Pilgrim Dark Energy in f(T, T G ) cosmology
Surajit Chattopadhyay, Abdul Jawad, Davood Momeni, Ratbay Myrzakulov
aa r X i v : . [ g r- q c ] J un Pilgrim Dark Energy in f ( T, T G ) cosmology Surajit Chattopadhyay, ∗ Abdul Jawad, † Davood Momeni, ‡ and Ratbay Myrzakulov § Pailan College of Management and Technology,Bengal Pailan Park, Kolkata-700 104, India. Department of Mathematics, Lahore Leads University, Lahore, Pakistan. Eurasian International Center for Theoretical Physics and Department of General Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan. (Dated: September 25, 2018)We work on the reconstruction scenario of pilgrim dark energy” (PDE) in f ( T, T G ). InPDE model it is assumed that a repulsive force that is accelerating the Universe is phantomtype with ( w DE < −
1) and it so strong that prevents formation of the black hole. Weconstruct the f ( T, T G ) models and correspondingly evaluate equation of state parameterfor various choices of scale factor. Also, we assume polynomial form of f ( T, T G ) in terms ofcosmic time and reconstruct H and w DE in this manner. Through discussion, it is concludedthat PDE shows aggressive phantom-like behavior for s = − f ( T, T G ) gravity. I. INTRODUCTION
Accelerated expansion of the current universe is well established through observational studies(Perlmutter et al. 1999; Bennett et al. 2003; Spergel et al. 2003; Tegmark et al. 2004; Abazajianet al. 2004, 2005; Allen et al. 2004). It is believed that this expansion is due to missing energycomponent, also dubbed as Dark Energy (DE) characterized by negative pressure. Reviews on DEinclude Padmanabhan (2005), Copeland et al. (2006), Li et al. (2011) and Bamba et al. (2012a),Nojiri and Odintsov (2007). The ΛCDM model, the simplest DE candidate, is consistent very wellwith all observational data. However, it has the following two weak points as enlisted in Nesseriset al. (2013): The Λ has fine tuning amount and good marginal adoption with cosmologicalobservations in large scales. This motivates the researchers in proposing the wide range of more Corresponding author ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] complex generalized cosmological models of DE which have been discussed in Copeland et al.(2006)and Bamba et al. (2012a,2012b).In recent years, the holographic dark energy (HDE) (Li, 2004,2011), is based on holographicUniverse idea and it is one of the interesting and powerful candidates for the DE. and its densityis given by (Li, 2004) ρ Λ = 3 c m p L − (1)where L is the IR cutoff, m p = (8 πG ) − / is the reduced Planck mass. IR cut-offs L parameter isconsideredin various ways in numerious articles: s the Hubble horizon H − , particle horizon, thefuture event horizon, the Ricci scalar curvature radius. Also it was proposed a generalization ofholographic models in which the cut-off parameter has a more general form (S. i. Nojiri and S. D.Odintsov, 2006a). The last generalized holographic dark energy scenario has been investigated fromdifferent point of the views,specially the stability under small perturbations. In all of these models,the cut-off length scale is proportional to the causal length that considered to the perturbationsof the flat spacetime . Further more, it is suggested that phenomenon of matter collapse can beavoided through the existence of appropriate repulsive force. In the present set up of cosmologicalscenario, this can be only prevented through phantom-like DE which contains enough repulsiveforce. Different attempts have been made in this way, e.g., reduction of BH mass though phantomaccretion phenomenon (Babichev et al. (2004); Martin-Moruno 2008; Jamil et al. 2008; Babichevet al. 2008; Jamil 2009; Jamil and Qadir 2011; Sharif and Abbas 2011;Jamil et al. 2011; Bhadraand Debnath 2012; Sharif and Jawad 2013) and the avoidance of event horizon in the presence ofphantom-like DE (Lobo 2005a, 2005b; Sushkov 2005, Sharif and Jawad 2014).Moreover, it is predicted that phantom DE with strong negative pressure can push the uni-verse