Effects of Anisotropy on Scalar Field Ghost Dark Energy and the Non-Equilibrium Thermodynamics in Fractal Cosmology

Transcription

1 Commun. Theor. Phys ) Vol. 68, No., October 1, 2017 Effects of Anisotropy on Scalar Field Ghost Dark Energy and the Non-Equilibrium Thermodynamics in Fractal Cosmology A. Najafi 1, and H. Hossienkhani 2, 1 Young Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iran 2 Department of Physics, Hamedan Branch, Islamic Azad University, Hamedan, Iran Received February 13, 2017; revised manuscript received July 10, 2017) Abstract Since the fractal cosmology has been created in early universe, therefore their models were mostly isotropic. The majority of previous studies had been based on FRW universe, while in the early universe, the best model for describing fractal cosmology is actually the anisotropic universe. Therefore in this work, by assuming the anisotropic universe, the cosmological implications of ghost and generalized ghost dark energy models with dark matter in fractal cosmology has been discussed. Moreover, the different kinds of dark energy models such as quintessence and tachyon field, with the generalized ghost dark energy in fractal universe has been investigated. In addition, we have reconstructed the Hubble parameter, H, the energy density, ρ, the deceleration parameter, q, the equations of state parameter, ω D, for both ghost and generalized ghost dark energy models. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field according to the evolution of generalized ghost dark energy density. Eventually, thermodynamics of the cosmological apparent horizon in fractal cosmology was investigated and the validity of the Generalized second law of thermodynamics GSLT) have been examined in an anisotropic universe. The results show the influence of the anisotropy on the GSLT of thermodynamics in a fractal cosmology. PACS numbers: Sf, d, x DOI: / /68//553 Key words: anisotropic universe, fractal cosmology, ghost DE, generalized ghost DE, non-equilibrium thermodynamics, generalized second law najafi.amin@znu.ac.ir hossienhossienkhani@yahoo.com c 2017 Chinese Physical Society and IOP Publishing Ltd 1 Introduction Some recent observations on the accelerated expansion behaviour of the universe have been studied. [1 2] Many researchers believe that dark energy DE), which is characterized by negative pressure, may be the main candidate for the cosmic acceleration. [3 6] Recently many DE models have been investigated. Among various models, the new model of DE called Veneziano ghost dark energy GDE) [7 8] was supposed to exist to solve the U1) problem in low-energy effective theory of QCD, [9 11] but it is completely decoupled from the physical sector. [12] One clarification is that there are some DE models where the ghost plays the role of DE see, e.g., Ref. [13]) and becomes a real propagating physical degree of freedom subjected to some severe constraints. In Ref. [1], the author discussed that the contribution of the Veneziano QCD ghost field to the vacuum energy is not exactly of order H and a subleading term H 2 appears because the vacuum expectation value of the energy-momentum tensor is conserved in isolation. [15] It was argued that the vacuum energy of the ghost field can be written as H + OH 2 ), where the subleading term H 2 in the GDE model might play a crucial role in the early stage of the universe evolution, acting as the early DE. [16] The GDE model in the framework of Brans Dicke cosmology and its stability against perturbations was investigated in Ref. [17]. In recent years the correspondence between DE model and scalar field models has been widely extended by many authors. [18] These connections motivate us to reconstruct scalar field and potential from GGDE model of fractal cosmology. The pioneer study on the deep connection between gravity and thermodynamics was done by Jacobson [19] who showed that the