Recently, Tsallis, Rényi, and Sharma-Mittal entropies have widely been used to study the gravitational and cosmological setups. We consider a flat FRW universe with linear interaction between dark energy and dark matter. We discuss the dark energy models using Tsallis, Rényi, and Sharma-Mittal entropies in the framework of Chern-Simons modified gravity. We explore various cosmological parameters (equation of state parameter, squared sound of speed ) and cosmological plane (ωd-ωd′, where ωd′ is the evolutionary equation of state parameter). It is observed that the equation of state parameter gives quintessence-like nature of the universe in most of the cases. Also, the squared speed of sound shows stability of Tsallis and Rényi dark energy model but unstable behavior for Sharma-Mittal dark energy model. The ωd-ωd′ plane represents the thawing region for all dark energy models.
Research Institute for Astronomy & Astrophysics of Maragha1/5237-81. Introduction
In the last few years, remarkable progress has been achieved in the understanding of the universe expansion. It has been approved by current observational data that the universe undergoes an accelerated expansion. The observations of type Ia Supernovae (SNeIa)[1–4], large scale structure (LSS) [5–8], and Cosmic Microwave Background Radiation (CMBR) [9, 10] determined that the expansion of the universe is currently accelerating. There is also consensus that this acceleration is generally believed to be caused by a mysterious form of energy or exotic matter with negative pressure so called dark energy (DE) [11–21].
The discovery of accelerating expansion of the universe is a milestone for cosmology. It is considered that 95% of our universe is composed of two components, that is DE and dark matter [16]. The dark matter constitutes about 25% of the total energy density of the universe. The existence of the universe is proved by astrophysical observation but the nature of dark matter is still unknown. Mainly the DE is also a curious component of our universe. It is responsible for current accelerating universe and DE is entirely different from baryonic matter. DE constitutes almost 70% of the total energy density of our universe.
In order to describe the accelerated expansion phenomenon, two different approaches have been adopted. One is the proposal of various dynamical DE models such as family of Chaplygin gas, holographic dark energy, quintessence, K-essence, and ghost [16]. A second approach for understanding this strange component of the universe is modifying the standard theories of gravity, namely, general relativity (GR). Several modified theories of gravity are f(R),f(T) [17], f(R,T) [18], and f(G) [19], where R is the curvature scalar, T denotes the torsion scalar, T is the trace of the energy-momentum tensor, and G is the invariant of Gauss-Bonnet.
Holographic DE (HDE) model is favorable technique to solve DE mystery which has attracted much attention and is based upon the holographic principle that states the number of degrees of freedom of a system scales with its area instead of its volume. In fact, HDE relates the energy density of quantum fields in vacuum (as the DE candidate) to the infrared and ultraviolet cut-offs. In addition, HDE is an interesting effort in exploring the nature of DE in the framework of quantum gravity. Cohen et al. [22] reported that the construction of HDE density is based on the relation with the vacuum energy of the system whose maximum amount should not exceed the black hole mass. Cosmological consequences of some HDE models in the dynamical Chern-Simons framework, as a modified gravity theory, can be found in [23].
By considering the long term gravity with the nature of spacetime, different entropy formalism has been used to observe the gravitational and cosmological effects [24–29]. The HDE models such as Tsallis HDE (THDE) [27], Rényi HDE model (RHDE) [28], and Sharma-Mittal HDE (SMHDE) [29] have been recently proposed. In the standard cosmology framework and from the classical stability view of point, while THDE is not stable [27], RHDE is stable during the cosmic evolution [28] and SMHDE is stable only whenever it becomes dominant in the world [29]. In the present work, we use the Tsallis, Sharma-Mittal, and Rényi entropies in the framework of dynamical Chern-Simons modified gravity and consider an interaction term. We investigate the different cosmological parameters such as equation of state parameter and the cosmological ωd-ωd′ plane where ωd′ shows the evaluation with respect to lna. We also investigate the squared sound speed of the HDE model to check the stability and the graphical approach.