towards the big rip singularity where all the physical objects lose the gravitational boundsand finally dispersed. This prediction supports the phenomenon of the avoidance of BH forma-tion and motivated Wei (2012) in constructing PDE model. He pointed out different possibletheoretical and observational ways to make the BH free phantom universe with Hubble horizonthrough PDE parameter. Further, Sharif and Jawad (2013a, 2013b, 2014) have analyzed thisproposal in detail by chosing different IR cutoffs through well-known cosmological parametersin flat and non-flat universes. This model has also been in different modified gravities (Sharifand Rani (2014); Sharif and Zubair (2014)). Another direction that one can follow to explainthe acceleration is to modify the gravitational sector itself, acquiring a modified cosmological dy-namics. However, note that apart from the interpretation, one can transform from one approachto the other, since the crucial issue is just the number of degrees of freedom beyond GeneralRelativity and standard model particles (Kofinas et al. 2014; Sahni and Starobinsky, 2006). De-tailed review in this modified gravity approach is available in references like Nojiri and Odintsov(2007), Tsujikawa (2010) ,Cai et al.(2009) and Clifton et al. (2012). In the majority of mod-ified gravitational theories, one suitably extends the curvature based Einstein-Hilbert action ofGeneral Relativity. However, an interesting class of gravitational modification arises when onemodifies the action of the equivalent formulation of General Relativity based on torsion (Kofi-nas and Saridakis, 2014). Inspired by the f ( R ) modifications of the Einstein-Hilbert Lagrangian, f ( T ) modified gravity has been proposed by extending T to an arbitrary function (Ferraro andFiorini,2007). Diffrent aspects of the f ( T ) has been investigated in the literature (Aslam et al.2013; Bamba et al. (2012a,2012b,2013a,2013b,2013c,2014a,2014b,2014c,2014d); Farooq et al. 2013;Houndjo et al. (2012, 2013); Jamil et al. (2013a,2013b,2012a-2012i); Cai et al. 2011; Momeni etal.(2011,2012);Rodrigues et al. (2013a,2013b); Setare et al. 2011) . The simplest modified gravityis obtained by replacing R → f ( R ),is so called f ( R ) gravity. This kind of the modified gravitieshave several interesting extensions(for a recent review see (Nojiri and Odintsov (2014)). The nextmodification is inspired from the string theory. It comes from the Gauss- Bonnet term G and widelyinvestigated in the literature (Capozziello et al. 2014; Nojiri and Odintsov 2011a,2011b,2011c; No-jiri et al. 2010; Bamba et al. 2010a,2010b; Cognola et al. 2009; Capozziello et al. 2009;Capozzielloet al. 2013; Bamba et al. 2008; Nojiri et al. 2007b; Cognola et al. 2007,2008; Nojiri and Odintsov2007a,c,d,2006e; Nojiri at al. 2006b,2006c,; Cognola et al. 2006a,2006b; Nojiri and Odintsov2005a,2005b,2005c,2006d; Nojiri et al. 2005a,2005b; Nojiri et al. 2002; Lidsey et al. 2002; Cveticat al. 2002; Noji et al. 2009;Nojiri and Odintsov 2008; Nojiri and Odintsov 2003 ).The main motivation to consider GB models is they are motivated from the string theory.In low limit of string theory these higher order curvature corrections appered. Motivated by f ( R ) gravity,we can introduce the f ( G ) gravity ,proposed by Nojiri and Odintsov (2005). Thismodification of the Einstein gravity unified dark matter and dark energy (Cognola et al. 2006) in asame scenario and in a consistent way and also in relation to the gauge/gravity proposal(Lidsey etal. 2002a,2002b). To include the higher GB terms in f(T) gravity and motivated from the f ( R, G )model, recently 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 Gauss-Bonnetterm) (Kofinas and Saridakis, 2014). This theory also belongs to a novel class of gravitationalmodification. Cosmological applications for this gravity have also been presented in detailed.To realize the role of DE in modified gravity,a very useful technique proposed by (Nojiri andOdintsov 2006e,2005a, 2006d) and it extended for several