gravitational Einstein equation can be derived from the relation between the horizon area and entropy, together with the Clausius relation δq = T δs. Further studies on the connection between gravity and thermodynamics have been investigated in various gravity theories. [20 22] In the cosmological context, attempts to disclose the connection between Einstein gravity and thermodynamics were carried out in Refs. [23 27]. It was shown that the differential form of the Friedmann equation in the Friedmann Robertson Walker FRW) universe can be written in the form of the first law of thermodynamics on the apparent horizon. [28 32] Another theoretical approach was made by Calcagni [33 3] for a power-counting renormalizable field theory living in a fractal space-time and consequently fractal cosmology was developed. Historically the first appearance of fractal cosmology was in Andrei Linde s paper. [35] For an overview of fractal cosmology one can see Ref. [36]. The

2 55 Communications in Theoretical Physics Vol. 68 action in this model is Lorentz covariant and the metric space-time M, ϱ), which is equipped with a Stieltjes measure ϱ. Recently, it was shown that the Friedmann equations in a fractal cosmology can be transformed to the Clausius relation, but a treatment with non-equilibrium thermodynamics of space-time is needed. [37] Thermodynamic properties of the apparent horizon in fractal universe was investigated. [38] The relation between the DE and DM in FRW fractal universe have been investigated by authors in Refs. [39 0]. It were found that some large-angle anomalies appear in CMB radiations which violate the statistical isotropy of the universe. [1] Plane Bianchi models which are homogeneous but not necessarily isotropic) seem to be the most promising explanation of these anomalies. Jaffe, et al. [2] investigated that removing a Bianchi component from the WMAP data could account for several large-angle anomalies leaving the universe to be isotropic. Some Bianchi type DE models in general relativity have been discussed by researchers. [3 ] Singh and Agrawal [5] studied some Bianchi type III cosmological models in scalar tensor theory. More details about the Bianchi models were studied by many authors. [6 8] So, the contribution of the fractal component is significantly observed only in the early stages of the evolution of the universe. Thus, it would be worthwhile to explore DE models in the context of fractal cosmology in Bianchi type I BI) models. The present paper is organized as follows: at first, we introduce the basic equations in the framework of fractal cosmology in BI models. In Sec. 3, we present the DE content of a universe governed by fractal gravity with BI models. We reconstruct the potential and the dynamics of the fractal scalar field according to the results we obtained for the GGDE model. We can establish a correspondence between our model and the quintessence and tachyon scalar field models. In Sec., we review the nonequilibrium properties of fractal universe in BI model and we investigate the validity of the GSL of thermodynamics for this scenario. The numerical results are discussed and concluding remarks are summarized in Sec Metric and Fractal Cosmology in a BI Model The fractal properties of quantum gravity theories in n dimensions have been explored in several contexts. Assuming that matter is minimally coupled with gravity, the total action of Einstein gravity in a fractal space-time is given by [33 3] S = S g + S m, 1) where S g is S g = M p 2 dϱx) gr 2Λ η µ v µ v), 2) 2 M S m = dϱx) gl m, 3) is the matter action. Here g is the determinant of the metric tensor g µν, Mp 2 = 8πG is reduced Planck mass, R is Ricci scalar, Λ is the bare cosmological constant, and the term proportional to η has been added because v, similar the other geometric field g µν, is now dynamical. Note that ϱ is Lebesgue Stieltjes measure generalizing the