Reconstructing interacting entropy-corrected holographic scalar field models of dark energy in non-flat universe

Transcription

1 arxiv:0.3634v physis.gen-ph] 9 Jan 0 Reonstruting interating entropy-orreted holographi salar field models of dark energy in non-flat universe K. Karami,, M.S. Khaledian, Mubasher Jamil 3 Department of Physis, University of Kurdistan, Pasdaran St., Sanandaj, Iran Researh Institute for Astronomy & Astrophysis of Maragha RIAAM, Maragha, Iran 3 Center for Advaned Mathematis and Physis CAMP, National University of Sienes and Tehnology NUST, Islamabad, Pakistan September 7, 07 Abstrat Here we onsider the entropy-orreted version of the holographi dark energy model in the non-flat universe. We obtain the equation of state parameter in the presene of interation between dark energy and dark matter. Moreover, we reonstrut the potential and the dynamis of the quintessene, tahyon, K-essene and dilaton salar field models aording to the evolutionary behavior of the interating entropy-orreted holographi dark energy model. PACS numbers: x, Pp KKarami@uok.a.ir MS.Khaledian@uok.a.ir mjamil@amp.nust.edu.pk

2 Introdution The present aeleration of the universe expansion has been well established through numerous and omplementary osmologial observations ]. A omponent whih is responsible for this aelerated expansion usually dubbed dark energy DE. However, the nature of DE is still unknown, and people have proposed some andidates to desribe it for a good review see, 3] and referenes therein. The holographi DE HDE is one of interesting DE andidates whih was proposed based on the holographi priniple 4]. Aording to the holographi priniple, the number of degrees of freedom of a physial system should sale with its bounding area rather than with its volume 5] and it should be onstrained by an infrared ut-off 6]. By applying the holographi priniple to osmology, one an obtain the upper bound of the entropy ontained in the universe 7]. Following this line, Li 8] suggested the following onstraint onitsenergydensity ρ Λ 3 MPL, theequality signholding onlywhentheholographi bound is saturated. In this expression is a numerial onstant, L denotes the IR ut-off radius and M P = 8πG / is the redued Plank Mass. The HDE models have been studied widely in the literature 9, 0,, ]. Obviously, in the derivation of HDE, the blak hole entropy S BH plays an important role. As is well known, usually, S BH = A/4G, where A L is the area of horizon. However, in the literature, this entropy-area relation an be modified to 3] S BH = A 4G αln A 4G β, where α and β are dimensionless onstants of order unity. These orretions an appear in the blak hole entropy in loop quantum gravity LQG 4]. They an also be due to thermal equilibrium flutuation, quantum flutuation, or mass and harge flutuations for review see 4] and referenes therein. Using the orreted entropy-area relation, the energy density of the entropy-orreted HDE ECHDE an be obtained as 4] ρ Λ = 3 M P L αl 4 lnm P L βl 4, where α and β are dimensionless onstants of order unity. In the speial ase α = β = 0, the above equation yields the well-known HDE density. Sine the last two terms in Eq. an be omparable to the first term only when L is very small, the orretions make sense only at the early stage of the universe. When the universe beomes large, ECHDE redues to the ordinary HDE 4]. Reonstruting the holographi and agegraphi salar field models of DE is one of interesting issue whih has been investigated in the literature5, 6, 7]. The holographi and the agegraphi DE models are originated from some onsiderations of the features of the quantum theory of gravity. On the other hand, the salar field models suh as quintessene, tahyon, K-essene and dilaton are often regarded as an effetive desription of an underlying theory of DE 7]. The salar field models an mimi osmologial onstant at the present epoh and an give rise to other observed values of the equation of state parameter ω reent data indiate that ω lies in a narrow strip around ω = ω Λ = and is onsistent with being below this value 8]. They an also alleviate the fine tuning and oinidene problems 8]. Therefore it beomes meaningful to reonstrut the salar

