Cosmic Age Problem Revisited in the Holographic Dark Energy Model ∗ Jinglei Cui, Xin Zhang

Physics Letters B 690 (2010) 233–238

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Physics Letters B

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Cosmic age problem revisited in the holographic dark energy model ∗ Jinglei Cui, Xin Zhang

Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China article info abstract

Article history: Because of an old quasar APM 08279 + 5255 at z = 3.91, some dark energy models face the challenge of Received 18 April 2010 the cosmic age problem. It has been shown by Wei and Zhang [H. Wei, S.N. Zhang, Phys. Rev. D 76 (2007) Received in revised form 18 May 2010 063003, arXiv:0707.2129 [astro-ph]] that the holographic dark energy model is also troubled with such a Accepted 20 May 2010 cosmic age problem. In order to accommodate this old quasar and solve the age problem, we propose in Available online 24 May 2010 this Letter to consider the interacting holographic dark energy in a non-ﬂat universe. We show that the Editor: T. Yanagida cosmic age problem can be eliminated when the interaction and spatial curvature are both involved in Keywords: the holographic dark energy model. Holographic dark energy © 2010 Elsevier B.V. All rights reserved. Cosmic age problem

1. Introduction of quantum gravity could be taken into account. It is commonly believed that the holographic principle [16] is just a fundamental The fact that our universe is undergoing accelerated expansion principle of quantum gravity. Based on the effective quantum field has been confirmed by lots of astronomical observations such as theory, Cohen et al. [17] pointed out that the quantum zero-point type Ia supernovae (SNIa) [1], large scale structure (LSS) [2] and energy of a system with size L should not exceed the mass of a 3 4 2 cosmic microwave background (CMB) anisotropy [3].Itisthemost black hole with the same size, i.e., L Λ LMPl, where Λ is the accepted idea that this cosmic acceleration is caused by some kind ultraviolet (UV) cutoff of the effective quantum field theory, which of negative-pressure matter known as dark energy whose energy is closely√ related to the quantum zero-point energy density, and density has been dominative in the universe. The combined anal- MPl ≡ 1/ 8π G is the reduced Planck mass. This observation re- ysis of cosmological observations indicates that the universe today lates the UV cutoff of a system to its infrared (IR) cutoff. When we consists of about 70% dark energy, 30% dust matter (cold dark take the whole universe into account, the vacuum energy related matter plus baryons), and negligible radiation. The famous cos- to this holographic principle can be viewed as dark energy (its en- mological constant λ introduced first by Einstein is the simplest ergy density is denoted as ρΛ hereafter). The largest IR cutoff L is candidate for dark energy. However, the cosmological constant sce- chosen by saturating the inequality, so that we get the holographic nario has to face the so-called “fine-tuning problem” and “cosmic dark energy density coincidence problem” [4]. Many dark energy models have been = 2 2 −2 proposed, while the nature of dark energy is still obscure. Besides ρΛ 3c MPl L (1) quintessence [5], a wide variety of scalar-field dark energy models where c is a numerical constant characterizing all of the uncer- have been studied including k-essence [6], hessence [7], phan- tainties of the theory, and its value can only be determined by tom [8],tachyon[9],quintom[10], ghost condensate [11],etc.In observations. If we take L as the size of the current universe, say, addition, there are other proposals on dark energy such as interact- − the Hubble radius H 1, then the dark energy density will be close ing dark energy models [12], brane world models [13], Chaplygin to the observational result. However, Hsu [18] pointed out that this gas models [14], Yang–Mills condensate models [15], and so on. yields a wrong equation of state for dark energy. Subsequently, The dark energy problem is essentially an issue of quantum Li [19] suggested to choose the future event horizon of the uni- gravity, owing to the concern of the vacuum expectation value of verse as the IR cutoff of this theory. This choice not only gives a some quantum fields in a universe governed by gravity. However, reasonable value for dark energy density, but also leads to an ac- by far, we have no a complete theory of quantum gravity yet. So, it celerated universe. Moreover, the cosmic coincidence problem can seems that we have to consider the effects of gravity in some effec- also be explained successfully in this model, provided that the in- tive quantum field theory in which some fundamental principles flation lasts for more than 60 e-folds. Most recently, a calculation of the Casimir energy of the photon field in a de Sitter space is * Corresponding author. performed [20], and it is a surprising result that the Casimir en- E-mail address: [email protected] (X. Zhang). ergy is indeed proportional to the size of the horizon (the usual

