New Agegraphic Pilgrim Dark Energy in f(t, T G ) Gravity

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1 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 2, August 1, 2015 New Agegraphic Pilgrim Dark Energy in f(t, T G ) Gravity Abdul Jawad 1, and Ujjal Debnath 2, 1 Department of Mathematics, COMSATS Institute of Information Technology, Lahore-54000, Pakistan 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah , India (Received February 2, 2015; revised manuscript received May 6, 2015) Abstract In this work, we briefly discuss a novel class of modified gravity like f(t, T G) gravity. In this background, we assume the new agegraphic version of pilgrim dark energy and reconstruct f(t, T G) models for two specific values of s. We also discuss the equation of state parameter, squared speed of sound and w DE-w DE plane for these reconstructed f(t, T G) models. The equation of state parameter provides phantom-like behavior of the universe. The w DE-w DE plane also corresponds to ΛCDM limit, thawing and freezing regions for both models. PACS numbers: Kd, x, k Key words: f(t, T G ) gravity, Pilgrim DE models, cosmological parameter 1 Introduction Nowadays it is strongly believed that the universe is experiencing an accelerated expansion. Recent observations from type Ia supernovae [1 2] in associated with Large Scale Structure [3] and Cosmic Microwave Background anisotropies [4 5] have provided main evidence for this cosmic acceleration. The main theory responsible for this scenario is the theory of dark energy (DE). This mysterious DE with negative pressure leads to this cosmic acceleration. It is believed that our present Universe is made up of about 4% ordinary matter, about 74% DE and about 22% dark matter. Recent WMAP data analysis [4,6] also gives us the confirmation of this acceleration. Over the past decade there have been many theoretical models for mimicking the DE behaviors, such as the simplest (just) cosmological constant in which the equation of state is independent of the cosmic time and which can fit the observations well. This model is the so-called ΛCDM, containing a mixture of cosmological constant Λ and cold dark matter (CDM) whose equation of state (EoS) parameter w = 1. [7] Although this model has a good agreement with observational data but it suffers several difficulties such as fine tuning and coincidence problem. [1,8] Further observations show that the EoS of DE w is likely to cross the cosmological constant boundary 1 (or phantom divide), i.e. w is larger than 1 in the recent past and less than 1 today. [9 10] The well known scalar field model, the quintessence [11 12] with a canonical kinetic term, can only evolve in the region of 1 < w < 1/3, whereas the model of phantom with negative kinetic term can always lead to w < 1. There are other DE candidates namely tachyonic field, DBI-essence, hessence, k-essence, dilaton DE, jawadab181@yahoo.com; abduljawad@ciitlahore.edu.pk ujjaldebnath@gmail.com c 2015 Chinese Physical Society and IOP Publishing Ltd etc which drive acceleration of the universe. [13 17] Also reviews on DE include in Refs. [18 20]. In recent times, considerable interest has been stimulated in explaining the observed DE by the holographic DE (HDE) model. [21 22] This arises from the holographic principle, [23] which states that the number of degrees of freedom related directly to entropy scales with the enclosing area of the system. [24 25] It was shown that effective local quantum field theories greatly overcount degrees of freedom because the entropy scales extensively for an effective quantum field theory in a box of size L with UV cut-off Λ. Cohen et al., [26] showed that in quantum field theory, short distance cut-off Λ is related to long distance cut-off L due to the