3 field models from some DE models possessing some signifiant features of the LQG theory, suh as ECHDE and entropy-orreted agegraphi DE ECADE models. An interesting feature of entropy-orreted DE is that it permits suessive aelerationdeeleration phase transitions. Moreover the osmi oinidene problem is resolved and the universe eventually tends to de Sitter expansion 9]. Here our aim is to investigate the orrespondene between the entropy-orreted version of the interating HDE model with the quintessene, tahyon, K-essene and dilaton salar field models in the non-flat universe. These orrespondenes are essential to understand the onnetion of various salar field models of DE with the ECHDE. This paper is organized as follows. In Setion, we obtain the equation of state parameter for the interating ECHDE model in a non-flat universe. In Setions 3-6, we suggest a orrespondene between the interating ECHDE and the quintessene, tahyon, K-essene and dilaton salar field models in the presene of a spatial urvature. We reonstrut the potentials and the dynamis for these salar field models, whih desribe aelerated expansion of the universe. Setion 7 is devoted to onlusions. Interating ECHDE and DM in non-flat universe Within the framework of the standard FRW osmology, ds = dt a t dr kr r dω, 3 for the non-flat FRW universe ontaining the ECHDE and DM, the first Friedmann equation takes the form H k a = ρ 3MP Λ ρ m, 4 where k = 0,, represent a flat, losed and open FRW universe, respetively. Also ρ Λ and ρ m are the energy density of ECHDE and DM, respetively. Observational evidenes have implied that our universe is not a perfetly flat universe and that it possesses a small positive urvature 0]. Besides, as usually believed, an early inflation era leads to a flat universe. This is not a neessary onsequene if the number of e-foldings is not very large ]. Additionally the parameter Ω k disussed below represents the ontribution in the total energy density from the spatial urvature and is onstrained as < Ω k < with 95% onfidene level by urrent observations ]. It has been shown that a non-zero positive urvature parameter k allows for a boune, thereby preventing the osmi singularities without violating the null energy ondition ρ p 0 3]. From Eq. 4, we an write where we have used the following definitions Ω m = Ω k, 5 Ω m = ρ m ρ r = ρ m 3M PH, = ρ Λ ρ r = ρ Λ 3M PH, Ω k = k a H. 6 3

4 The reent observational evidene provided by the galaxy luster Abell A586 supports the interation between DE and DM 4]. This motivates us to onsider the interation between ECHDE and DM. Hene ρ Λ and ρ m do not onserve separately and the energy onservation equations for ECHDE and DM are ρ Λ 3Hω Λ ρ Λ = Q, 7 ρ m 3Hρ m = Q, 8 where following 5], we hoose Q = Γρ Λ as an interation term and Γ = 3b H Ω k is the deay rate of the ECHDE omponent into DM with a oupling onstant b. Although this expression for the interation term may look purely phenomenologial but different Lagrangians have been proposed in support of it 6]. Note that in Eq., taking L as the size of the urrent universe, for instane, the Hubble sale, the resulting energy density is omparable to the present day DE. However, as found by Hsu 7], in that ase, the evolution of the DE is the same as that of DM dust matter, and therefore it annot drive the universe to aelerated expansion. The same appears if one hooses the partile horizon of the universe as the length sale L 8]. To obtain an aelerating universe, Li 8] proposed that for a flat universe, L should be the future event horizon R h and Huang and Li ] argued that for the non-flat ase, the IR ut-off L should be defined as L = a sinn k y, 9 k where y = R h a = dt r t a = dr. 0 0 kr Here R h is the radial size of the event horizon measured in the r diretion and L is the radius of the event horizon measured on the sphere of the horizon ]. For a flat universe, L = R h. The last integral in Eq. 0 has the expliit form as r 0 dr = kr k sinn k r = From definition ρ Λ = 3M P H and using Eq., we get sin r, k =, r, k = 0, sinh r, k =. γ where γ = L = H Taking time derivative of Eq. 9 and using yields /, 3 M P L αlnm P L β ]. 3 γ / L = osn k y, 4 4

