{\sigma }+3b^2{\Omega }_D(1+u)}.$$

$${\Omega }_D^{\prime }={\Omega }_D((2\delta -4)(1-F)+2(1+q)).$$

$${\dot{R}}_p=H{R}_p+1,$$

$${\rho }_D=B{R}_p^{2\delta -4},$$

$${\dot{\rho }}_D={\rho }_D(2\delta -4)H(1+F),$$

$${\omega }_D=-1-b^{2}q(1+u)-(\frac{2\delta -4}{3})(1+F).$$

Full text

I Introduction

Cosmological observations such as type Ia supernova Riess ; Riess1 , WMAP Bennett ; Bennett1 and Large Scale Structure (LSS) Verde ; Verde1 ; Verde2 indicate that our Universe is now experiencing a phase of accelerated expansion roos . This accelerated phase of the Universe expansion is describable by a mysterious source of energy (DE) whose energy density $\rho_{DE}$ and pressure $P_{DE}$ filling about $0.73$ percent of the Universe, while $\rho_{DE}+3P_{DE}<0$ meaning that its equation of state (EoS) parameter, $\omega_{DE}={P_{DE}}/{\rho_{DE}}$, must satisfy the condition $\omega_{DE}<-{1}/{3}$ Bamba:2012cp ; Nojiri:2010wj ; Capozziello:2011et ; Capozziello:2010zz ; Bamba:2015uma ; Cai:2015emx ; Nojiri:2017ncd ; shey1 ; shey2 ; shey3 ; shey4 ; shey5 ; shey6 ; shey7 ; shey8 ; shey9 ; shey10 . An interesting attempt to find a physical origin for DE is called the holographic dark energy (HDE) making a relation between the system entropy and the UV cutoff (or equally the energy density of quantum fields in vacuum) Cohen99 .

Since gravity is a long-range interaction, some physicists tried to study the cosmological evolution by using the generalized entropy formalism in which the Bekenstein entropy (as the backbone of the HDE hypothesis Cohen99 ) is not always met non2 ; non15 ; non16 ; non17 ; non18 ; non19 ; non20 ; non21 ; non22 ; non23 ; non13 ; non4 ; non5 ; non6 ; non7 ; non8 ; non9 ; non10 ; non11 ; non12 ; non14 ; eb ; eb1 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 10a ; 11 ; 11a ; 11b ; cite1 ; cite2 . Recently, using generalized entropy formalisms, some new HDE models have been introduced THDE ; smm ; epjcr . Tsallis holographic dark energy (THDE) is one of these new attempts, based on Tsallis entropy Tsallis , which can provide suitable description for the current universe in various cosmological setups THDE1 ; THDE2 ; THDE3 ; THDE4 ; Sar . It is also worth mentioning that although THDE is not stable at the classical level THDE , a more comprehensive study on its stability may consider the global features of the metric perturbations Stabil1 ; Stabil2 .

On the other hand, observations admit a mutual interaction between DE and DM Cai ; Wei ; Wei1 ; Kh ; Kh1 ; Kh2 ; Kh3 ; Kh4 ; Kh5 ; Kh6 ; Abdollahi2016 ; Abdollahi2017 ; Abdollahi2017a , and the $CMB+BAO+SN+H_{0}$ data suggests that the sign of mutual interaction has probably been changed during the cosmic evolution in the $0.45\leq z\leq 0.9$ interval Cai . This result motivates physicists to consider sign-changeable interactions, including the deceleration parameter Wei ; Wei1 ; Kh ; Kh1 ; Kh2 ; Kh3 ; Kh4 ; Kh5 ; Kh6 ; Abdollahi2016 ; Abdollahi2017 ; Abdollahi2017a . Although the FLRW metric is a powerful tool in modeling the universe, and is in accordance with the cosmological principle roos , it does not compatible with the early universe anisotropy, and also the anisotropy of the cosmos in scales smaller than $100$-Mpc and also roos . Thus, a comprehensive study on the cosmic evolution should consider the anisotropy effects. The Bianchi models are some interesting attempts to model the anisotropy of the early universe khani1 ; khani2 ; khani3 ; khani4 ; khani5 ; khani6 ; khani7 ; khani8 ; khani9 ; khani10 ; khani11 ; khani12 ; khani13 ; khani14 ; khani15 ; khani16 . Motivated by the above arguments, we are going to study the cosmic evolution of a Bianchi type I (BI) model filled by DE and DM interacting with each other throughout a sign-changeable interaction. In our models, THDE with various IR cutoffs is used to model DE.

