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Introduction
Nowadays it is well known that cosmological models must describe an accelerated expansion of the Universe at the present era [1-3]. To achieve this, sources of matter capable of generating this acceleration are considered, which are commonly dubbed dark energy [4].
A cosmological constant A is an important candidate for dark energy providing a good explanation for the current acceleration. But the cosmological constant faces some problems [5, 6] such as, the mismatch between the expected value of the vacuum energy density and the energy density of the cosmological constant, and the lock of an explanation of why densities of dark energy and dark matter are of same order at present while they evolve in rather different ways. So, as an alternative, dynamic dark energy models have been proposed and analyzed in the literature, highlighting the Holographic Dark Energy Models [7-12], these originate from the holographic principle in Cosmology [13]. The holographic principle asserts that the number of relevant degrees of freedom of a system dominated by gravity must vary along with the area of the surface bounding the system [14]. According to this principle, the vacuum energy density can be bounded [15] as p x ≤ M2 p L-2, where p x is the dark energy density (the vacuum energy density), M p is the reduced Planck mass, and L is the size of the region (i.e IR cutoff). This bound implies that, the total energy inside a region of size L, should not exceed the mass of a black hole of the same size. From effective quantum field theory, an effective IR cutoff can saturate the length scale, so that the dark energy density can be written as ρx = 3c2 M2 p L-2[16], where c is a dimensionless parameter, and the factor 3 is for mathematical convenience. In the Holographic Ricci Dark Energy Model, L is given by the average radius of the Ricci scalar curvature |R|-1/2, so in this case the density of the Holographic Dark Energy (hereafter, abbreviated as HDE) is px ∞ R.
In a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe, the Ricci scalar of the spacetime is given by R = 6(Ḣ + 2H 2), where H(t) = ά (t)/a(t) is the Hubble expansion rate of the universe in terms of the scale factor a, where the dot denotes the derivative with respect to the cosmic time t. In this sense, the authors of reference [7] introduced the following generalization:
where α and β are constants to be determined. This model works fairly well in fitting the observational data, and it is a good candidate to alleviate the cosmic coincidence problem [8-11,17].
Basic Equations
In the framework of General Relativity we consider a homogeneous, isotropic and flat universe scenario through the FLRW metric [18]
where (t, r, θ, Ф) are comoving coordinates. Friedmann's equations in this context are written as
where p is the total energy density, p is the total pressure and 8πG = c = 1 is assumed. Also, the conservation of the total energy-momentum tensor is given by [18]
Holographic Dark Energy Model
We studied a scenario that contains baryons, radiation, cold dark matter and HDE, i.e. ρ = ρ b + ρ r + ρ c + ρ x and ρ = ρ b + ρ r + ρ c + ρ x. In addition, we consider a barotropic equation of state for the fluids, ρ í = ω i p i with ω b = 0, ω r = 1/3, ω c =0ωx=ω and By including a phenomenological interaction in the dark sector, we split the conservation equation (5) in the following equations.
where prime denotes a derivative with respect to ln a3 and r represents the interaction function between cold dark matter and the HDE. From Eqs. (1) and (3) we obtain
Given that radiation and baryons are separately conserved, we have p
r ∞ α-4 and p
b ∞ α-3. From here it is easy to realize that p”
b
= -p’
b
= p”
b
and p”
r
=
On the other hand, in the study of HDE scenarios usually it is only considered the dark sector, since these predominate in the current universe. Also, it is possible to analyze a HDE scenario with two different approaches, the first one considers a variable state parameter for the HDE or assuming a parameterization as shown in [11], while the second approach considers an interaction term between the dark components [8,12,19]. We work in the last approach.
For convenience, we denote the energy density of the dark sector as p d:= p c + p x. Then, by combining equations (6) - (7) we obtain
where the submipt 0 denotes a current value. Notice that the Eq. (8) can be easily solve when Γ = Γ (p
d, p'd, p, p'). In the literature (see [20, 21] and its references) scenarios have been studied where only the dark components of the Universe are considered and a phenomenological interaction between them is included. It is usual to choose scenarios of interaction with a linear term, or linear combinations of the dark components [22]. For example, terms of interaction of the form were studied: Γα = αp
c+βp
x, Γb = α p'c+β p'x, Γ
c
= αp
c
p
x/p, Td = p
2
c/p, Γe = p
2
x/p, among others [20, 21]. Scenarios with linear interaction of type Γ ∞ p
c and Γ ∞ p
x, are particular cases studied in [22-24]. In the reference [23], the authors studied the interaction between dark matter and holographic dark energy, with an interaction term of the form Γ ∞ p, con p = p
x
, p = p
c
and p = p
x +p
c, and obtained a second order differential equation for H. While that in [24], the authors studied the interaction of dark matter and holographic dark energy with ω = ω(r), where r = p
c
/p
x
. Then, they obtained the interaction term Γ = Γ(p, p'), and finally, p
i = p
i (α) and ω = ω(α). It is so that in this work we consider the following types of linear interactions [20-22]:
The energy density of the dark sector
We can convenient rewrite Eq. (8) as
including the three interaction types of our interest where the values of the constants b i, b 2, b 3 and b 4 are shown in Table 1. The general solution of Eq. (9) is:
Tabla 1 De_nition of the constants b 1, b 2, b 3 and b 4 in terms of the model's parameters for the studied interactions.
