DGP Cosmological model with generalized Ricci dark energy

The braneworld model proposed by Dvali, Gabadadze and Porrati (DGP) leads to an accelerated universe without cosmological constant or other form of dark energy for the positive branch $(\epsilon =+1)$. For the negative branch $(\epsilon =-1)$ we have investigated the behavior of a model with an holographic Ricci-like dark energy and dark matter, where the IR cutoff takes the form $\alpha H^2 + \beta \dot{H}$, being $H$ the Hubble parameter and $\alpha$, $\beta$ positive constants of the model. We perform an analytical study of the model in the late-time dark energy dominated epoch, where we obtain a solution for $r_cH(z)$, where $r_c$ is the leakage scale of gravity into the bulk, and conditions for the negative branch on the holographic parameters $\alpha$ and $\beta$, in order to hold the conditions of weak energy and accelerated universe. On the other hand, we compare the model versus the late-time cosmological data using the latest type Ia supernova sample of the Joint Light-curve Analysis (JLA), in order to constraint the holographic parameters in the negative branch, as well as $r_cH_0$ in the positive branch, where $H_0$ is the Hubble constant. We find that the model has a good fit to the data and that the most likely values for $(r_cH_0, \alpha, \beta)$ lie in the permitted region found from an analytical solution in a dark energy dominated universe. We give a justification to use holographic cut-off in 4D for the dark energy in the 5 dimensional DGP model. Finally, using the Bayesian Information Criterion we find that this model it is disfavored compared with the flat $\Lambda$CDM model.


INTRODUCTION
The acceleration in the expansion of the universe during recent cosmological times, first indicated by supernova observations [1] and also supported by the astrophysical data obtained from WMAP, indicates the existence of a dark fluid with negative pressure, which have been identified as dark energy due to its unknown nature. Other non conventional approaches have advocated extra dimensions inspired by string and superstring theories. One of these models that have lead to an accelerated universe without cosmological constant or other form of dark energy is the braneworld model proposed by Dvali, Gabadadze, and Porrati (DGP) [2], [3], [4] (for reviews, see [5] and [6]). In a cosmological scenario, this approach leads to a late-time acceleration as a result of the gravitational leakage from a 3-dimensional surface (3-brane) to a fifth extra dimension on Hubble distances.
It is a well known fact that the DGP model has two branches of solutions: the selfaccelerating branch and the normal one. The self accelerating branch leads to an accelerating universe without invoking any exotic fluid, but present problems like ghost [7]. Nevertheless, the normal branch requires a dark energy component to accommodate the current observations [8], [9]. Extend models of gravity on the brane with f(R) terms have been investigated to obtain self acceleration in the normal branch [10]. Solutions for a DGP brane-world cosmology with a k-essence field were found in [11] showing big rip scenarios and asymtotically de Sitter phase in the future.
In the present work we explore, in the framework of the holographic dark energy models [12], [13], [14], based on the holographic principle [15], which is believed to be a fundamental principle for the quantum theory of gravity, a DGP cosmology. Based on the validity of the effective quantum field theory, Cohen et al [12] suggested that the total energy in a region of size L should not exceed the mass of a black hole of the same size, which The largest L is chosen by saturating the this bound so that we obtain the holographic dark energy (HDE) density where c is a free dimensionless O(1) parameter that can be determined by observations.
Taking L as the Hubble radius H = H −1 0 this ρ Λ is comparable to the observed dark energy density, but gives wrong EoS for the dark energy [13].
For higher dimensional space-times, the holographic principle in cosmological scenarios has been formulated considering the maximal uncompactified space of the model, i.e. in the bulk, leading to a crossing of phantom divide for the holographic dark energy, in 5D two-brane models [16]. Other investigation shows that when IR cut-off is the event horizon the vacuum energy would end up with a phantom phase with an inevitable Big Rip singularity [17].
Recently, a modified holographic dark energy model has been formulated using the mass of black holes in higher dimensions and the Hubble scale as IR cutoff [18]. Using the future event horizon as IR cutoff, it was found in that the EoS of the holographic dark energy can cross phantom divide [19]. The inclusion of a Gauss-Bonnet term in the bulk and an holographic energy density have been explored in [20], obtaining a late time acceleration consistent with observations. In the same approach, but using a Ricci dark energy, scenarios free of future singularities were found in [21].
Our aim in this work is to investigate a DGP model of a flat universe filled with an holographic Ricci dark energy [22] and dark matter. Using the SNIa data set and the Hubble parameters for different redshifts we constraint the holographic parameters and also for the parameter r c H 0 , where r c is the characteristic scale of the DGP model given by which sets a length beyond which gravity starts to leak out into the bulk.
In the next section we treat the DGP model with null curvature and a barotropic equation of state. In section III we work in the late-time phase universe and solve numerically a differential equation for E = H/H 0 , where H is the Hubble parameter and we show the table with the best estimates for the holographic parameters and figures of the confidence regions which was obtained marginalizing some variables for each branches. In section IV we expose the main calculation to use the SNIa data set and the Hubble parameters for different redshifts and we display the covariance matrix for each branches. In section V we change the time treat and now consider a dominating dark energy density, the cosmic evolution is then driving by only one fluid and we present how the future evolution will behave in terms of the redshift obtaining analytical expressions for the Hubble parameter and the scale factor, observing a "Big Rip" type singularity for positive branch. Finally we make the conclusions in the section VI.