Casimir energy in a cavity is inversely proportional to the size of Ωm + ΩΛ + Ωk = 1. (4) the cavity), in agreement with the holographic dark energy model. Now, let us consider some interaction between holographic Up to now, the holographic dark energy model has been tested dark energy and matter: by various observational data including SNIa [21],SNIa+ BAO + CMB [22,23], X-ray gas mass fraction of galaxy clusters [24],dif- ρ˙ + 3Hρ = Q , (5) ferential ages of passively evolving galaxies [25], Sandage–Leob m m test [26], and so on [27]. These analyses show that the holographic ρ˙Λ + 3H(ρΛ + pΛ) =−Q , (6) dark energy model is consistent with the observational data. How- ever, Wei and Zhang [28] used some old high redshift objects where Q denotes the phenomenological interaction term. Owing (OHROs) to test the holographic dark energy model and found that to the lack of the knowledge of micro-origin of the interaction, the original holographic dark energy model can be ruled out un- we simply follow other work on the interacting holographic dark = less a lower Hubble constant (e.g., h = 0.56) is taken. So, according energy and parameterize the interaction term generally as Q + to Ref. [28], there is a cosmic age crisis in the holographic dark 3H(αρΛ βρm), where α and β are the dimensionless coupling energy model. constants. For reducing the complication and the number of pa- = In fact, many dark energy models are in the face of such a cos- rameters, one often considers the following three cases: (i) β 0, = = = + mic age problem. In history, the cosmic age problem has been and thus Q 3αHρΛ, (ii) α β,andthusQ 3αH(ρΛ ρm), = = focused in cosmology for several times. At present, the cosmic and (iii) α 0, and thus Q 3β Hρm. Note that in these three age crisis coming from some OHROs appears again in cosmolog- cases, according to our convention, α > 0(orβ>0) means that ical models, even though dark energy is involved in the mod- dark energy decays to matter. Moreover, it should be pointed out els. In cosmology there is a very basic principle that the uni- that α < 0(orβ<0) will lead to unphysical consequences in verse cannot be younger than its constituents. So, if the age of physics, since ρm will become negative and ΩΛ will be greater some astronomical object (at some redshift) is measured accu- than 1 in the far future. So, in the present Letter, we only con- rately, then it can be used to test cosmological models according sider the physically reasonable situations, namely, α > 0orβ>0 to this simple age principle. Now, there are some OHROs discov- in the above three cases. In the rest of this section, we will formu- ered, for example, the 3.5 Gyr old galaxy LBDS 53W091 at redshift late the model generally (by using both α and β), but in the next z = 1.55 [29] and the 4.0 Gyr old galaxy LBDS 53W069 at red- section we will only consider the above three simpler cases due to shift z = 1.43 [30]. In particular, the old quasar APM 08279 + 5255 the aforementioned reason. at redshift z = 3.91 is an important one, which has been used as From the definition of holographic dark energy (1),wehave a “cosmic clock” to constrain cosmological models. Its age is es- c2 timated to be 2.0–3.0Gyr[31]. These three OHROs at z = 1.43, Ω = , (7) Λ 2 2 1.55 and 3.91 have been used to test many dark energy mod- H L els, including the ΛCDM model [32], the general EoS dark en- or equivalently, ergy model [33], the scalar-tensor quintessence model [34],the = √c f (R) = R2 − R2 model [35], the DGP braneworld model [36],the L . (8) 0 H ΩΛ power-law parameterized quintessence model [37], the Yang–Mills condensate model [38], the holographic dark energy model [28], Thus, we easily get the agegraphic dark energy model [39], and so on. These investiga- ˙ ˙ c H ΩΛ tions show that the two OHROs at z = 1.43 and 1.55 can be easily L˙ =− √ + . (9) accommodated in most dark energy models, whereas the OHRO H ΩΛ H 2ΩΛ = at z 3.91 cannot, even in the ΛCDM model [32] and the holo- Following Ref. [40], in a non-flat universe the IR cutoff length graphic dark energy model [28]. scale L takes the form In this Letter, we revisit the cosmic age problem in the holographic dark energy model. We consider an interacting holographic L = ar(t), (10) dark energy model in a non-flat universe. We will show that the age crisis in the original holographic dark energy model can be and r(t) satisfies avoided when the interaction and the spatial curvature are in- r(t) +∞ volved in the holographic dark energy model. dr dt √ = . (11) 1 − kr2 a(t) 2. The holographic dark energy model with spatial curvature 0 t and interaction Consequently, we have In this section we describe the interacting holographic dark +∞ +∞ √ √ energy in a non-flat universe. In a spatially non-flat Friedmann– 1 dt 1 da r(t) = √ sin k = √ sin k . (12) Robertson–Walker (FRW) universe, the Friedmann equation reads k a k Ha2 t a(t) 3M2 k 2 2 = + − Pl 3M H ρΛ ρm , (2) Eq. (10) leads to another equation about r(t), namely, Pl a2 − where ρ = 3c2 M2 L 2 is the holographic dark energy density, and L c Λ Pl r(t) = = √ . (13) ρm is the energy density of matter. We define a ΩΛ Ha 2 k H0 ρΛ ρm Combining Eqs. (12) and (13) yields Ω =− = Ω ,Ω= ,Ω= , (3) k 2 2 k0 Λ m H a aH ρc ρc +∞ √ √ dt c k = 2 2 = √ where ρc 3MPl H is the critical density of the universe, thus we k arcsin . (14) have a ΩΛaH t J. Cui, X. Zhang / Physics Letters B 690 (2010) 233–238 235

