Modeling holographic dark energy with particle and future horizons

Abstract In this work we confront the particle horizon and future horizon as candidates to model the dark energy within a holographic context for a flat Friedmann-Lemaitre-Robertson-Walker universe. Under these considerations we can find that the model admits a genuine big rip singularity when the dark energy density is sketched by the future horizon, in consequence the resulting parameter state can cross to the phantom regime. For the particle horizon case the cosmological fluid can emulate ordinary matter. Additionally, the coincidence parameter has a decreasing behavior for the future horizon. On the other hand, from the interacting scheme for cosmological fluids we obtain that in the dark energy dark matter interaction, the dark energy fluid will have negative entropy production, therefore the second law of thermodynamics can not be guaranteed in this sector.


I. INTRODUCTION
If we consider that general relativity is the correct description of gravity on cosmological scales, it is well known that the observed expansion of the universe can be obtained from any homogenous and isotropic cosmological model, moreover, recent observations revealed that such expansion presents a late time accelerated phase [1]. The scale factor of a homogeneous and isotropic universe will expand with accelerated rate always that the pressure and density of the cosmological fluid obey the following relation, p < −(1/3)ρ.
However, the acceleration could arise from a more general form of dark energy with the same characteristic of having a negative pressure, which has a repulsive gravitational effect.
The simplest explanation for this accelerated expansion is provided by the cosmological constant but until now it has not been possible to provide a clear justification for the very small value of its magnitude. These and another kind of difficulties have led over the years to consider possible extensions or modifications of general relativity, from including a scalar field (see for instance Ref. [2]) to increasing the number of dimensions of spacetime, see the Chapter 27 of Ref. [3] for an interesting review on this subject.
An interesting approach for the dark energy problem is given by the holographic principle, which establishes that in quantum field theory the ultraviolet cut-off is related to the infrared cut-off, and this is imposed by the formation of a black hole [4]. A complete review for holographic dark energy models can be found in [5]. If ρ is the density caused by the ultraviolet cut-off, then the total energy in a region of size L must be of the order of the mass of a black hole of the same size, therefore such density must be of the form ρ ∼ L −2 . Several works have considered L −1 as the Hubble scale since the resulting density is comparable with the present day dark energy value [5][6][7][8][9][10]. However, the Hubble scale is not the unique option to consider for the size L, see the Refs. [11,12] where the particle and future horizons are considered as generalized holographic dark energy models for a specific f (R) gravity model and for the unification between the inflationary stage and late time epoch in a scalar field cosmological model. In Ref. [13], a holographic dark energy model based on the future horizon is explored in order to find a possible relation between the quantum entanglement and the dark energy. On the other hand, in Ref. [14] with the use of the apparent horizon, a holographic dark energy model is constructed for a curved universe, resulting that such approximation is adequate to describe the late time evolution of the universe.
In this paper we explore two different possibilities for the characteristic length, L, to be considered in the expression of the holographic dark energy density in order to establish which of the two options is a better candidate to describe this sector of the universe. As we will see later, the future horizon presents more similarities with the dark energy behavior in several aspects, being one of the most interesting that in this case the model admits a genuine big rip singularity (phantom scenario). It is interesting to have as final fate of the universe this scenario since is not ruled out by the observational data [15,16]. It is important to point out that also under the election of the future horizon, the model seems to have an alleviation for the cosmological coincidence problem and on the other hand, the dark energy density becomes dominant over other possible components of the universe at late times. As we will see, for the particle horizon case the parameter state, ω, can take positive values, therefore this characteristic length is less viable to describe the dark energy content of the universe. Finally, when the interacting scheme is considered, despite the temperature for the dark energy sector has a positive value, the entropy production is always negative.
The organization of this work is the following: In Section II we provide some general aspects of the holographic dark energy for a spatially flat FLRW universe. We write some well-known results in the holographic scheme for the dark energy density and discuss the future singularity admitted by the model when the future horizon is considered. At the end of the section we briefly discuss the behavior of the dark energy density when the model approaches the far future. In Section III we consider an universe with dark energy and dark matter content with no interaction between them. From this description it is possible to see that the coincidence parameter has a decreasing behavior at late times, which is in agreement with observations. In Section IV we adopt the standard thermodynamics point of view. The computed temperature for the particle horizon and future horizon remains positive but in the future horizon case has a divergent behavior in the far future, as expected, the adiabaticity condition is obtained for non interacting fluids. In Section V we consider the interacting scheme between dark energy and dark matter. In this description a negative entropy production is obtained for the dark energy fluid. In Section VI we write the conclusions of our work.