cosmological scenarios. Consider thefirst Friedmann equation of a type of modified gravity in the following form: κ ρ DE = Σ Σ n i = n A ( R, G, ... ) ∂ n f ( R, G, ... ) ∂R n ∂G n ... (2)In the above equation , f ( R, G, ... ) stands for the modified gravity action. For a model of DE,forexample the holographic DE, in general ρ DE = ρ DE ( H, ˙ H , .. ). Also, implicitly we able to write it as ρ DE = ρ DE ( R, G, ... ). So, if we can solve the following partial differential equations for f ( R, G, ... ),indeed we reconstructed the modified gravity for this type of DE. Also, the reconstruction schemeworks if we assume that in any cosmological epoch , a ( t ) is given (for a review see Nojiri andOdintsov 2011c). Our aim in this work is to reconstruct f ( T, T G ) for PDE in flat FRW Universe.Here, we also check cosmological aspects of this theory in flat FRW universe. It is interestingto mention that we provide the correspondence scenario of newly proposed gravity theory as wellas dynamical DE model (PDE).The plan of the paper is given as follows: Next section contains the reconstruction scheme. Weprovide the construction of reconstructed f ( T, T G ) = ˜ f models and corresponding EoS parameterwith respect to PDE parameter ( s ). In section (II) , we have reported a reconstruction approachthrough power-law form of scale factor. In section (III) , we present reconstruction scheme witha choice of Hubble rate H leading to unification of matter and dark energy dominated universe.In section (IV) , we choose the scale factor in the “intermediate” form and reconstruct f andsubsequently w DE . A bouncing scale factor in power law form is considered in section (V) and areconstruction scheme is reported. An analytic form of f is assumed in section (VI) and Hubbleparameter is reconstructed without any choice of scale factor. The paper is concluded in section (VII) . II. RECONSTRUCTION SCHEME FOR f ( T, T G ) THROUGH POWER-LAW SCALEFACTOR
A new and valid generalization of f ( T ) in the modified gravity models as f ( T, T G ) based on Tand equivalent to Gauss-Bonnet term T G in the teleparallel , is quite different from their counter-parts f ( T ) and f ( R, G ) in Einstein gravity (Kofinas et al., 2014). In f ( T, T G ) gravity S = 12 Z d x e F ( T, T G ) + Z d x e L m , (3)where in God-given natural units c = 1 , κ ≡ πG = 1 and L m is Lagrangian of the matter fields.For a spatially flat cosmological FRW metric ds = − dt + a ( t ) (cid:16) d~x.d~x (cid:17) . (4)We obtain: T = 6 H (5) T G = 24 H ( ˙ H + H ) (6)where H = ˙ a/a is the Hubble parameter and dots denote differentiation with respect to t . Fried-mann equations in the usual form are H = 13 ( ρ m + ρ DE ) (7)˙ H = −
12 ( ρ m + ρ DE + p m + p DE ) (8)Kofinas et al. (2014) modified the Eqs. (7) and (8) by defining the energy density and pressure ofthe effective dark energy sector as ρ DE = 12 (cid:16) H − f + 12 H f T + T G f T G − H ˙ f T G (cid:17) (9) p DE = 12 h − H + 3 H ) + f −
4( ˙ H + 3 H ) f T − H ˙ f T − T f T G +23 H T G ˙ f T G + 8 H ¨ f T G (cid:21) (10)Standard matter and dark energy are conserved separately, i.e. the evolution equations are˙ ρ m + 3 H ( ρ m + p m ) = 0 (11)˙ ρ DE + 3 H ( ρ DE + p DE ) = 0 (12)The first property of PDE is ρ Λ & m p L − (13)To implement Eq. (13), Wei (2012) set PDE in the simplest way as ρ Λ = 3 n m − sp L − s (14)where n and s are both dimensionless constants. From Eqs. (13) and (14) we have L − s & m s − p = l − sp , where l p is the reduced Plank length. Since L > l p , one requires s ≤ w Λ < − L and the simplest cut-off is the Hubble horizon L = H − .The PDE mentioned above would now be studied in f ( T, T G ) gravity proposed recently byKofinas et al. (2014). As we are going to consider PDE in f ( T, T G ) gravity in Eq. (9) we use( L = 1 /H ) ρ DE = ρ Λ = 3 n m − sp (cid:18) tm (cid:19) − s (17)Reconstruction scheme is a way to solve the following second order partial differential equation for f ( T, T G ): ρ DE ( T, T G ) = T − f ( T, T G ) + T ∂∂T f ( T, T G ) + 12 G ∂∂G f ( T, T G ) (18) − A ( T, T G ) ∂ ∂G f ( T, T G ) − B ( T, T G ) ∂ ∂T ∂T G f ( T, T G )There is no simple solution for this partial differential equation for a given set of functions { A, B } .But sometimes for a choice of scale factor a ( t ) we can solve it.We consider the scale factor in the form a ( t ) = a t m (19)where m >
