The scalar field in the framework of the holographic principle

Let us consider the simplest model of non-minimally coupled scalar field with potential in the presence of holographic dark energy (vacuum energy). The action for the scalar field with matter in a general background is given by

where x ² = 8ϖG, S _m is the action for the matter fields which in the present case includes the usual baryonic and dark matter. Initially we consider the limit of dark energy dominance and neglect the baryonic and dark matter contribution. We are considering the flat Friedmann-Robertson-Walker (FRW) metric with signature (-,+,+,+). The Friedmann equation including the energy contribution from the holographic principle [40] is

where

and p _A is the energy contribution from the holographic principle. The equation of motion for the scalar field is given by

and the effective equation of state resulting from the time evolution of this model is

Quintessence Solutions

To solve the equations (2) and (4), we can propose the following functions

Replacing these expressions in (2) it is found that under the restriction

λφ ₀ = -2,

the following relation takes place (here we use ϰ ²= M _p ² )

And from the equation of motion (4) one finds

(3p - 1) φ₀ ² - 2t₀ ² V₀ = 0. (8)

The last two equations give us the initial values and in terms of the parameters

φ₀ ² = 2M² _p [(1 - α) p + β], (9)

and

t₀ ²V₀= M² _p (3p - 1) [(1 - α)p + β]. (10)

The scalar field and the potential keep the same functional dependence on time as in the simplest case of the canonical scalar field on FRW background with the advantage, in the present case, that there exist phantom solutions (leading to BR singularities) without resorting to ghost degrees of freedom. Using the initial value φ₀ one finds the power p as

In absence of scalar field this equation gives the usual restriction for accelerated expansion [40], β > α - 1, and with the scalar field the Eq. (11) gives more possibilities including negative values of p leading to super acceleration, depending on the relation between φ₀ and Mp. Thus, the conditions for accelerated expansion (p > 1) take the form

or

where y = qJMp. Note that these conditions take place whether y is greater than or less than 1.

Big Rip solutions

Let's consider the following solutions

with q > 0, which lead to Big Rip singularity at t = t_c. Replacing (14) in the Eqs. (2) and (4) one finds the same restriction, λ = -2/φ₀, and the relations

and

and the expression for q in terms of φ₀ is given by

where y = φ ₀ /Mp. The effective equation of state from (5) takes the value

From (15) and (16) follows that, given q > 0, we don't need to resort to ghost degrees of freedom, since φ² ₀ > 0 whenever (1 - α) q - β > 0, which implies that V₀ > 0. Thus, the standard canonical scalar field in the vacuum background generated by the holographic principle can describe an accelerated expansion with EoS bellow the phantom divide, i.e.W _eff < -1^.

The matter contribution and critical points

In this case the cosmological equations take the form

with the scalar potential given by

V = V₀е^iφ (20)

where p _m and p _m are the density and pressure of the matter contribution respectively (baryonic and dark matter), p_A is the pressure corresponding to the vacuum energy. Both type of energy contributions are modeled by ideal fluids that obey the independent continuity equations

p _m + 3H(p _m + p _m ) = 0, p _A + 3H (p _A +p _A ) = 0 (21)

additionally, we have the equation of motion for the scalar field given by (4) which is not independent of (18) and (19). Here we assume that the equation of state for matter, w _m = pm/pm is constant, which allows the integration of the conservation equation, giving pm = Pm0 a^{-3(1 + W} _m ^). To understand some dynamical properties of the model we will consider the autonomous system and analyze the properties of the critical points. The dynamical variables can be deduced from (18) and will be defined as

which lead to the following restriction from Eq. (18)

x ² + y² + Ω_m.+ Ω_A = 1 (23)

and from Eqs. (4), (19) and (21), it is straightforward to derive the following equations

where N = ln a is the slow-roll variable and y = 1 + W_m. Note that if one sets α = β = 0, then the Eqs. (21)-(25) reduce to the autonomous system for the uncoupled quintessence scalar [33]. The effective equation of state is given by

The following are the fixed points of the dynamical system (23)-(25).

where

By setting a = β = 0 we recover the critical points corresponding to the minimally coupled scalar field. The critical points characterize different cosmological scenarios, depending on the parameters, α β, y, λ. Here we illustrate some cases that involve the effects of the holographic density:

The point PI:

The corresponding eigenvalues are

If we take a = 1, then w _eff = -1 and Ω_m = 0. Then the energy becomes dominated by the holographic component and the point is a de Sitter solution with eigenvalues (0,-3), which corresponds to marginally stable fixed point. On the other hand, if we take a = 3βy/2, then Weff = y - 1 = w _m with the energy density dominated by matter Ω_m = 1, and eigenvalues

which corresponds to a saddle point for 0 < y < 2. The case a = β = 0, which corresponds to the standad quintessence scalar, gives the known matter dominated solution (Ωm = 1) with w _eff = w _m [33]. In this last case the eigenvalues become giving a saddle point for 1 < y < 2 . The de Sitter solution in this point is due to the holographic component.

The point P2:

The eigenvalues are

If we take a = 3β, then Ωm = 0 and the solution becomes dominated by the scalar field (Q _φ = 1) and is stable only under the conditions λ < - > β, < y<2. Note that w _eff = 1 for any value of the parameters, including the case of the quintessence scalar field corresponding to a = β = 0 (P2 = (-1,0), giving also Ωm = 0), with the eigenvalues given by (-3(Y -2), 3 + λ), which lead to unstable (λ > -6) or saddle point (λ < -6) for 0 < y < 2. The difference between these two cases is that the inclusion of the holographic component can lead to stable fixed point. There is also a stable solution dominated by matter (Ω_m = 1, Ω _φ + Ω_A = 0) if one sets

In this case the scalar field and holographic contribution cancel each other, which is not of interest since one of these density parameters should be negative.