II. HOLOGRAPHIC DARK ENERGY
In the framework of the spatially flat FLRW geometry, we define the Hubble rate as where a is the scale factor and the dot represents a derivative with respect to time. By means of the Friedmann constraint, we can write 3H 2 = ρ, being ρ the energy density of the cosmological fluid. Since we are interested in a general holographic description, we will assume the conventional formula for the Hubble rate as where c is a positive constant for an expanding universe and L is the cosmological length scale. For this radius we will focus on two possibilities, the particle and the future horizons denoted by L p and L f , respectively, which are given by the following expressions If we consider the Eqs.
(1)-(3), we can obtain [11,12] d dt where the sign +(−) corresponds to the particle (future) horizon. From the previous expression we can solve for the Hubble ratė whose solution is given by being t 0 some initial time. Using the above equation we can write the following expression for the scale factor the pressure of the fluid and its density respectively, together with the Friedmann equations and the Eq. (6), the parameter state takes the form In general, c is a constant given in the interval 0 < c < 1 [20], therefore we can have the following possibilities 1 + 1/c > 1 and 1 − 1/c < 0, it is important to point out that in each case the parameter state given in Eq. (8) will represent an ordinary matter behavior or a phantom fluid, respectively. Besides, for the choice c = 1 in the above equation results, ω − = −1, which represents a cosmological constant evolution, this result is also obtained in Ref. [7], however, note that only under the election of the future horizon, the model resembles a cosmological constant evolution. If we consider the future horizon case in the expression (6) we have in the previous expression one gets which represents a genuine big rip singularity for t = t s according to the classification given in Ref. [21], besides t s = t 0 + 1/[(1/c − 1)H(t 0 )] > t 0 , by means of the Friedmann equations ρ ∼ H 2 and p ∼ 2Ḣ + 3H 2 , therefore as t → t s we have ρ → ∞ and p → ∞ simultaneously. t s represents a finite time in the future at which the singularity will take place. The generalities of a big rip singularity within the holographic context was studied in Ref. [17]. In Ref. [10] can be found that within the framework of holographic dark energy a singular behavior can be induced for the Hubble rate when the spatial curvature is included, however the singularity obtained is not a genuine big rip but instead a Type III future singularity. It is important to point out that given the quantum nature of the holographic scheme, the effects coming from quantum gravity near the singularity could help to avoid the big rip fate for the universe [5,18], in other words, the phantom scenario of the universe could be only a transient stage.
By considering the standard expression for the redshift, 1 + z = a(t 0 )/a(t), from the scale factor given in Eq. (7) we can write where α ± is a constant defined as α ± := 1 ± 1/c. By replacing the previous equation in the expression (6), we obtain the Hubble rate as a function of the redshift as follows where In order to describe the cosmological fluid we will consider the particle and future horizons as holographic cut-off, therefore the conventional formula for the dark energy density becomes where p and f denotes the particle (future) horizon, respectively. The previous expression can be written as a function of the redshift as follows where the Eqs.
(1) and (12) were considered together with the Friedmann constraint and we have defined the constant ρ ± (0) := 3c 2 H 2 (0). It is important to point out an interesting feature of the density expression given in Eq. (14), if we consider the future horizon case we can see that the exponent will be negative, therefore as the model evolves to the future this density increases, ρ − (z → −1) → ∞, in other words, the dark energy density becomes dominant over other matter components. This characteristic for the dark energy is identified in the well known cosmological coincidence problem and it is corroborated by observational data [19]. On the other hand, for the particle horizon case, as the model evolves to the future the density tends to zero.