0. Subsequently Hubble parameter H and its time derivative ˙ H are H = mt , ˙ H = − mt (20) T = 6 m t , T G = 24( − m ) m t ; ˙ T = − m t ; ˙ T G = − − m ) m t (21)Considering Eqs. (9), (17) and (21) we get the following differential equation (cid:18) t s − m (cid:19) d fdt + t s (cid:18) m − m − (cid:19) dfdt + t s f − m t − s + 6 m s n = 0 (22)This differential equation has an exact solution given by the following expression: f ( t ) = C t m − + √ − m +25 m + C t m − − √ − m +25 m + (23)120 (cid:0) − m s + m s + m s (cid:1) n t − s − (cid:0)(cid:0) s − (cid:1) m + s + − s (cid:1) ( m − m t (cid:0)(cid:0) s − (cid:1) m + s + − s (cid:1) (cid:0) − + m (cid:1) It is possible to write f ( T, T G ) in the following form: f ( T, T G ) = C T / √ G − T ! − T TG − T − / / r T TG − T + T ( TG − T ) (24)+ C T / √ T G − T ! − T TG − T − / − / r T TG − T + T TG − T − T G − / T ) (( T G − / T ) [3 / − T T G − T ) s − / (cid:18) − T T G − T (cid:19) s + (cid:18) − T T G − T (cid:19) s ] n T / √ T G − T ! − s + 245 (( − / s + 8 / s ) T + T G ( − s + 4 + s ) T T G )(3 T G − T ) − × (8 T s + 3 s T G − s T + 12 T G − sT G ) − ( − T + 15 T G ) − T − Based on the choice of the scale factor we have ˙
H < w Λ = − − s ˙ H H < − s <
0. It was clearlyestablished in Wei (2012) that for PDE • EoS w Λ goes asymptotically to − • w Λ < − • w Λ never crosses the phantom divide w = 1 in the whole cosmic history.In order to verify whether consideration of PDE in the f ( T, T G ) gravity, which is a class of modifiedgravity, leads to results consistent with that of Wei (2012) we consider both 0 < s ≤ s <
0. In particular we consider s = 2 and s = −
2. Solving Eq. (22) for the said two cases wehave two solutions for f ( t ) as:˜ f ( s = 2) = 12( − m ) m (cid:0) − n (cid:1) ( − m ) t + t (cid:16) − m − √ m ( − m ) (cid:17) [ C + C × t √ m ( − m ) i (25)and ˜ f ( s = −
2) = 12( − m ) (cid:16) m − m + n t − m (cid:17) m t + t (cid:16) − m + √ m ( − m ) (cid:17) [ C + C × t − √ m ( − m ) i (26)It may be noted that from now onwards ˜ f would denote the reconstructed f . Using solution (25)in Eq. (10) we get the modified p DE for s = 2 as function of tp DE ( s = 2) = 1192( − m ) m ( − m ) t ( − − ξ ) h m (cid:16) − n t ξ − C t m + ξ (cid:17) + 384 m t ξ (cid:0) − t + n (2 + 11 t ) (cid:1) − m t ξ (cid:0) − t + n (8 + 15 t ) (cid:1) + 25 C m t m + ξ (158 + 45 t − ξ ) + +5 C t m + ξ ( − ξ ) ( − − t + 4(2+ t ) ξ + ( − t ) ξ (cid:1) + 3 C m t m + ξ ( t ( − ξ ) − − ξ ) ξ )) − C mt m + ξ (2382 − ξ (962 + ξ ( −
179 + 8 ξ )) + t (8396 + ξ ( − ξ ( −
50 + 3 ξ )))) − C ( − m ) t m/ (cid:0) m − m (5 t + 2(6 + ξ )) − (9 + ξ )( − ξ )(3 + ξ ) + t ( −
53 + ( − ξ ) ξ )) + m ( t (1259 + 65 ξ )+ 10(104 + ξ (41 + 3 ξ ))) + m ( − ξ )(37 + ξ (22 + ξ )) + t ( − ξ ( −
138 + 7 ξ )))) + m (cid:16) t ξ (cid:0) − t + n (2 + t ) (cid:1) − C t m + ξ × ( − ξ (1532 + 3( −
53 + ξ ) ξ ) + t ( − ξ (739 + 21 ξ ))))] (27)Using solution (26) in Eq. (10) we get the modified p DE for s = − tp DE ( s = −
2) = t ( − − ξ ) − m ) m ( − m )( −
13 + 7 m ) h − C m t m + ξ + 25344 m t ξ × ( − t ) − m t ξ ( − t ) + 5376 m t ξ ( − t ) + 5760 n t ξ × ( −
10 + 7 t ) − mn t ξ ( −
260 + 97 t ) − m t ξ (cid:0) −
52 + 104 t + 63 n t (cid:1) + 25 C m t m + ξ (1496 + 315 t − ξ ) + m (cid:16) − n t ξ (220 + 17 t ) − C × t m + ξ ( − ξ ) (cid:0) − − t + 4(2 + t ) ξ + ( − t ) ξ (cid:1)(cid:17) + C m t m + ξ (3 t × ( − ξ ) − ξ ( −
219 + 7 ξ ))) − C m t m + ξ ( − ξ (23594 + 3 ξ ( −
566 + 7 ξ )) + t ( − ξ (7708 + 147 ξ ))) − C m t m + ξ × (10(10159 + ξ ( − − ξ ) ξ )) + t (194934 + ξ ( − ξ ( −
89+ 3 ξ )))) − C m (65 + m ( −
74 + 21 m )) t m/ (cid:0) m − m (5 t + 2(6 + ξ )) − (9 + ξ )( − ξ )(3 + ξ ) + t ( −
