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Holographic Dark Energy in a Cyclic Universe

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Holographic Dark Energy in a Cyclic Universe 中国科技论文在线 http://www.paper.edu.cn Eur. Phys. J. C 52, 693–699 (2007) DOI 10.1140/epjc/s10052-007-0408-2 THE EUROPEAN PHYSICAL JOURNAL C Regular Article – Theoretical Physics Holographic dark energy in a cyclic universe Jingfei Zhang1,a,XinZhang2,b, Hongya Liu1,c 1 School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, P.R. China 2 Kavli Institute for Theoretical Physics China, Institute of Theoretical Physics, Chinese Academy of Sciences (KITPC/ITP-CAS), P.O. Box 2735, Beijing 100080, P.R. China Received: 28 July 2007 / Revised version: 11 August 2007 / Published online: 13 September 2007 − © Springer-Verlag / Societ`a Italiana di Fisica 2007 Abstract. In this paper we study the cosmological evolution of the holographic dark energy in a cyclic uni- verse, generalizing the model of holographic dark energy proposed by Li. The holographic dark energy with c<1 can realize a quintom behavior; namely, it evolves from a quintessence-like component to a phantom- like one. The holographic phantom energy density grows rapidly and dominates the late-time expanding phase, helping to realize a cyclic universe scenario in which the high energy regime is modified by the effects of quantum gravity, causing a turn-around (and a bounce) of the universe. The dynamical evolution of holo- graphic dark energy in the regimes of low energy and high energy is governed by two differential equations, respectively. It is of importance to link the two regimes for this scenario. We propose a link condition giving rise to a complete picture of holographic evolution of a cyclic universe. The astronomical observations over the past decade imply nite future time, at which all bound objects will be torn that our universe is currently dominated by dark energy, apart. which leads to an accelerated expansion of the universe In the phantom scenario, generically, there exist two (see e.g. [1–7]). The combined analysis of cosmological ob- space-time singularities in the universe; one is the initial servations suggests that the universe is spatially flat and “big bang” singularity, the other is the future “big rip” sin- consists of about 70% dark energy, 30% dust matter (cold gularity. The space-time singularities are disgusting for the dark matter plus baryons), and negligible radiation. The majority of theorists; thus a mechanism for avoiding the basic characteristic of dark energy is that its equation of initial and future singularities are attractive for physicists state parameter w (the definition of w is w = p/ρ,whereρ and cosmologists. An effective way for eliminating the sin- is the energy density and p is the pressure) has a nega- gularities is to introduce a ρ2 term with a negative sign to tive value (w<−1/3). The most obvious candidate for the Friedmann equation if the energy is very high. Such dark energy is the cosmological constant λ [8] for which a modified Friedmann equation with a phantom energy w = −1 (for reviews see e.g. [9–14]). However, the cos- component leads to a cyclic universe scenario in which the mological constant always suffers from the “fine-tuning” universe oscillates through a series of expansions and con- and “cosmic coincidence” problems. Another candidate for tractions. Phantom energy can dominate the universe to- dark energy is the energy density associated with a dynam- day and drive the current cosmic acceleration. Then, as the ical scalar field; a slowly varying, spatially homogeneous universe expands, it becomes more and more dominant and component. Typical examples of such a type of dark en- its energy density becomes very high. When the phantom ergy are the so-called “quintessence” [15–24] and “phan- energy density reaches a critical value, a very high energy tom” [25, 26] models. Quintessence dark energy provided density, the universe reaches a state of maximum expan- by a canonical scalar field has an equation of state w>−1, sion, which we call the “turn-around” point, and then it while the phantom energy associated with a scalar field begins to recollapse, according to the modified Friedmann with a negative-kinetic energy has an equation of state equation. The contraction of the universe makes the phan- w<−1. It is remarkable that for phantom dark energy tom energy density dilute away and the matter density in this scenario all the energy conditions in general rela- dominate. Once the universe reaches its smallest extent, tivity (including the weak energy condition) are violated. the matter density hits the value of the critical density, the Due to such a supernegative equation of state, the phan- modified Friedmann equation leads