Inconsistences in Interacting Agegraphic Dark Energy Models
aa r X i v : . [ h e p - t h ] S e p Inconsistences in Interacting Agegraphic Dark EnergyModels
Cheng-Yi Sun ∗ ,a and Yu Song aa Institute of Modern Physics, Northwest University,Xian 710069, P.R. China.
December 14, 2018
Abstract
It is found that the origin agegraphic dark energy tracks the matter in the matter-dominated epoch and then the subsequent dark-energy-dominated epoch becomes impos-sible. It is argued that the difficulty can be removed when the interaction between theagegraphic dark energy and dark matter is considered. In the note, by discussing threedifferent interacting models, we find that the difficulty still stands even in the interactingmodels. Furthermore, we find that in the interacting models, there exists the other seriousinconsistence that the existence of the radiation/matter-dominated epoch contradicts theability of agegraphic dark energy in driving the accelerated expansion. The contradictioncan be avoided in one of the three models if some constraints on the parameters hold.
PACS: 95.36.+x, 98.80.Qc, 98.80.-k
Key words: agegraphic dark energy, interacting, track
Increasing evidence suggests that the expansion of our universe is being accelerated [1, 2, 3].Within the framework of the general relativity, the acceleration can be phenomenally attributedto the existence of a mysterious exotic component with negative pressure, namely the dark energy[4, 5]. However, we know little about the nature of dark energy. The most nature, simple and ∗ [email protected]; [email protected] ρ D = 3 n M p T . (1)Here M p = (8 πG ) − / and T is chosen to be the age of our universe T = Z t dt ′ = Z a daHa , (2)where a is the scale factor of our universe, H ≡ ˙ a/a is the Hubble parameter and a dot denotesthe derivative with respect to cosmic time. However, it is found that the agegraphic dark energyproposed in [6] tracks the matter in the matter-dominated epoch [7]. This can be understood easily[9]. In the matter-dominated epoch, a ∝ t / . Then we have ρ D ∝ a − . Since the energy densityof matter ρ m ∝ a − , ρ D tracks ρ m in the matter-dominated epoch and the dark-energy-dominatedepoch becomes impossible. This is of course unacceptable.Two ways out of the difficulty are suggested. The first one is to replace T with T + δ [6],where δ is a constant with dimension of time. The second one is the so-called new agegraphicdark energy by replacing T with the conformal time η [7]. Both of the ways change Eq.(1). In[10], it is argued that the difficulty in the origin version can also be removed when the interactionbetween the agegraphic dark energy and matter is considered.In note, the interacting agegraphic dark energy models with three different forms of interactionare considered respectively. We find that in the model with interaction proportional to the energydensity of matter, ρ D still tracks ρ m in the matter-dominated epoch. Furthermore, we find thatthe existence of the matter-dominated epoch contradicts the ability of agegraphic dark energy indriving the accelerated expansion even in the interacting models.The paper is organized as follows. In the next section, we will discuss the difficulties ofthe agegraphic dark energy. In Sec.3, we will recall the analysis in [10] which tells us thatthe agegraphic dark energy will not track the matter during the matter-dominated epoch if theinteraction between agegraphic dark energy and dark matter is considered. In Sec.4, we willshow that there exists the other inconsistence in the interacting models. Finally, Conclusions andDiscussions are given. 2 Inconsistences in Agegraphic Dark Energy Model
