Holography and holographic dark energy model

The holographic principle is used to discuss the holographic dark energy model. We find that the Bekenstein-Hawking entropy bound is far from saturation under certain conditions. A more general constraint on the parameter of the holographic dark energy model is also derived. PACS numbers: 98.80.Cq, 98.80.Es, 04.90.+e Holography and holographic dark energy model 2 The studies of black hole thermodynamics tell us that the total entropy of matter inside a black hole cannot be greater than the Bekenstein-Hawking entropy, which is one quarter of the area of the event horizon of the black hole measured in Planck unit. Bekenstein proposed a universal entropy bound S ≤ 2πER for a weakly selfgravitating physical system with total energy E and size R in an asymptotically flat four dimensional spacetime [1] in 1981. ’t Hooft and Susskind later introduced the Bekenstein-Hawking entropy as the holographic entropy bound [2]. Based on the earlier works, Bousso proposed the covariant entropy bound in [3]. The conjectured Antide-Sitter/Conformal Field Theory duality provided the evidence of the existence of a holographic principle in quantum gravity although we are far from understanding quantum gravity [4]. According to the holographic principle, under certain conditions all the information about a physical system inside a spatial region is encoded in its boundary instead of its volume. Fischler and Susskind (FS) discussed the application of the holographic principle to cosmology by considering our universe inside a particle horizon [5]. They found that the holographic bound was violated for a closed universe. To solve this problem, Bak and Rey then replaced the particle horizon by the apparent horizon [6], and Rama found that the holographic bound with the particle horizon could be satisfied in a closed universe by adding negative pressure matter [7]. There are other modifications to the original FS version of holographic principle [8]. The holographic principle was also used to discuss the number of e-foldings of inflation [9] and constrain dark energy models [10]. The discussion of holographic principle in the context of Brans-Dicke theory was first considered in [11]. The type Ia supernova observations suggest that the Universe is dominated by dark energy with negative pressure which provides the dynamical mechanism of the accelerating expansion of the Universe [12, 13]. The simplest candidate of dark energy is the cosmological constant. However, the unusual small value of the cosmological constant is a big challenge to theoretical physicists. The idea of holography may be used to solve the cosmological constant problem [14, 15]. Cohen, Kaplan and Nelson proposed that for any state with energy E in the Hilbert space, the corresponding Schwarzschild radius Rs ∼ E is less than the infrared (IR) cutoff L [14]. Under this assumption, a relationship between the ultraviolet (UV) cutoff ρ 1/4 D and the IR cutoff is derived, i.e., 8πGLρD/3 ∼ L [14]. So the holographic dark energy density is ρD = 3c d 8πGL2 , (1) where c is the speed of light and d is a constant of the order of unity. Hsu found that the holographic dark energy model based on the Hubble scale as the IR cutoff won’t give an accelerating universe [16]. This does not mean that the form ρD ∼ H won’t never work for dark energy model building. The model ρD = Λ + 3c dH/(8πG) derived from the re-normalization group models of the cosmological constant can explain the accelerating expansion of the Universe [17]. Ito discovered a viable holographic dark energy model by using the Hubble scale as the IR cutoff with the use of non-minimal coupling to scalar field [18]. More recently, a dark energy model ρD ∼ H with the Holography and holographic dark energy model 3 interaction between dark energy and dark matter was proposed [19]. In [20], Li showed that the holographic dark energy model based on the particle horizon as the IR cutoff won’t give an accelerating universe either. However, Li found that the holographic dark energy model based on the event horizon as the IR cutoff works [20, 21]. The model was also found to be consistent with current observations [22]. The holographic dark energy model in the framework of Brans-Dicke theory was discussed in [23]. Some speculations about the deep reasons of the holographic dark energy were considered by several authors [24]. The holographic dark energy model was also discussed in [25, 26]. For a fluid with constant equation of state p = wρ, the second law of thermodynamics tells us that a relationship ρ ∼ σ between the entropy density σ and the energy density ρ. If we apply the Bekenstein-Hawking entropy bound, then we will get an upper bound on ρ. If we apply the energy bound Rs ≤ L, then we will get an upper bound on entropy. The consistency between the entropy bound derived from the energy bound and the holographic bound needs to be checked. Furthermore, if we use the Bekenstein entropy bound, a lower bound on ρ is obtained. Therefore, a detailed discussion about the effects of those arguments on the holographic dark energy model is needed. In this paper, we first review the holographic dark energy model [20, 21, 25], then we apply the second law of thermodynamics and the holographic principle to discuss the consistency of those bounds. For a general perfect fluid, there is no relationship between σ and ρ, but the comoving entropy density is a constant. This fact is used to prove that the Bekenstein-Hawking entropy bound is far from saturation for the holographic dark energy model. We use the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric ds = −cdt + a(t) [ dr 1− k r2 + r 2 dΩ ] . (2) For a null geodesic, we have ∫ t0 t1 c dt a(t) = ∫ r1 0 dr √ 1− kr2 ≡ f(r1), (3) where