limit set by forming a black hole, i.e., the total energy of the system with size L should not exceed the mass of the same size black hole, i.e., L 3 ρ Λ Lm 2 p where ρ Λ is the quantum zero-point energy density caused by UV cut-off Λ and m p denotes the reduced Planck mass (m 2 p = 1/8πG). The IR cut-off L is chosen by saturating the inequality so that the holographic DE density can be obtained as Ref. [27] in the form ρ Λ = 3c 2 m 2 pl 2 where c is a constant. The most simple choice of L is the Hubble horizon H 1. The another choice of L is the particle horizon. Finally, Li [28] found that L might be the future event horizon of the universe. On the basis of the holographic principle, several other authors have studied the holographic models of DE. [26,29 30] Another approach to explore the accelerated expansion of the universe is the modified theories of gravity. Einstein s theory of gravity may not describe gravity at very high energy. In this case cosmic acceleration would arise not from DE as a substance but rather from the dynamics of modified gravity. [31] The simplest alternative to

2 146 Communications in Theoretical Physics Vol. 64 general relativity is Brans Dicke scalar-tensor theory. [32] But among the most popular modified gravities which may successfully describe the cosmic speed-up is f(r) gravity (which is obtained by replacing R f(r)). Motivated by the string theory, another theory proposed as gravitational DE is scalar Gauss Bonnet gravity (f(g) gravity) [33] which comes from the Gauss Bonnet term G that is closely related with low-energy string effective action. However, most devoted works is being made in f(t ) gravity which based on torsion in recent years. It is made on Einstein s original idea of Teleparallel Equivalent of General Relativity (TEGR). [34] The proposal of TEGR evolves curvature-less Weitzenböck connection instead of the torsion-less Levi Civita connection. Also, it is mentioned here that f(t ) does not coincide with f(r). However, TEGR coincides with general relativity at the level of equations. That is why, it produces different modification classes and many authors have been investigated different cosmological implications in f(t ) gravity which are new, viable and interesting (references therein [37] ). Inspired by the f(r) modifications of the Einstein Hilbert Lagrangian, f(t ) modified gravity has been proposed [35] by extending T to an arbitrary function. To include the higher Gauss Bonnet (GB) terms in f(t ) gravity and motivated from the f(r, G) model, recently a novel class of gravitational modification f(t, T G ) has been constructed on the basis of T (old quadratic torsion scalar) and T G (new quartic torsion scalar T G that is the teleparallel equivalent of the GB term). [36 37] Implications of this gravity in Cosmology have been studied in various scenarios. [38 40] Recently, the correspondences between several candidates of DE models and modified gravities are very challenging subject in cosmological phenomena. This scenario help us in analyzing the role of DE in modified gravity. The correspondence phenomenon a very useful technique which was proposed by Nojiri and Odintsov [41 46] and it extended for several cosmological scenarios. Till now, several authors [47 50] have discussed the correspondence between different DE models and their cosmological implications. One can assume the first Friedmann equation of a type of modified gravity in the following way κ 2 ρ DE = Σ Σni=nA(R, G,...) n f(r, G,...) R n1 G n2. In the above equation, f(r, G,...) stands for the modified gravity action. For a model of DE, in general, ρ DE = ρ DE (H, Ḣ,...). Also, we can also implicitly write it as ρ DE = ρ DE (R, G,...). So, if we can solve the corresponding partial differential equations