Considering the flat Friedmann-Robertson-Walk universe with the agegraphic dark energy andpressureless matter, the corresponding Friedmann equation is H = 13 M p ( ρ m + ρ D ) . (3)By defining ρ c = 3 H M p , Ω m = ρ m ρ c , Ω D = ρ D ρ c , (4)we may rewrite the Friedmann equation as1 = Ω m + Ω D . (5)And from Eq.(1), we can easily find that Ω D = n H T . (6)The conservation laws of the agegraphic dark energy and matter are respectively˙ ρ m + 3 Hρ m = 0 , (7)˙ ρ D + 3 H (1 + w D ) ρ D = 0 , (8)where w D is the equation-of-state (EoS) parameter of the agegraphic dark energy. Taking thederivative of Eq.(1) with respect to the cosmic time t , we can get˙ ρ D + H √ Ω D n ρ D = 0 . (9)Comparing the equation with Eq(8), we get [6] w D = − n p Ω D . (10)From Eq.(6) and using Eqs.(3), (4), (7) and (9), we can have [6]Ω ′ D = Ω D (1 − Ω D ) (cid:16) − n p Ω D (cid:17) , (11)where a prime denotes the derivative with respect to the e-folding time N = ln a . The evolutionof Ω D governed by Eq.(11) has been analyzed in Ref.[6] and it is found that the agegraphic darkenergy model works well [6]. 3owever, it is found in [6, 7] that there exists an implicit inconsistence in the agegraphicdark energy model. In the matter-dominated epoch with Ω D ≪
1, the Friedmann equationapproximately becomes H ≃ M p ρ m . (12)Together with Eq.(7), we have a ∝ t / . Then we have H = 23 t . Substituting the result into Eq.(6), we haveΩ DI = 9 n . (13)Hereafter we use the subscript I to denote the initial value of Ω D when deep in the matter-dominated epoch. It can be easily checked that Eq.(13) is also a critical point of Eq.(11). In fact,there exist the other two critical points of Eq.(11),Ω D = 0 , (14)Ω D = 1 . (15)Obviously, Eq.(14) is unstable. Since Eq.(13) is the value of Ω D deep in the matter-dominatedepoch, then we have Ω DI = 9 n ≪ . (16)With this result, it can be checked easily that Eq.(13) is an attractor, while Eq.(15) is unstable.This implies Ω D → n ≪ a → + ∞ and the subsequent dark-energy-dominated epochbecomes impossible.Furthermore, we find that Eq.(13) indicates the other serious problem. This can be easilyshown as follows. From Eq.(16), we have n ≪ . On the other hand, Eq.(10) tells us that the necessary condition for agegraphic dark energy todrive the accelerated expansion is n > . (17)So the matter-dominated epoch contradicts the ability of agegraphic dark energy in driving theaccelerated expansion. 4 similar conclusion is also given in Ref.[8]. By extending the discussion of agegraphic darkenergy model to include the radiation-dominated epoch, the author in [8] noted that the boundimposed on the fractional dark energy density parameter Ω D < . n < . (18)The contradiction between Eqs(17) and (18) is obvious. Then it is interesting for us to explorewhether the contradiction can be solved when the interaction is involved. In [10], it is shown that the agegraphic dark energy will not track the matter when the interactionbetween the agegraphic dark energy and matter is considered. In the section, we will recall theanalysis in [10]. In the next section, we will show that, actually, there still exist some inconsistenceseven in the interacting models.The conservation laws of the agegraphic dark energy and matter are respectively [10]˙ ρ m + 3 Hρ m = Q, (19)˙ ρ D + 3 H (1 + w D ) ρ D = − Q, (20)where Q denotes the phenomenological interaction term. Comparing Eq.(9) and Eq.(20), we have[10] w D = − n p Ω D − Q Hρ D . (21)If Q = 0, the equation reduces to Eq.(10). From Eqs.(3), (9) and (19), we have¨ aa = − πG (cid:16) ρ m − ρ D + 2 √ Ω D n ρ D − QH (cid:17) . (22)From Eq.(6) and using Eqs.(3), (4), (9) and (19), we can obtain the evolving equation of Ω D with interaction [6] Ω ′ D = Ω D h (1 − Ω D ) (cid:16) − n p Ω D (cid:17) − Q M p H i . (23)If Q = 0, this equation reduces to Eq.(11).In [10], Eq.(23) has been solved numerically with the initial condition Ω D = 0 .
