Ghost Quintessence in Fractal Gravity

PRAMANA c Indian Academy of Sciences Vol. 84, No. 4 — journal of April 2015 physics pp. 503–516

Ghost quintessence in fractal gravity

HABIB ABEDI1 and MUSTAFA SALTI2,∗ 1Department of Physics, University of Tehran, North Kargar Ave, Tehran, Iran 2Physics Department of Science Faculty, Dicle University, 21280, Diyarbakir Turkey ∗Corresponding author. E-mail: [email protected]

MS received 24 October 2013; revised 24 February 2014; accepted 13 May 2014 DOI: 10.1007/s12043-014-0850-2; ePublication: 15 November 2014

Abstract. In this study, using the time-like fractal theory of gravity, we mainly focus on the ghost dark energy model which was recently suggested to explain the present acceleration of the cosmic expansion. Next, we establish a connection between the quintessence scalar field and fractal ghost dark energy density. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field (the fractal quintessence) according to the evolution of ghost dark energy density.

Keywords. Ghost dark energy; quintessence; fractal gravity.

PACS Nos 04.50.+h; 95.35.+d

1. Introduction

Recent cosmological observations give very important evidences in favour of the present acceleration of cosmic expansion. It is commonly believed that our Universe has a phase transition [1] from decelerating phase to accelerating phase and expands with accelerat- ing velocity. To explain this acceleration, in the context of modern cosmology, we need an antigravity fluid with negative pressure [2]. This interesting feature of the Universe is caused by the mysterious dark components: dark energy, dark matter, and dark radiation. The cosmological constant, as vacuum energy density, is the best instrument to identify this nature of the Universe. This, with the equation-of-state parameter ω =−1, repre- sents the earliest and the simplest theoretical candidate for dark energy, but causes some constraints like fine-tuning and cosmic-coincidence puzzle [3]. The former cosmologists question why the vacuum energy density is so small, [4] and the latter ones explain why the vacuum energy and dark matter are nearly equal [5]. Furthermore, according to Type Ia supernovae observations, it is now known that the time-varying dark energy models give better fits compared with a cosmological constant [2] and in particular, the value of the equation-of-state parameter of dark energy (ωD) gives three different phases: vac- uum (ωD =−1), phantom (ωD < −1) and quintessence (ωD > −1). Also, many other

Pramana – J. Phys., Vol. 84, No. 4, April 2015 503 Habib Abedi and Mustafa Salti candidates (tachyon, K-essence, quintessence, dilaton, Chaplygin gas, modified gravity) have been proposed to explain the nature of dark energy [6,7], but yet the nature of dark Universe is completely unknown [8]. A good review about the dark energy problem, including a survey of some theoretical models, is given by Li et al in 2011 [9]. In previous works, a very interesting interpretation on the origin of dark energy is sug- gested, without new degrees of freedom, with the dark energy of the right magnitude to obtain the observed expansion [10–14]. Among various models, the new model of dark energy called Veneziano ghost dark energy [15] is supposed to solve the U(1)A problem in low-energy effective theory of QCD [16–18], but it is completely decoupled from the physical sector [19–21]. The Veneziano ghost field is unphysical in the quantum field the- ory in Minkowski space-time, but exhibits an important non-trivial physical influence in the expanding Universe and this remarkable effect gives rise to the vacuum energy density ∼ 3 ∼ −3 4 ∼ −33 ∼ ρD HQCD (10 eV) (with H 10 eV and QCD 100 eV we have the right magnitude for the force accelerating the Universe) [22]. This numerical coincidence is noteworthy and also means that the ghost energy model overcomes the problem of fine tuning. On the other hand, scalar fields can be regarded as an effective way of describing the dark Universe which naturally arise in particle physics, including the string/M theory and supersymmetric field theories. Hence scalar fields are expected to reveal the dynam- ical mechanism and the nature of the dark Universe [2]. Fundamental theories such as string/M theory provide many possible scalar field candidates, but they do not predict their potential V(φ)uniquely. In this work, we are interested to consider the ghost dark energy model as the under- lying theory of dark energy in time-like fractal gravity and how the low-energy effective scalar field model can be used to describe it. For this purpose, we reconstructed the poten- tial and the dynamics of the fractal quintessence according to the results obtained for the ghost dark energy. We can establish a correspondence between the ghost dark energy and quintessence scalar field in time-like fractal gravity, and describe ghost dark energy effectively by making use of quintessence.