for f(r, G,...), indeed we reconstructed the modified gravity for this type of DE. Also, the reconstruction scheme works if we assume that in any cosmological epoch, a(t) is given (for a review see Refs. [46]). For this purpose, many authors [49 50] get taken up with the idea that a power law for the scale factor versus time as in the following form a(t) = a 0 t m, (1) where a 0 > 0, m > 0 and cosmic time t measures in Gyr. The power law for the scale factor gives a good fit to the supernova data in a flat Universe. If m > 1, this is an accelerating Universe and thus can give a not bad fit. [51] Motivated by above discussion, we explore the correspondence of new agegraphic version of Pilgrim DE (PDE) with newly proposed modified gravity called f(t, T G ). In the present paper, we explore the correspondence of new agegraphic version of PDE with newly proposed modified gravity called f(t, T G ). We extract the f(t, T G ) models via two different values of PDE parameter (s = 2, 2). Further, we analyze these models through different cosmological parameters. We discuss the f(t, T G ) models in the next section and summarize our results in the last section. 2 New Agegraphic Pilgrim Dark Energy f(t, T G ) Models The flat FRW universe is given by ds 2 = dt 2 + a 2 (t)(dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 ), (2) here, a(t) is cosmic scale factor and measures the expansion of the universe. Also, the total modified gravitational action in f(t, T G ) gravity takes the form S = 1 2κ 2 d 4 x e f(t, T G ), (3) where e = det(e a µ) = g and κ 2 = 8πG. This theory is different from both f(t ) and f(r, G) ones. [36] Further, the quantities T and T G can be obtained through vierbein and are given by [36] T = 6H 2, T G = 24H 2 (Ḣ + H2 ), (4) where dot represents the derivation with respect to cosmic time and H = ȧ/a known as the Hubble parameter. The Friedmann equations in case of FRW geometry can be obtained by taking variation of the total action S + S m as follows [36] f 12H 2 f T T G f TG + 24H 3 f TG = 2κ 2 ρ m, f 4(Ḣ + 3H2 )f T 4Hf T T G f TG + 2 3H T Gf TG + 8H 2 ftg = 2κ 2 p m, where f T and f TG denote the partial differentiation of function f with respect to their arguments, respectively. Hence, Friedmann equations can be re-written in usual form H 2 = κ2 3 (ρ m + ρ DE ), (5) Ḣ = κ2 2 (ρ m + ρ DE + p m + p DE ), (6) where ρ DE and p DE are given by ρ DE = 1 2 (6H2 f + 12H 2 f T + T G f TG 24H 3 f TG ), (7) p DE = 1 2 [ 2(2Ḣ + 3H2 ) + f 4(Ḣ + 3H2 )f T

3 No. 2 Communications in Theoretical Physics 147 4Hf T T f TG + 2 3H T ] Gf TG + 8H 2 ftg. (8) Now assume that the normal matter and DE are conserved separately, so the continuity equations are ρ m + 3H(ρ m + p m ) = 0, (9) ρ DE + 3H(ρ DE + p DE ) = 0. (10) Wei [52] proposed PDE based on speculation that the repulsive force contributed by the phantom-like DE (w Λ < 1) is strong enough to prevent the formation of the black hole. If this speculation is true, the energy bound could be violated, namely the total energy in a box of size L could exceed the mass of a black hole of the same size, i.e., ρ Λ L 3 m 2 pl. Therefore, the first property of Pilgrim DE is ρ Λ m 2 pl 2. (11) To implement this and based of the speculation, Wei [52] has defined PDE as follows ρ Λ = 3n 2 m 4 s p L s, (12) where n and s are both dimensionless constants. From Eqs. (11) and (12), we have L 2 s m s 2 p = lp 2 s, where l p is the reduced Planck length. Since L > l p, one requires s 2. (13) It was stated in Ref. [52] that to obtain the EoS for PDE, we should choose a particular cut-off L. In the present work, we choose future event horizon and conformal age of the universe. The analysis of PDE model in apparent and event horizons has been also discussed in Refs. [53 54]. The new agegraphic