Diagnosing holographic dark energy models with statefinder hierarchy

We apply a series of null diagnostics based on the statefinder hierarchy to diagnose different holographic dark energy models including the original holographic dark energy, the new holographic dark energy, the new agegraphic dark energy, and the Ricci dark energy models. We plot the curves of statefinders $S^{(1)}_3$ and $S^{(1)}_4$ versus redshift $z$ and the evolutionary trajectories of $\{S^{(1)}_3, \epsilon\}$ and $\{S^{(1)}_4, \epsilon\}$ for these models, where $\epsilon$ is the fractional growth parameter. Combining the evolution curves with the current values of $S^{(1)}_3$, $S^{(1)}_4$, and $\epsilon$, we find that the statefinder $S^{(1)}_4$ performs better than $S^{(1)}_3$ for diagnosing the holographic dark energy models. In addition, the conjunction of the statefinder hierarchy and the fractional growth parameter is proven to be a useful method to diagnose the holographic dark energy models, especially for breaking the degeneracy of the new agegraphic dark energy model with different parameter values.


Introduction
Dark energy (DE) with negative pressure was considered as an exotic component causing the Universe to a stage of accelerating expansion and has been widely studied [1]. Because of the lack of knowledge about the nature of DE, physicists constructed a host of viable theoretical DE models. The ΛCDM model consisting of the cosmological constant (Λ) and the cold dark matter (CDM) is the simplest one, in which DE has the equation of state w = −1. And this elegant model is even defined as a criterion in several cosmological observations. However, the cosmological constant scenario has to face the so-called "fine-tuning problem" and "coincidence problem". Furthermore, different observational data are in tension with one another to some extent when constraining parameters of the ΛCDM model. Under such circumstances, the possibility that w is dependent on time cannot be excluded. At the present, a number of dynamical DE models have been suggested, such as quintessence [2], quintom [3], k-essence [4], Chaplygin gas [5], and so on.
In face of numerous DE models, it is important to discriminate various models. Sahni et al. [6] introduced the statefinder diagnostic {r, s}, which is a geometrical diagnosis in a modelindependent manner. The statefinder parameter pair {r, s} contains the third-derivative of a(t), where a(t) is the scale factor of the Universe. Since different DE models exhibit different evolution trajectories in the r-s plane, and especially can be separated distinctively with the values of {r 0 , s 0 }, the statefinder can be used to diagnose different DE models [7]. Besides, other diagnostics, such as Om and Om3 [8][9][10], were also used to distinguish the DE models. In the previous work [11], we compared the holographic DE models by using the statefinder pair {r, s}. Here, the holographic DE models include the original holographic dark energy (HDE) [12], the new holographic dark energy (NHDE) [13], the new agegraphic dark energy (NADE) [14], and the Ricci dark energy (RDE) [15], which were all proposed based on the holo-and give the specific expressions of them which contain variables Ω de and w dependent on redshift z, where Ω de is the fractional density of DE (Ω de ≡ ρ de /3M 2 p H 2 ) and w is the equation of state (EOS) of DE (w ≡ p de /ρ de ). The growth rate of perturbations is briefly described in the second part.

The statefinder hierarchy
In this paper, we consider a spatially flat Friedmann-Robertson-Walker (FRW) universe containing dark energy and matter. The Friedmann equation is where H =ȧ/a is the Hubble parameter (the dot denotes the derivative with respect to time t), M 2 p = (8πG) −1 is the reduced Planck mass, ρ de and ρ m are the energy densities for dark energy and matter, respectively.
The scale factor of the Universe, a(t)/a 0 = (1 + z) −1 , can be Taylor expanded around the present epoch t 0 as follows: where with a(t) (n) = d n a(t)/dt n . Various derivatives of a(t) have been described historically by other quantities. A 2 is the negative value of the deceleration parameter q, and A 3 is the statefinder r [6,21] or the jerk j [22]. In addition, A 4 and A 5 are the snap s and the lerk l [22,23], respectively. For the ΛCDM model, we can easily get: where Ω m ≡ ρ m /3M 2 p H 2 is the fractional density of matter. The statefinder hierarchy, S n , is defined as [18]: The reason for this redefinition is to peg the statefinder at unity for ΛCDM during the cosmic expansion, This equation defines a series of null diagnostics for ΛCDM when n ≥ 3. By using this diagnostic, we can distinguish easily the ΛCDM model from other DE models. Because of Ω m = 2 3 (1 + q) for ΛCDM, when n ≥ 3, statefinder hierarchy can be rewritten as: where the superscript (1) is to discriminate between S (1) n and S n . Obviously, S (1) n | ΛCDM = 1 for ΛCDM and S (1) 3 is statefinder r [6,21]. In this paper, we use the statefinders S (1) 3 and S (1) 4 to diagnose the holographic type DE models. We give the specific expressions of S (1) 3 and S (1) 4 using the variables Ω de and w dependent on redshift z: where the prime denotes the derivative with respect to x = ln a.