7, and thereasonable evolution of Ω D has been shown that the agegraphic dark energy is negligible in thepast and eventually dominates the evolution of our universe. Then it seems to be reasonable toconclude that the inconsistence in the non-interacting agegraphic dark energy has been removed inthe interacting models. However, as shown below, we find that there still exist the inconsistencesin the interacting agegraphic dark energy models.5 Inconsistences in Interacting Agegraphic Dark EnergyModels
In the section, owing to the lack of the knowledge of micro-origin of the interaction, we simplyconsider three forms of the interaction Q = 3 βHρ m , αHρ D , γH ( ρ m + ρ D ) , (24)which are used in the literature most often [10, 11, 12, 13, 14, 15]. Here β, α and γ are positiveconstants. Q = 3 βH ρ m Firstly, we consider the interacting agegraphic dark energy model with Q = 3 βHρ m . Then Eq.(23)reads Ω ′ D = Ω D (1 − Ω D ) h − β ) − n p Ω D (cid:17)i . (25)And the conservation law of matter reads˙ ρ m + 3 Hρ m = 3 Hβρ m . (26)From the above equation, we can easily obtain ρ m ∝ a − − β ) . (27)Then, in the matter-dominated epoch with Ω D ≪
1, from Eq.(12), we have a ∝ t − β ) ⇒ H = 23(1 − β ) 1 t . (28)Substituting the equation into Eq.(6), we haveΩ DI = h − β ) n i . (29)This is the initial value of Ω D in the model with Q = 3 βHρ m when deep in the matter-dominatedepoch. It can be checked easily that Eq.(29) is also a critical point of Eq.(25). And the othertwo critical points of Eq.(25) are given in Eqs.(14) and (15) respectively. Since deep in thematter-dominated epoch Ω DI ≪ , (30)6hen we can find that Eq.(29) is an attractor of Eq.(25) while the other two critical points ofEq.(25) are unstable. This implies Ω D → h − β ) n i ≪ a → + ∞ and the subsequentdark-energy-dominated epoch becomes impossible.Furthermore, we find that the contradiction between the matter-dominated epoch and theability of agegraphic dark energy in driving the accelerated expansion also exists in the interactingmodel. Let us show it. From Eq.(22), in the matter-dominated epoch, approximately we have¨ aa ≃ − πG − β ) ρ m , since ρ D ≪ ρ m . Then we must have β < , (31)since, if β > , the expansion of the universe during the matter-dominated epoch would beaccelerated and then the observed large scale structure of the universe could not be formed.Substituting β < into Eq.(29), we have Ω DI > n . (32)Then together with Eq.(30), we have n ≪ , (33)On the other hand, in the dark-energy-dominated epoch, since Ω D ≃ ρ m ≪ ρ D , fromEq.(22) we have ¨ aa ≃ − πG (cid:16) n − (cid:17) ρ D , (34)where Q = 3 βHρ m has been used. Then in order for the agegraphic dark energy to drive theaccelerated expansion, we must have n > . Obviously, this result contradicts Eq.(33). So the contradiction between the matter-dominatedepoch and the ability of agegraphic dark energy in driving the accelerated expansion still standsin the interacting model with Q = 3 βHρ m . Q = 3 αH ρ D Secondly, we consider the case of Q = 3 αHρ D . Then Eq.(23) readsΩ ′ D = Ω D h (1 − Ω D ) (cid:16) − n p Ω D (cid:17) − α Ω D i . (35)7n the matter-dominated epoch, Eq.(19) reads approximately˙ ρ m + 3 Hρ m ≃ , (36)since ρ D ≪ ρ m . Then in the matter-dominated epoch, approximately we have ρ m ∝ a − , (37)So from Eqs.(12) and (37), we have a ∝ t ⇒ H = 23 t . (38)Substituting the equation into Eq.(6), we haveΩ DI = 9 n . (39)Then, when deep in the matter-dominated epoch, we have the same initial value of Ω D as inthe non-interacting model. But the case is different. Here, due to the interaction, Eq.(39) isnot the critical point of the evolving equation (35). Then it seems that the tracking behaviorof agegraphic dark energy during the matter-dominated epoch is eliminated, and eventually theagegraphic dark energy will become dominated. However, we find this problem still stands. Asin the non-interacting model, here we also haveΩ DI = 9 n ≪ . (40)Then, using this result, from Eq.(35), we findΩ ′ D < , for Ω DI ≤ Ω D ≤ . (41)Here we have used α >