DE (NADE model) model is another alternate choice of DE which is the improved form of agegraphic DE (ADE). [55 57] Here, we use conformal age of the universe (of NADE model) as an IR cutoff of PDE which is defined as follows L = η = t 0 d t a( t). (14) With this IR cutoff, the PDE model is known as new agegraphic version of PDE. By using the scale factor Eq. (1), we can obtain H = m t, Ḣ = m t 2, T = 6m2 24( 1 + m)m3 t 2, T G = t 4, T = 12m2 t 3, T 96( 1 + m)m3 G = t 5. (15) Equations (1), (7), (12), (14), and (15) produce the following f(t, T G ) solutions for s = 2, 2 f(s = 2) = c 5 (2 m 1 ( T /2 + 2c6 (2 m 1+( T + /2 ( ( 3( 1 + 8m) (2 m 1) 1/2( T ) 1/4 T G T G T G m 2 + (2 m 1) 0.5+2m( T ) ))( 0.5+2m/2 ( 1 + m)n 2 2(2 m 1) 5/2( T ) 5/4 ) 1 ( 1 + 3m), (16) T G T G f(s = 2) = c 7 (2 m 1 ( T /2 + c8 (2 m 1+( T + /2 ( ( + 3( 1 + 8m) m 2 (1 3m) 1 T G T G ( + (2 m 1) 4 2m( T ) ) (4 2m)/2 n 2 (( 1 + m) 3 ( 9 + 7m)) 1))( 2(2 m 1) 2( T )) 1, (17) T G T G where l ± = 0.5 ( 5 + 5m ± ( 1 + m)( m)). Fig. 1 (Color online) Plot of f versus t for s = 2 with n = 2.2 (red), n = 2.4 (green), n = 2.6 (blue). Fig. 2 (Color online) Plot of f versus t for s = 2 with We plot the above reconstructed two functions versus its argument t for two specific values of PDE parameter s = 2, 2 as shown in Figs. 1 and 2. Also, we choose three different values of n = 2.2, 2.4, 2.6 and other constants

4 148 Communications in Theoretical Physics Vol. 64 are c 5 = 0.5, c 6 = 0.2, c 7 = 0.5, c 8 = 0.2, m = 2. It can be observed from Fig. 1 (for s = 2) that the reconstructed function f(t, T G ) shows increasing behavior with the passage of time. Also, we observe that f(t, T G ) increases upto finite time with negative values to zero level and finally it increases from zero to infinite value as time goes on. But for s = 2, f decreases upto finite time and then increases as time goes on but it keeps always positive as shown in Fig. 2. In this scenario, the pressure turns out to be for the two cases of s = 2, 2 are p DE (s = 2) = ((48( 1 + 3m)( 1 + n)n) 1 )t 3 0.5ζ ( c 5 ( 1 + 3m)(125m (3 + ζ) 5m 2 (18 + 5ζ + 10n) + n( ζ + 3(17 + 5ζ)n) + m( 83 + ζ + ( ζ 75n)n))t m + t 0.5ζ (192m 5 n 2 t 1+2m 16c 6 ( 3 + ζ)t δ + c 6 ( ζ)nt δ + 6n 4 t(8 + 9t 2m ) 6n 3 t( t 2m ) 3m 4 t 2m ( 24n 2 t + 64n 3 t + 125c 6 t δ ) + 3n 2 (32t + 12t 1+2m c 6 ( ζ)t δ ) + m 3 (48(6 + n ( n))t 12n 2 (89 + 6n( n))t 1+2m 5c 6 ( ζ 30n) t δ ) + m 2 ( 6(6 + n( n))t + 6n 2 (198 + n( n))t 1+2m + c 6 ( ζ + n( ζ + 225n))t δ ) m( c 6 ( ζ)t δ + c 6 ( ζ)nt δ + 144n 4 t(1 + 4t 2m ) 216n 3 t(2 + 5t 2m ) + 3n 2 (96t + 140t 1+2m c 6 ( ζ)t δ )))), (18) where ζ = ( 1 + m)( m) and δ = 0.5(1 + 5m + ζ). p DE (s = 2) = (48( 1 + m) 3 ( 1 + 3m)( 9 + 7m)( 1 + n)n) 1 t χ (c 7 ( 1 + m) 3 ( 1 + 3m)( 9 + 7m)(125m (3 + ζ) 5m 2 (18 + 5χ 1/2 + 10n) + n( ζ + 51n + 15ζn) + m( 83 + ζ + ( ζ 75n)n))t 9m/2 + t 0.5ζ ( 6( 1 + 3m)n 4 (3( 7 + 5m)( 1 + 8m)t ( 1 + m) 3 ( 9 + 7m) t 0.5+2m ) + 6( 1 + 3m)n 3 (( 1 + 8m)( 21 + m(11 + 4m))t ( 1 + m) 3 ( 9 + 7m)t 0.5+2m ) + ( 1 + m) 3 ( 9 + 7m)t 2m (36m 2 ( m) t 16c 8 ζt χ + 47c 8 mζt 0.5ζ + 28c 8 m 2 ζt χ 75c 8 m 3 ζt 0.5ζ c 8 ( 1 + m)( 1 + 3m)( m(7 + 25m))t ψ ) ( 1 + m) 3 ( m)nt 2m (102m 2 ( 1 + 8m) t 29c 8 ζt ψ + 97c 8 mζt ψ 30c 8 m 2 ζ t ψ c 8 ( 1 + 3m)(105 + m( m))t ψ + 3n 2 (4( 5 + m)( 1 + m)( 3 + 2m)( 1 + 3m) ( 1 + 8m)t ( 1 + m) 3 ( 9 + 7m)(16 + 3m( m( 1 + 8m)))t 0.5+2m + c 8 ( 1 + m) 3 ( 1 + 3m)( 9 + 7m)( m + 5ζ)t ψ )))), (19) where χ = 0.5( 5 4m δ) and ψ = 0.5(5m + δ 1/2 ). 