The paper is organized as follows. In the next section, we present the general remarks on the BI universe filled by mutually interacting DE and DM. The results of considering the Hubble horizon as the IR cutoff is investigated in sections  III. The cases using the event and particle horizons as the IR cutoffs are studied in sections  IV) and  V, respectively. Two other famous IR cutoffs, including the GO Li20041 ; Li20042 and Ricci Li2004 ; Li2004a ; Gao cutoffs, will be addressed in sections. VI and VII, respectively. The last section is devoted to a brief summary.

II GENERAL FRAMEWORK

The BI metric, which includes the anisotropy of early cosmos, is written as khani1 ; khani2 ; khani3 ; khani4 ; khani5 ; khani6 ; khani7 ; khani8 ; khani9 ; khani10 ; khani11 ; khani12 ; khani13 ; khani14 ; khani15 ; khani16

[TABLE]

where $A(t)$, $B(t)$ and $C(t)$ are functions of cosmic time. Hence, the FLRW metric is recovered whenever $A=B=C$ khani1 ; khani2 ; khani3 ; khani4 ; khani5 ; khani6 ; khani7 ; khani8 ; khani9 ; khani10 ; khani11 ; khani12 ; khani13 ; khani14 ; khani15 ; khani16 . For this metric, $i$) $V^{3}\equiv ABC$ denotes the spatial volume. $ii$) $a=(ABC)^{\frac{1}{3}}$ is defined as the average scale factor. $iii$) $H=\frac{1}{3}(H_{1}+H_{2}+H_{3})$ is the generalized mean Hubble parameter, where $H_{1}=\frac{\dot{A}}{A}$, $H_{2}=\frac{\dot{B}}{B}$ and $H_{3}=\frac{\dot{C}}{C}$ are called the directional Hubble parameters in the directions of $x,y$ and $z$ axes, respectively khani1 ; khani2 ; khani3 ; khani4 ; khani5 ; khani6 ; khani7 ; khani8 ; khani9 ; khani10 ; khani11 ; khani12 ; khani13 ; khani14 ; khani15 ; khani16 .

Now, consider a BI universe filled by a pressureless source (with energy density $\rho_{m}$) and a DE candidate with $T^{\mu}_{\nu}=diag[\rho_{D},-\omega_{D}\rho_{D},-\omega_{D}\rho_{D},-\omega_{D}\rho_{D}]$ where $\rho_{D}$ and $\omega_{D}(\equiv\frac{p_{D}}{\rho_{D}})$ represent the energy density and EoS parameter of dark energy, respectively, and $p_{D}$ denotes the DE pressure. In this manner, the corresponding Friedmann equations (the BI equations) take the form Hossienkhani ; Hossienkhani11 ; Hossienkhani12

[TABLE]

in which $m_{p}^{2}=1/(8\pi G)$ and $\sigma$ are the Planck mass and the shear scalar, respectively. Moreover, $\sigma^{2}=1/2\sigma_{ij}\sigma^{ij}$ while $\sigma_{ij}=u_{i,j}+\frac{1}{2}(u_{i;k}u^{k}u_{j}+u_{j;k}u^{k}u_{i})+\frac{1}{3}\theta(g_{ij}+u_{i}u_{j})$ is called the shear tensor, which describes the rate of distortion of the matter flow, and $\theta=3H=u^{j}_{;j}$ denotes the scalar expansion. Defining the critical density $\rho_{cr}=3m_{p}^{2}H^{2}$, and introducing the dimensionless density parameters

[TABLE]

the first BI equation can be rewritten as

[TABLE]