II. DGP MODEL
For an homogeneous and isotropic universe described by the FLRW metric the field equation is given by [3], [4] (with 8πG = c = 1) where a is the cosmic scale factor, ρ is the total cosmic fluid energy density on the brane.
The parameter ǫ = ±1 represents the two branches of the DGP model. It is well known that the solution with ǫ = +1 represent the self-accelerating branch, since even without dark energy the expansion of the universe accelerates, and for late times the Hubble parameter approaches a constant, H = 1/r c . In the previous investigation, ǫ = −1 has been named as the normal branch, where acceleration only appears if a dark energy component is included.
By considering the null curvature case, the Eq.(3) becomes and the week energy condition (WEC) implies r c H ≥ ǫ. If cosmic fluid satisfy a barotropic equation of state p = ωρ, the conservation equation is given bẏ From Eqs. (4) and (5), we obtain an expression for equation of state parameter ω in terms of the Hubble parameter which is given by According to Eq.
0 for both cases (r c H) −1 ≶ 1 and we notice now that the WEC is strict. By using the deceleration parameter defined by we can write From this above equation we notice that r c H = 1 implies a divergence, so we only consider the case r c H = 1 in the rest of the work. For today we obtain that r c H 0 has the following expression In the next section, the condition r c H 0 > 1 is used from the beginning when the this free parameter is constrained from observational data.

A. Dynamics of the model
We shall consider an holographic Ricci dark energy, so the holographic energy density takes the form where α and β are positive constants. This type of holographic dark energy works fairly well in fitting the observational data. Nevertheless, a global fitting on the parameters of this model using a combined cosmic observations from type Ia supernovae, baryon acoustic oscillations, Cosmic Microwave Background and the observational Hubble data do not favor the holographic Ricci dark energy model over the ΛCDM model [23]. For the far future, the EoS behaves like a quintom model, crossing the phantom barrier [24], [25]. The statefinder diagnostic of this model, in the framework of general relativity, indicates that interactions in the dark sector are favored [26]. In was found that without giving a priori some specific model for the interaction function, this can experience a change of sign during the cosmic evolution [27].
For a spatially flat FRW universe composed by the holographic dark energy as well as a matter component (dark and baryon matter), the Friedmann equation (4) in the DGP cosmology has the form (with units) where the subscripts "h" and "m" stand for holographic dark energy and matter respectively.
The pressureless matter scale in the usual way, so ρ m = ρ m0 a −3 , where ρ m0 is the present-day value of the matter density in the Universe.
Inserting the expressions for ρ h and ρ m at eq. (11) and reorganizing terms, we have We change the derivative of H with respect to time to the scale factor asḢ = (dH/da)ȧ = (dH/da)aH, to obtain the differential equation Dividing the eq. (13) by the Hubble constant H 0 , defining the parameter density Ω m0 ≡ , changing of variable from the scale factor to the redshift, and defining the dimensionless Hubble parameter as E ≡ H/H 0 , the differential equation (13) becomes We solve numerically this differential equation with the initial condition E(z = 0) = 1, and for both branches, ǫ = ±1. We consider Ω m0 = 0.27, and the values of (α, β, r c H 0 ) are estimated and constrained using the cosmological observations of type Ia Supernovae and the Hubble parameter.