standard D-dimensional measure d D x. So, ϱ is absolutely continuous, it can be written as dϱ = d D x)vx). The scaling dimension of ϱ is [ϱ] = Dα D, where α > 0 is a positive parameter. In what follows we shall find exact solutions of BI spacetime in fractal universe. Highly irregular sets with scale-dependent dimension do not satisfy the absolute continuity hypothesis, and they would be excluded a priori if one had defined the model starting from dϱ = d D x)vx) rather than dϱ = d D x. This is why a general formulation in terms of a Lebesgue Stieltjes action is preferable over a Lebesgue action with weight v. We can choose vx) = x Dα 1) + M D1 α), ) where M is a constant mass, and we expect M to be about the Planck mass and X = t or X = x). The metric g µν and the scalar v are independent degrees of freedom, which constitute the composite geometric structure metric and fractal) of M. Equation ) is inspired by results in classical mechanics, according to which integrals on fractals can be approximated by Weyl or fractional integrals which, in turn, are particular Lebesgue Stieltjes integrals. At small space-time scales, the weight v vx) x Dα 1) dominates over the constant term, while at large scales or late times it is negligible. This is true simply by construction, and independently from renormalization issues. Therefore, at least the phenomenological effectiveness of the model is guaranteed. In the coordinates and positive semi-definite is vx) = D 1 i=0 v i x i ), v i x i ) 0. 5) Using the standard Lebesgue measure dϱ = d D x)qx), one can formally re-express the measure as q i x i ) : = x i dx i v i x i ). 6) These distributions, which we call geometric coordinates, have anomalous scaling qλx) = fλ, x) under a dilation x λx, contrary to the standard Lebesgue measure where d D λx) = λ D d D x. The line element of BI spacetime is given by ds 2 = dt 2 A 2 dx 2 B 2 dy 2 C 2 dz 2, 7) where A, B, and C are cosmic scale factors. This metric is the simplest anisotropic and homogeneous cosmological model, which, upon having equal scale factors becomes the flat Friedmann Robertson Walker FRW) metric. Such space-times have an Abelian symmetry group of translations with Killing vector fields ξ = x, y, z ). All

3 No. Communications in Theoretical Physics 555 the structural constants of such a symmetry group are of course zero. To find the equations of motion we need the variations δ g = 1 2 g µν gδg µν, δγ ρ µν = 1 2 gρτ µ δg ντ + ν δg µτ τ δg µν ), δr µν = ξ δγ ξ µν ν δγ ξ ξν, δr = R µν + g µν µ ν )δg µν, 8) where ν U µ = ν U µ Γ ξ µνu ξ is the covariant derivative of a vector U µ. Using Eqs. ), 5), 6), and 8) and with the help of dϱ = d D x)vx), we can obtain the Einstein equation δs/δg µν = 0 as R µν 1 2 g v µνr 2Λ) + g µν v µ ν v v 1 ] +η[ 2 g µν ξ v ξ v µ v ν v = 8πGT µν, 9) where T µν = 2 g δ v S m δ v g µν = 2 gv δs m δg µν. The 00 and ii components of the Einstein equation 9) give ȦḂ AB + AĊ AC + ḂĊ BC + A A + Ḃ B + Ċ ) v C v 1 2 η v2 = 8πGρ + Λ, 10) 2 Ä 3 A + B B + C ) 1 ȦḂ C 3 AB + AĊ AC + ḂĊ ) + v BC v + 1 A 3 A + Ḃ B + Ċ ) v C v 1 2 η v2 = 8πGp Λ, 11) where ρ and p are the total energy density and pressure of the ideal fluid composing the universe, respectively. We evaluate the quantity v by simple tensorial calculation as v = v A/A + Ḃ/B + Ċ/C) v. The corresponding average scale factor a, the Ricci scalar R, the mean Hubble parameter H, and the traceless shear tensor σ in BI universe are [9] a = ABC) 1/3, 12) R = 2 3Ḣ + 2H2 ) + σ 2), 13) H = 1 Ȧ 3 A + Ḃ B + Ċ ), C 1) 2σ 2 A ) 2 Ḃ ) 2 Ċ ) 2 = + + 3H 2, A B C 15) where σ 2 = 1/2σ ij σ ij in which σ ij = u i,j u i;ku k u j + u j;k u k u i ) θg ij + u i u j ) is the shear tensor, which describes the rate of distortion of the matter flow and that matter is a perfect fluid with energy-momentum tensor T µν = ρ + p)u µ u ν pδ ν µ and u is the fluid relativistic velocity g µν u µ u ν = 1). Thus, the BI equations can be re-written in form 3H 2 σ 2 + 3H v v 1 2 η v2 = 8πGρ + Λ, 16) 2Ḣ 3H2 σ 2 + v v + H v v 1 2 η v2 = 8πGp Λ, 17) then the above equations take the following form Ḣ + H σ2 + v 2v + H 2 v v η v2 = πg 3 ρ + 3p) + Λ 3. 