Taking the derivative of Eq. (14) with respect to t,onecanget For convenience, we introduce the dimensionless cosmic age ∞ 2 ˙ ˙ ΩΛ H k ΩΛ H dz − = + H + . (15) ≡ = 2 2 Tcos(z) H0t(z) , (22) c a 2ΩΛ H (1 + z )E(z ) z Let us combine Eqs. (5) and (6), and then we have (ρ˙Λ + ρ˙m) + where E(z) ≡ H(z)/H , and for the holographic dark energy model 3H(ρΛ + ρm + pΛ) = 0, which is equivalent to the equation 0 ˙ ˙ it is given by the solutions of Eqs. (19) and (20). At any redshift, (ρc − ρk) + 3H(ρc − ρk + pΛ) = 0. From this equation, we can ob- the age of the universe should be larger than, or at least equal to, tain the form of pΛ: the age of the OHRO, namely T (z) T (z) ≡ H t (z), where ˙ ˙ cos obj 0 obj 1 H a tobj(z) is the age of the OHRO at redshift z. Following Ref. [28],we pΛ =− 2 ρc + 2 ρk − ρc + ρk. (16) 3H H a define a dimensionless quantity, the ratio of the cosmic age and the OHRO age, On the other hand, from Eqs. (1), (8) and (9), we find that ∞ ˙ ˙ Ω H Tcos(z) − − dz ˙ = Λ + + τ (z) ≡ = H 1t 1(z) . (23) ρΛ 2ρΛ H . (17) 0 obj + 2ΩΛ H Tobj(z) (1 z )E(z ) z Furthermore, substituting Eqs. (16) and (17) into Eq. (6),weobtain So, the condition Tcos(z) Tobj(z) is translated into τ (z) 1. From ˙ ˙ H Eq. (23), it is easy to see that given the age of OHRO tobj(z),the ΩΛ + 2 (ΩΛ − 1) + 3H(ΩΛ − 3 + Ωk) =−3H(αΩΛ + βΩm). H lower H0, the higher τ (z); given the Hubble constant H0,the smaller t (z),thelarger (z). (18) obj τ In the work of Wei and Zhang [28], the original holographic Combining this equation with Eq. (15), we eventually obtain the dark energy model (neither spatial curvature nor interaction is in- following equations governing the dynamical evolution of the involved) has been examined by using the three OHROs, the old teracting holographic dark energy in a non-flat universe, galaxy LBDS 53W091 at redshift z = 1.55, the old galaxy LBDS 53W069 at redshift z = 1.43, and the old quasar APM 08279 + 1 d H Ω =− Λ 5255 at redshift z = 3.91. It is found in Ref. [28] that the former + H/H0 dz H0 1 z two OHROs, the old galaxy LBDS 53W091 at redshift z = 1.55 and + 2 + 2 = − + Ωk0(1 z) + + − − Ωk0(1 z) the old galaxy LBDS 53W069 at redshift z 1.43, can be easily ac- ΩΛ 3 2 3αΩΛ 3β(1 ΩΛ 2 ) × (H/H0) (H/H0) commodated, but the last one, the old quasar APM 08279 + 5255 2Ω at redshift z = 3.91, cannot be accommodated in the model. In Λ the present Letter, we extend the holographic dark energy model 2 ΩΛ Ωk0(1 + z) + + , (19) to involving the spatial curvature and the interaction, as described 2 2 c (H/H0) in the previous section, and we shall examine whether the OHRO, the old quasar APM 08279+5255 at redshift z = 3.91, is consistent dΩΛ with such a sophisticated holographic dark energy model. dz For the age of the OHRO at z = 3.91, following Ref. [28],we 2 = ΩΛ(1 − ΩΛ) ΩΛ Ωk0(1 + z) use the lower bound estimated, tobj(3.91) 2.0 Gyr. For