New infrared cut-off for the holographic scalar fields models of dark energy

Introducing a new infrared cut-off for the holographic dark-energy, we study the correspondence between the quintessence, tachyon, K-essence and dilaton energy density with this holographic dark energy density in the flat FRW universe. This correspondence allows to reconstruct the potentials and the dynamics for the scalar fields models, which describe accelerated expansion.

The present observational data favor an equation of state for the dark energy with parameter very close to that of the cosmological constant. The next simple model proposed for dark energy is the quintessence (see [9], [10], [11]), a dynamical scalar field which slowly rolls down in a flat enough potential. The equation of state for a spatially homogeneous quintessence scalar field satisfies w > −1 and therefore can produce accelerated expansion. This field is taken to be extremely light which is compatible with its homogeneity and avoids the problem with the initial conditions.
Other scalar field models proposed to explain the nature of the dark energy, are related with K-essence models based on scalar field with non-standard kinetic term [12], [13]; string theory fundamental scalars known as tachyon [14] and dilaton [15]; scalar field with negative kinetic energy, which provides a solution known as phantom dark energy [16]. Another class of dark energy models involve non-standard equations of state [17], [18] (for a review on above mentioned and other approaches, see [6]). In all this models of scalar fields the cosmological dynamics is defined once the potential is proposed.
Another alternative to the solution of the dark energy problem, is related with some facts of the quantum gravity theory known as the holographic principle ( [19,20,21,22,23]). This principle emerges as a new paradigm in quantum gravity and was first put forward by t' Hooft [20] in the context of black hole physics and later extended by Susskind [23] to string theory. According to the holographic principle, the entropy of a system scales not with it's volume, but with it's surface area. In the cosmological context, the holographic principle will set an upper bound on the entropy of the universe [24]. In the work [22], it was suggested that in quantum field theory a short distance cut-off is related to a long distance cut-off (infra-red cut-off L) due to the limit set by formation of a black hole, namely, if is the quantum zero-point energy density caused by a short distance cut-off, the total energy in a region of size L should not exceed the mass of a black hole of the same size, thus L 3 ρ Λ ≤ LM 2 p . Thus, if we take the whole universe into account, then the vacuum energy related to this holographic principle is viewed as dark energy, usually called holographic dark energy. The largest L allowed is the one saturating this inequality so that we get the holographic dark energy density where c 2 is a numerical constant and M −2 p = 8πG. In this paper we use a new IR cut-off proposed in [25] for the holographic dark energy where H =ȧ/a is the Hubble parameter and α and β are constants which must satisfy the restrictions imposed by the current observational data. Besides the fact that the underlying origin of the holographic dark energy is still unknown, the inclusion of the time derivative of the Hubble parameter may be expected as this term appears in the curvature scalar (see [26]), and has the correct dimension. This kind of density may appear as the simplest case of more general f (H,Ḣ) holographic density in the FRW background. By other hand, contrary to the IR cut-off given by the event horizon [27], this model avoids the causality problem. The coincidence problem may also be solved as will be clear from the behavior of the scale parameter a. In this paper, using the correspondence with the holographic dark energy density we find the explicit form of the fields and potentials for the quintessence, tachyon, K-essence and dilaton fields, which determine the cosmological dynamics of this models.

The model
and integrating this equation with respect to the cosmological time t we obtain which gives rise to the power-law expansion a ∝ t β/(α−1) . Similar expression for H was obtained in [27] using the future event horizon as the infrared cut-off, and in [28] in the small t limit of an infrared cut-off that depends on local and non-local quantities. By other hand, from the conservation equation and using a barotropic equation of state for the holographic energy and pressure densities p Λ = ω Λ ρ Λ , we obtain an expression for the EoS parameter ω Λ which express ω Λ in terms of the constants α and β. In order to obtain accelerated expansion, the constants α and β must satisfy the restrictions followed from the Eq.

Correspondence with scalar field models
In this section we establish a correspondence between our proposal for the holographic density and various scalar field models, by comparing the holographic density with the corresponding scalar field model density and also equating the equations of state for this models with the EoS parameter given by (2.5). In this work we are not considering the contributions from matter and radiation to the Friedman equation, and for this reason the solution presented here differs from the one presented in the work [25]. Therefore we will not fix the constants here and will impose conditions on α and β dictated by the existence in each model of attractor solutions giving accelerated expansion.
In the flat Friedman background the energy density and pressure density of the scalar field are given by [6] The equation of state parameter for the scalar field is given by which compared with the holographic EoS parameter (2.5) gives the equatioṅ which together with the equation can be solved to obtain the explicit expressions for the scalar field and the poten- and A detailed analysis of the cosmological dynamics of an exponential potential is given in [6]. This potential can produce an accelerated expansion provided that β/(α − 1) > 1 (see Eq. (3.3)) and also has cosmological scaling solutions [9]. In the context of the phase-space analysis as presented in ( [6]), the exponential potential for the scalar field has attractor solutions which give rise to an accelerated expansion if α and β satisfy (α − 1)/β < 1 which is the same condition required by the power-law accelerated expansion.

