Research Article | Open Access
A Study on Some Special Forms of Holographic Ricci Dark Energy in Fractal Universe
We have considered the modified and the extended holographic Ricci dark energy models (MHRDE and EHRDE) in fractal universe. We have assumed a time-like fractal profile , where . We have reconstructed the Hubble parameter , the energy density, the equations of state parameter and the deceleration parameter for both MHRDE and EHRDE.
Motivated by the virtues and problems of Horava-Lifshitz gravity, Calcagni  formulated an effective quantum field theory with two key features: (i) power-counting renormalizability is obtained when the fractal behaviour is realized at structural level; (ii) Lorentz invariance is maintained. Calcagni  proposed a field theory which lives in fractal space-time and argued this to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal and discussed its implications for quantum gravity, cosmology, and the cosmological constant. Lemets and Yerokhin  demonstrated the key features and motivations of fractal cosmology models. Sheykhi et al.  investigated the thermodynamical properties of the apparent horizon in a fractal universe. Karami et al.  investigated the holographic, the new agegraphic, and the ghost dark energy models in the framework of fractal cosmology, and they also discussed the behavior of the equations of state parameter . In the present work, we are going to investigate the behavior of the modified holographic Ricci dark energy (MHRDE) and the extended holographic Ricci dark energy (EHRDE) models in fractal universe.
The holographic dark energy (HDE) model, proposed by Li , is inspired by the holographic principle , which states that the maximum number of degrees of freedom in a volume should be proportional to the surface area . Applying the novel principle that (where the infrared (IR) cut-off is encoded by the scale , corresponds to the quantum zero-point energy density, and represents the Planck mass, with being the Newton's gravitational constant) within the cosmological context implies that the dark energy density of the universe takes the same form as the vacuum energy; that is, . Using the largest as the one saturating the above inequality, it turns out to be HDE, whose density is given by , where is a numerical factor. Many IR cut-offs have been considered in recent works, for example, the large scale of the universe, the Hubble horizon, the particle horizon, the event horizon or generalized IR cut-off, and so forth [8–12]. Studies on HDE from various points of view include [13–18]. Gao et al.  have proposed a HDE model, where the future event horizon is replaced by the inverse of the Ricci scalar curvature, and this model is named as “Ricci dark energy model” (RDE). This model does not only avoid the causality problem and is phenomenologically viable, but also solve the coincidence problem of dark energy . The Ricci curvature of FRW universe is given by . The RDE proposed in  has energy density which is proportional to the Ricci scalar curvature; that is, . Some significant works on RDE are [21–31]. In the present work, we are considering two special modified forms of holographic RDE: (i)modified holographic Ricci dark energy (MHRDE) [8, 10, 32, 33], whose energy density is given by , where and are free constants;(ii)extended holographic Ricci dark energy (EHRDE) [10, 34], whose density has the form , where is the reduced Planck mass and and are constants to be determined. If , then we get back to RDE.
In the present work, we will consider separately that the fractal universe is filled with MHRDE and EHRDE, respectively. We will reconstruct the Hubble parameter for both cases. Based on the reconstructed Hubble parameters, we will investigate the behavior of the equation of state parameter and the deceleration parameter .
2. The Dark Energy Models in the Fractal Universe
The total action of Einstein gravity in a fractal space-time is given by  where the gravitational part of the action is given by and the matter part of the action is given by
Here, is the determinant of the dimensionless metric , and are, respectively, the cosmological constant and the Ricci scalar. Moreover, is the fractional function, and is the fractal parameter. The standard measure is replaced with a Lebesgue-Stieltjes measure . Taking the variation of the action given in (1) with respect to the FRW metric , we obtain the Friedmann equations in a fractal universe as 
The curvature parameter corresponds to a flat, closed, and open universe, respectively. In the present work, we will consider as flat universe. The continuity equation in a fractal universe takes the form
It is clear that for , the standard Friedmann equations are recovered. We further assume that only the time direction is fractal while spatial slices have usual geometry. Assuming a time-like fractal profile , where (with ) is the fractal dimension. The Friedmann equations given in (4) in the absence of the cosmological constant can be written as while the continuity equation given in (5) takes the form In (6), we replace by and , respectively, for flat FRW universe (), and we get nonlinear differential equations on . For MHRDE, we have Using (9) in continuity equation given in (8), we get the pressure for the MHRDE model as follows:
Since we cannot have analytical solution of the differential equations, we solve them numerically.
We plot the reconstructed Hubble parameter in Figures 1 and 2 for MHRDE and EHRDE, respectively. In the figures for MHRDE the solid, dashed, and dot-dashed lines correspond to , , and , respectively. For EHRDE, the solid, dashed, and dot-dashed lines correspond to , , and , respectively. For all figures, we have considered and .
In Figure 1, we observe that is a decreasing function of cosmic time for MHRDE. In Figure 2, we observe that for in EHRDE, is a decreasing function of . However, for the other combinations, is decreasing up to , and then it is increasing gradually. Next we plot the time derivatives of the reconstructed Hubble parameter in Figures 3 and 4. For MHRDE (Figure 3), we find that is increasing with and staying at negative level throughout. Although for EHRDE (Figure 4) we observe that is increasing with , we find that for and the is changing its sign at .
In Figure 5, we find that is decreasing with . Instead, in Figure 6, we observe that is increasing with . The rate of increasing is less for than for the other combinations. The negative pressure for MHRDE is decreasing with time as shown in Figure 7. Instead, for EHRDE, the negative pressure is increasing for and (Figure 8). The EoS parameters and are plotted, respectively, in Figures 9 and 10. In Figure 9, we observe that for all combinations of and . This indicates a quintessence-like behavior. In Figure 10, we observe that transits from to for and at . This indicates a transition from quintessence to phantom. Thus, the EoS parameter has a quintom-like behavior in these two cases. For , we have .
In the present work, we have considered, modified, and extended holographic Ricci dark energy in fractal universe. We have assumed a time-like fractal profile , where . We have reconstructed Hubble parameter for both of the dark energy candidates in fractal universe. Under the reconstructed Hubble parameters, we have reconstructed dark energy candidates in fractal universe, and we have observed that the modified holographic dark energy density is decreasing with cosmic time , and extended holographic dark energy density is increasing with time. The equation of state parameter for modified holographic dark energy has a quintessence-like behavior and for extended holographic dark energy has a quintom-like behavior. We have seen accelerated expansion of the universe through the deceleration parameter.
The first author Surajit Chattopadhyay acknowledges the research Grant under Fast Track Programme for Young Scientists provided by the Department of Science and Technology (DST), Government of India. The Grant number is SR/FTP/PS-167/2011.
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Copyright © 2013 Surajit Chattopadhyay et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.