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A Holographic Dark Energy Model from Ricci Scalar Curvature

A Holographic Dark Energy Model from Ricci Scalar Curvature
Changjun Gao? and Xuelei Chen?
The National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China

You-Gen Shen?
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China and Joint Institute for Galaxy and Cosmology of SHAO and USTC, Shanghai 200030, China (Dated: February 2, 2008) Motivated by the holographic principle, it has been suggested that the dark energy density may be inversely proportional to the area of the event horizon of the Universe. However, such a model would have a causality problem. In this paper, we propose to replace the future event horizon area with the inverse of the Ricci scalar curvature. We show that this model does not only avoids the causality problem and is phenomelogically viable, but also naturally solves the coincidence problem of dark energy.

arXiv:0712.1394v3 [astro-ph] 21 Jan 2008



Ever since the discovery of the dark energy [1, 2], cosmologists are confronted with two fundamental problems: (1) the ?ne tuning problem and (2) the coincidence problem [3]. The ?ne tuning problem is the following: the simplest form of dark energy is the cosmological constant introduced by Einstein. However, this is also the vacuum energy in quantum ?eld theory. The estimated size of the vacuum energy ρ ? ρp where ρp is the Plank density, this is greater than the observed value ρ ? 10?123 ρp by some 123 orders of magnitude. The coincidence problem is the following: the density of the dark energy and matter evolves di?erently as the Universe expands, yet they are comparable today, this is a great coincidence if there is not some internal connection between the two. Recent advance in the studies of black hole theory and string theory may provide an inspiration for solving these problems. It was realized that in quantum gravity, the entropy of a system scales not with its volume, but with its surface area L2 , this is the so called holographic principle [4]. To see how this principle could help solve the cosmological constant problems, we note that in the Einstein equation, G?ν = 8πT?ν + Λg?ν , where we use the unit system with c=G= =1, the cosmological constant Λ is the inverse of some length squared, [Λ] ? l?2 , and to be consistent with observations, l must be of the same order as the present cosmological scale. It is then proposed [5] that an unknown vacuum energy could be present, and its density is proportional to the Hubble scale lH ? H ?1 . In this way, the ?ne tuning problem is solved, and the coincidence problem relaxed. Unfornately, in this case, the equation t of state for dark energy is zero and the Universe is decelerating. Alternatively, the particle horizon lP H = a 0 dt/a could be used as the length scale [6]. However, as S. Hsu [7] and M. Li [8] pointed out, the equation of state for this dark energy model is greater than ?1/3, so it could not explain the observed acceleration of the Universe. In view of this, M. Li [8] proposed that the future event horizon of the Universe to be used as length l. This holographic dark energy model is successful in ?tting the current observations. The underlying origin of the holographic dark energy is still unknown. Furthermore, the model also has some conceptual problems. As R. Cai [9] pointed out, an obvious drawback concerning causality appears in this proposal. Event horizon is a global concept of space-time. However, the density of dark energy is a local quantity. Why is a local quantity determined by a global quantity? Inspired by the holographic dark energy models, in this paper we propose taking the average radius of Ricci scalar curvature, R?1/2 , as the length l. So we have the dark energy ρX ∝ R. In the following we shall call our model the Ricci dark energy model. We ?nd this model works well not only in explaining the current acceleration of the Universe, but also in understanding the coincidence problem. Moreover, in this model the presence of event horizon is not presumed, thus the causality problem is avoided.

? Electronic ? Electronic

address: gaocj@bao.ac.cn address: xuelei@cosmology.bao.ac.cn ? Electronic address: ygshen@center.shao.ac.cn


The metric of the Friedmann-Robertson-Walker Universe is given by ds2 = ?dt2 + a (t)

dr2 + r2 dθ2 + r2 sin2 θdφ2 , 1 ? kr2


where k = 1, 0, ?1 for closed, ?at and open geometries respectively. In this we have adopted the convention of a0 = 1, where the subscript 0 denotes the value at present time (zero redshift). The Friedman equation is H2 = 8π 3 ρi ?

k a2


where H ≡ a/a is the Hubble parameter, dot denotes the derivative with respect to the cosmic time t, and summation ˙ runs over the non-relativistic matter, radiation and other components of matter. The Ricci scalar curvature is given by k ˙ R = ?6 H + 2H 2 + 2 a , (3)