5 where osn k y osy, k =, =, k = 0, oshy, k =. Taking time derivative of Eq. and using and 4, one an obtain 5 HρΛ ρ Λ = γ γ αh 3 M P ] / osn k y. 6 Substituting Eq. 6 in 7 gives the equation of state EoS parameter of the interating ECHDE as ω Λ = b Ωk γ αh 3γ 3 MP ] / osn k y. 7 Note that as we already mentioned, at the very early stage when the universe undergoes an inflation phase, the orretion terms in the ECHDE density beome important. After the end of the inflationary phase, the universe subsequently enters in the radiation and then matter dominated eras. In these two epohs, sine the universe is muh larger, the entropy-orreted terms to ECHDE, namely the last two terms in Eq., an be safely ignored. Therefore if we set α = β = 0, then from Eq. 3 γ = and Eq. 7 reovers the EoS parameter of the ordinary HDE 8] ω Λ = Ωk 3 b osn k y. 8 3 In next setions, we suggest a orrespondene between the interating ECHDE model with the quintessene, tahyon, K-essene and dilaton salar field models in the non-flat universe. 3 Entropy-orreted holographi quintessene model Quintessene is desribed by an ordinary time dependent and homogeneous salar field φ whih is minimally oupled to gravity, but with a partiular potential Vφ that leads to the aelerating universe. The ation for quintessene is given by 3] S = d 4 x g ] gµν µ φ ν φ Vφ. 9 The energy momentum tensor of the field is derived by varying the ation 9 with respet to g µν : T µν = δs g δg µν, 0 5

6 whih gives ] T µν = µ φ ν φ g µν gαβ α φ β φvφ. The energy density and pressure of the quintessene salar field φ are as follows ρ Q = T 0 0 = φ Vφ, p Q = T i i = φ Vφ. 3 The EoS parameter for the quintessene salar field is given by ω Q = p Q ρ Q = φ Vφ φ Vφ. 4 From 4 for ω Q < /3, we find that the universe aelerates when φ < Vφ. Here we establish the orrespondene between the interating ECHDE senario and the quintessene DE model, then equating Eq. 4 with the EoS parameter of interating ECHDE 7, ω Q = ω Λ, and also equating Eq. with, ρ Q = ρ Λ, we have φ = ω Λ ρ Λ, 5 Vφ = ω Λρ Λ. 6 Substituting Eqs. and 7 into Eqs. 5 and 6, one an obtain the kineti energy term and the quintessene potential energy as follows φ = 3M P H b Ω k Vφ = 3M P H 3γ γ αh 3 M P b Ω k 3γ γ αh 3 M P ] / osn k y ], 7 ] / osn k y ]. 8 From Eqs. 7 one an obtain the evolutionary form of the quintessene salar field as a φa φa 0 = M P 3b Ω k a 0 Ω Λ γ αh Ω ] Λ Ω Λ / osn k y ] / da γ 3 MP a, 9 where a 0 is the sale fator at the present time. 6

7 The above integral annot be taken analytially. But during the early inflation era when the orretion terms make sense in the ECHDE density, the Hubble parameter H is onstant and a = a 0 e Ht. Hene the Hubble horizon H and the future event horizon R h = a dt t will oinide i.e. R a h = H = onst. On the other hand, sine an early inflation era leads to a flat universe we have L = R h = H = onst. Also from Eqs. and 5 we have = and osn k y =. Therefore during the early inflation era, Eq. 9 redues to φa = φa 0 a 3 bm P ln. 30 a0 For the late-time universe, i.e. = and Ω k = 0, the universe beomes large and ECHDE redues to the ordinary HDE. In this ase L = R h H and H onst. Now by setting γ = α = β = 0 and osn k y =, the Hubble parameter from Eqs. and 4 an be obtained as H = H 0 H0 t t 0, 3 where H 0 is the Hubble parameter at the present time. After integration of Eq. 3 with respet to t, the sale fator an be obtained as ] a = a 0 H 0 t t 0. 3 Using the above relation, one an rewrite Eq. 3 as Finally for the late-time universe, Eq. 9 yields H = H 0 a a φa = φa 0 M P 3b / a ln. 34 ] a0 4 Entropy-orreted holographi tahyon model In reent years, a huge interest has been devoted in studying the inflationary model with the help of tahyon field. The tahyon field assoiated with unstable D-branes might be responsible for osmologial inflation in the early evolution of the universe, due to tahyon ondensation near the top of the effetive salar potential 9]. Also the tahyoni matter ould suggests some new form of DM at late epoh 30]. The tahyon field has emerged as a possible soure of the DE. A rolling tahyon has an interesting EoS whose parameter smoothly interpolates between and 0 3]. This disovery motivated to take DE as the dynamial quantity, i.e. a variable osmologial onstant and model inflation using tahyons. The effetive Lagrangian density of tahyon matter is given by 3] L = Vφ µ φ µ φ. 35 7