the holo- =− 2 + 2 2 graphic dark energy model, since the main goal of this Letter is 1 + z c (H/H0) to probe the effects of spatial curvature and interaction in fight- + 2 + 2 3αΩ + (1 z) Ωk0 + 3β(1 − Ω − Ωk0(1 z) ) ing against the cosmic age crisis, we keep the values of c and Λ (H/H )2 Λ (H/H )2 + 1 − 0 0 . Ω fixed in the whole Letter. We take c = 0.8 and Ω = 0.28 − m0 m0 1 ΩΛ that are consistent with the observational constraint results of (20) Ref. [23]. For decreasing the complication, let us close some parameters in turn. We shall consider the following three cases: These two equations can be solved numerically, and the solutions, (a) the model of holographic dark energy with spatial curvature ΩΛ(z) and H(z), determine the expansion history of the universe but without interaction (namely, Ω = 0butQ = 0), denoted as in the holographic dark energy model. k0 KHDE; (b) the model of holographic dark energy with interaction The holographic dark energy model with spatial curvature and but without spatial curvature (namely, Q = 0butΩ = 0), de- interaction described in this section has been strictly constrained k0 noted as IHDE; (c) the model of holographic dark energy with both in Ref. [23] by using the current observational data including the interaction and spatial curvature (namely, Q = 0 and Ω = 0), de- SNIa Constitution data, the shift parameter of the CMB given by k0 noted as KIHDE. Next, let us discuss the use of the Hubble constant the five-year WMAP observations, and the BAO measurement from − − H = 100h km s 1 Mpc 1. Based on the HST key project, Freedman the SDSS. The main fitting results were summarized as Table I and 0 et al. [41] give the result h = 0.72 ± 0.08. However, recently, many Figs. 1–5 of Ref. [23]. In the following discussions, we restrict the authors argue for a lower Hubble constant, say, h = 0.68 ± 0.07 values of parameters to the observational constraint results derived (2σ ) [42]. Moreover, the final result of the 15-year HST program by Ref. [23]. Note that our definition of Ω , α and β are different k given by Sandage et al. [43] is h = 0.623 ± 0.063 which has at- from that of Ref. [23] by a minus sign. tracted more and more attention. Furthermore, when the holographic dark energy model is fitted via observational data (SNIa + 3. Testing the model with the OHRO BAO + CMB), a lower value of h (h ∼ 0.65) is obtained [23] (the = Theageoftheuniverseatredshiftz is given by latest fit value is h 0.686 [44]). It should also be mentioned that the result of the 7-year WMAP observations (WMAP + BAO + H0)is ∞ + h = 0.704 0.013 [45], which is derived based on a ΛCDM model. In dz −0.014 t(z) = . (21) this Letter, we follow Ref. [23] and take h = 0.64 that is the lower + (1 z )H(z ) boundofFreedmanetal.[41]. We will also extend our discussion z 236 J. Cui, X. Zhang / Physics Letters B 690 (2010) 233–238

