arXiv:0712.1394v4 [astro-ph] 25 Dec 2008 A Holographic Dark Energy Model from Ricci Scalar Curvature Changjun Gao, Fengquan Wu, 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: November 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 avoid the causality problem and is phenomenologically viable, but also naturally solves the coincidence problem of dark energy. Our analysis of the evolution of density perturbations show that the matter power spectra and CMB temperature anisotropy is only slightly affected by such modification. PACS numbers: 98.80.Cq, 98.65.Dx I. INTRODUCTION Ever since the discovery of the dark energy [1, 2], cos- mologists are confronted with two fundamental problems: (1) the fine tuning problem and (2) the coincidence prob- lem (see e.g. [3]). The fine tuning problem is the fol- lowing: the simplest form of dark energy is the cosmo- logical constant introduced by Einstein. However, the vacuum energy in quantum field theory has exactly the same property, and the estimated size of the vacuum en- ergy is ρ ≃ ρ p where ρ p ∼ m 4 p is the Plank density. It is greater than the observed value ρ ≃ 10 -123 ρ p by some 123 orders of magnitude, so extreme fine tuning of the vacuum energy is required. The coincidence problem is the following: the density of the dark energy and matter evolves differently as the Universe expands, yet they are comparable today, this is an incredibly great coincidence if there is not some internal connection between the two. An important advance in the studies of black hole the- ory and string theory is the suggestion of the so called holographic principle, which may provide some clue for solving these problems. It is realized that in quantum gravity, the entropy of a system scales not with its vol- ume, but with its surface area L 2 [4]. To see how this could help solve the cosmological constant problems, we note that in the Einstein equation, G μν =8πGT μν + Λg μν , the cosmological constant Λ is the inverse of some length squared, [Λ] ∼ l -2 , and to be consistent with ob- servations, l must be of the same order as the present cosmological scale. It is then proposed [5] that an un- known vacuum energy could be present, and according to the holographic principle its density is proportional to the Hubble scale l H ∼ H -1 . In this model the fine tun- ing problem is solved as the scale of dark energy is de- termined not by Planck length but by cosmological scale, and the coincidence problem is also all alleviated. Unfor- tunately, the effective equation of state for such vacuum energy is zero and the Universe is decelerating. Alterna- tively, the particle horizon size l PH = a  t 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 still 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 the characteristic length l. This holographic dark energy model and its interacting versions are successful in fitting the current observations [9]. However, the underlying origin of the holographic dark energy is still unknown. Furthermore, the model also has some serious conceptual problems. As R. Cai [10] 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 lo- cal quantity. Why should a local quantity be determined by a global one? Also puzzling is that the present value of dark energy is determined by the future evolution of the Universe, thus posing a challenge to the concept of casuality. Furthermore, for a spatially flat Friedmann- Robertson-Walker Universe, it is well-known that the fu- ture event horizon exists if and if the Universe is accel- erating. So in order to interpret the cosmic acceleration, the holographic dark energy model itself has presumed the acceleration. Inspired by the holographic dark energy models, in this paper we propose to consider another possibility: the length l is giving by the average radius of Ricci scalar curvature, R -1/2 , so that we have the dark energy ρ X ∝ R. In the following we shall call this model the Ricci dark energy model, and investigate its phenomenological properties. We find that this model works fairly well in fitting the observational data, and it could also help us to understand the coincidence problem. Moreover, in this model the presence of event horizon is not presumed, so the causality problem is avoided. In the next section we describe the model and its cos- mic expansion history. Section III is devoted to the