2.1 Equation of State Parameter In the present scenario, the EoS parameter takes the from w PDE = p DE /ρ DE. This parameter can be discussed for PDE model conformal age of the universe as follows: Figures 3 (for s = 2) and 4 (for s = 2) represent the behavior of EoS parameter for new agegraphic PDE model. Figure 3 shows that the EoS parameter starts from high phantom values approaches to lower phantom region for all n = 2.2, 2.4, 2.6. However, the EoS parameter remains in the phantom region with time lapses. Figure 4 indicates that the EoS parameter also exhibits phantom-like nature of the universe for all three cases of n and always remains in this era. Hence, in these cases of PDE parameter, we observe the EoS parameter favors the PDE phenomenon for all cases of n and s. 2.2 ω DE -ω DE Plane The ω DE -ω DE plane is firstly proposed by Caldwell and Linder [58] which is used to discuss the dynamical property of DE models, where ω DE is the evolutionary form of ω DE with prime indicates derivative with respect to ln a. They used this method for analyzing the behavior of quintessence scalar field DE model. They observed that ω DE -ω DE plane for quintessence model with scalar field potential asymptotically approaching to zero. Also, they pointed out that this plane can be divided into two categories of thawing and freezing regions. The thawing region is described as (ω Λ > 0, ω Λ < 0) while freezing region as (ω Λ < 0, ω Λ < 0). This analysis has also been extended for other DE models by several authors. [59 65] In the present scenario, we also develop the w DE -w DE plane for the reconstructed PDE models (s = 2, 2) as shown in Figs. 5 and 6. From these planes, we can observe that the w DE -w DE planes provide the ΛCDM (where (w DE, w DE) = ( 1, 0)) for all cases of m. The thawing as well as freezing regions are also obtained through these planes in all cases of n and s.

5 No. 2 Communications in Theoretical Physics 149 While, f(t, T G ) decreases upto finite time and then increases as time goes on but it keeps always positive for the case s = 2 as shown in Fig. 2. Fig. 3 (Color online) Plot of w PDE (s = 2) versus t with Fig. 5 (Color online) Plot of w PDE-w PDE for s = 2 with Fig. 6 Plot of w PDE-w PDE for s = 2 with n = 2.2 (red), n = 2.4 (green), n = 2.6 (blue). Fig. 4 (Color online) Plot of w PDE (s = 2) versus t with 3 Summary In this work, we have discussed the f(t, T G ) modified gravity in FRW universe. We have assumed the new agaegraphic version of PDE where L is chosen as the conformal time. Next we have reconstructed f(t, T G ) models for assuming the power law form of scale factor. The reconstructed function f(t, T G ) shows increasing behavior with the passage of time as shown in Fig. 1 (for s = 2) that t. Also, we observe that f(t, T G ) increases upto finite time with negative values to zero level and finally it increases from zero to infinite value as time goes on. It has been observed from Fig. 3 that the EoS parameter starts from high phantom values approaches to lower phantom region for all n = 2.2, 2.4, 2.6. However, the EoS parameter has remained in the phantom region with time lapses, while it may be approaches after a long time. In Fig. 4, the EoS parameter also exhibits phantom-like nature of the universe for all three cases of n and always remains in this era. Hence, in these cases of PDE parameter, we observe the EoS parameter favors the PDE phenomenon for all cases of n and s. In the present scenario, we also develop the w DE w DE for the reconstructed PDE models (s = 2, 2) as shown in Figs. 5 and 6. From these planes, we can observe that the w DE -w DE planes meet the ΛCDM limit (where (w DE, w DE) = ( 1, 0)) for all cases of n. The thawing as well as freezing regions are also obtained through these planes in all cases of n and s. References [1] A.G. Riess, et al., Astron. J. 116 (1998) [2] S. Perlmutter, et al., Astrophys. J. 517 (1999) 565. [3] M. Tegmark, et al., Phys. Rev. D 69 (2004) [4] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175.

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