53 + ( − ξ ) ξ )) + m ( t (1259 + 65 ξ ) + 10 × (104 + ξ (41 + 3 ξ ))) + m ( − ξ )(37 + ξ (22 + ξ )) + t ( − ξ ( − ξ )))) + m (cid:16) n t ξ (12 + 13 t ) + C t m + ξ (32856 − ξ (13871 + ξ × ( − ξ )) + t (125843 + ξ ( − ξ ( −
825 + 74 ξ )))))] (28)where ξ = p
65 + m ( −
74 + 25 m ). Using Eqs. (27) and (28) the EoS for PDE i.e. w DE = p DE n m − sp ( tm ) − s is obtained through reconstructed ˜ f for s = 2 and s = − w DE ( s = 2) = 172 m ( − m ) n t − (7+ ξ ) h C ξt m + ξ − C mξt m + ξ + 278 C m ξ × t m + ξ − C m ξt m + ξ + 15 C ξt m + ξ − C mξt m + ξ + 15 C m ξ = t m + ξ + C ( − m ) t m/ (cid:0) − m + 26 ξ + (19 + 3 ξ ) t + 5 m − (88 + 5 ξ + 5 t ) − m (533 + 51 ξ + (52 + 5 ξ ) t )) + C ( − m ) t m + ξ (218+ 19 t + m ( − t ) + 5 m (88 − m + 5 t ))) − m t (5+ ξ ) ( − t + 9 m n t + 2 n (2 + t ) − m (cid:0) − t + n (4 + 13 t ) (cid:1) ) (cid:3) (29)and w DE ( s = −
2) = 172 mξ n t ( − − ξ ) h − C m ξt m + ξ + 5239 C m ξt m + ξ − C m × ξt m + ξ + 2921 C m ξt m + ξ − C m ξt m + ξ − C m ξt m + ξ + 547 × C m ξt m + ξ − C m ξt m + ξ + 105 C m ξt m + ξ − t (5+ ξ ) × (cid:0) − m ) m ( −
13 + 7 m ) − − m ) m ( −
13 + 7 m ) t − m ( −
8+ 3 m )) n t + 3( − m )( −
14 + m ( − m )) n t (cid:1) − C m (65 + m ( −
74+ 21 m )) t m + ξ ( − − t + m (533 + 5 m ( −
88 + 25 m − t ) + 52 t )) − C m × (65 + m ( −
74 + 21 m )) t m/ (cid:0) −
218 + 125 m − ξ − (19 + 3 ξ ) t − m (88+ 5 ξ + 5 t ) + m (533 + 51 ξ + (52 + 5 ξ ) t ))] (30)Firstly we include the discussion of reconstructed f models corresponding to PDE parameter s = 2 , −
2, respectively. We plot f ( s = 2 , −
2) versus cosmic time as well as cosmic scale factor m > and . In FIG. , it is observed that ˜ f ( s = 2) shows decreasing behaviorfrom very high value and approaches to zero versus cosmic time in the range 2 ≤ m ≤ .
3. However,for m > .
3, it shows decreasing behavior initially, becomes flat for a short interval of time, andthen exhibits increasing behavior. FIG. indicates that ˜ f ( s = −
2) increases with cosmic time fromvery low value, becomes flat for a glimpse of time interval and then decreases for 2 ≤ m ≤ . . < m ≤ .
5, it increases but approaches to zero after short interval of time. For 2 . < m , f ( s = −
2) increases with cosmic time from very low value, becomes flat for a glimpse of timeinterval and then increases.We also plot EoS parameter versus cosmic time corresponding to the same values of PDEparameter for three different values of m as shown in FIG. and . It can be observed from FIG. that EoS parameter starts from dust-like matter, passes the quintessence-like and vacuum DE0 t m f Ž H s = L FIG. 1: Plot of reconstructed f (Eq. (25)) fromPDE taking s = 2 and m >
1. Also, n = 3 , C =0 . , C = 0 . t m - - f Ž H s =- L FIG. 2: Plot of reconstructed f (Eq. (26)) fromPDE taking s = − m >
1. Also, n =3 , C = 0 . , C = 0 . - - - - t w P D E H s = L FIG. 3: Plot of the reconstructed EoS parameter(Eq. (29)) for s = 2. - - - - - t w P D E H s = - L FIG. 4: Plot of the reconstructed EoS parameter(Eq. (30)) for s = − era and then goes towards phantom DE era. The w DE crosses phantom boundary at t ≈ . quintessence to phantom i.e. behaves like quintom . This plot also represents thatEoS parameter attains more reliable phantom era which possesses the ability for prevention of BHformation. FIG. represents that EoS parameter remains in the phantom DE era forever. Thetrajectories of EoS parameter corresponding to different values of m starts from higher phantomvalues and goes towards less phantom values. Also, it is observed that it never meets or crosses1the − phantom . It is worthwhile to mention herethat EoS parameter corresponding to both cases of PDE parameter correspond to phantom eraof the universe which is a favorable sign to PDE conjecture. However, the reconstructed modelcorresponing to s = 2 is more attractive as compare to s = − w DE < − w DE = − s = − w DE < − → − −
1. Moreover, s = − H < s <
0. Hence, it is observed that the results stated for PDE in Einsteingravity are in close agreement with those in the framework of modified gravity under consideration.