III. NON INTERACTING COSMOLOGICAL FLUIDS
From now on, we will denote by ρ 1 the dark energy density and ρ 2 will describe a dark matter density, we will also assume the corresponding parameter state for the dark matter as ω 2 = 0. For an universe composed by dark energy and dark matter, the Friedmann constraint reads where the dark energy density is described by the holographic expression given in Eq. (14) and ρ 2 (0) is an appropriate constant. The corresponding continuity equations for the densities are given as follows where the prime stands for a derivative with respect to z, additionally we have being r ± (z) the coincidence parameter, which is defined as the ratio between the densities for dark matter and dark energy, r ± (z) = ρ 2 (z)/ρ ± 1 (z) and we have defined the constant r ± (0) := ρ 2 (0)/ρ ± 1 (0). In order to be in agreement with observational data, the coincidence parameter must decrease as the universe expands [19,22], note that as we approach to the far future (z = −1), the coincidence parameter given in Eq. (18) can have an increasing (decreasing) behavior, which only depends on the election of the particle (future) horizon 1 . This is consistent with the result obtained for the holographic dark energy density in the previous section. The coincidence parameter (18) can be written in terms of the scale factor, one gets in each case since 0 < c < 1, we have the limits r + (a → ∞) → ∞ and r − (a → ∞) → 0. From the previous results we can observe that the coincidence parameter problem is solved in the context of holographic dark energy under the election of the future horizon.

IV. THERMODYNAMICS
In standard cosmology, for a perfect fluid we have the following temperature evolution The previous equation is valid always that the Gibbs integrability condition holds together with the number (n) and energy conservation. If we consider a barotropic equation of state in the temperature evolution we can write where T (0) is an integration constant. If in the previous expression we consider the parameter state given in Eq. (8), we obtain for the temperature from the last equation we can obtain the following condition, T + (z → −1) → 0, and for the future horizon we have a divergent behavior as we approach the far future. Given that ρ − has an increasing behavior together with T − , we will assume that when the dark energy is described by the future horizon, this thermodynamics description is consistent.
Within the context of standard thermodynamics, the problem of the entropy and temperature for the phantom regime has been widely studied, since the positivity of both quantities can not be guaranteed simultaneously unless a negative chemical potential is introduced by hand [24][25][26]. This problem was solved recently in Ref. [27] by the introduction of dissipative effects in the framework of irreversible thermodynamics.
On the other hand, the Gibbs equation reads where V is the Hubble volume given by V (a) = V (a(t 0 ))(a(t)/a(t 0 )) 3 , therefore dV /V = 3(a(t)/a(t 0 )) −1 d(a(t)/a(t 0 )) = 3Hdt and also we have considered a barotropic equation of state, yielding and by means of the continuity equation for the density ρ 1 one gets therefore we can see that, S = constant, and this is independent of the election of the particle (future) horizon for the dark energy density ρ 1 , this feature is generally obtained for non interacting systems.

V. INTERACTING COSMOLOGICAL FLUIDS
For two interacting cosmological fluids, the continuity equations for both densities given in the expressions (16) and (17) must be written as follows where the Q-terms determine the interaction between the dark energy and dark matter.
Observational data seems to indicate that Q > 0 [22,28], this means that there exists an energy flow from dark energy fluid to dark matter one [29]. By following a similar procedure as the one described for non interacting fluids, we can establish