This paper is organized as follows. In Section 2, we provide the basics of Chern-Simons modified gravity. In Section 3, we observe the equation of state parameter (EoS), cosmological plane, and squared sound speed for THDE model. Sections 4 and 5 are devoted to finding the cosmological parameter, cosmological plane, and squared of sound speed for RHDE and SMHDE models, respectively. In the last section, we conclude the results.
2. Dynamical Chern-Simons Modified Gravity
In this section, we give a review of dynamical Chern-Simons modified gravity. The action which describes the Chern-Simons modified gravity is given as(1)S=116πG∫νd4x-gR+l4θR∗ρσμνRρσμν-12gμν∇μθ∇νθ+Vθ+Smat,where R represents the Ricci scalar, R∗ρσμνRρσμν is a topological invariant called the Pontryagin term, l is a coupling constant, θ shows the dynamical variable, Smat represents the action of matter, and V(θ) is the potential term. In the case of string theory, we use V(θ)=0. By varying the action equation with respect to gμν and the scalar field θ, we get the following field equations:(2)Gμν+lCμν=8πGTμν,gμν∇μ∇νθ=-l64πR∗ρσμνRρσμν.Here, Gμν and Cμν are Einstein tensor and Cotton tensor, respectively. The Cotton tensor Cμν is defined as(3)Cμν=-12-g∇ρθερβτ(μ∇τRβν)+∇σ∇ρθR∗ρμνσ.The energy-momentum tensor is given by(4)T^μνθ=∇μθ∇νθ-12gμν∇ρθ∇ρθ,Tμν=ρ+puμuν+pgμν,where Tμν shows the matter contribution and T^μνθ represents the scalar field contribution, while P and ρ represent the pressure and energy density, respectively. Furthermore, uμ=(1,0,0,0) is the four velocity. In the framework of Chern-Simons gravity, we get the following Friedmann equation:(5)H2=13ρm+ρd+16θ˙2,where H=a˙/a is the Hubble parameter and the dot represents the derivative of a with respect to t and 8πG=1. For FRW spacetime, the pony trying term RR∗ vanishes identically; therefore, the scalar field in (2) takes the following form:(6)gμν∇μ∇νθ=gμν∂ν∂μθ=0. We set θ=θ(t) and get the following equation:(7)θ¨+3Hθ˙=0,which implies that θ˙=ba-3, b is a constant of integration. Using this result in (5), we have(8)H2=13ρm+ρd+16b2a-6.
We consider the interacting scenario between DE and dark matter and thus equation of continuity turns to the following equations:(9)ρ˙m+3Hρm=Q,(10)ρ˙d+3Hρd+pd=-Q.Here, ρd is the energy density of the DE, ρm is the energy density of the pressureless matter, and Q is the interaction term. Basically, Q represents the rate of energy exchange between DE and dark matter. If Q>0, it shows that energy is being transferred from DE to the dark matter. For Q<0, the energy is being transferred from dark matter to the DE. We consider a specific form of interaction which is defined as Q=3Hd2ρm and d2 is interacting parameter which shows the energy transfers between CDM and DE. If we take d=0, then it shows that each component, that is, the nonrelativistic matter and DE, is self-conserved. Using the value of Q in (9) we have(11)ρm=ρm0a-31-d2,where ρm0 is an integration constant. Hence, (10) finally leads to the expression for pressure as follows:(12)pd=-d2ρm+ρd+ρ˙d3H,
The EoS parameter is used to categorize the decelerated and accelerated phases of the universe. This parameter is defined as(13)ω=pρ.If we take ω=0, it corresponds to nonrelativistic matter and the decelerated phase of the universe involves radiation era 0<ω<1/3. ω=-1,-1<ω<-1/3, and ω<-1 correspond to the cosmological constant, quintessence, and phantom eras respectively. To analyze the dynamical properties of the DE models, we use ω-ω′plane_ [30]. This plane describes the evolutionary universe with two different cases, freezing region and thawing region. In the freezing region the values of EoS parameter and evolutionary parameter are negative (ω<0 and ω′<0), while for the thawing region, the value of EoS parameter is negative and evolutionary parameter is positive (ω<0 and ω′>0). In order to check the stability of the DE models, we need to evaluate the squared sound speed which is given by(14)vs2=dpdρ=dp/dtdρ/dt.The sign of vs2 decides its stability of DE models, when vs2>0, the model is stable; otherwise, it is unstable.