18) Now, by taking the variation of the action 1) with respect to the scalar v, we get R 2Λ = 16πGL m η2v v + µ v µ v). 19) Comparing Eqs. 18), 19), and using Eqs. 12), 13), 1), and 15), we can obtain 6Ḣ 12H2 2σ v + ηv v 3 v 2 ) v = 8πGρ + p). 20) Combining Eqs. 16), 17), and 20), the differential equation for the mean Hubble parameter has the form 2Ḣ 3H2 σ 2 v v 2H v v η v2 + ηv v = 0. 21) In order to obtain the BI equation in terms of the relative densities it is necessary to enter anisotropy energy density parameter the same way as Ω σ = σ 2 /3H 2. Its definition determines the continuity equation. [33] In fact, let δs m = 1 d xv gt µν δg µν + d x gl m δv. 22) 2 Using the properties of the Levi Civita connection, Γ α µν = 1/2)g αβ µ g νβ + ν g µβ β g µν ), and the definition of the covariant derivative of a rank-2 tensor, µ T µ ν = µ T µ ν + Γ µ µτ T τ ν Γ τ µνt µ τ = 1 g µ gt µ ν ) 1 2 νg µτ )T µτ. 23) Finally, it obtains the continuity equation µ vt µ ν ) ν vl m = 0. 2) The continuity equation 2), contracted with u µ, is [33 3] ρ + 3H + v ) ρ + p) = 0, 25) v where we used the definition of proper-time derivative, d/dt = u µ µ, T0 0 = ρ, and Ti i = p. We assume a power law form of the fractional function v is v = t β, where β = 1 α) is the fractal dimensional. Then BI equations 16) and 17) in the absence of the cosmological constant can be written as 3H 2 σ 2 3Hβ 1 t 2 ηβ2 t 21+β) = 8πGρ, 26)

4 556 Communications in Theoretical Physics Vol. 68 2Ḣ 3H2 σ 2 + 2βH t β1 + β) t ηβ2 t 21+β) = 8πGp, 27) again for Eq. 21) 2Ḣ 3H2 σ 2 + 2βH t β1 + β) t 2 ηβt 21+β) 1 + 2β 3Ht) = 0, 28) and the continuity equation ρ + 3H β ) ρ + p) = 0. 29) t Here we have β = 0 in the infrared regime), β = 2 in the ultraviolet regime). It is important to mention here that the mean Hubble parameter H in Eqs. 26), 27), 28), and 29) is given by Eq. 1). In the ultraviolet regime β = 2), the BI 26) and 27) yield 3H 2 σ 2 6H t 2Ḣ 3H2 σ 2 + H t 2ηt 6 = 8πGρ, 30) 6 t 2 2ηt 6 = 8πGp. 31) We define, as usual, the fractional energy densities such as Ω = ρ 3H 2, Ω σ = σ2 3H 2, Ω f = η ) Ht 3Ht 5, 32) where Ω f is the fractal relative density. Thus, the first BI equation 30) can be written 1 Ω σ = Ω f + Ω. 33) The above equation shows that the sum of the energy density parameters approaches 1 at late times. So at late times the universe becomes flat. Therefore for a sufficiently large period of time, this model predicts that the anisotropy of the universe will damp out and the universe will become isotropic. This result also shows that in the early time, i.e. during the radiation and matter dominated era, the universe is anisotropic and then it approaches to isotropy as its energy density starts to be dominated by DE. When the infrared regime and an isotropic universe is assumed, Ω m + Ω D = 1, then the model has only one free parameter, Ω D. The current best fit value from cosmological observations is Ω D = 0.73 ± 0.0 in the flat case. [50] The gravitational constraint, which is given by Eq. 28) in the ultraviolet regime yields 2Ḣ 3H2 σ 2 + H 6 t t 2 2η 5 3Ht) = 0. 3) t6 Solution of these equations gives us the following form Ht) = η Ω σ + 5Ω σ ) 1F Ωσ +5Ω σ 8 ; Ω σ ; 3η t ) 1 + Ω σ )t 5 1F Ω σ+5ω σ 8 ; Ω σ ; 3η t ) Ωσ 1 + Ω σ )t The expression for ρ as 8πGρ = 8η1 + Ω σ) Ω σ )1 Ω σ 81 + Ω σ )t ) 1 + Ω σ ) 2 t 6 η Ω σ + 5Ω σ ) 1F Ωσ+5Ω σ 8 ; Ω σ ; 3η t ) 21 + Ω σ )t 6 1F Ω σ +5Ω σ 8 ; Ω σ + 3η1 Ω σ) Ω σ + 5Ω σ ) 1F Ωσ+5Ω σ 8 ; Ω σ ; 3η t ) Ω σ ) 2 t 10 ; 3η t ). 35) 1F Ω σ +5Ω σ 8 ; Ω σ ; 3η t ) 2. 36) Fig. 1 a) Represents the evolution of the Hubble parameter Ht) as a function of time in fractal cosmology BI with Ω σ = 0.001, while b) represents Ht) for the different values of Ω σ.