2. Correspondence between ghost and quintessence

We assume the ghost dark energy is accommodated in a flat Friedmann–Robertson– Walker ds2 =−dt2 + a2(t) dr2 + r2(dθ 2 + sin2 θdφ2) . (1) Here a(t) is the cosmic scale factor and it measures the expansion of the Universe. In four-dimensional time-like fractal gravity, we have [23,24]

S = SG + Sm, (2) where 1 √ S = dξ −g[R − η∂ υ∂μυ], (3) G 2κ2 μ √ Sm = dξ −g£m, (4)

504 Pramana – J. Phys., Vol. 84, No. 4, April 2015 Ghost quintessence in fractal gravity where g, R and £m are the determinants of metric gμν , Ricci scalar and the matter part of total Lagrangian, respectively. Also, we have κ2 = 8πG, where G denotes the grav- itational constant. υ and η are two quantities known as the fractal function and fractal parameter, respectively. It is important to mention here that dξ(x) is the Lebesgue– Stieltjes measure generalizing the standard four-dimensional measure d4x. The dimension of ξ is [ξ]=−Dα, where α is a positive parameter. The theory of fractal gravity is power- counting, renormalizable, free from ultraviolet divergence and a Lorentz invariant [25]. Recently, Calcagni [23,24] worked on the quantum gravity in a fractal Universe and has discussed cosmology in that framework. Considering a time-like fractal profile [24] υ = t−β (where β = 4(1 − α) is the fractal dimension) in four-dimensional (D = 4) fractal gravity, we recover the following Friedmann equation: ηβ 2 1 H 2 − βHt−1 + = (ρ + ρ ), (5) 2(β+1) 2 G m 6t 3MP where ρG and ρm are the densities of the ghost dark energy and dark matter inside the Universe, respectively. Here, we assume a pressureless dark matter pm = 0, MP is the −2 = =˙ reduced Planck mass (MP 8πG) and H a/a is the Hubble parameter. Accordingly, it is known that β = 0 describes the infrared regime while β = 2 implies the ultraviolet regime. On the other hand, the continuity equation is written as ρ˙ + (3H − βt−1)(ρ + p) = 0, (6) where ρ and p are the total energy and pressure density, respectively. Nonetheless, the gravitational constraint [24] in a flat fractal Friedmann–Robertson– Walker space-time is 3η βH β(β + 1) ηβ(2β + 1) H˙ + 3H 2 + 2 + − − = 0. (7) t2β t t2 t2β+2 Note that in the infrared regime, eq. (5) gives the corresponding relation in Einstein’s theory of general relativity (there is no gravitational constraint). Also, the gravitational constraint in the ultraviolet regime becomes 3η 2H 6 10η H˙ + 3H 2 + 2 + − − = 0. (8) t4 t t2 t6 Solving this equation gives [24] 22η (15/4; 17/4; 3η/2t4) H(t) =−2t−1 − , (9) 13t5 (11/4; 13/4; 3η/2t4) 11 13 3η a3(t) = t−6 ; ; . (10) 4 4 2t4

Here  (also denoted as 1F1 or M) is Kummer’s confluent hypergeometric function of the first kind: +∞ (b) (a + n) xn (a; b; x) ≡ . (11) (a) (b + n) n! n=0

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Defining the following dimensionless density parameters, ρ  = G , (12) G 2 2 3H MP ρ  = m , (13) m 2 2 3HMP β H ηβ  = − , (14) f H 2 t 6t2(β+1) we can rewrite the fractal Friedmann equation as

1 = G + m + f. (15)

Here, the parameter f represents the fractal contribution to the total density. Moreover, the Friedman equation can also be rewritten in a very elegant form as i ≡ 1, (16) i=G,m,f where

i ≡ (G,f,m). (17)

2.1 Non-interacting case

For this case, the conservation equations read as β ρ˙ + 3H − ρ = 0, (18) m t m β ρ˙ + 3H − (1 + ω )ρ = 0, (19) G t G G where ωG = pG/ρG. The ghost energy density is proportional to the Hubble parame- ter [26]