3. Tsallis Holographic Dark Energy
The definition and derivation of standard HDE density are given by ρd=3c2mp2/L2, where mp2 represents reduced Plank mass and L denotes the infrared cut-off. It depends upon the entropy area relationship of black holes, i.e., S~A~L2, where A=4πL2 represents the area of the horizon. Tsallis and Cirto [31] showed that the horizon entropy of the black hole can be modified as(15)Sδ=γAδ,where δ is the nonadditivity parameter and γ is an unknown constant [31]. Cohen at al. [22] proposed the mutual relationship between IR (L) cut-off, system entropy (S), and UV (Λ) cut-off as(16)L3Λ3≤S3/4.After combining (15) and (16), we get the following relation:(17)Λ4≤γ4πδL2δ-4,where Λ4 is vacuum energy density and ρd~Λ4. So, the Tsallis HDE density [29] is given as(18)ρd=BL2δ-4.Here, B is an unknown parameter and IR cut-off is taken as Hubble radius which leads to L=1/H, where H is Hubble parameter. The density of Tsallis HDE model along with its derivative by using (18) becomes(19)ρd=BH4-2δ,ρ˙d=B4-2δH3-2δH˙.Here, H˙ is the derivative of Hubble parameter w.r.t. t. The value of H˙ is calculated in terms of z using a=1/1+z which is given as follows.(20)dHdz=1/2ρm01-d21+z31-d2+b21+z61-1/3B4-2δH3-2δH1+zInserting these values in (12) yields(21)pd=13-3d2ρm0a-31-d2-BH2-2δ3H2+4-2δH˙. The EoS is obtained from (13):(22)ωd=pdρd=-1-d2ρm0a-31-d2H2δ-4B+2δ-4H˙3H2.The plot of ωd versus z is shown in Figure 1. In this parameter and further results, the function H(z) is being utilized numerically. The other constant parameters are mentioned in Figure 1. The trajectory of EoS parameter remains in quintessence region at early, present, and latter epoch.
Plot of ωd versus z for THDE model where δ=1.1, ρm0=1, d2=0.001, B=-1.3, b=0.5.
The square of the sound speed is given by(23)vs2=16Bδ-2a4H3H˙9d2d2-1ρm0a3d2H2δa˙-2Bδ-2a4H×3H2H˙-2δ-1H˙2+HH¨.
The plot of squared sound speed versus z is shown in Figure 2 for different parametric values. This graph is used to analyze the stability of this model. We can see that vs2>0, for -0.6<z<1 which corresponds to the stability of THDE model. However, the model shows instability for z<-0.6.
Plot of vs2 versus z for THDE model where δ=1.1, ρm0=0.8, d2=0.001, B=-1.3, b=0.5.
Taking the derivative of the EoS parameter with respect to lna, we get ωd′ as follows.(24)ωd′=13Ba4H6-3d2ρm03d2H2δ3d2-1Ha˙+2δ-4H˙+2Bδ-2×a4H2-2H˙2+HH¨The graph of ωd versus ωd′ is shown in Figure 3, for which ωd′ depicts positive behavior. Hence, for ωd<0, the evolution parameter shows ωd′>0, which represents the thawing region of evolving universe.
Plot of ωd versus ωd′ for THDE model where δ=1.1, ρm0=1, d2=0.001, B=-1.3, b=0.5.