The point P3: his point gives the same results

with eigenvalues

and the stability for the scalar field dominated solution is reached under the conditions, 0< y<2. The quintessence case (α = β = 0), giving P3= (1,0)) is unstable for λ < and saddle for λ > , given that 0 < y < 2. The matter dominated solution that takes place for y = 2 (α- 3β + 1), is stable under the restrictions

The point P4: First we note that for the case , and independently of the other parameters (β, y, λ), w _eff = -1 and Ω _m = 0, Ω _φ = 0, Ω_A = 1. Due to its complexity to calculate the eigenvalues, we analyzed the case y = 1, and have found that the point is an attractor if for any λ ≠ 0. This de Sitter attractor is due exclusively to the holographic component. For the quintessence scalar field a = β = 0, the fixed point takes the values giving the stable scaling solution with w _eff = w _m for λ² > 3y. One case of quintessence solution takes place if one sets α = β = 2/λ². In this case we find with This point is an attractor with 0 < Ω_m, Ω _φ , Ω_A < 1 and -1 < w _eff < 0 if and 0 < y < λ²/3. Another stable quintessence solution is obtained by taking a = y/3 and β = 2/λ², giving . This point is an attractor with

where all the density parameters are in the interval 0 < Ωm, Ω ₉ , Ω _A < 1 whenever and and

The point P5 as in the case of the point P4 contains a de Sitter attractor (w _eff = -1) dominated by the holographic component (Ωa = 1) for a = 1, and the stability results from the restriction y = 1 and β > 2/3 (λ² + 3) for any λ ≠ 0. This critical point contains also the quintessence solutions described for the point P4 with the same stability properties.

The point P6: For the scalar quintessence field, taking a = β = 0, this critical point takes the values and becomes stable node dominated by the scalar field with w _eff = -1 + λ² /3, provided that λ < and y > λ² /3. By setting α = 1 in this point we find

This point satisfies all the conditions for an attractor solution with accelerated expansion, but the analytical expression for the conditions on the parameters are very large and we limit ourselves here to the two numerical examples: taking λ = , β = 0.9, y = 1.05, it is found w _eff = -0.785, Ωm = 0.18, Ω _φ = 0.11 and Ωa = 0.71. Taking λ = 3, β = 0.7, y = 1.065, one finds w _eff = -0.999, Ω _m = 0.0007, Ω _φ = 0.00008 and Ω_A = 0.999. The above expressions for the main physical parameters simplify if one sets β = 2/λ ² . In this case we obtain

w _eff = -1 + λ²/3, Ω _m = 0, Ω _φ = 1, Ω_A = 0

This scalar field dominated critical point is an attractor if

By taking a = 1 and this point gives a de Sitter saddle point dominated by the holographic component (Ω_A = 1, Ω _m = 0, Ω _φ = 0) with eigenvalues (-3,0).

The point P7 presents the same properties as the point P6, leading to stable de Sitter and quintessence attractors.

Non-minimal coupling in the framework of the holographic principle

Here we assume the generalization of the holographic principle in the presence of non-minimally coupled scalar field. Let us start with the following action for a dark energy dominated universe

Variation with respect to the metric, and assuming that the scalar field <p has only time dependence, gives the following modified Friedman Eqs. in the flat FRW background [44, 45].

which corresponds to the (00) component of the variation with respect to the metric, and for the (11) component it is obtained

where H is the Hubble parameter, and Pa, Pa are the energy density and pressure of the holographic dark energy. The equation of motion of the scalar field is the modified Klein-Gordon equation

Replacing the holographic density (3) in (3), leads to the following Friedmann equation

Although the above equations may include the potential, in this case we show that the effect of accelerated expansion can be obtained without the need to introduce a potential. Due to the non-minimal coupling from (30), the effective gravitational coupling can be expressed as

where G is the constant Newtonian coupling. The relative time variation of the gravitational coupling, obtained from (34) is given by

An appropriate solution of the equations (32) and (33) should give the relative variation of the gravitational coupling consistent with current observational bounds.

The equation (33) simplifies under the restriction a = 1, and has an exact solution for the scalar field for H = H₀ = const., corresponding to de Sitter expansion with w _eff = -1.

Setting a = 1 and H = H₀ the equations (32) and (33) become

If one assumes the scalar field of the form

then, the Eqs. (36) and (37) have two solutions corresponding to

Of special interest is the solution with ξ = 1/6 corresponding to the conformal coupling of the scalar field. This de Sitter solution is not possible in absence of the holographic energy density. So, the non-minimally coupled scalar field in the framework of the holographic energy density can produce the same effect as the cosmological constant, leading to a de Sitter expansion when the coupling constant takes the conformal value ξ = 1/6 and the scalar field evolves as φ = φ₀e^ηHot.

For the solution (38), and using Eq. (35) we find the current value of as

Here we used 8ϖG= M^-2 _p (M _p is the Planck mass) and H₀ is the current value of the Hubble parameter. For the case of the conformal coupling (ξ= , η =-1), if we take φ₀ /M _p ≃10^-3 and t ~ H ₀ ^-1 , then the relative variation of the gravitational coupling is of the order of , clearly satisfying the observational bounds.