5 No. Communications in Theoretical Physics 557 Fig. 2 The evolution of ρ as a function of t in fractal anisotropic universe with Ω σ = Fig. 3 The evolution of q = 1 Ḣ/H2 as a function of t in fractal anisotropic universe. We have taken Ω σ = and the FRW model. In Fig. 2, we observe that for both η = ±1, ρ decreases with a steep slope as a function of t. From Fig. 3 we see that q approaches to 1 at early times and then either decreases for η = 1) or increases for η = 1) with increasing t. This shows that for the case of η = 1 the accelerated expansion of universe presently dominated by DE is just a transient phenomenon. Reference [51] established on independent observational data, including SNe Ia, CMB and BAO. It was shown that the acceleration of universe expansion reached its maximum value, then it decreased. As we expected in Fig. 3 for the case η = 1, the universe at early times transit from an acceleration phase to the matter dominated epoch in the future. Moreover, we have a cosmic acceleration to deceleration phase at t = Finally we have considered, the model isotropize. The measure of the anisotropy is described by σ/h, describe the magnitude of the spacetime shear per the average expansion rate. However, as shown in Fig., during the inflationary phase, we find that the effect of shear cannot become as large as the Hubble parameter and the shear decreases at the end of the inflation. 3 Bianchi Type I and Dark Energy in Fractal Universe In this section, we consider a universe filled with the DE density ρ D and the pressureless DM p m = 0. By using of Eq. 29), the energy equations for DE and DM in the ultraviolet regime reduce to ρ D + 3H 2 t ) 1 + ω D )ρ D = 0, 37) ρ m + 3H 2 ) ρ m = 0, 38) t where ω D = p D /ρ D is the EoS parameter of the noninteracting DE. In the following, we mainly focus on two special forms of DE in BI models: the ghost DE and the generalized ghost DE. 3.1 Ghost Dark Energy The ghost energy density is proportional to the Hubble parameter [35] ρ D = αh, 39) Fig. The evolution of the anisotropy parameter σ/h with respect to the scale factor a in a fractal cosmology with σ 0 = [52] It increases during the anisotropic inflationary phase. Then it falls rapidly down to OσH). Time evolutions H, ρ, and q for both the η = 1 and η = 1 are plotted in Figs. 1, 2, and 3, respectively. In Fig. 1, we observe that for the case of η = 1, Ht) is a decreasing function of cosmic time t for fractal universe with BI while for η = 1 its H increases. This is an important feature that distinguishes between this model where α is a constant with dimension [energy] 3, and roughly of order of Λ 3 QCD where Λ QCD 100 MeV. Taking time derivative of Eq. 39) yields ρ D = Ḣ H ρ D. 0) Substituting the above result into Eq. 37), one can find the EoS parameter ω D of the GDE in BI universe: ω D = 1 + Ḣt 2H 3H 2 t, 1) where H is given by Eq. 35). It is found that ω D < 1 for the case of η = 1, which have phantom-like be-

6 558 Communications in Theoretical Physics Vol. 68 haviour, while η = 1 leads to ω D > 1, which corresponds to the quintessence phase, which is shown in Fig. 5. Consequently, this model does not favor the phenomenon of fractal cosmology. On the other hand, ω D at early time t 0) behaves like the cosmological constant, i.e. ω D = 1. Moreover, Fig. 5 shows that for a given time t, ω D decreases when the Ω σ increases. So anisotropy parameter increases the phantom phase. It can also be observed that the EoS parameter indicates the presence of less phantom energy at initial time and converges towards strong phantom region at late times. This behavior strongly favors the GDE phenomenon and also indicates that there is a possibility of Big-Rip singularity. We notice that in the general relativity v = β = 0), ω D a0) = 1) 0.78, which is 2σ off according to WMAP. [5] Fig. 5 Time evolution of EoS parameter of the fractal GDE in the BI universe against t for different values of the anisotropy energy density parameter Ω σ. a) Corresponds to η = 1 while b) corresponds to η = Generalized Ghost Dark Energy In the following, we present basic scenario of noninteracting generalized ghost dark energy GGDE) with cold dark matter CDM) in an anisotropic universe. It is proposed [16] that the contribution of the term H 2 in the ordinary GDE may be useful in describing the early evolution of the universe, which is defined as follows ρ D = αh + βh 2, 2) where β is another constant with dimension [energy] 2. In addition, for the dimensionless density parameter of the GGDE, one can obtain Ω D = ρ D = α + βh ρ cr 3Mp 2 H. 3) Differentiating Eq. 2), we perform ρ D = Ḣα + 2βH). ) Using Eqs. 37), 2), and ), we obtain the EoS parameter as ω D = 1 + Ḣt1 + 2ξH) 1 + ξh)2h 3H 2 t), 5) where ξ = β/α. Setting ξ = 0, ω D reduces to the respective relation in the absence of interaction obtained in Eq. 1). We have plotted ω D of GGDE versus t numerically for both the η = 1 and η = 1 by taking initial value of ξ = 0.5 as shown in Fig. 6. Fig. 6 Time evolution of EoS parameter of the fractal GGDE in the BI universe against t and different values of the anisotropy energy density parameter Ω σ, for H 0 = 72, ξ = 0.5. η = 1 a) and η = 1 b).