where the integration constants C i and C 2 are given by
where H
0, Ω
c0
, and Ωx0 are the current values of the Hubble parameter, the density parameters for dark matter and HDE (i.e.) Ω
i0
= p
i0
/3H2
0 with i = {c,x}), respectively. The coefficients in eq. (10) are A = 
B=
,as well as
The state parameter of the HDE
The state parameter of the HDE corresponds to the ratio ω=
Using the expression (7) in Eq. (6), and the linear interactions Γ, we find
where à = (2α - 3/β)(A + p
bo
),
= 2(α - 2/β)(B + p
ro
),
1,2
= C
1,2(3βλ1,2 + 2α) and the constant coefficients D
i
are shown in table 2.
Tabla 2 De_nition of the constants D 1, D 2, D 3 and D 4 in terms of the model's parameters for the studied interactions.
In the limit to the future, α → ∞, the expression (12) remains as ω =
for λ1 > λ 2 > 0, while for λ 2 > λ 1 > 0, we have ω =
The coincidence and deceleration parameters
To examine the problem of cosmological coincidence, we define r = p c /p x . Then, using pc = p d - p x , together with the expression (7), we find
Then, for all our interactions we get r(α → ∞) =
- 1, a constant that depends on the interaction parameters, where λi = max{λ 1, λ 2} for λi > 0.
On the other hand, the deceleration parameter q is a dimensionless measure of the cosmic acceleration in the evolution of the universe. It is defined by q = -
[18]. Using (10), we obtain
Given the expressions (12)-(14), hereinafter we use the following values for the parameters [3]: Ωb0 = 0,0484, Ωr0 = 1,25 x 10-3, Ω co = 0,258, Ω x0 = 0,692, H 0 = 67,8 km s-1 Mpc-1, and ω Λcdm = -1. In addition, (α1 , β 1-0,0076,0) and (α2,/β 2) = (0,0074, 0) [20, 21] are considered. It is very important to emphasize that the interaction models between dark energy and dark matter [20,21,25] are based on the premise that no known symmetry in Nature prevents or suppresses a non-minimal coupling between these components, therefore, this possibility should be investigated in the light of observational data (see, for example [26]). In some classes of these interaction models, the coincidence problem can be greatly alleviated when compared to ACDM. Thus, several interaction models have been proposed with both analytical and numerical solutions [20,21,25-27].
Note that in equation (6), Γ>0 indicates a transfer of dark matter to dark energy and Γ<0 indicates otherwise. It is so, that in the Fig. 1, we analyze the behavior of the interaction terms for each model. It is shown that model 1 and 2 undergo a sign change in that function, while model 3 does not. The change of sign in the interaction term highlights the domain of one of the different types of matter in each epoch of evolution of the universe (fundamentally late universe). Thus, models 1 and 2 are useful for our study of the evolution of the universe.
FIGURA 1 Evolution of interaction term without dimensions for holographic interaction models. The orange, green and brown lines represent Models 1, 2 and 3, respectively.
In Fig. 2 we show the evolution of the coincidence and deceleration parameters in term of the redshift z, where α(z) = (1 + z)-1 The blue line represents ACDM, the orange line the model r1 with (α,β) = (0,86,0,46) and the green line the model Γ2 with (α,β) = (1,01,0,45). In the cases shown for the HDE models with interaction Γ1 and Γ2, the problem of cosmological coincidence is alleviated, given that the coincidence parameter r tend asymptotically to a positive constant. Besides, we note that the HDE models resemble the ACDM model, in the evolution of both parameters, noting only differences in quickness of falling of deceleration parameter value. However contrasting this with figure 1, i.e., taking into account the characteristics of interaction model, model 2 is the one that best describes the evolution of the late universe, the last two stages being dominated by dark components. It goes from a time dominated by matter (Γ<0) to a dominated by dark energy (Γ>0), in our case this dark energy is of holographic type.
Final Remarks
A theoretical model was developed according to the current components of the Universe, such as baryons, radiation, cold dark dark and HDE, with interaction in the dark sector, obtaining for the HDE, the functions ω(z), r(z) and q(z). The proposed model was compared graphically with ACDM, using referential values for the HDE parameters and the given interactions.
In the near future we expect to contrast the present scenarios with the observational data (SNe Ia, CC, BAO, CMB), using Bayesian statistics.