III. RECONSTRCTION SCHEME FOR UNIFICATION OF MATTER DOMINATEDAND ACCELERATED PHASES
We consider that the Hubble rate H is given by (Nojiri and Odintsov, 2006d,2006e) H ( t ) = H + H t . (31)that leads to a ( t ) = C e H t t H and due to this choice of Hubble parameter, the PDE takes theform ρ Λ = 3 n t − s ( H t + H ) s (32)When t << t , in the early Universe and H ( t ) ∼ H t , the Universe was filled with perfect fluidwith EOS parameter as w = 1 + H . On the other hand, when t >> t the Hubble parameter H ( t ) is constant H → H and the Universe seems to de-Sitter . So, this form of H ( t ) providestransitio from a matter dominated to the accelerating phase. In Eq. (9) we use (31) and we getthe following ρ DE = 12 (cid:2) − f [ t ] + (cid:0) ( H + H t ) (cid:0) − t (cid:0) H (2 + ( − H ) H ) + H H (21 + H ( −
73 + 40 H )) t + H (cid:0) − H + 60 H (cid:1) t + H ( −
13 + 40 H ) t + 10 H t (cid:1) f ′ [ t ] + ( H + H t ) (cid:0) H × H t ( − H t ) + H ( − H t )) (cid:0) H (cid:0) H + H t ( − H t ) + H ( − H t ) (cid:1)(cid:1)(cid:1) + t f ′′ [ t ] (cid:1) n H t (cid:0) H + H t ( − H t ) + H ( − H t ) (cid:1) o − (cid:21) . (33)In the above as well as in the subsequent differential equations f ′ [ t ] and f ′′ [ t ] denote the firstand second order derivatives of f respectively with respect to t . If we consider ρ Λ = ρ DE as2available in Eqs. (32) and (33), we get a differential equation that can not be solved analyticallyfor f . Hence, we solve it numerically to have f graphically. Using the same approach as inthe previous section, we have reconstructed EoS parameter and showed it graphically. In FIGs. and , we have plotted ˜ f for s = 2 and −
2, respectively. In case of s = 2, we have taken n = 0 . , H = 2 (red) , . , . H = 0 .
5. For s = −
2, we have taken n = 4 , H = 3 (red) , . , . H = 0 . t f Ž H s = L FIG. 5: Plot of reconstructed ˜ f taking s = 2 for H ( t ) = H + H t . - - - - t f Ž H s = - L FIG. 6: Plot of reconstructed ˜ f taking s = − H ( t ) = H + H t . - - - - - - t Ω D E FIG. 7: Plot of the reconstructed EoS parameterfor s = 2. - - - - t Ω D E FIG. 8: Plot of the reconstructed EoS parameterfor s = − s = 2, we have observed increasing pattern of ˜ f and in case of s = −
2, it is exhibitingdecreasing pattern. We have plotted the EoS parameters corresponding to s = − and , respectively. In case of s = 2, the EoS parameter is ≥ − s = −
2, the EoS parameter is crossing − H = 3 . f ( T, T G ) throughconsideration of PDE can attain phantom era when s = −
2. This is consistent with the behavior ofPDE that leads to purely phantom era when considered in Einstein gravity with s = −
2. However,in f ( T, T G ) gravity, it can go beyond phantom for our choice H ( t ) = H + H t . A. On unification of inflation with DE in f ( T, T G ) In this short subsection our aim is to realize f ( T, T G ) to unify the early inflationary epochwith the late time de-Sitter era. We mention here that the unification of inflation with DE inmodified gravities firstly examined for f ( R ) gravity. It was proposed in Nojiri-Odintsov model(Nojiri and Odintsov 2003), which was subsequently generalized to more realistic versions (Nojiriand Odintsov 2007d; Cognola et al. 2008). One important problem in early Universe is singularityand it investigated later (Nojiri and Odintsov 2008). Indeed it has been shown that there is a classof non-singular exponential gravity to unify the early- and late-time accelerated expansion of theUniverse (Elizalde et al. 2011). For a review see (Nojiri and Odintsov 2011c).For our f ( T, T G ) case, if we consider the inflationary solution for H = H t then we have an exactsolution of f as ˜ f = 1 t (cid:20) n ( − t ) − s −
14 + s ) s − t + C t −√ + C t √ (cid:21) (34)This is an inflationary solution. We denote by j = 7 − s > , ℓ = n − s ) s , so we obtain:˜ f = 1 t h ℓ t j − t + C t −√ + C t √ i (35)At inflationary (early) Universe, when t << t ,the dominant part of the ˜ f is written as follows:˜ f ∼ C t − (7+ √ . (36)Since in this limit, T ∼ H t , T G ∼ H ( H − t (37)So, the reconstructed f ( T, T G ) for inflationary era is written as the following: f ( T, T G ) = C h − T G r T i √ . (38)4So, f ( T, T G ) produces also the inflationary (phantom) solutions as well.At late time,i.e. in the de-Sitter epoch when H ( t ) ∼ H , we use of this fact that : T ∼ H , T G ∼ H , ˙ f T G = ˙ T G f T G ,T G + ˙ T f T G ,T | H = H = 0 (39)so the reconstruction scheme gives us: f ( T, T G ) = √ T F (cid:18) T G √ T (cid:19) − T + 6 − s/ n m − sp s − T s/ . (40)Where F denotes an arbitrary function. The above f ( T, T G ) reproduces de-Sitter (late time) epoch. IV. RECONSTRUCTION SCHEME FOR INTERMEDIATE SCALE FACTOR