7 No. Communications in Theoretical Physics 559 Figure 6 shows that the EoS parameter of the GGDE behaves like the GDE model at early time see Fig. 5), which behaves like the cosmological constant and the slope is greater than the previous similar behaviour. From this figure we see that ω D of the fractal GGDE model cannot cross the phantom divide for the case of η = 1. But for the case of η = 1, it shows that the EoS parameter of the GGDE starts from 1 and then drops down. With the increase in anisotropy parameter, one can see the EoS parameter will increase towards to higher values in phantom region, which corresponds to ω D < 1. Moreover, the range of the present values of ω D is consistent with observations data. [55] It showed constraints on the EoS parameter, for example: ω D = from Planck+WP+BAO, ω D = from Planck+WP+Union 2.1 and ω D = from Planck+WP+SNLS. Now, we intend to classify our study into two types of the model: quintessence and tachyon models in a non-isotropic universe. 3.3 Quintessence Field Reconstruction of GGDE in Fractal Universe In the following we will construct the GGDE quintessence models, connecting the quintessence scalar field with the GGDE. The energy density and pressure of the quintessence DE model are given by [3,56] ρ ϕ = 1 2 ϕ 2 + V ϕ), 6) p ϕ = 1 2 ϕ 2 V ϕ). 7) Then, we can easily obtain the scalar potential and the kinetic energy term as V ϕ) = 1 ω ϕ ρ ϕ, 2 8) ϕ 2 = 1 + ω ϕ )ρ ϕ, 9) where ω ϕ = p ϕ /ρ ϕ. In order to implement the correspondence between GGDE and quintessence scaler field, we identify ρ ϕ = ρ D and ω ϕ = ω D. Inserting Eqs. 2) and 5) in Eqs. 8) and 9) we reach V ϕ) = αh1 + ξh) ϕ = Ḣαt1 + 2ξH) 22 3Ht), 50) Ḣαt1 + 2ξH) ) 1/2. 51) 22 3Ht) Using ϕ = Hϕ, one can rewrite Eq. 51) as H ϕ αt1 + 2ξH) ) 1/2 =. 52) 2H2 3Ht) Consequently, we can easily obtain the evolutionary form of the field by integrating the above equation obtained as a 1 H ϕa) ϕ1) = αt1 + 2ξH) da, 53) a 2H2 3Ht) 1 where we take a 0 = 1 for the present time and H is given by Eq. 35). For next calculations we assume that ξ = 0.1. In this way we establish a relation between the GGDE in BI and a quintessence field, then we reconstruct the potential and the dynamics of the quintessence field, which describe fractal cosmology. The evolution of the GGDE quintessence fractal scalar filed, Eq. 51), for four different values of Ω σ is plotted in Fig. 7. It illustrates that the ϕt) either decreases or increases with time t for the cases η = ±1, respectively. The curve is also shifted to the larger value of ϕt) with decreasing increasing) Ω σ for η = 1 η = 1), respectively, as it can be seen from the diagram. The evolution of the GGDE quintessence fractal potential, Eq. 50), with respect to the time for different values of the Ω σ is plotted in Fig. 8. It illustrates that for a given Ω σ, V ϕ) decreases with increasing t and the curve is shifted to the smaller values of V ϕ) with increasing decreasing) the Ω σ for η = 1 η = 1), respectively. Also, we see from these figures that the cosmic evolution trends are quite similar for these four anisotropy energy density parameters. Fig. 7 The figures indicate the quintessence scalar field of GGDE, Eq. 51), versus the time t for η = ±1 and ξ = 0.1.