Table 1 Table 2 The ratio τ (3.91) ≡ Tcos(3.91)/Tobj(3.91) for different Ωk0 in the KHDE model with The values of Tcos(3.91) and τ (3.91) in the IHDE models with c = 0.8, Ωm0 = 0.28 c = 0.8andΩm0 = 0.28. and h = 0.64.

Ωk0 0.04 0.02 −0.02 −0.04 −0.06 −0.1 Case I (β = 0) α 0.02 0.06 0.10 0.15

Tcos(3.91) 0.1131 0.1135 0.1143 0.1146 0.1150 0.1157 Tcos(3.91) 0.1172 0.1246 0.1335 0.1475 τ (3.91) 0.864 0.866 0.872 0.875 0.878 0.883 τ (3.91) 0.894 0.951 1.019 1.126 Case II (α = β) α 0.01 0.02 0.03 0.05

Tcos(3.91) 0.1194 0.1253 0.1316 0.1456 τ (3.91) 0.912 0.957 1.005 1.111 Case III (α = 0)β 0.01 0.03 0.05 0.07

Tcos(3.91) 0.1177 0.1259 0.1346 0.1440 τ (3.91) 0.899 0.961 1.028 1.099

Fig. 1. The dimensionless cosmic age Tcos(z) in the KHDE model. For the curves, we ﬁx c = 0.8, Ωm0 = 0.28 and h = 0.64, and take Ωk0 = 0.02, 0.04, −0.02 and −0.04. The dots represent the dimensionless age of the old quasar APM 08279 + 5255 at z = 3.91, Tobj, under the assumption tobj = 2.0 Gyr. The blue dot corresponds to h = 0.64 and the pink one corresponds to h = 0.72. (For interpretation of colors in this ﬁgure, the reader is referred to the web version of this Letter.)