Next, we consider the following scale factor (Barrow et al., 2006) a ( t ) = exp ( At m ) , < m < . (41)The scale factor and Hubble parameter is suitably chosen so that it is consistent with the interme-diate expansion: H ( t ) = Amt m − . (42)The scale factor is necessary to perform the analysis and therefore working with a hypothetical scalefactor may not be consistent with the inflationary scenario. Hence we picked the intermediate scalefactor which is also consistent with astrophysical observations (Barrow et al., 2006). Subsequentlywe have the following differential equation3 n (cid:18) t − m Am (cid:19) − s = 12 (cid:18) A m t − m − f [ t ] + tf ′ [ t ] − m + t ( − m + Amt m ) f ′ [ t ]( − m ) ( − m (3 + 4 At m ))+ t (cid:0)(cid:0)
20 + m (9 + 16 At m ) − m (27 + 20 At m ) (cid:1) f ′ [ t ] − t ( − m (3 + 4 At m )) f ′′ [ t ] (cid:1) ( − m ) ( − m (3 + 4 At m )) ! (43)Solving Eq. (43) numerically and plotting reconstructed ˜ f in FIGs. and for s = − s = 2. We observe that for s = 2 as well as s = −
2, the reconstructed f ( T, T G ) model isdisplaying decreasing pattern. It is noted from FIG. (for s = 2) that the reconstructed w DE crosses phantom boundary at t ≈ .
15 and hence behaves like quintessence. However, in Fig. (for s = − w DE < − f ( T, T G )5 - ´ - ´ - ´ - ´ t f Ž H s = L FIG. 9: Plot of reconstructed ˜ f taking s = 2 forintermediate scale factor. - - - - - - t f Ž H s = - L FIG. 10: Plot of reconstructed ˜ f taking s = − - - - - t Ω D E FIG. 11: Plot of the reconstructed EoS parame-ter for s = 2 for intermediate scale factor. - - - - - - - t Ω D E FIG. 12: Plot of the reconstructed EoS parame-ter for s = − pertains to phantom era of the universe and hence it is consistent with the behavior of PDE inEinstein gravity for s = −
2. It may be noted (for s = 2 case) that we have taken n = 0 . , A = 10and red, green and blue lines correspond to m = 0 . , . , .
40 respectively. On the otherhand, for s = −
2, we have taken n = 4 , A = 0 . m = 0 . , . , .
30 respectively.6
V. RECONSTRUCTION SCHEME FOR BOUNCING SCALE FACTOR
Inflation is a solution for flatness problem in big-bang cosmology. Bounncing scenario predicts atransitionary inflationary Universe,in which the Universe evoles from a contracting epoch (
H <
H > a ( t ) reaches a local minima. So thecosmological solution is non-singular. In GB gravity,bouncing solutions widely studied in literature(Bamba et al. 2014a,2104b; Odintsov et al. 2014; Makarenko et al. 2014; Amoros et al. 2013;Nojiri et al. 2003; Lidesy et al. 2002a,2002b).This scale factor takes the following form (Myrzakulov and Sebastiani, 2014) a ( t ) = a + α ( t − t ) n , H ( t ) = 2 nα ( t − t ) n − a + α ( t − t ) n , n = 1 , , ... (44)where a , α are positive (dimensional) constants and n is a positive natural number. The time ofthe bounce is fixed at t = t . When t < t , the scale factor decreases and we have a contractionwith negative Hubble parameter. At t = t , we have the bounce, such that a ( t = t ) = a , andwhen t > t the scale factor increases and the universe expands with positive Hubble parameter.It should be mentioned that for sake of simplicity (without any loss of generality) we have taken n in the power law form as well as in the PDE density (Eq. 14). For this choice of scale factor, weget the following differential equation14 " n ( t − t ) − n α ( a + ( t − t ) n α ) + ( t − t ) (cid:0) a + ( t − t ) n α (cid:1) f ′ [ t ] a ( − n ) − t − t ) n α − t − t ) (cid:0) a + ( t − t ) n α (cid:1) f ′ [ t ] a − a n + ( t − t ) n α − f [ t ] (cid:0) ( t − t ) (cid:0)(cid:0) a ( − n )( − n ) − a ( −
20 + 3 n (9 + 2 n ))( t − t ) n α + 10( t − t ) n α (cid:1) × f ′ [ t ] − ( t − t ) (cid:0) a ( − n ) − t − t ) n α (cid:1) (cid:0) a + ( t − t ) n α (cid:1) f ′′ [ t ] (cid:1)(cid:1) (( − n ) ( a (2 − n ) × +2( t − t ) n α (cid:1) ) − i = 32 s n ( t − t ) − n (cid:0) a + ( t − t ) n α (cid:1) nα ! − s . (45)Solving Eq. (45) numerically and plotting reconstructed ˜ f in FIGs. and , we observe thatthe reconstructed f ( T, T G ) is displaying increasing pattern for s = 2. However, for s = −
2, thereconstructed f ( T, T G ) is displaying increasing pattern and is tending to 0 at late stage of theuniverse. It is also noted from FIG. (for s = 2) that the reconstructed w DE → − w DE > −
1. However, it never crosses phantom boundary and hence behaves like quintessencethroughout. However, in FIG. , it can be observed that for s = − w DE < − s = − f ( T, T G ) pertains to phantom era of the universe and7 t f H s = L FIG. 13: Plot of reconstructed ˜ f taking s = 2for bounce with power-law scale factor. - - - - t f H s = - L FIG. 14: Plot of reconstructed ˜ f taking s = − - - - - t Ω D E FIG. 15: Plot of the reconstructed EoS param-eter for s = 2 for bounce with power-law scalefactor. - - - - - t Ω D E FIG. 16: Plot of the reconstructed EoS parame-ter for s = − hence it is consistent with the behavior of PDE in Einstein gravity for s = −
2. It may benoted that for both of the cases, we have taken n = 2 , A = 10 and red, green and blue linescorrespond to m = 0 . , . , .