As discussed above, we introduce a holographic dark energy component, which is proportional to the inverse of squared Ricci scalar curvature radius, ρX = 3α 8π k ˙ H + 2H 2 + 2 a ∝ R, (4)

3 where α is a constant to be determined. The factor 8π before α is for convenience in the following calculations. The corresponding energy-momentum tensor can be written as:

T?ν = (ρX + pX ) U? Uν + pX g?ν , where U? is the 4-velocity of the co-moving observer. pX is the pressure of dark energy. Setting x = ln a, we can rewrite the Friedmann equation as follows H2 = 8π 3k ?2x (α ? 1) e + ρm e?3x + ρr e?4x 3 8π 1 dH 2 +α + 2H 2 , 2 dx



where ρm and ρr term are the contributions of non-relativistic matter and radiation, respectively. We introduce the scaled Hubble expansion rate h ≡ H/H0 , then the above Friedman equation becomes h2 = (α ? 1) ?k0 e?2x + ?m0 e?3x + ?r0 e?4x + 1 dh2 α + 2h2 , 2 dx


where ?k0 , ?m0 and ?r0 are the relative density of the curvature, non-relativistic matter and radiation in the present Universe, and the dark energy relative density is denoted by ?X , with ?k0 + ?m0 + ?r0 + ?X0 = 1. Solving Eq.(7), we obtain h2 = ??k0 e?2x + ?m0 e?3x + ?r0 e?4x + 2 α ?m0 e?3x + f0 e?(4? α )x , 2?α


where f0 is an integration constant. On the right hand side of Eq.(8), the last two terms come from the dark energy, ρX =
2 α ?m0 e?3x + f0 e?(4? α )x . 2?α


Thus the Ricci dark energy has one part which evolves like non-relativistic matter (? e?3x ), and another part which is slowly increasing with decreasing redshift.

3 Substituting the expression of ρX into the conservation equation of energy, pX = ?ρX ? we obtain the pressure of dark energy pX = ? 2 1 ? 3α 3 f0 e?(4? α )x .

1 dρX , 3 dx



There are two constants α and f0 to be determined in the expressions of ρX and pX . Taking into account of the present density of dark energy ?X and its present pressure w0 ?X , we can express α and f0 in terms of present day value of cosmological parameters using Eq.(9) and Eq.(11): ?m0 α + f0 = ?X0 , 2?α We then obtain f0 = 3w0 ?2 X0 , 3w0 ?X0 ? ?m0 α= 2?X0 . ?m0 + ?X0 ? 3w0 ?X0 (13) ? 1 2 ? 3α 3 f0 = w0 ?X0 . (12)

0 –0.2 –0.4 –0.6 w –0.8 –1 –1.2 –1 0 1 2 z 3 4 5 6

FIG. 1: Evolution of the equation of state w for the Ricci dark energy as a function of redshift z, for the following parameter values: ?k0 = 0, h0 = 0.73, ?m0 = 0.27, ?r0 = 8.1 · 10?5 h?2 , ?X0 = 0.73, w0 = ?1. 0

As an example, we consider the cosmology model with the following values of parameters: ?k0 = 0, h0 = 0.73, ?m0 = 0.27, ?r0 = 8.1 · 10?5 h?2 , ?X0 = 0.73, w0 = ?1, which are consistent with current observations [10]. Then we have 0 f0 ? 0.65 and α ? 0.46. We plot the evolution of the equation of state w ≡ pX /ρX for this model in Fig. 1. At high redsh?ts, the equation of state is nearly zero, so the Ricci dark energy behaves just like dark matter, with ρX /ρm ? 0.29. The equation of state w approaches -1 at z ? 0. In the distant future, the equation of state approaches ?1.12, The Universe evolves into the phantom dominated epoch [11]. For this model, the equation state crosses -1, so it is a “quintom” [12]. In Fig.2, we plotted the evolution of densities, log ρ, for radiation (cross line), non-relativistic matter (solid line) and dark energy (circle line) with ln a 1 . It shows that in this model the densities of non-relativistic matter and dark energy were always comparable with each other in the past Universe, and the acceleration began at low redshift, so the coincidence problem is solved. We can also see that dark energy made negligible contribution in the epoch of radiation dominated Universe, hence the Big-Bang Nucleosynthesis (BBN) model does not need any revision. The Universe speeds up at ln a ? 0 when dark energy dominates over dark matter.