It is worthwhile mentioning that the shear scalar is describable, using the average Hubble parameter, as $\sigma^{2}=\sigma_{0}^{2}H^{2}$ in which $\sigma_{0}^{2}$ is a constant khani1 ; khani2 ; khani3 ; khani4 ; khani5 ; khani6 ; khani7 ; khani8 ; khani9 ; khani10 ; khani11 ; khani12 ; khani13 ; khani14 ; khani15 ; khani16 . In the presence of a mutual interaction ($Q$) between the cosmos sectors, the continuity equation is decomposed as Wei ; Wei1

[TABLE]

Following Wei ; Wei1 , we assume $Q=3b^{2}qH\rho_{D}(1+u)$ where $b^{2}$ is coupling constant, $u=\frac{\rho_{m}}{\rho_{D}}$ and $q$ is the deceleration parameter defined as

[TABLE]

Thus, whenever the universe phase expansion is changed, the interaction sign is also changed. Indeed, for $Q<0$ ($Q>0$), there is an energy flow from DM (DE) to DE (DM). At the classical level, for a stable model, the sound speed square ${v}^{2}_{s}$ is positive Peebles20031 . For DE candidate, which controls the current universe dynamics, it is evaluated as

[TABLE]

A new set of completely geometrical parameters $\{r,s\}$, dubbed the statefinder, has been introduced by Sahni et al Sahni

[TABLE]

which obviously show the statefinder pair depend only on the scale factor and it’s time derivatives up to the third order. Combining the above relation with each other, we can also write

[TABLE]

Another way to analysis the cosmic evolution has been introduced in Ref. Caldwell , based on the set of parameters $\{\omega_{D},{\omega}^{\prime}_{D}\}$ (prime denotes derivative respect to $x=lna$). This approach works in the $\omega_{D}-{\omega}^{\prime}_{D}$ plane, in which ${\omega}^{\prime}_{D}>0$ and $\omega_{D}<0$ present the thawing region, while ${\omega}^{\prime}_{D}<0$ and $\omega_{D}<0$ present the freezing region Caldwell .

III Sign-changeable interacting THDE with HUBBLE RADIUS AS IR CUTOFF IN BI MODEL

THDE is defined as THDE

[TABLE]

leading to

[TABLE]

where $B$ is an unknown parameter, if we consider the Hubble radius $H^{-1}$ as the IR cutoff $L$. The time derivative of relation (14) leads to

[TABLE]

combined with the time derivative of the first Friedmann equation and Eq. (7) to reach

[TABLE]

where

[TABLE]

By inserting equation (16) in relation (9), the deceleration parameter $q$ is found out

[TABLE]

We can also find the EoS parameter, by combining Eqs. (15), (8) and (18) as

[TABLE]

With the help of Eqs. (6), (15) and the $\dot{\Omega}_{D}={\Omega}^{\prime}_{D}H$ relation, one reaches

[TABLE]

in which prime denotes derivative with respect to $\ln(a)$. Using the time derivative of Eqs.(18) and (19), we can find $v_{s}^{2}$ and the ($r,s$) pair as the statefinder parameters. Since these expressions are too long, we do not present them here, study the evolution of these quantities via figures. The model behavior has been depicted in Figs. (1-9) for the initial conditions $\Omega_{D}(z=0)=0.73$, $H(z=0)=67$, $b^{2}=.01$ and $\Omega_{\sigma}=.001$. These figures indicate that although the model is unstable at the classical level ($v_{s}^{2}<0$), it can cover the current accelerated universe for $\omega_{D}\geq-1$.

IV Sign-changeable interacting THDE with Event horizon AS IR CUTOFF IN BI MODEL

If the future event horizon, defined as

[TABLE]

leading to

[TABLE]

is used as the IR cutoff ($L=R_{h}$), then

[TABLE]

By combining the time derivative of the above equation with Eq.(22), we find

[TABLE]

where $F=(\frac{3\Omega_{D}H^{2\delta-2}}{B})^{\frac{1}{-2\delta+4}}$.