ρG = λH. (20) 3 ∼ Here λ is a constant of order QCD and QCD 100 MeV is the QCD mass scale. Taking a time derivative in both sides of relation (20) and using Friedmann eq. (5), we obtain ηβ 2(β + 1) βH ρ ρ˙ = λ(2H −βt−1)−1 − − G (3H −βt−1)(1+ω +) , G 2β+3 2 2 G 3t t 3MP (21) where ρ  = m . (22) ρG Inserting this relation in continuity eq. (19) and using expressions of dimensionless density parameters, we find ηβ 2(β+1) − β − − β − − 3H 2t2β+3 Ht2 3H t (1 G f) ωG =−1 + . (23) − β − − β 3H t G 2 t

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In the fractal infrared regime, considering eq. (23), we get IR = − −1 ωG (G 2) . (24) → =−1 Here, one can easily see that at an early time (t 0) where G 1wehaveωG 2 , while at a later time (t →∞) where G → 1 the fractal ghost dark energy mimics a cosmological constant, i.e., ωG =−1. Furthermore, in the ultraviolet regime, we get 4η − 2 − − 2 − − UV 2 7 2 3H (1 G  ) ωUV =−1 + H t Ht t f , (25) G − 2 − − 2 3H t G 2 t where 2 H η UV = − , (26) f H 2 t 3t6

n 15 3η +∞ ( +n) 4 (17 )( 11 ) 4 2t 2 22η 4 4 n=0 ( 17 +n) n! 4 H(t) =− − m . (27) 5 3η t 13t +∞ 11 + 15 13 ( 4 m) 2t4 ( 4 )( 4 ) m=0 13 + m! ( 4 m) UV In figure 1, we plotted [27] the time evolution of the equation-of-state parameter ωG . UV From this figure we see that ωG of the fractal ghost dark energy model cannot cross the phantom divide. This figure also shows that, at a later time (t →∞), the equation-of-state UV =∼ 1 parameter of the ghost dark energy in the fractal ultraviolet regime yields ωG 3 , which UV implies the ultrarelativistic matter behaviour. On the other hand, ωG at an early time → UV = (t 0) behaves like the pressureless dark energy, i.e., ωG 0, where its equation-of- state behaves like the dust. Furthermore, it is known that the equation-of-state parameter needs to be less than −1/3, but we have positive values in the ultraviolet regime. In conclusion, one can say that, in the ultraviolet regime, the fractal ghost dark energy model

UV G 0.8

0.6

0.4

0.2

0.0 t 1 2 3 4 5 Figure 1. Time evolution of the equation-of-state parameter of fractal ghost dark energy in the ultraviolet regime, (eq (27)), for η = λ = 1andG = 0.72.

Pramana – J. Phys., Vol. 84, No. 4, April 2015 507 Habib Abedi and Mustafa Salti does not cross the phantom line. On the other hand, the recent astronomical data from Planck-2013 [28], SNe-Ia [29], WMAP [30], SDSS [31] and X-ray [32] strongly suggest that our Universe is spatially flat and dominated by an exotic component with negative pressure called the dark energy [4,6,33]. Hence, in the ultraviolet regime, the fractal ghost dark energy model does not give valid results. Now, we are in a position to establish the significance between the ghost dark energy and quintessence scalar field. First, we assume the quintessence scalar field model to be the effective underlying theory. The action for quintessence is defined as [3] 1 √ S =− d4x −g[gμν ∂ φ∂ φ + 2V(φ)], (28) Q 2 μ ν where V(φ)is the potential of quintessence. The quintessence field is defined by an ordi- nary time-dependent and homogeneous scalar which is minimally coupled to gravity, but with a particular potential that leads to the accelerating Universe [5]. Taking a variation of the action (28) with respect to the inverse metric tensor gμν yields the energy–momentum tensor of the quintessence field: 1 T Q = ∂ φ∂ φ − g gλδ∂ φ∂ φ − g V(φ). (29) μν μ ν 2 μν λ δ μν Therefore, for the quintessence scalar field, energy and pressure densities are found as [34,35] φ˙2 ρ = + V(φ), (30) Q 2 φ˙2 p = − V(φ), (31) Q 2 and, the equation-of-state parameter of the quintessence is given as