We have neglected phase transitions, new degrees of freedoms and transitions from non-relativistic to relativistic particles at high temperature, etc. These would not make qualitative di?erence in the result.


200 150 100 logp 50 0 –50 –100 –40 –20 lna 0 20

FIG. 2: Evolution of the radiation density (cross line), non-relativistic matter density (solid line) and Ricci dark energy density (circle line) log ρ as the function of ln a.

0.4 0.2 0 q –0.2 –0.4 –0.6 0.2 0.4 0.6 0.8 1 z 1.2 1.4 1.6 1.8 2

FIG. 3: Evolution of the deceleration parameter with redshift.

In Fig.3, we plotted the evolution of deceleration parameter, q≡ 1 2 1+ 3ptot ρtot = 1 3pX , + 2 2ρX + 2ρm (14)

where ptot , ρtot denote the total pressure and density of the Universe. It shows that Universe speeds up at z ? 0.55. This is consistent with the joint analysis of SNe+CMB data zT = 0.52 ? 0.73 [13]. In Fig.4, we plotted the evolution of age of the Universe 1 H0
1 1+z



dx . h


Three circles denote the ages of three old objects, LBDS 53W091 (z = 1.55, t = 3.5 Gyr) [14], LBDS 53W069 (z = 1.43,t = 4.0 Gyr) [15] and APM 08279+5255 (z = 3.91, t = 2.1 Gyr) [16]. H. Wei and S. N. Zhang [17] recently pointed that by considering the ages of these three old high redshift objects, the holographic dark energy model can be ruled out. However, as shown in Fig.4, our model is free of this age problem.


14 12 10 t8 6 4 2 0 1 2 z 3 4 5

FIG. 4: Age of the Universe with redshift. Three circles denote the ages of three old objects, LBDS 53W091 , LBDS 53W069 and APM 08279+5255 .



We have shown that if we replace the future event horizon in the holographic dark energy model with the Ricci scalar curvature radius, i.e., ρX ∝ R, then the resulting Ricci dark energy model is viable phenomenologically. The model is free of both the age problem and the causality problem, which plagues the holographic dark energy problem. We have only illustrated our model with one simple example. For di?erent cosmological parameters, the model is also slightly di?erent. We plan to make further investigation on the allowed range of parameters and their impact on model behavior in future work. In our model the ?ne tuning problem is avoided, because the dark energy is not associated Planck or other high energy physics scale, but with the size of space-time curvature. More interestingly, the coincidence problem is solved in this model: as the dark energy is proportional to R, it is small during radiation dominated era. Later, it is always comparable to the size of the non-relativistic matter. The change from deceleration to acceleration happens near ln a ? 0 with plausible model parameters. Like the holographic dark energy model, our model is still incomplete: it is not known what kind of physical mechanism could provide such a contribution. Indeed, one can not yet write out the e?ective action for this dark energy form, but have to start from the Friedmann equation. Nevertheless, unlike models associated with the event horizon, in our model the dark energy is determined by local quantities, thus it is more hopeful to construct some model which reproduces the desired behavior. For example, it is well-known that for non-minimally coupled scalar ?eld, 2φ ∝ Rφ, hence it might still be possible to ?nd a model in which the desired behavior is mimiced. Finally, although we have been motivated by the holographic principle, the Ricci dark energy does not have to be connected with the holographic principle. The dark energy of the Ricci scalar form may also arise from other reasons.

We thank Professors Miao Li, Ronggen Cai, and Chongming Xu for helpful discussions. This work is supported by the National Science Foundation of China under the Distinguished Young Scholar Grant 10525314, the Key Project Grant 10533010, and Grant 10575004; by the Chinese Academy of Sciences under grant KJCX3-SYW-N2; and by the Ministry of Science and Technology under the National Basic Sciences Program (973) under grant 2007CB815401.

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