IV. COSMOLOGICAL PROBES
We test the viability of the model and constrain its free parameters (α, β, r c H 0 ) using the type Ia Supernovae (SNe Ia) observations and the Hubble parameter H(z) measured at different redshifts. We compute their best estimated values, the goodness-of-fit of the model to the data and the credible regions by a χ 2 function minimization, to constrain their possible values with levels of statistical confidence.

A. Type Ia Supernovae
We use the "Union2.1" SNe Ia data set (2012) from "The Supernova Cosmology Project" (SCP) composed by 580 type Ia supernovae [28]. Let us consider the definition of luminosity distance d L in a flat FRW cosmology where E(z, α, β, r c H 0 ) is given by the numerical solution of the differential equation (14) and 'c' is the speed of light given in units of km/sec. The theoretical distance moduli for the k-th supernova with redshift z k is defined as where m and M are the apparent and absolute magnitudes of the SNe Ia respectively, the superscript 't' stands for "theoretical". We construct the statistical χ 2 function as where µ k is the observational distance moduli for the k-th supernova, σ 2 k is the variance of the measurement and n is the amount of supernova in the data set (n = 580). The Hubble constant H 0 is marginalized assuming a constant prior distribution function. suggest. The χ 2 function is defined as where H obs  [32]; E(z, α, β, r c H 0 ) is given by the numerical solution of the differential equation (14) and σ Hi is the standard deviation of each H obs i datum.
We construct the total χ 2 t function that combine the SNe and H(z) data sets together, as where χ 2 SNe and χ 2 H are given by expressions (17) and (18) respectively. We numerically minimize it to compute the "best estimates" for (α, β). The minimum value of the χ 2 function gives the best estimated values and measures the goodness-of-fit of the model to data. We use also the definition of "χ 2 function by degrees of freedom", χ 2 d.o.f. , defined as where n is the number of total combined data used and p the number of free parameters estimated.
Once computed the best estimated values for (α, β, r c H 0 ) through the minimization of the χ 2 t function (19), we compute also the covariance matrices C + and C − for the positive and negative branches, ǫ ± 1 respectively, for the parameters (α, β, r c H 0 ), given in this order for the rows and columns in the matrices, The solution for the scale factor yields Notice that a(t) have a singularity at the time t s given by if the exponent β/(1 − α) is positive. From the expression for the acceleration For this case, when t −→ t s then a −→ ∞, ρ −→ ∞ and |p| −→ ∞ so the singularity is classified as Type I, a"Big Rip" singularity [33]. For the parameter r c H 0 , we find a very large dispersion on their possible values, even when it is constrained jointly with H 0 . Since H 0 alone is well constrained in our results, then it turns out that the dispersion is due to the leakage length scale r c only. So, the SNe + H(z) observations are not able to set more useful constraints on r c in the present work. above.
For the negative branch, ǫ = −1, we find that α and β have a positive covariance but both of them have a negative covariance with r c H 0 . This means that as α increases its value, β increases too, but r c H 0 decreases. We find the following linear relationship among the parameters: 0.517α + 0.2 = β, 39 − 27α = r c H 0 and 49 − 51β = r c H 0 . The negative covariance between α and r c H 0 , as well as the fact that they can have positive values only, gives the constraint 0 < r c H 0 < 57 at 3σ, that is a much better constraint than in the case of ǫ = +1.
We find also a negligible covariance of H 0 with the other three parameters (see the covariance matrix C − and figure 2), indicating a negligible effect of the value of H 0 on the estimation and inference about the values of (α, β, r c H 0 ). Also, from the combined joint credible regions of H 0 with α and β we obtain the constrains: 0 < α < 55 and 0 < β < 29 at 4σ of confidence level. Given that ǫ = −1 is the normal branch, i. e., it is not a self-accelerated branch like ǫ = +1, then the values of (α, β) are larger in this branch than in the positive one. This is expected because larger values of (α, β) implies a larger contribution of the holographic dark energy density, that is needed in the negative branch to accelerate the universe.
From the values for the parameters α and β which best fit the cosmological data, we have found that for the late time phase, where the holographic dark energy density dominates, the normal branch with ǫ = −1 leads to a de Sitter like expansion in the far future. On the other hand, the cosmological scenario in the positive branch presents a Big Rip singularity of Type I, according to the classification given in [33].