4. Rényi Holographic Dark Energy Model
We consider a system with W states with probability of getting ith state Pi and satisfying the condition Σi=1WPi=1. Rényi and Tsallis entropies are defined as(25)S=1δlnΣi=1WPi1-δ,ST=1δΣi=1WPi1-δ-Pi,where δ≡1-U, where U is a real parameter. Now, combining the above equations, we find their mutual relation given as(26)S=1δln1+δST.This equation shows that S belongs to the class of most general entropy functions of homogenous system. Recently, it has been observed that Bekenstein entropy, S=A/4, is in fact Tsallis entropy which gives the expression(27)S=1δln1+δA4,which is the Rényi entropy of the system. Now for the RHDE, we focus on WMAP data for flat universe. Using the assumption ρddv∝Tds, we can get RHDE density(28)ρd=3C2H28π1+δπ/H2.Considering the term 8π=1 and substituting in (28), we get the expression for density as(29)ρd=3C2H21+δπ/H2.Now, dH/dz is given by the following.(30)dHdz=1/2ρm01-d21+z31-d2+b21+z61-2c2H2z2+δπ-c2H4/H2+δπ2H1+zThe pressure for this case is obtained as(31)pd=-d2ρm0a-31-d2+c2H2-3H2πδ+H2-22πδ+H2H˙πδ+H22.The expressions for EoS parameter ωd can be evaluated from (12) as follows:(32)ωd=πδ+H2-d2ρm0a-31-d23c2H4-3H2πδ+H2+22πδ+H2H˙3H2πδ+H22.
Figure 4 shows the plot of ωd versus z. The trajectory of EoS parameter evolutes the universe from quintessence region towards the ΛCDM limit. The squared sound speed of this RHDE model is given by using (13) as(33)vs2=3H1-d2d2ρm0a-31-d2πδ+H226c2H32πδ+H2H˙-13H22πδ+H2πδ+H2×H˙6π2δ2H2+9πδH4+3H6+4π2δ2H˙+HH¨πδ+H2×2πδ+H2.
Plot of ωd versus z for RHDE model where δ=1.1, ρm0=0.8, d2=0.001, c=0.1, b=0.05.
The graph of squared speed of sound is shown in Figure 5 versus z. In this case, we have vs2>0 for all ranges of z, which shows the stability of RHDE model at the early, present, and latter epoch of the universe.
Plot of vs2 versus z, for RHDE model where δ=1.1, ρm0=0.8, d2=0.001, c=0.1, b=1.5.
The expression for ωd′ is evaluated as(34)ωd′=13c2a4H6πδ+H22-d2ρm0a3d2πδ+H223Ha˙-1+d2×πδ+H2-2aH˙2πδ+H2+2c2a4H24π2δ2+8πδH2+2H4H˙2-2Hπδ+H22πδ+H2H¨.In Figure 6, we plot the EoS parameter with its evolution parameter to discuss ωd-ωd′ plane for RHDE model. The graph shows that, for ωd<0, the evolutionary parameter remains positive at the early, present, and latter epoch. This type of behavior depicts the thawing region of the evolving universe.
Plot of ωd versus ωd′ for RHDE model where δ=1.1, ρm0=0.8, d2=0.001, c=0.1, b=0.05.
5. Sharma-Mittal Holographic Dark Energy Model
From the Rényi entropy, we have the generalized entropy content of the system. Using (26), Sharma-Mittal introduced a two-parametric entropy which is defined as(35)SSM=11-rΣi=1WPi1-δ1-r/δ-1,where r is a new free parameter. We can observe that Rényi and Tsallis entropies can be recovered at the proper limits; using (25) in (35), we have(36)SSM=1R1+δSTR/δ-1,where R≡1-r. Using the argument that Bekenstein entropy is the proper candidate for Tsallis entropy by using S=A/4, where A is horizon entropy, we get the following expression:(37)SSM=1R1+δA4R/δ-1,and the relation of UV (Λ) cut-off, IR (L) cut-off, and system horizon (S) is given as follows.(38)Λ4∝SL4
Now, taking L≡1/H=A/4π, then the energy density of DE given by Sharma-Mittal [29] is considered as(39)ρd=3c2H48πR1+δπH2R/δ-1,where c2 is an unknown free parameter. Using 8π=1 in above equation, we get the following expression for energy density(40)ρd=3c2H4R1+δπH2R/δ-1.The differential equation of H is given by the following.(41)dHdz=1/2ρm01-d21+z31-d2+b21+z61+c2π1+δπ/H2R/δ-1-2c2H2/R1+δπ/H2R/δ-1H1+zThe pressure can be evaluated by energy conservation (11) as follows:(42)pd=-d2ρm0a-31-d2-c231+πδ/H2R/δ-1H4R-2πH˙1+πδH2R/δ-1+41+πδ/H2R/δ-1H2H˙R.The EoS parameter for this model is given by(43)ωd=2c2π1+πδH2R/δ-1-2H2H˙R1+πδH2-1R/δ-d2Rρm0a-31-d23c2H41+δπ/H2R/δ-1-1.The plot of ωd versus z is shown in Figure 7. The EoS parameter represents the quintessence nature of the universe. The square of the sound speed is evaluated as(44)vs2=16c2HH˙-π1+πδ/H2R/δ-1+2H2/R1+πδ/H2R/δ-1×-3d2H-1+d2ρm0a-31-d2+2c2HR6H2H˙+4H˙2+2HH¨-1πδ+H221+πδH2R/δ3H2H˙πδ+H2-πR+2πδ+2H2+2H˙π2R-2δR-δ-2H˙2πR-2δH2+2H4+HH˙πδ+H2×-πR+2πδ+2H2H¨.
