A Study on Some Special Forms of Holographic Ricci Dark Energy in Fractal Universe

Abstract

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 𝑣=𝑡^−𝛽, where 𝛽=4(1−𝛼). We have reconstructed the Hubble parameter 𝐻, the energy density, the equations of state parameter 𝑤, and the deceleration parameter 𝑞 for both MHRDE and EHRDE.

1. Introduction

Motivated by the virtues and problems of Horava-Lifshitz gravity, Calcagni [1] 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 [2] 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 [3] demonstrated the key features and motivations of fractal cosmology models. Sheykhi et al. [4] investigated the thermodynamical properties of the apparent horizon in a fractal universe. Karami et al. [5] 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 𝜔. 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 [6], is inspired by the holographic principle [7], which states that the maximum number of degrees of freedom in a volume should be proportional to the surface area [8]. Applying the novel principle that 𝐿³𝜌_Λ≤𝐿𝑀²_𝑃 (where the infrared (IR) cut-off is encoded by the scale 𝐿, 𝜌_Λ corresponds to the quantum zero-point energy density, and 𝑀_𝑃⁻²=8𝜋𝐺 represents the Planck mass, with 𝐺 being the Newton's gravitational constant) within the cosmological context implies that the dark energy density of the universe 𝜌_𝑥 takes the same form as the vacuum energy; that is, 𝜌_Λ=𝜌_𝑥 [8]. Using the largest 𝐿 as the one saturating the above inequality, it turns out to be HDE, whose density is given by 𝜌_Λ=3𝑐²𝑀²_𝑝𝐿⁻² [6], where 𝑐 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. [19] 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 [20]. The Ricci curvature of FRW universe is given by 𝑅=−6(𝐻̇+2𝐻²+𝑘/𝑎²). The RDE proposed in [19] has energy density which is proportional to the Ricci scalar curvature; that is, 𝜌_Λ∝𝑅. 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 𝜌_MHRDE=(2/(𝜉−𝜂))(𝐻̇+(3𝜉/2)𝐻²), where 𝜉 and 𝜂 are free constants;(ii)extended holographic Ricci dark energy (EHRDE) [10, 34], whose density has the form 𝜌_EHRDE=3𝑀²_𝑝(𝜉𝐻²+𝜂𝐻̇), where 𝑀²_𝑝 is the reduced Planck mass and 𝜉 and 𝜂 are constants to be determined. If 𝜉=2𝜂, 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 𝐻 for both cases. Based on the reconstructed Hubble parameters, we will investigate the behavior of the equation of state parameter 𝜔 and the deceleration parameter 𝑞.

2. The Dark Energy Models in the Fractal Universe

The total action of Einstein gravity in a fractal space-time is given by [4] 𝑆=𝑆_𝐺+𝑆_𝑚, (1) where the gravitational part of the action is given by 𝑆_𝐺=1(𝑥)√16𝜋𝐺𝑑𝜚−𝑔𝑅−2Λ−𝜔𝜕_𝜇𝑣𝜕^𝜇𝑣, (2) and the matter part of the action is given by 𝑆_𝑚=𝑑𝜚(𝑥)√−𝑔ℒ_𝑚. (3)

Here, 𝑔 is the determinant of the dimensionless metric 𝑔_𝜇𝜈, Λ and 𝑅 are, respectively, the cosmological constant and the Ricci scalar. Moreover, 𝑣 is the fractional function, and 𝜔 is the fractal parameter. The standard measure 𝑑⁴𝑥 is replaced with a Lebesgue-Stieltjes measure 𝑑𝜚(𝑥). Taking the variation of the action given in (1) with respect to the FRW metric 𝑔_𝜇𝜈, we obtain the Friedmann equations in a fractal universe as [4] 𝐻²+𝑘𝑎²𝑣̇+𝐻𝑣−𝜔6𝑣̇²=8𝜋𝐺3Λ𝜌+3,𝐻̇+𝐻²−𝐻𝑣̇𝑣+𝜔3𝑣̇²−12□𝑣𝑣=−8𝜋𝐺6+Λ𝜌+3𝑝3. (4)

The curvature parameter 𝑘=0,1,−1 corresponds to a flat, closed, and open universe, respectively. In the present work, we will consider 𝑘=0 as flat universe. The continuity equation in a fractal universe takes the form ̇𝑣̇𝜌+3𝐻+𝑣𝜌+𝑝=0. (5)

It is clear that for 𝑣=1, 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 𝑣=𝑡^−𝛽, where 𝛽=4(1−𝛼) (with (0<𝛼≤1)) is the fractal dimension. The Friedmann equations given in (4) in the absence of the cosmological constant can be written as 𝐻²+𝑘𝑎²𝛽+𝐻𝑡−𝜔𝛽²6𝑡^2(𝛽+1)=8𝜋𝐺3̇𝜌,𝐻+𝐻²𝛽−𝐻+𝛽2𝑡𝛽+12𝑡²+𝜔𝛽²3𝑡^2(𝛽+1)=−8𝜋𝐺6𝜌+3𝑝, (6)(7) while the continuity equation given in (5) takes the form ̇𝛽𝜌+3𝐻−𝑡𝜌+𝑝=0. (8) In (6), we replace 𝜌 by 𝜌_MHRDE and 𝜌_EHRDE, respectively, for flat FRW universe (𝑘=0), and we get nonlinear differential equations on 𝐻. For MHRDE, we have 𝜌̇_MHRDE=2̇̈𝐻.𝜉−𝜂3𝜉𝐻𝐻+ (9) Using (9) in continuity equation given in (8), we get the pressure 𝑝_MHRDE for the MHRDE model as follows: 𝑝_MHRDE=2−3𝜉−𝜂2𝜉𝐻²−̇𝑡𝐻+̇𝐻̈.𝛽−3𝑡𝐻3𝜉𝐻𝐻+ (10)