φ˙2 − 2V(φ) ωQ = . (32) φ˙2 + 2V(φ) ˙2 − 1 It is seen that the Universe accelerates for φ

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By making use of eq. (33) we find ηβ 2(β+1) − β − − β − − 3H 2t2β+3 Ht2 3H t (1 G f) φ˙ = . (35) 2 2 −1 − β − − β (3H MPG) 3H t G 2 t Now, we define a new variable x = ln a, (36) which helps us to rewrite the result (35) in another form. Using this new variable, we can write d d = H , (37) dx dt and we get φ˙ = Hφ , (38) where prime denotes derivative with respect to the new variable x. Hence, integrating eq. (35) with respect to the new variable x yields ηβ 2(β+1) − β − − β − − a 2 2β+3 2 3H (1 G f) da 3H t Ht t φ(a) − φ(a0)= . (39) 2 − β β a0 a 1 − − − (3MPG) 3H t G 2 t

Here, we have set a0 = 1 for the present value of the scale factor. The analytical form of the potential in terms of the fractal ghost field cannot be determined due to the complexity of the equations involved. However, it can be calculated numerically. For simplicity, we can focus on the infrared fractal profile, i.e., β = 0. Thus, we have √ a da  (1 −  ) φ (a) − φ (a ) = M G G IR IR 0 3 P − (40) a0 a 2 G and 2 2 − 3H MPG(G 3) VIR(φ) = . (41) 2(G − 2) On the other hand, considering the ultraviolet fractal profile (β = 2), we get 4η − 2 − − 2 − − UV ˙ H 2t7 Ht2 3H t (1 G f ) φUV = (42) 2 2 −1 − 2 − − 2 (3H MPG) 3H t G 2 t and ⎡ ⎤ 4η − 2 − 3H − 2 (1 −  − UV) = 2 2 ⎣ − H 2t7 Ht2 t G f ⎦ VUV(φ) 3H MPG 1 , − β − − 2 2 3H t G 2 t (43)

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UV where f and H are given by eqs (26) and (27), respectively. Finally, one can find the evolutionary form of the quintessence field as t 4η − 2 − − 2 − − UV H 2t7 Ht2 3H t (1 G f ) φUV(t) = φUV(0) + dt. (44) 2 2 −1 − 2 − − 2 0 (3H MPG) 3H t G 2 t

2.2 Interacting case In this section, we generalize our investigation to the interacting case. Hence, in the presence of interaction, the conservation equations read as β ρ˙ + 3H − ρ = Q, (45) m t m β ρ˙ + 3H − (1 +˜ω )ρ =−Q. (46) G t G G Here, we have introduced Q to define mutual interaction between two principal compo- nents of the Universe [36–40]. Negative values of Q correspond to energy transfer from dark matter to dark energy sector, and vice versa for positive values of Q. It was reported recently that this interaction is observed in the Abell Cluster A586 showing a transition from dark energy to dark matter sector and vice versa [41,42]. Moreover, this event may effectively appear as self-conserved dark energy, with a non-trivial equation-of-state mim- icking quintessence or phantom, as in the XCDM scenario [43–45]. Nevertheless, the significance of this interaction is not clearly identified [46]. In general we choose the following expression for the interaction term:

2 2 Q = 3ξ H(ρG + ρm) = 3ξ HρG(1 + ) (47) with ξ being a coupling parameter between dark energy and dark matter [47,48], although more general terms can be used. The sign of ξ 2 indicates the direction of energy transi- tion. The case ξ = 0 represents the non-interacting fractal Friedmann–Robertson–Walker model, while ξ = 1 represents the complete transfer of energy from dark energy sector to dark matter sector. In some cases, ξ 2 is taken in the range [0, 1] [49]. Galactic clusters and CMB observations show that the coupling parameter ξ 2 < 0.025, i.e., a small but positive constant of order of unity [50,51]. The case of negative coupling parameter is avoided due to the violation of gravitational thermodynamic laws. Inserting eqs (21) and (47) in eq. (46) we find

− − (2H − β ) 1(3H − β ) 1 ηβ 2(β + 1) ω˜ =−1 − t t G 2 2β+3 H 1 − HG 3H t 2H − β t βH β − − H 2(1 −  −  ) 3H − t2 G f t β 1 −  −  +3ξ 2H 2 2H − 1 + G f . (48) t G