to a “bounce”, making tomcomponentleadstoa“bigrip”singularityatafi- the universe once again begin to expand. The idea of an oscillating universe was first proposed by a e-mail: [email protected] Tolman in the 1930’s [27, 28]. In recent years, Steinhardt, b e-mail: [email protected] Turok and collaborators [29–33] proposed a cyclic model of c e-mail: [email protected] the universe as an alternative to the inflation scenario, in 转载 中国科技论文在线 http://www.paper.edu.cn 694 J. Zhang et al.: Holographic dark energy in a cyclic universe which the cyclicity of the universe is realized in the light of ergy becomes a phantom, this scenario will involve a big two separated branes. The cyclic scenario discussed in this rip singularity in the far future unless the Friedmann equa- paper is distinguished from the Steinhardt–Turok cyclic tion gets a quantum gravity correction in the high energy scenarios, in that the phantom energy plays a crucial role. regime as shown in (1). Thus, phantom-like holographic In the oscillating or cyclic models, the principal obstacles dark energy can play a crucial role in realizing a cyclic against success come from the problems of black holes and universe scenario, and inversely, a cyclic universe can en- entropy. For the discussions of the problems of black holes dow the holographic dark energy with peculiar features. In and entropy in the cyclic scenario, see e.g. [34–36]. this paper, we shall study the cyclic universe with a holo- Usually, the phantom energy density becomes infinite graphic phantom and investigate the characteristic of the in a finite time, leading to the big rip singularity. How- holographic dark energy in such a cyclic universe. ever, we expect that an epoch of quantum gravity sets in According to the holographic principle, the number before the energy density reaches infinity. Therefore, we of degrees of freedom for a system within a finite region arrive at the notion that quantum gravity governs the be- should be finite and should be bounded roughly by the havior of the universe both at the beginning and at the area of its boundary. In the cosmological context, the holo- end of the expanding universe, where the energy density graphic principle will set an upper bound on the entropy of is enormously high. This high energy density physics may the universe. Motivated by the Bekenstein entropy bound, lead to modifications to the Friedmann equation, such as in it seems plausible that one may require that for an effect- loop quantum cosmology [37–40] and braneworld scenar- ive quantum field theory in a box of size L with UV cut- 3 3 ios [41–43], which causes the universe to bounce when it is off Λ, the total entropy should satisfy S = L Λ ≤ SBH ≡ 2 2 small, and to turn around when it is large. At high energy πMplL ,whereSBH is the entropy of a black hole with the densities, we employ the modified Friedmann equation same size L. However, Cohen et al. [46] pointed out that to saturate this inequality some states with Schwartzschild 8πG ρ H2 = ρ 1 − , (1) radius much larger than the box size have to be counted 3 ρc in. As a result, a more restrictive bound, the energy bound, has been proposed to constrain the degrees of freedom of where H =˙a/a is the Hubble parameter, G is the New- the system, requiring the total energy of a system with size ton gravity constant, and ρc is the critical energy dens- L not to exceed the mass of a black hole with the same size, ity set by quantum gravity, distinguished from the usual 3 4 3 2 √ namely, L Λ = L ρΛ ≤ LM . This means that the max- critical density 3M 2 H2 (where M =1/ 8πG is the re- pl pl pl imum entropy is in the order of S3/4. When we take the duced Planck mass). This modified Friedmann equation BH whole universe into account, the vacuum energy related to can be derived from the effective theory of loop quan- this holographic principle is viewed as dark energy, usually tum cosmology [37–40], and also from the braneworld dubbed holographic dark energy. The largest IR cut-off L scenario [43]. In loop quantum cosmology, the critical is chosen by saturating the inequality, so that we get the energy density can be evaluated as ρc ≈ 0.82ρpl,where −2 76 4 holographic dark energy density ρpl = G =2.22 × 10 GeV is the Planck density. In the braneworld scenario, ρc =2σ,whereσ is the brane tension, 2 2 −2 ρΛ =3c M L , (2) and a negative sign in (1) can arise from a second time-like pl dimension; but that gives difficulties with closed time-like where c is a numerical constant (note that c>0isas- paths. In models motivated by the Randall–Sundrum sce- sumed), and as usual Mpl is the reduced Planck mass. nario [41, 42], the most natural energy scale of the brane Hereafter, we will use the unit Mpl = 1 for convenience.
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