40 respectively. On the other hand, for s = − n = 4 , a = 10 . , α = 10 . n = 6 , , VI. RECONSTRUCTION THROUGH A SEMI ANALYTIC FORM OF f We assume that f ( T, T G ) realizes the form f ( T, T G ) ≡ f = b + b t + b t + b t + ... (46)For this choice of f , Eq. (9) takes the form ρ DE = (cid:20) − b − b t − b t − b t + 6 H + ( b + t (2 b +3 b t )) HH ′ + ( b + t (2 b +3 b t )) H ( H + H ′ ) H ′ (2 H + H ′ )+ HH ′′ − H ( b +3 b t ) H ( H ′ ( H + H ′ ) + HH ′′ ) − b + t (2 b +3 b t )) ( H ′ +4 H H ′′ +6 HH ′ H ′′ + H ( H ′ + H ))) H ′ (2 H + H ′ )+ HH ′′ ) (cid:21) (47)which, on setting equal to ρ Λ gives rise to a differential equation on H that is solve numericallyto generate the reconstruction of H and subsequently EoS parameter w DE = − − H H that areplotted in Figs. 17-20. We have set the coefficients of the polynomial as b = 0 . , b = 0 . , b =0 . , b = 0 .
2. In this reconstruction through an assumed polynomial form of f , the Hubbleparameter H has been reconstructed and the reconstructed ˜ H has been examined for its first timederivative and it is observed that ˙˜ H < s = ± w DE < − s = ± w DE is consistent with the basic property ofpilgrim dark energy. VII. CONCLUDING REMARKS
Pilgrim dark energy (PDE) model is studied in this paper and Hubble horizon has been usedas an IR cutoff. The basic assumption of this model is that phantom acceleration prevents theformation of the BH. The said PDE is considered in a modified gravity f ( T, T G ), which has beenconstructed by Kofinas and Saridakis (2014) 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 Gauss-Bonnet term).We have compiled our work in two phases: Firstly, we have assumed different scale factors such as a ( t ) = a t m , H = H + H t , a ( t ) = exp ( At m ) and a ( t ) = a + α ( t − t ) n . We have reconstructed f and subsequently w DE in this scenario. Secondly, we have assumed analytic function such f = b + b t + b t + b t and reconstructed Hubble parameter and w DE without any choice ofscale factor.Throughout the study, we have considered s = − s = 2, separately. We have observedthat s = −
2, as described in PDE (Wei, 2012), seems more realistic choice for s than s = 2 and9 - ´ - ´ - ´ - ´ - ´ - ´ - ´ t H Ž H s = L FIG. 17: Plot of reconstructed ˜ H for s = 2. - ´ - ´ - ´ - ´ t H Ž H s = - L FIG. 18: Plot of reconstructed ˜ H for s = − - - - - t w D E H s = L FIG. 19: Plot of the reconstructed EoS parame-ter for s = 2. - - - - - t w D E H s = - L FIG. 20: Plot of the reconstructed EoS parame-ter for s = − this outcome of the present reconstruction work is consistent with Wei (2012). Moreover, it hasbeen observed that the reconstructed w DE , irrespective of choices of scale factor or a choice of f ,exhibit a more aggressive phantom-like behavior for s = − s = 2. This result also matchesthe study of Wei (2012). Hence, it is finally concluded that PDE, when considered in f ( T, T G )gravity is capable of attaining the phantom phase of the universe. Acknowledgment
Sincere thanks are due to the anonymous reviewer for constructive suggestions. The first author0(SC) wishes to acknowledge the financial support from Department of Science and Technology,Govt. of India under Project Grant no. SR/FTP/PS-167/2011.
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