Inserting Eq.(24) in Eq.(8), we easily reach at

[TABLE]

Additionally, combining the time derivative of Eq.(2) with Eqs.(24) and (7), one obtains

[TABLE]

and

[TABLE]

Using the $\Omega_{D}$ expression and Eq.(24), we find

[TABLE]

Although $v_{s}^{2}$, $r$ and $s$ can be found out by by taking the time derivative of Eqs.(25) and (27), since they are too long relations, we do not present them here, and only plot them. In Figs. (10-18), the model behavior is depicted for the initial conditions $\Omega_{D}(z=0)=0.73$, $H(z=0)=67$, $b^{2}=.1$, $B=2.4$ and $\Omega_{\sigma}=.001$ indicating that, unlike the previous case, the phantom behaviors is unavoidable.

V Sign-changeable interacting THDE with Particle horizon AS IR CUTOFF IN BI MODEL

The particle horizon, introduced as Li2004

[TABLE]

is considered as the IR cutoff in this section. In this manner, Eq.(13) leads to

[TABLE]

and

[TABLE]

the time derivative of $\rho_{D}$.

Combining Eq.(31) with the conservation law (8), one obtains

[TABLE]

In addition, by using Eqs. (2), (31) and (9), the deceleration parameter $q$ is found out as

[TABLE]

Moreover, one can insert $\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}}$ in Eq.(31) to reach at

[TABLE]

The same as the previous sections, since the expressions of $v_{s}^{2}$, $r$ and $s$ are too long, we only plot them and do not write them here. In Figs. (19-27), the system parameters have been plotted by employing the initial conditions $\Omega_{D}(z=0)=0.73$, $H(z=0)=67$, $\delta=2.4$, $B=2.4$ and $\Omega_{\sigma}=.001$.

VI Sign-changeable interacting THDE with GO horizon AS IR CUTOFF IN BI MODEL

Bearing Eq. (13) in mind, and employing the GO cutoff Li20041 ; Li20042 , the energy density of THDE is given by

[TABLE]

rewritten as

[TABLE]

Now, by using relation (9), we have

[TABLE]

In addition, inserting the time derivative of Eq.(2) in Eq.(7), one obtains

[TABLE]

combined with

[TABLE]

to reach at

[TABLE]

The EoS parameter $\omega_{D}$) of THDE is found out as

[TABLE]

by combining Eq.(38) with Eq.(8). Figs. (28-36) show the behavior of the model parameters during the cosmic evolution by presuming $\Omega_{D}(z=0)=0.73$, $H(z=0)=67$, $\alpha=.8$,$\beta=.5$, $b^{2}=.01$ as the initial conditions.

VII Sign-changeable interacting THDE with Ricci horizon AS IR CUTOFF IN BI MODEL

Let us use the Ricci cutoff Gao for describing cosmological parameters. It leads to

[TABLE]

where $\lambda$ is the unknown HDE constant as usual Li2004 ; Gao , for THDE.

By rewriting Eq.(43) as

[TABLE]

and using relation (9), we get

[TABLE]

Moreover, combining Eqs.(38) with (39), one finds

[TABLE]

Following the recipe of the previous section, the EoS parameter is calculated as

[TABLE]

In Fig. (37-38) the model parameters have been plotted for the initial conditions $\Omega_{D}(z=0)=0.73$, $H(z=0)=67$, $\delta=1$, $\lambda=1.5$ and $\Omega_{\sigma}=.001$ and $\Omega_{\sigma}=.001$.

VIII Closing remarks

A Bianchi type I universe was considered filled by DM and DE interacting with each other throughout a sign-changeable interaction. In our study, THDE plays the role of DE and various IR cutoffs, including the Hubble, event and particle horizons as well as the GO and Ricci cutoffs, have been used to study the evolution of the universe. We tried to present a comprehensive study by addressing diverse parameters such as $q$, $H$, $r$, ${\omega}^{\prime}_{D}$ and etc. Although suitable dynamics can be obtained for the models, the classical stability analysis ($v_{s}^{2}$) shows that the models are always unstable at the $z\rightarrow-1$ limit. It is also worthwhile mentioning that the models may show stability by themselves at the classical level for the current universe (where $z\rightarrow 0$), depending on the unknown parameters of model such as $\delta$.

Acknowledgements.

We thank Shiraz University Research Council. This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran. In addition, the work of KB was partially supported by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds for Fukushima University Faculty (18RI009).