The growth rate of perturbations
The fractional growth parameter (z) [24] can also be used as a null diagnostic, which is defined as where f (z) = d ln δ/d ln a describes the growth rate of the linear density perturbation [25], where w either is constant, or varies slowly with time. For the ΛCDM model, γ 0.55 and (z) = 1 [25,26]. However, for other models, the values of γ and (z) depart from ΛCDM. For this reason, the fractional growth parameter (z) can be combined with the statefinders to define a composite null diagnostic (CND) {S n , } [18]. Obviously, we have {S n , } = {1, 1} for ΛCDM.

Holographic dark energy models
Based on the holographic principle, the dark energy density is defined as ρ de = 3c 2 M 2 p L −2 [12,27], where c is an    (1) 4to (min), and ∆ 0 = 0 (max)− 0 (min) within one model. introduced numerical constant characterizing some uncertainties in the effective quantum field theory, and L is the infrared (IR) cutoff in the theory. A series of DE models originating from the holographic principle were proposed. In this paper, we focus on the following models: HDE, NHDE, NADE, and RDE, and we describe them briefly in this section.

The HDE model
In the HDE model [12], ρ de = 3c 2 M 2 p L −2 , and L is the future event horizon given by Here, the prime is used to differentiate the integration variable from the lower limit in the integral. Note that throughout the paper, prime in integrals plays the same role as here. In this model, Ω de is described by the differential equation where the prime denotes differentiation with respect to ln a, and w is given by

The NHDE model
In 2012, the HNDE model in light of the action principle was proposed [13], in which the dark energy density reads where d is a numerical parameter, and For the NHDE model, Ω de and w can be given by w =λL 2 − 2da 2 whereL ≡ H 0 L,λ ≡ λ/H 2 0 , and E = H/H 0 .

The NADE model
In the NADE model [14], ρ de = 3n 2 M 2 p η −2 , where n is a numerical parameter introduced, and the IR cutoff is provided by the conformal time η, In this case, Ω de is the solution of the following differential equation where the prime denotes differentiation with respect to ln a, and w is given by

The RDE model
In the RDE model [15], the IR cutoff L is connected to Ricci scalar curvature, R = −6( H +2H 2 ). So Ricci dark energy density is where α is a dimensionless coefficient. Accordingly, one can get Ω de and w of RDE: where f 0 = 1 − 2 2−α Ω m0 is an integration constant.
Firstly, the evolutions of S (1) 3 versus redshift z for the holographic DE models are plotted in Fig. 1, and those of ΛCDM are also shown for comparison. We can see that in the low-redshift region the holographic DE models can easily be differentiated from the ΛCDM model, although in the highredshift region they all but the NHDE model are nearly degenerate with the ΛCDM model. Furthermore, the difference between various values of parameter in one model can be directly identified for HDE and RDE in the low-redshift region, and for NHDE in the range of z > 0.5. However, for NADE, the cases with different parameter values degenerate highly in both the low-redshift and the high-redshift region. Note that the S (1) 3 diagnostic for the holographic DE models has been discussed in our previous work [11], and we repeat the relevant discussion in this paper for making the paper selfcontained.
For breaking the degeneracy of NADE, in this paper we take into account S (1) 4 from the statefinder hierarchy diagnostic [18], which includes the fourth-derivatives of a(t). In Fig. 2, the evolutions of S (1) 4 versus redshift z for the holographic DE models are plotted, and those of ΛCDM are also shown for comparison. From Fig. 2, on one hand, the differences between the holographic DE models and the ΛCDM model become clearer in the low-redshift region, although the curves of HDE and RDE degenerate with those of ΛCDM in the high-redshift region but which is slighter than that of Fig. 1. On the other hand, it is important to see that the degeneracy in the NADE model with different parameter values appearing in Fig. 1 is broken, and for HDE, NHDE, and RDE models the cases with different parameter values can be discriminated more evidently in comparison with those of Fig. 1.
The same conclusion can also be drawn from Table I, in which we show the today's values of statefinders, S (1) 3to and S (1) 4to , and the differences of them, ∆S (1) 3to and ∆S (1) 4to , for the holographic DE models, where ∆S (1) 3to = S (1) 3to (max) − S (1) 3to (min) and ∆S (1) 4to = S (1) 4to (max) − S (1) 4to (min) within one model. The current values of the statefinders also play an important role in diagnosing different DE models. In Table I, we can see that the differences between different parameter values in one model are magnified through S (1) 4to , because the values of ∆S (1) 4to are remarkably bigger than those of ∆S (1) 3to for most cases. For the NADE model, S (1) 4to = 0.09, only slightly larger than ∆S (1) 3to = 0, which also indicates a weak degeneracy for different parameter values.
For further comparing the statefinders S (1) 3 and S (1) 4 , we make comparisons of the holographic DE models and the ΛCDM model in the S (1) 3 (z) plots (Fig. 3) and in the S (1)