0. Eq.(41) tells us that Ω D will never become larger than n and willapproach a value less than Ω DI = n as ln a → + ∞ , and consequently the subsequent dark-energy-dominated epoch is impossible.Now let us show whether the contradiction between the matter-dominated epoch and theability of agegraphic dark energy in driving the accelerated expansion exists in the case. Eq.(40)implies n ≪ . (42)On the other hand, in the dare-energy-dominated epoch, since Ω D ≃ ρ D ≫ ρ m , fromEq.(22), approximately we have ¨ aa ≃ − πG (cid:16) − n − α (cid:17) ρ D . n < α. (43)Together with Eq.(43), we have 22 + 3 α < n ≪ . (44)Then in the interacting model, we can remove the contradiction between the matter-dominatedepoch and the ability of agegraphic dark energy in driving the accelerated expansion if Eq.(44)holds. Q = 3 γH ( ρ D + ρ m ) Finally, we consider the case of Q = 3 γH ( ρ D + ρ m ). In the case Eq.(23) readsΩ ′ D = Ω D h (1 − Ω D ) (cid:16) − n p Ω D (cid:17) − γ i . (45)In the matter-dominated epoch, since ρ D ≪ ρ m , the conservation law of matter reads approxi-mately ˙ ρ m + 3 H (1 − γ ) ρ m ≃ . (46)Then we have ρ m ∝ a − − γ ) . (47)Together with Eq.(12), in the matter-dominated epoch we have a ∝ t − γ ) ⇒ H = 23(1 − γ ) 1 t . (48)Substituting the equation into Eq.(6), we haveΩ DI = h − γ ) n i . (49)Due to the interaction, Eq.(49) is not the critical point of Eq.(45). However, since in the matter-dominated epoch Ω DI = h − γ ) n i ≪ , (50)then from Eq.(45) we find Ω ′ D < , for Ω DI ≤ Ω D ≤ . (51)9ere γ > Q = 3 αHρ D , Eq.(51) tells us that Ω D will approacha value smaller than Ω DI = h − γ ) n i as ln a → + ∞ and the subsequent dark-energy-dominatedepoch becomes impossible.Furthermore, we find that as in the case of Q = 3 βHρ m , in the model with Q = 3 γH ( ρ D + ρ m ),the matter-dominated epoch contradicts the ability of agegraphic dark energy in driving theaccelerated expansion, too. Let us show it. In the matter-dominated epoch, since ρ D ≪ ρ m , fromEq.(22), approximately we have ¨ aa ≃ − πG − γ ) ρ m , (52)Then we must have γ < , (53)in order for the expansion of the universe to be decelerated to form the large scale structure duringthe matter-dominated epoch. Using Eqs.(53) and (50) we have n ≪ . (54)On the other hand, in the dark-energy-dominated epoch, since ρ D ≫ ρ m and Ω D ≃
1, fromEq.(22), approximately we have ¨ aa ≃ − πG (cid:16) − n − γ (cid:17) ρ D , (55)where Q = 3 γH ( ρ D + ρ m ) ≃ γHρ D has been used. Then the necessary condition for agegraphicdark energy to drive the accelerated expansion is n >
22 + 3 γ . (56)Together with Eq.(53), we have n > . (57)Roughly, it seems that Eq.(57) may not contradict Eq.(54). But the contradiction between Eq.(57)and Eq.(18) is obvious. Here we note that the condition (18) imposed by BBN on the agegraphicdark energy model is not effected by the interaction between dark energy and matter, sinceboth agegraphic dark energy and matter are negligible during the radiation-dominated epoch.So, in the interacting agegraphic dark energy model with Q = 3 γH ( ρ D + ρ m ), the existence ofthe radiation-dominated epoch contradicts the ability of agegraphic dark energy in driving theaccelerated expansion. 10 Conclusions and Discussions