Fig. 2. The dimensionless cosmic age Tcos(z) in the IHDE2 model. For the curves, we by taking some higher values of h into account (say, we will also ======fix c 0.8, Ωm0 0.28 and h 0.64, and take α β 0.01, 0.02, 0.03 and 0.05. consider h 0.72, the central value of Freedman et al. [41], which The dots represent the dimensionless age of the old quasar APM 08279 + 5255 at is high enough for our discussion, since it is even higher than the z = 3.91, Tobj, under the assumption tobj = 2.0 Gyr. The blue dot corresponds to = = upper bound of WMAP 7-year result). Note that Tobj(3.91) = 0.131 h 0.64 and the pink one corresponds to h 0.72. (For interpretation of colors in this figure, the reader is referred to the web version of this Letter.) is obtained according to tobj(3.91) = 2.0 Gyr and h = 0.64. First, we test the KHDE model. The current observational constraint result of the KHDE model is [23]: −0.02 Ωk0 0.02 (1σ ). the value of τ (3.91) obtained is 1.005. The cosmic age Tcos versus When we take Ωk0 = 0.02, we find τ (3.91) = 0.866, less than 1; redshift z in the case of IHDE2 is also displayed in Fig. 2.Thisfig- when we take Ωk0 =−0.02, we obtain τ (3.91) = 0.872, still less ure shows explicitly that the age problem can be overcomed when than 1. So, we find that the spatial curvature is hard to help solve the interaction is involved in the holographic dark energy model. the cosmic age crisis for the holographic dark energy model. From However, it should be pointed out that the parameter values mak- the above example, we find that the value of τ inaclosedspaceis ing τ (3.91)>1actuallyexceedthe2σ regions given by Ref. [23]. greater than that in an open space. Thus, let us increase the value Therefore, if we confine our discussions in the parameter space of |Ωk0| in a closed space geometry in order to see whether the constrained by current observational data, the problem is not so problem can be solved in some extremal cases. Our efforts can be easy as it looks. Nevertheless, it is found in Ref. [23] that when si- found in Table 1. In this table, we see that even the value of Ωk0 multaneously considering the interaction and spatial curvature in is taken to be −0.1, the value of τ derived is merely 0.883, far the holographic dark energy model, the parameter space is ampli- from solving the cosmic age problem. In addition, we also plot the fied, especially, the ranges of α (or β)andΩk0 are enlarged by 10 Tcos(z) curves for the KHDE model in Fig. 1. It can be explicitly times comparing to the IHDE and KHDE models. Based on this fact, seen from this figure that the cosmic age problem is still acute in it can be expected that the age problem could be solved when the the KHDE model. Therefore, the conclusion is that the cosmic age interaction and spatial curvature are both taken into account. crisis cannot be avoided by only considering the spacial curvature Now, let us consider the KIHDE model. For simplicity, in our in the holographic dark energy model. discussion we fix Ωk0 =−0.06. The three phenomenological in- For the IHDE model, we consider the aforementioned three ca- teraction cases are the same as in the IHDE model. Since the ses: (I) β = 0, named IHDE1; (II) α = β, named IHDE2; (III) α = 0, parameter space of the KIHDE model is greatly amplified, the in- named IHDE3. To see how the interaction influences the cosmic teraction parameter can be chosen to be some large values. For age in the holographic dark energy model, we calculate the age for example, for the KIHDE2 case, one can choose α = 0.05 that is al- these three cases in Table 2. From this table, we see that with the lowed by current observations, then the result τ (3.91) = 1.137 is increase of the interaction parameter α or β, the cosmic age Tcos obtained. Some typical examples for all the three cases are shown also increases. It is clear that the value of τ (3.91) can be greater in Table 3, where the values of the interaction parameters are than 1 when the value of α (or β) is large enough. For exam- takenwithinthe2σ ranges of the observational constrains given ple, for the case of IHDE2 (Case II), when α is taken to be 0.03, by Ref. [23]. It is explicitly shown that the cosmic age problem J. Cui, X. Zhang / Physics Letters B 690 (2010) 233–238 237

Table 3 Table 4 The values of Tcos(3.91) and τ (3.91) in the KIHDE models with c = 0.8, Ωm0 = 0.28, The values of Tobj(3.91) corresponding to different h and tobj(3.91). Ωk0 =−0.06 and h = 0.64. htobj(3.91) Gyr Tobj(3.91) Case I (β = 0) α 0.10.2 0.64 2.00.131 Tcos(3.91) 0.1375 0.1782 2.10.137 τ (3.91) 1.049 1.360 3.00.196 Case II (α = β) α 0.05 0.1 0.72 2.00.147 2.10.154 Tcos(3.91) 0.1489 0.1993 3.00.220 τ (3.91) 1.137 1.521 Case III (α = 0)β 0.05 0.1

Tcos(3.91) 0.1364 0.1623 tremal cases such as a much larger possible age of the quasar τ (3.91) 1.041 1.239 tobj with a larger h. Consider the upper limit of the quasar age, tobj(3.91) = 3.0 Gyr. For this extreme case, when h = 0.64, we have Tobj(3.91) = 0.196; when h = 0.72, we have Tobj(3.91) = 0.220; see also Table 4. So, we have to admit that for the limit case of tobj(3.91) = 3.0Gyrandh = 0.72 the age problem cannot be solved yetevenintheKIHDEmodel.