For EHRDE we have 𝜌̇_EHRDE=6𝜉𝐻𝐻̇+3𝜂𝐻̈. (11) Using (11) in continuity equation given in (8), we get the pressure 𝑝_EHRDE for the EHRDE model as follows: 𝑝_EHRDE=−3𝜉𝐻²𝐻̇+̇𝐻̈+𝜂3𝑡2𝜉𝐻𝐻+𝜂.𝛽−3𝑡𝐻 (12)

Since we cannot have analytical solution of the differential equations, we solve them numerically.

We plot the reconstructed Hubble parameter 𝐻 in Figures 1 and 2 for MHRDE and EHRDE, respectively. In the figures for MHRDE the solid, dashed, and dot-dashed lines correspond to (𝜉=0.5,𝜂=1.12), (𝜉=0.5,𝜂=1.14), and (𝜉=0.5,𝜂=1.17), respectively. For EHRDE, the solid, dashed, and dot-dashed lines correspond to (𝜉=1,𝜂=2), (𝜉=1.01,𝜂=2), and (𝜉=1.05,𝜂=2.01), respectively. For all figures, we have considered 𝛼=0.8 and 𝜔=0.02.

Figure 1 

Reconstructed Hubble parameter 𝐻 for MHRDE in fractal universe.

Figure 2 

Reconstructed Hubble parameter 𝐻 for EHRDE in fractal universe.

In Figure 1, we observe that 𝐻 is a decreasing function of cosmic time 𝑡 for MHRDE. In Figure 2, we observe that for (𝜉=1.01,𝜂=2) in EHRDE, 𝐻 is a decreasing function of 𝑡. However, for the other combinations, 𝐻 is decreasing up to 𝑡≈1.2, and then it is increasing gradually. Next we plot the time derivatives of the reconstructed Hubble parameter ̇𝐻 in Figures 3 and 4. For MHRDE (Figure 3), we find that ̇𝐻 is increasing with 𝑡 and staying at negative level throughout. Although for EHRDE (Figure 4) we observe that ̇𝐻 is increasing with 𝑡, we find that for (𝜉=1,𝜂=2) and (𝜉=1.05,𝜂=2.01) the ̇𝐻 is changing its sign at 𝑡≈1.3.

Figure 3 

Time derivative of reconstructed Hubble parameter ̇𝐻 for MHRDE in fractal universe.

Figure 4 

Time derivative of reconstructed Hubble parameter ̇𝐻 for EHRDE in fractal universe.

In Figure 5, we find that 𝜌_MHRDE is decreasing with 𝑡. Instead, in Figure 6, we observe that 𝜌_EHRDE is increasing with 𝑡. The rate of increasing is less for (𝜉=1.01,𝜂=2) 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 (𝜉=1,𝜂=2) and (𝜉=1.05,𝜂=2.01) (Figure 8). The EoS parameters 𝑤_MHRDE=𝑝_MHRDE/𝜌_MHRDE and 𝑤_EHRDE=𝑝_EHRDE/𝜌_EHRDE are plotted, respectively, in Figures 9 and 10. In Figure 9, we observe that 𝑤_MHRDE>−1 for all combinations of 𝜉 and 𝜂. This indicates a quintessence-like behavior. In Figure 10, we observe that 𝑤_EHRDE transits from >−1 to <−1 for (𝜉=1,𝜂=2) and (𝜉=1.05,𝜂=2.01) at 𝑡≈1.3. This indicates a transition from quintessence to phantom. Thus, the EoS parameter has a quintom-like behavior in these two cases. For (𝜉=1.01,𝜂=2), we have 𝑤_EHRDE>−1.

Figure 5 

Density of MHRDE in fractal universe.

Figure 6 

Density of EHRDE in fractal universe.

Figure 7 

Pressure of MHRDE in fractal universe.

Figure 8 

Pressure of EHRDE in fractal universe.

Figure 9 

EoS parameter for MHRDE in fractal universe.

Figure 10 

EoS parameter for EHRDE in fractal universe.

The deceleration parameter 𝑞=(1/2)(1+3𝑝/𝜌) for the two models has been plotted in Figures 11 and 12, and in both cases it stays at a negative level, which indicates accelerated expansion of the universe.

Figure 11 

Deceleration parameter for MHRDE in fractal universe.

Figure 12 

Deceleration parameter for EHRDE in fractal universe.

3. Conclusion

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 𝑣=𝑡^−𝛽, where 𝛽=4(1−𝛼). 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 𝑡, 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.

Acknowledgment

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.