A. Entropy behavior
In this section we will provide a general description of the entropy behavior for the dark cosmological sector [30] obtained from Eq. (29).
• Dark matter sector: For this case we have that ω 2 = 0, therefore from the Euler relation we can establish that S 2 > 0 and T 2 > 0, using the r.h.s. of Eq. (29), but considering the change, z → t, one gets therefore for the dark matter sector we have a growth in the entropy.
• Dark energy sector: In this case we can have two possibilities for the l.h.s. of Eq. (29): T 1 < 0, dS 1 /dt > 0 and T 1 > 0, dS 1 /dt < 0, but since we are in the interacting scheme, if the entropy of the dark matter sector growths, the corresponding entropy to the dark energy must decrease, then the first possibility is discarded if we consider the obtained previous result, therefore with positive temperature, then for the dark energy sector the entropy decreases while for the dark matter sector it increases. On the other hand, given that the parameter state, ω − 1 , can take values in the phantom region, we can have the following situation for the phantom case if we consider the Euler relation (30), the product T − 1 S − 1 is negative, based on the previous results we have, S − 1 < 0, and this is the aforementioned positivity problem of the entropy and temperature for the phantom universe.
Note that from the Eq. (29) we can have As pointed out in Ref. [23], the symmetries of the FLRW spacetime allow only scalar dissipation, i.e., absence of energy flux due to heat flow, therefore the total entropy of the system fulfills the second law of thermodynamics always that T 2 < T 1 . However, until now the determination of both temperature values remains as a challenge for modern cosmology.

VI. FINAL REMARKS
In this work we consider a holographic description to model the dark energy in a spatially flat FLRW universe. This description is based on a comparison between the particle horizon and future horizon. Under this construction it was obtained a general solution for the Hubble rate, the information of each horizon can be identified in this general solution by the election of its corresponding sign, +, for the particle horizon and, −, for the future horizon. According to the election of the sign and the value of the constant c, which enters in the conventional holographic formula, the Hubble rate has several limit cases as the model approaches the far future, this is an important difference between this model and the ΛCDM model. Specifically, when the future horizon case is considered, the model admits a genuine big rip singularity.
By considering a barotropic equation of state we can extract the general form of the parameter state, ω, which also has the information of each horizon through the election of the appropriate sign, in a consistent way, when the future horizon is elected, the parameter state can take values in the phantom regime, ω < −1, additionally, for the specific value c = 1, the model resembles a cosmological constant evolution.
In order to have a clear visualization of the behavior of all quantities, we introduced the cosmological redshift in each of them. In this way, from the Friedmann equations we can see that the density of the dark energy content has an increasing behavior as the model approaches the far future when the future horizon is considered. This is the desired behavior for the dark energy since the observations also indicates this growth and it is known as the cosmological coincidence problem. On the other hand, for the particle horizon case, the density has a diluting behavior.
If we consider that the universe besides the dark energy also contains a dark matter fluid, we can provide a description when these fluids can interact or not. By adopting the second case we can construct the corresponding coincidence parameter and again, we have a decreasing behavior when the future horizon is considered and in complementary way, when the coincidence parameter is written in terms of the scale factor, we can see that as the universe expands the coincidence parameter tends to zero. From a thermodynamics point of view, the temperature for the dark energy fluid increases and can also have a divergent behavior for z = −1. On the other hand, for the particle horizon case we can have a phase of cooling down. For both cases the entropy takes a constant value. When we consider an interacting scheme between the dark energy and dark matter we can observe that in such case the entropy production for the dark energy sector will be negative despite its temper-ature is positive. Within the framework of irreversible thermodynamics this problem can be solved with the introduction of dissipative cosmological effects as it was done in Ref. [27].
In general grounds, the particle horizon is not a good candidate to model the dark energy content of the universe. One might think that in such case it could be a good alternative to model dark matter, if we consider a cosmological model with dark energy (ρ − 1 (z)) and dark matter (ρ + 1 (z)), then the Friedmann constraint will be given by by definition the coincidence parameter is r(z) = ρ + 1 (z)/ρ − 1 (z) ∼ (1 + z) 4/c , therefore r(z → −1) → 0, which is in agreement with observational data. Additionally, for the particle horizon we have, ω + 1 = −1 + 2(1 + 1/c)/3, if we consider the value c = 1 we obtain, ω + 1 = 1/3, and for c = 1/2 the fluid behaves as stiff matter. These values for the parameter state can not describe an accelerating universe.
It is important to mention that the analysis of the parameters of the model with the use of observational data is beyond the scope of this work. However, in the general descriptions of the cosmological quantities obtained for both horizons, we have considered only those cases in which it is well known that some previously constrained parameters of the model are consistent with observations. Finally, in Ref. [5] can be found that the fluctuations for the holographic dark energy are stable.