4. Conclusion

In this Letter, we have revisited the cosmic age problem in the holographic dark energy model. The cosmic age problem brought by the old quasar APM 08279 + 5255 has caused trouble to many cosmological models, and the holographic dark energy model is not an exception either [28]. In order to accommodate the old quasar APM 08279 + 5255 in the holographic dark energy model, we propose to consider the interaction between dark energy and matter in the model. We have shown that the quasar indeed can be accommodated in the holographic dark energy model when an appropriate interaction strength is chosen. Taking the current observational constraints [23] into account, we have demonstrated

Fig. 3. The dimensionless cosmic age Tcos(z) in the KIHDE2 model. For the curves, that both interaction and spatial curvature should be simulta- = = = we fix c 0.8, Ωm0 0.28 and h 0.64. The sophisticated case of interacting holo- neously involved in the holographic dark energy model. It has graphic dark energy in a non-flat universe is represented by the blue dash-dotted =− = = been shown that if such a sophisticated case is considered the curve(wherewetakeΩk0 0.06 and α β 0.05), and other cases such as HDE + (black curve), KHDE (red dashed curve) and IHDE (green dotted curve) are the spe- quasar APM 08279 5255 can be accommodated and the cos- cial cases in this framework. The dots represent the dimensionless age of the old mic age problem can thus be avoided in the holographic dark + = = quasar APM 08279 5255 at z 3.91, Tobj, under the assumption tobj 2.0Gyr. energy model. The price of solving the age problem in this way = = Thebluedotcorrespondstoh 0.64 and the pink one corresponds to h 0.72. is also apparent, i.e., the model involves too many free param- (For interpretation of colors in this figure, the reader is referred to the web version eters, which may weaken the plausibility of the model, to some of this Letter.) extent. It is well known that the consideration of interaction in the can be successfully solved in the KIHDE model. For clarity, we plot holographic dark energy can be used to avoid the future big-rip singularity caused by c 1 [23,46].InthisLetterwehavepro- the curves of Tcos(z) in Fig. 3. This figure shows a direct compar- < ison of HDE, KHDE, IHDE, and KIHDE (the Case II of interaction vided another advantage for the consideration of interaction in the is taken as an example in this figure). It should be noted that holographic dark energy, i.e., the interaction between dark energy α = 0.05 is not allowed in the IHDE model but is allowed in the and matter can also be used to avoid the age problem caused by KIHDE model, from the viewpoint of observation. From Fig. 3,we the old quasar. So, our result can be viewed as a further support also see that the age problem can be evaded in the KIHDE even to the interacting holographic dark energy model. a much larger value of h is taken, for instance, when h = 0.72, Of course, we have to confess that the age problem would still exist if some extreme cases are taken into account, say, a much we get Tobj(3.91) = 0.147, Tcos(3.91) = 0.149, and thus τ > 1in this case. Moreover, when a larger age of the quasar is taken, say, larger possible age of the quasar tobj with a larger h.Itisre- markable that the age of the old quasar APM 08279 + 5255 has tobj(3.91) = 2.1 Gyr, the age problem can also be overcame in the not been measured accurately yet, and the age problem caused by KIHDE model; in Table 4 one can find the values of Tobj(3.91) cor- this quasar has troubled many dark energy models (including the responding to tobj(3.91) = 2.1Gyrforh = 0.64 and 0.72. Therefore, the cosmic age crisis can be avoided in the holographic dark en- ΛCDM model). It is expected that the future accurate measure- ergy model when the interaction and spatial curvature are both ment on the age of this old quasar would eliminate the cosmic taken into account. Nevertheless, we have to admit that the price age crisis in dark energy models. for solving the age problem has been paid by the holographic dark energy model, i.e., there are too many free parameters have to be Acknowledgements considered in the model. This would inevitably weaken, to some extent, the plausibility of the model. Of course, to be honest, it should also be confessed that This work was supported by the Natural Science Foundation of the age problem would still exist if one considers some ex- China under Grant Nos. 10705041 and 10975032. 238 J. Cui, X. Zhang / Physics Letters B 690 (2010) 233–238

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