Plot of ωd versus z for SMHDE where δ=1.1, ρm0=0.01, d2=0.001, c=0.01, b=0.4, R=7.
In Figure 8, we draw vs2 versus z which shows the unstable behavior of the SMHDE model as vs2<0 at early, present, and latter epoch.(45)ωd′=-131+πδ/H2R/δ-12H61πδ+H222H22-2πδ+H22+1+πδH22R/δπ2R-2δδ+2πR-2δH2-2H4+1+πδH2R/δ×-π2R2+Rδ-4δ2-2πR-4δH2+4H4H˙2+πδ+H2×1+πδH2R/δ-1H-2πδ+H2+1+πδH2R/δ×-πR+2πδ+2H2H¨+3d2-1+d2c2ρm0Ra-31-d2H2×1+πδH2R/δ-1+2d2ρm0Ra-31-d2c2πδ+H2πR-2δ-2H2×1+πδH2R/δ+2πδ+H2H˙
Plot of vs2 versus z for SMHDE where δ=1.1, ρm0=0.8, d2=0.001, c=0.8, b=0.05, R=7.
Figure 9 shows the plot of ωd-ωd′ plane to classify the dynamical region for the given model. We can see that ωd′>0 for ωd<0, which indicates the thawing region of the universe.
Plot of ωd versus ωd′ for different values of δ for SMHDE where δ=1.1, ρm0=0.01, d2=0.001, c=0.01, b=0.4, R=7.
6. Conclusion
In this paper, we have discussed the THDE, RHDE, and SMHDE models in the framework of Chern-Simons modified theory of gravity. We have taken the flat FRW universe, and linear interaction term is chosen for the interacting scenario between DE and dark matter. We have evaluated the different cosmological parameters (equation of state parameter and squared sound speed), ωd-ωd′ cosmological plane. The trajectories of all these models have been plotted with different constant parametric values.
We have summarized our results in Table 1.
Summary of the cosmological parameters and plane.
DE models
ωd
vs2
ωd-ωd′
THDE
quintessence-to-vacuum
partially stability
thawing region
RHDE
quintessence-to-vacuum
stability
thawing region
SMHDE
quintessence
un-stable
thawing region
Jawad et al. [32] have explored various cosmological parameters (equation of state, squared speed of sound, Om-diagnostic) and cosmological planes in the framework of dynamical Chern-Simons modified gravity with the new holographic dark energy model. They observed that the equation of state parameter gives consistent ranges by using different observational schemes. They also found that the squared speed of sound shows a stable solution. They suggested that the results of cosmological parameters show consistency with recent observational data. Jawad et al. [33] have also considered the power law and the entropy corrected HDE models with Hubble horizon in the dynamical Chern-Simons modified gravity. They have also explored various cosmological parameters and planes and found consistent results with observational data. Nadeem et al. [34] have also investigated the interacting modified QCD ghost DE and generalized ghost pilgrim DE with cold dark matter in the framework of dynamical Chern-Simons modified gravity. It is found that the results of cosmological parameters as well as planes explain the accelerated expansion of the universe and are compatible with observational data.
However, the present work is different from the above-mentioned works in which we have recently proposed DE models along with nonlinear interaction term and found interesting and compatible results regarding current accelerated expansion of the universe.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work of H. Moradpour has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) under research project No. 1/5237-8.
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