Holographic tachyon model
The tachyon field has been proposed as the source of dark energy [29], [30] and may be described by effective field theory corresponding to some sort of tachyon condensate with an effective Lagrangian density given by [31], [32].
In a flat FRW background the energy density ρ and the pressure density p are given where V (φ) is the tachyon potential. From Eqs. where we assumed the integration constant equal to zero. Therefore, using Eq. (3.10) and the expression (2.2) for H we obtain for the tachyon potential in terms of the scalar field (3.11) this inverse square potential also corresponds to the potential obtained for scaling solutions in the context of brane world cosmology [33], [34]. Considering the phasespace analysis, note that for the tachyon system with the inverse square potential in the case of a scalar-field dominated solution (see [6], [35]), the condition for accelerated expansion translates into the condition or equivalently −1/9(1 + √ 10) < (α − 1)/β < 1/9(−1 + √ 10), which gives the only viable late-time attractor solution ( [6], [36]). Note that this restriction is consistent with the one imposed by the power-law accelerated expansion (α − 1)β < 1 and with (α−1)β < 1/2, so that the square root is well defined . An holographic correspondence with the tachyon, phantom and Chaplygin gas models using the event horizon, was poposed in [37], [38], [39].
Holographic K-essence model The scalar field model known as k-essence is also used to explain the observed latetime acceleration of the universe. It is well known that k-essence scenarios have attractor-like dynamics, and therefore avoid the fine tuning of the initial conditions for the scalar field ( [40]). This kind of models is characterized by non-standard kinetic energy terms, and are described by a general scalar field action which is a function of φ and X = −1/2∂ µ φ∂ µ φ, and is given by [41] where p(φ, X) corresponds to a pressure density and usually is restricted to the Lagrangian density of the form p(φ, X) = f (φ)g(X). Based on the analysis of the low-energy effective action of string theory (see [41] for details) the Lagrangian density can be transformed into From the energy momentum-tensor for this Lagrangian density, follows the next expression for the energy density of the field φ (see [41]) And the equation of state using (3.15) and (3.16) is given by Equating this parameter with the holographic EoS parameter (2.5) ω K = ω Λ we find the solution for X X = 1 3 3β − α + 1 2β − α + 1 (3.18) which shows that X is constant. The condition X < 2/3 which gives rise to an accelerated expansion, translates into α − 1 < β. The equation (3.18) can be solved to obtain the expression for the scalar field in the flat FRW background where we have taken the integration constant φ 0 equal to zero.
Using the correspondence between K-essence and holographic energy densities, Eqs where we used the Eq. (3.19) for φ. Hence, as a result of our holographic-K-essence correspondence one obtains the k-essence potential f (φ) given by (3.20), which is a result of the power-law expansion.

Holographic dilaton field
This model is described by the pressure (Lagrangian) density where c is a positive constant and X = 1/2φ 2 . This model appears from a fourdimensional effective low-energy string action [42] and includes higher-order kinetic corrections to the tree-level action in low energy effective string theory. The correspondence between the dilaton energy density given by ρ D = −X + 3ce λφ X 2 (see ρ D = −X + 3ce λφ X 2 = 3M 2 p αH 2 + βḢ (3.22) and the correspondence with the holographic dark energy equation of state is written as Solving this equation with respect to φ and integrating with respect to t we find replacing Xe λφ from Eq. (3.23) and using Eqs. can express λM p in terms of α and β.

Discussion
We propose an infrared cut-off for the holographic dark energy model, which includes a term proportional toḢ. Contrary to the holographic dark energy based on the event horizon, this model depends on local quantities, avoiding in this way the causality problem. In the case of dark energy dominance, the power-law expansion appears as the solution to the Friedman equations and this avoids conflict with the coincidence problem. Our proposal automatically generates scalar-field potentials which give rise to scaling solutions in a FRW cosmological background. We found in the case of quintessence that the potential has an exponential form which has attractor solutions giving rise to accelerated expansion for β/(α−1) > 1; for the tachyon field, the potential has an inverse squared form, which contains a late-time attractor solution in the region of the constants, satisfying 3.13. The potential for the k-essence holographic correspondence has also an inverse squared dependence on the field given by 3.20, and the model 3.15 with this potential, has an attractor solution with accelerated expansion for β/(α − 1) > 1 (see [41]). We also considered the dilaton condensate without potential, and used the correspondence to find the form of the scalar field and established the region for the constants in which we can expect scaling solutions giving rise to accelerated expansion. In conclusion, with the help of this proposal we have reconstructed the potentials for some scalar field models of dark energy and all this favors the proposed infrared cut-off as a viable phenomenological model of holographic density.