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Before establishing a correspondence between the quintessence and ghost dark energy, we want to consider a constant fractal profile (υ = constant) to start some discussions using eq. (48). For a constant fractal profile, using eq. (48), we get 1 2ξ 2 ω˜ G =− 1 + . (49) 2 − G G

Here we can see that in the late time (t →∞) where G → 1, the equation-of- state parameter of interacting fractal ghost dark energy crosses the phantom regime 2 (ω˜ G =−1 − 2ξ < −1). For the present time where G = 0.72, the phantom regime crossing can be achieved if we choose ξ 2 > 0.1. In addition to these conclusions, eqs (45) and (46) show that the interaction term is a function of a quantity with units of inverse time multiplied by the energy density and it is important to mention here that, an ideal interaction term must be evolved from quantum gravity [2]. Next, for the time-like infrared profile, we get  + 2ξ 2 ˜ IR| = G ωG ξ 2>0 . (50) G(G − 2) We know that the sign of ξ 2 indicates the direction of energy transfer. Hence, we may also consider ξ 2 < 0 which describes energy transition from the dark matter to the dark energy sector. It is easy to show that for ξ 2 < 0, eq. (50) becomes  − 2ξ 2 ˜ IR| = G ωG ξ 2<0 . (51) G(G − 2)

Figure 2 shows that for the present time, taking G = 0.72, the phantom region ˜ IR − 2 crossing (ωG < 1) can be achieved provided ξ > 0 which is consistent with recent 2 ˜ IR − 1 observations [52,53]. At the same time, figure 3 shows that ξ < 0 leads to ωG > 3

for 2 0 IR G

1.0 1.5 2.0

2

4

6

8

Figure 2. The equation-of-state parameter of the fractal ghost dark energy in the 2 infrared regime for ξ > 0 (eq. (50)). Here we have taken G = 0.72.

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for 2 0 IR G 8

6

4

2

1.0 1.5 2.0

Figure 3. The equation-of-state parameter of the fractal ghost dark energy in the 2 infrared regime for ξ < 0 (eq. (51)). Here we have taken G = 0.72. for the present time. This result indicates that the Universe is in deceleration phase at the present time which is ruled out by recent observations [28–32]. Furthermore, in figure 3, we see that the equation-of-state parameter takes values greater than 1 for some values of ξ. Generally, ξ 2 is taken in the range [0, 1]. ξ = 0 describes the non-interacting frac- tal Friedmann–Robertson–Walker model, while ξ = 1 represents the complete transfer of energy from dark energy sector to dark matter sector. If we ignore ξ>1 values in figure 3, we get meaningful results and can remove disturbing values of the equation-of- state parameter. For the ultraviolet fractal regime, we have − 2 −1 − 2 −1 UV (2H t ) (3H t ) 4η 2H ω˜ | 2 =−1 − G ξ >0 2 7 2 H 1 − HG H t t 2H − 2 t 2 −H 2(1 −  −  ) 3H − G f t 3ξ 2H 2 2 + (1 − f) 2H − , (52) G t and − 2 −1 − 2 −1 UV (2H t ) (3H t ) 4η 2H ω˜ | 2 =−1 − − G ξ <0 2 7 2 H 1 − HG H t t 2H − 2 t 2 −H 2(1− − ) 3H − G f t 3ξ 2H 2 2 − (1−f) 2H − . (53) G t

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Now, we implement a connection between the quintessence scalar field and interacting fractal ghost dark energy. In further calculations we assume that ξ 2 > 0. In this case, the potential and time-derivative of scalar field are found to be

β −1/2 1/2 2 MP(2H − ) (3HG) βH ηβ (β + 1) φ˙I = t − 1/2 2 2 2β+3 − β 1/2 − HG t 3H t (3H t ) 1 − β 2H t β + H 2(1 −  −  ) 3H − G f t β 1 −  1/2 −3ξ 2H 2 2H − f , (54) t G

2 β −1 2 3HM G(2H − ) ηβ (β + 1) V I(φ) = 3H 2M2 + P t P G 2 2β+3 − β − HG 3H t 2(3H t ) 1 − β 2H t β βH −H 2(1− − ) 3H − − G f t t2 β 1− +3ξ 2H 2 2H − f , (55) t G where superscript I denotes interacting case. Next, we consider the time-like infrared and ultraviolet fractal profiles to discuss special cases. For the infrared fractal profile, we have the following results:

I 2 2 dφIR 3MPG 2ξ = 1 − G − (56) dlna 2 − G G and 3H 2M2 2ξ 2 I = P G − + VIR(φ) 3 G . (57) 2(2 − G) G

Furthermore, the evolutionary form of the fractal quintessence field in the infrared regime is obtained by integrating eq. (49). The result is

a da 3M2 2ξ 2 φI (a) = φ (a ) + P G −  − . IR IR 0 − 1 G (58) a0 a 2 G G

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On the other hand, in the time-like ultraviolet fractal regime, the following results for the fractal quintessence are obtained: 2 −1/2 1/2 MP(2H − ) (3HG) 2H 4η φ˙I = t − UV 1/2 2 2 7 − 2 1/2 − HG t H t (3H t ) 1 − 2 2H t 2 +H 2(1− − UV) 3H − G f t 2 1 − UV 1/2 −3ξ 2H 2 2H − f (59) t G and 2 2 −1 3HM G(2H − ) 4η V I (φ) = 3H 2M2 + P t UV P G 2 7 − 2 − HG H t 2(3H t ) 1 − 2 2H t 2 −H 2(1 −  − UV) 3H − G f t 2H 2 1 − UV − + 3ξ 2H 2 2H − f , (60) 2 t t G UV where f and H are given by eqs (26) and (27), respectively. Furthermore, integrating eq. (59) with respect to the cosmic time t yields t M (2H − 2 )−1/2(3H )1/2 I = I + P t G φUV(t) φUV(0) 1/2 1 H 0 − 2 2 − G (3H t ) 1 − 2 2H t 2 2H × H 2(1 −  − UV) 3H − + G f t t2 4η 2 1 − UV 1/2 − −3ξ 2H 2 2H − f dt. (61) 2 7 H t t G

3. Concluding remarks

Considering the fractal ghost dark energy model as an effective description of the dark energy theory, and assuming the quintessence scalar field as pointing in the same scenario, it is exciting to discuss how the ghost energy density can be used to describe the fractal quintessence scalar field. In this work, using the fractal theory of gravity, we have established a connection between the ghost dark energy scenario and the quintessence scalar field model. If we consider the fractal quintessence scalar field model as an effective description of fractal ghost dark energy, we should be able to use the scalar field model to mimic the evolv- ing behaviour of the dynamical ghost dark energy and reconstruct this scalar field model

514 Pramana – J. Phys., Vol. 84, No. 4, April 2015 Ghost quintessence in fractal gravity according to the evolutionary behavior of the fractal ghost dark energy. Hence, with this interesting strategy, we have reconstructed the potential of the fractal ghost quintessence and the dynamics of the fractal field according to the evolution of ghost energy density. In the limiting case, we considered the time-like infrared and ultraviolet fractal profiles. Furthermore, we also would like to mention here that the results presented in this study can be generalized easily to other scalar fields such as phantom, tachyon, K-essence and dilaton. One can also extend the aforementioned discussion in this work to the non-flat fractal Friedmann–Robertson–Walker space-time.