8 560 Communications in Theoretical Physics Vol. 68 Reference [57] suggested the quintessence models can be divided into two types; thawing models and freezing models. Thawing models characterize scalar fields that evolve by ω D 1 while freezing models describe fields, which is evolved by ω D > 1. Thus both the GDE and GGDE quintessence of fractal cosmology, corresponding to case of η = 1, should be attributed to the thawing model, while the GDE quintessence of fractal cosmology, corresponding to case of η = 1, should be attributed to the freezing model. Fig. 8 The figures indicate the quintessence potential of GGDE, Eq. 50), versus the time t for ξ = Tachyon Field Reconstruction of GGDE in Fractal Universe The tachyon field is another approach for explaining DE. The tachyon energy density and pressure are [58 59] ρ T = T 0 0 = V ϕ) 1 ϕ 2, 5) p T The tachyon EoS parameter yields = Ti i = V ϕ) 1 ϕ 2. 55) ω T = ϕ ) To reconstruct the tachyon filed via the interacting GGDE, by Eq. 5) with Eq. 56), i.e. ω D = ω T, gives ϕ 2 = Ḣt1 + 2ξH) 1 + ξh)2h 3H 2 t). 57) Combining Eq. 5) with Eq. 57), the tachyon potential is obtained as Ḣt1 + 2ξH) ) 1/2 V ϕ) = αh1 + ξh) 1.58) 1 + ξh)2h 3H 2 t) Consequently, we can easily obtain the evolutionary form of the field by integrating Eq. 57) obtained as ϕa) ϕ1) = a 1 H t1 + 2ξH) ) 1/2 da H1 + ξh)2h 3H 2 t) a. 59) The GGDE tachyon potential and the evolution of the ghost tachyon scalar filed for different values of Ω σ is plotted in Figs. 9 and 10. It shows that the GGDE tachyon scalar field decreases increases) when the time increases for η = 1 η = 1), respectively. Fig. 9 The tachyon scalar field in terms of the time, with different values of Ω σ, and ξ = 0.1.

9 No. Communications in Theoretical Physics 561 Figure 9 also clears that for a given time, the GGDE tachyon scalar field decreases with increasing the anisotropy parameter Ω σ. In other words it shows that ϕt) goes down goes up) as the time increases for η = 1 η = 1), respectively, and the stronger the anisotropy parameter is, the slower the ϕt) changes as the time increases. In addition, for the case of η = 1 one can easily see that the corresponding evolutionary curves to different values of anisotropy parameter are overlapped. These show that the values Ω σ of anisotropy parameter nearly have no effect on the evolutionary trajectories of ϕt) when we take a fixed value of ξ. Note that the potential at early times shows a shape similar to an inverse powerlaw V ϕ) ϕ q with respect to the scalar field. Though the behavior have no effect on EoS parameter, the kind of potential was extremely considered in Ref. [60]. It has scaling solutions as well. Fig. 10 The tachyon potential of GGDE versus the time, with different values of Ω σ, and ξ = 0.1. Description of Non-Equilibrium Thermodynamics in Anisotropic Fractal Cosmology In this section, we review thermodynamics of the apparent horizon in fractal cosmology with anisotropic universe. It is suggested that according to the observational data of type Ia Supernovae, the GSL can be satisfied for the apparent horizon, and not for the event horizon. [61] The apparent horizon radius r A and the associated temperature T h for the BI universe is r A = 1 H, 60) T h = 1 2πr A, 61) where A = πr 2 is the apparent horizon area. [62] Considering the total energy E is given by E = ρv, where V = π/3r 3 A is the volume of the 3-sphere of radius r A. Because there is no change of the matter energy inside the apparent horizon, we have no term of volume change. Thus the total energy, E, can be expressed in terms of the energy density and volume as follows [38] de = V ρdt, 62) T h ds h = de. 63) Thus, we see that the first law of thermodynamics is satisfied on the apparent horizon. Considering Eqs. 60), 61), 62), 63), and ρ the continuity equation 29), we can get T h Ṡ h = πra 2 1 β ) ρ + p). 6) 3Ht Differentiating of BI Eqs. 26) and 60) with respect to the cosmic time t and using the continuity equation 29), one can obtain [ ṙ A r 3 A Hσ 2 + β H 2 t 2 Ḣ )] η t 6 β2 1 + β)rat 3 2β 3 = πgraρ 3 + p) H + β ). 65) 3t Combining Eq. 6) with Eq. 65) and using Eq. 60), one can obtain it as follows S h = 2π [ṙ A G 1 βr A 2t ) r 2 Aσ 2 β 2t 2 r2 A r3 A 6 β2 1 + β)ηt 2β 3] r A. 66) This is due to the non-equilibrium properties of fractal universe in BI model. For standard cosmology FRW) where σ = β = 0, we actually have no entropy production rate. [20] Next, we introduce the GSL of thermodynamics containing the matter field S in inside the apparent horizon, so the entropy of the universe inside the horizon can be related to its energy and pressure in the horizon by the Gibbs equation. [63] T in ds in = dρv ) + pdv = V dρ + ρ + p)dv. 67) Recall that in case of FRW k 0) as in our case apparent horizon r A = 1/ H 2 + k 2 /a 2.