8 The energy density and pressure for the tahyon field are as following 3] ρ T = Vφ φ, 36 p T = Vφ φ, 37 where Vφ is the tahyon potential. The EoS parameter for the tahyon salar field is obtained as ω T = p T ρ T = φ. 38 If we establish the orrespondene between the ECHDE and tahyon DE, then equating Eq. 38 with the EoS parameter of interating ECHDE 7, ω T = ω Λ, and also equating Eq. 36 with, ρ T = ρ Λ, we obtain φ = b Ω k γ αh Ω ] Λ Ω Λ / osn k y ], 39 3γ 3 MP Vφ = 3MP H b Ω k γ αh Ω ] Λ Ω Λ / osn k y ] /. 40 3γ 3 MP From Eq. 39, one an obtain the evolutionary form of the tahyon salar field as φa φa 0 = a a 0 da Ha b Ω k γ αh Ω ] Λ Ω Λ / osn k y ] /. 4 3γ 3 MP During the early inflation era L = R h = H = onst., Eq. 4 yields b a φa = φa 0 ln H, 4 a 0 where = H αlnm 3MP P H β ]. 43 For the late-time universe, i.e. =, Ω k = 0 and γ = α = β = 0, using Eq. 33 one an take the integral 4 as b 3 φa = φa 0 H 0 ] / a a 0 ]. 44 8

9 5 Entropy-orreted holographi K-essene model A model in whih the kineti energy term of the salar field appears in the Lagrangian in a non-anonial way is termed the K-essene model. Suh fields were originally used to model inflation, a senario alled K-inflation33]. Moreover the stable traker solutions for K-essene have been obtained i.e. solutions whih start from arbitrary initial onditions and reah to the same final state of osmi aeleration 34]. The purpose of introduing K-essene is to provide a dynamial explanation whih does not require the fine-tuning of initial onditions. It is also possible to have a situation where the aelerated expansion of the universe arises out of modifiations to the kineti energy of the salar fields. The K- essene is desribed by a general salar field ation whih is a funtion of φ and χ = φ /, and is given by 35, 36] S = d 4 x g pφ,χ, 45 where pφ, χ orresponds to a pressure density as and the energy density of the field φ is pφ,χ = fφ χχ, 46 ρφ,χ = fφ χ3χ. 47 The EoS parameter for the K-essene salar field is obtained as ω K = pφ,χ ρφ,χ = χ 3χ. 48 Equating Eq. 48 with the EoS parameter 7, ω K = ω Λ, we find the solution for χ ] χ = b Ωk γ 3γ αh 3 MP /osn k y ] ] 43b Ωk γ γ αh 3 MP /osn k y ]. 49 Using φ = χ and 49, we obtain the evolutionary form of the K-essene salar field as a da a 0 Ha φa = φa 0 b Ω k 3γ γ αh 3 M P 3b 4 Ω k γ γ αh 3 M P ] /osn k y ] ] /osn k y ] During the early inflation era, Eq. 50 redues to φa = φa 0 4 b / a ln, 5 H 4 3b a0 where is given by Eq. 43. For the late-time universe, i.e. L = R h H and H onst., using Eq. 33 one an take the integral 50 as ] φa = φa 0 b 3 3b 4 9 /.50 / a a 0 H 0. 5