In this note, we recall the inconsistence in the origin agegraphic dark energy model that theagegraphic dark energy tracks the matter in the matter-dominated epoch. And furthermore, wepoint out that there is the other more serious inconsistence in the model that the matter-dominatedepoch contradicts the ability of agegraphic dark energy in driving the accelerated expansion.Then, by considering three kinds of phenomenological interaction between the agegraphic darkenergy and matter, we analyze the interacting agegraphic dark energy models. We find that inthe dark energy model with interaction Q = 3 βHρ m , the agegraphic dark energy still tracksthe matter during the matter-dominated epoch, and the contradiction between the existenceof the matter-dominated epoch and the ability of the agegraphic dark energy in driving theaccelerated expansion also exists. In the models with Q = 3 αHρ D and Q = 3 αHρ D , it is stillimpossible for agegraphic dark energy to become dominated. And in the model with Q = 3 αHρ D the contradiction between the existence of the matter-dominated epoch and the ability of theagegraphic dark energy in driving the accelerated expansion can be avoided if the condition (44)holds. But in the model with Q = 3 γH ( ρ m + ρ D ), the ability of the agegraphic dark energy indriving the accelerated expansion contradicts the bound imposed by BBN on the agegraphic darkenergy.Then it seems that none of the three interacting agegraphic dark energy models can be takenas serious candidate for realistic dark energy. In Ref.[17], the authors studied the interactingagegraphic dark energy model by using a general form of interaction Q = 3 H ( αρ D + βρ m ). Inthe matter-dominated epoch with ρ D ≪ ρ m , the interaction reduces to Q ≃ Hβρ m . So β and n should satisfy the constraints (31) and (33) respectively, since the two constraints are obtainedby analyzing the model with Q = 3 Hβρ m during the matter-dominated epoch. Similarly, sincethe general interaction form reduces to Q ≃ Hαρ D in the dark-energy-dominated epoch, α and n should satisfy the condition (44). However, it can be checked easily that the values ofparameters used in Ref.[17] does not satisfy Eq.(44). Then there exist implicit inconsistences inthe interacting agegraphic dark energy models analyzed in Ref.[17], although the authors obtainedthe reasonable behaviors of Ω D and Ω m by solving the evolving equations numerically. This wouldnot be confusing or astonishing if we recall that even in the non-interacting agegraphic darkenergy model, the reasonable behavior of Ω D can be obtained by solving the evolving equationsnumerically with n = 3 and the initial condition Ω D = 0 . Acknowledgments
This work is supported by Research Fund for the Doctoral Program of Higher Education of Chinaunder Grant No. 20106101120023. 11 eferences [1] A. G. Riess et al. , Astron. J. 116, 1009 (1998), [astro-ph/9805201]; S. Perlmutter et al. ,Astrophys. J. 517, 565 (1999), [astro-ph/9812133].[2] D. N. Spergel et al. , Astrophys. J. Suppl. 148, 175 (2003), [astro-ph/0302209]; D. N. Spergel et al. , Astrophys. J. Suppl. 170, 377 (2007), [astro-ph/0603449].[3] M. Tegmark et al. , Phys. Rev. D 69, 103501 (2004), [astro-ph/0310723]; K. Abazajian etal. , Astron. J. 128, 502 (2004), [astro-ph/0403325]; K. Abazajian et al.et al.