References

[1] C Chawla, R K Mishra and A Pradhan, Eur. Phys. J. Plus 127, 137 (2012) [2] A Sheykhi and A Bagheri, Europhys. Lett. 95, 39001 (2011) [3] E J Copeland, M Sami and S Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006) [4] S Weinberg, Rev. Mod. Phys. 61, 1 (1989) [5] A Pasqua, A Khodam-Mohammadi, M Jamil and R Myrzakulov, Astrophys. Space Sci. 340, 199 (2012) [6] T Padmanabhan, Phys. Rep. 380, 235 (2003) [7] Y F Cai, E N Saridakis, M R Setare and J Q Xia, Phys. Rep. 493, 1 (2010) [8] S Chakraborty and A Biswas, Astrophys. Space Sci. 343, 791 (2013) [9] M Li, X D Li, S Wang and X Zhang, J. Cosmol. Astropart. Phys. 06, 036 (2009) [10] F R Urban and A R Zhitnitsky, J. Cosmol. Astropart. Phys. 0909, 018 (2009) [11] F R Urban and A R Zhitnitsky, Phys. Lett. B 688, 9 (2010) [12] F R Urban and A R Zhitnitsky, Phys. Rev.D80, 063001 (2009) [13] F R Urban and A R Zhitnitsky, Nucl. Phys. B 835, 135 (2010) [14] N Ohta, Phys. Lett. B 695, 41 (2011) [15] G Veneziano, Nucl. Phys. B 159, 213 (1979) [16] E Witten, Nucl. Phys. B 156, 269 (1979) [17] C Rosenzweig, J Schechter and C G Trahern, Phys. Rev.D21, 3388 (1980) [18] P Nath and R L Arnowitt, Phys. Rev.D23, 473 (1981) [19] K Kawarabayashi and N Ohta, Nucl. Phys. B 175, 477 (1980) [20] K Kawarabayashi and N Ohta, Prog. Theor. Phys. 66, 1789 (1981) [21] N Ohta, Prog. Theor. Phys. 66, 1408 (1981) [22] M Khurshudyan and A Khurshudyan, arXiv:1307.7859 [23] G Calcagni, Phys. Rev. Lett. 104, 251301 (2010) [24] G Calcagni, J. High Energy Phys. 03, 120 (2010) [25] K Karami, M Jamil, S Ghaffari and K Fahimi, arXiv:1201.6233 [26] R-G Cai, Q Su, Z-L Tuo and H-Bo Zhang, Phys. Rev.D84, 123501 (2011) [27] Wolfram Research Inc, Wolfram Mathematica 8.0, 2010 [28] Planck Collaboration, arXiv:1303.5072 [29] S Perlmutter et al, Astrophys. J. 517, 565 (1999) [30] C L Bennett et al, Astrophys. J. Suppl. 148, 1 (2003) [31] M Tegmark et al, Phys. Rev.D69, 103501 (2004) [32] S W Allen et al, Mon. Not. R. Astron. Soc. 353, 457 (2004) [33] M Malekjani, In. J. Mod. Phys D 22, 1350084 (2013) [34] M Jamil, K Karami and A Sheykhi, Int. J. Theor. Phys. 50, 3069 (2011) [35] J A E Carrillo, J M Silva and J A S Lima, Astronomy and Relativistic Astrophysics: New Phenomena and New States of Matter in the Universe, Proceedings of the Third Workshop (Joo Pessoa, Paraba, Brazil, 2007) pp. 183–192, arXiv:0806.3299 [gr-qc]

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[36] M Jamil, E N Saridakis and M R Setare, Phys. Rev.D81, 023007 (2010) [37] C Wetterich, Astron. Astrophys. 301, 321 (1995) [38] L Amendola, Phys. Rev.D60, 043501 (1995) [39] X Zhang, Mod. Phys. Lett. A 20, 2575 (2005) [40] T Gonzalez, G Leon and I Quiros, Class. Quant. Grav. 23, 3165 (2006) [41] O Bertolami, F Gil Pedro and M Le Delliou, Phys. Lett. B 654, 165 (2007) [42] M Jamil and M A Rashid, Eur. Phys. J. C 58, 111 (2008) [43] J Sola and H Stefancic, Phys. Lett. B 624, 147 (2005) [44] I L Shapiro and J Sola, Phys. Lett. B 682, 105 (2009) [45] J Grande, J Sola and H Stefancic, J. Cosmol. Astropart. Phys. 011, 0608 (2006) [46] C Feng, B Wang, Y Gong and R-K Su, J. Cosmol. Astropart. Phys. 9, 5 (2007) [47] L Amendola and D Tocchini-Valetini, Phys. Rev.D64, 043509 (2011) [48] M R Setare and M Jamil, Phys. Lett. B 690, 1 (2010) [49] H Zhang and Z H Zhu, Phys. Rev.D73, 043518 (2006) [50] K Ichiki and Y Y Keum, J. Cosmol. Astropart. Phys. 6, 5 (2008) [51] L Amendola, G C Campos and R Rosenfeld, Phys. Rev.D75, 083506 (2007) [52] B Wang, Y Gong and E Abdalla, Phys. Lett. B 624, 141 (2005) [53] B Wang, C Y Lin and E Abdalla, Phys. Lett. B 637, 357 (2005)

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