10 562 Communications in Theoretical Physics Vol. 68 From Gibbs equation, using the continuity equation 29), one can conclude [ T Ṡin = πraρ 2 + p) ṙ A r A H) 1 β )]. 68) 3Ht Now, we should survey the evolution of the total entropy S h + S in. By using Eqs. 6) and 68), we obtain T h Ṡin + S h ) = πraṙ 2 A ρ + p) 1 β ), 69) 3Ht where A = πh 2 is the apparent horizon area. Using Eqs. 26), 27), 60), 61), and considering ṙ A = Ḣ/H2 in flat BI universe, we obtain: Ṡ tot = πḣ GH 5 1 β ) 3Ht 2Ḣ 2σ2 βh β1 + β) t t 2 ηβ 2 t 21+β)), 70) where H is given by Eq. 35). The above equation shows the effects of the anisotropy on GSL in a fractal universe. Due to complicated expressions of the time variation of the total entropy we cannot definitely conclude about validity of GSL. However, we have drawn some inferences only from graphical analysis considering β = 2 and η = ±1. From Fig. 11 we see that for η = 1, the GSL can be fulfilled and this indicates validity of GSL for this model, while for η = 1, Ṡtot is always negative. Indeed, we again obtain the result of a GSL of non-equilibrium thermodynamics in fractal cosmology with an anisotropic universe by applying a constraint on the fractal parameter η. It is also observed that for a given time t, Ṡ tot decreases with increasing Ω σ. Fig. 11 a) Corresponds to the time derivative of the total entropy against t for fractal anisotropic model with η = 1 while b) shows the time derivative of the total entropy for η = 1, considering Ω 0 D = 0.69, β = 2, and H 0 = Conclusion and Discussion In this work, using the fractal theory of gravity, we have investigated the different models of cosmology in an anisotropic universe. We have discussed main features by choosing different values of the fractal parameter, η, and different values of the anisotropy energy density parameter Ω σ which are summarized as follows. Firstly, we reconstructed the field equations of fractal cosmology in an anisotropic universe. In the limiting case, we give the Hubble parameter, the energy density, the pressure and the deceleration parameter representation by considering BI model. These results can be seen from Figs. 1, 2, and 3. According to this model, the evolution of the accelerated expansion of the universe is faster than that of the FRW model. Furthermore, we have studied the evolution of the σ/h as shown in Fig.. Although the shear increases during the anisotropy inflation, it can fall at the end of the inflation. Moreover, we have seen that the anisotropy of the universe σ/h can decrease to O1) at the end of the isotropic inflation. Secondly, under the reconstructed Hubble parameters in BI model, we have evaluated the EoS parameter for both the ghost and generalized ghost DE models, and plotted them. The computation was performed under the limits of ultraviolet regime β = 2. It is shown that the ω D for both the ghost and generalized ghost DE at early time t 0) behaves like the cosmological constant, i.e. ω D = 1, and we have observed that the ω D is decreasing with increasing Ω σ, so the anisotropy parameter increases the phantom phase in fractal cosmology as shown in Figs. 5 and 6. It can be seen that for η = 1, the ω D has phantom-like behaviour while for η = 1, GDE behaves like DM at present times. We also investigated the GGDE models in the form of a quintessence scalar field in the context of fractal cosmology in an anisotropic universe. By using the values of the constant parameters, which are shown in Figs. 7, 8, 9, and 10, we have explored the behavior of potentials and dynamics of the scalar field corresponding to quintessence and tachyon GGDE. These figures illustrated that the ϕt) decreases and increases

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