10 6 Entropy-orreted holographi dilaton model The proess of ompatifiation of the string theory from higher to four dimensions introdues the salar dilaton field whih is oupled to urvature invariants. The oeffiient of the kinemati term of the dilaton an be negative in the Einstein frame, whih means that the dilaton behaves as a phantom-type salar field. The pressure Lagrangian density and the energy density of the dilaton DE model is given by 37] p D = χ e λφ χ, 53 ρ D = χ3 e λφ χ, 54 where and λ are positive onstants and χ = φ /. The EoS parameter for the dilaton salar field is given by ω D = p D = e λφ χ ρ D 3 e λφ χ. 55 Equating Eq. 55 with the EoS parameter 7, ω D = ω Λ, we find the following solution e λφ χ = b Ωk γ 3γ αh 3 MP 43b Ωk γ γ αh 3 M P then using φ = χ, we obtain ] /osn k y ] ] /osn k y ], 56 e λφ 4b ] Ω k Ω φ = Λ 4 γ 3γ αh 3 MP /osn k y ] ] 43b Ωk γ γ αh 3 MP /osn Integrating with respet to a, we get λ a da a 0 Ha e λφa = e λφa 0 4b Ω k 4 3γ γ αh 3 M P 43b Ωk γ γ αh 3 M P / k y ] ] /osn. 57 k y ] ] /osn k y ] Therefore the evolutionary form of the dilaton salar field is obtained as φa = λ ln e λφa 0 λ a da a 0 Ha b Ω k γ 3γ αh 3 MP 43b Ωk γ γ αh 3 M P / ] /osn k y ] / ] /osn k y ] During the early inflation era, Eq. 59 yields φa = λ ln e λφa 0 λ b / a ln, 60 H 4 3b a

11 where is given by Eq. 43. For the late-time universe, using Eq. 33 one an take the integral 59 as φa = λ ln λ b 3 43b e λφa 0 / a a 0 ] H Conlusions Here we onsidered the entropy-orreted version of the HDE model whih is in interation with DM in the non-flat FRW universe. The HDE model is an attempt for probing the nature of DE within the framework of quantum gravity ]. We onsidered the logarithmi orretion term to the energy density of HDE model. The addition of orretion terms to the energy density of HDE is motivated from the LQG whih is one of the promising theories of quantum gravity. Using this modified energy density, we obtained the EoS parameter for the interating ECHDE. We established a orrespondene between the interating ECHDE model with the quintessene, tahyon, K-essene and dilaton salar field models in the non-flat FRW universe. These orrespondenes are important to understand how various andidates of DE are mutually related to eah other. In the present ase, the orrespondene is established between ECHDE and various salar field models of DE. We adopted the viewpoint that these salar field models of DE are effetive theories of an underlying theory of DE. Thus, we should be apable of using these salar field models to mimi the evolving behavior of the interating ECHDE and reonstruting the salar field models aording to the evolutionary behavior of the interating ECHDE. We reonstruted the potentials and the dynamis of these salar field models, whih desribe aelerated expansion of the universe, aording to the evolutionary behavior of the interating ECHDE model. We also obtained the expliit evolutionary forms of the orresponding salar fields for the both of early inflation L = R h = H = onst. and late-time aeleration L = R h H and H onst. phases. For late-time aeleration, L is dynamial and L will ontribute in the above expressions and will yield the salar potentials for the DE. This suggests that the vauum energy that produed inflation at early osmi epoh and the one driving late-time osmi aeleration are fundamentally different. Hene the same salar field moves in different potentials at different times. Aknowledgements The authors thank the reviewers for valuable omments. The work of K. Karami has been supported finanially by Researh Institute for Astronomy & Astrophysis of Maragha RIAAM, Maragha, Iran. Referenes ] A.G. Riess, et al., Astron. J. 6, ; S. Perlmutter, et al., Astrophys. J. 57, ;

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