A MAGNIFIED GLANCE INTO THE DARK SECTOR: PROBING COSMOLOGICAL MODELS WITH STRONG LENSING IN A1689

The late cosmic acceleration, discovered by the Type IA supernovae (SN Ia) observations (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999), is the most intriguing feature of the universe. What gives origin to this phenomenon is a big puzzle in modern cosmology. There are two approaches that could drive the universe to an accelerated phase: an exotic component dubbed dark energy (DE, Copeland et al. 2006) and a modification of Einstein's gravity theory (Tsujikawa 2010). In the DE scenario, the natural and most simple model is the cosmological constant, Λ, associated with the vacuum energy, whose equation of state (EOS) parameter w, is equal to −1. There are several cosmological observations beyond SN Ia data, such as the baryon acoustic oscillations (BAO) and anisotropies of the cosmic microwave background (CMB) radiation, supporting the cosmological constant as the nature of DE (Weinberg et al. 2013). Nevertheless, there are theoretical problems associated with the cosmological constant: the fine-tuning problem, that is, its value is ∼120 orders of magnitude below the quantum field theory prediction, and the coincidence problem, that is, why the DE density is similar to that of dark matter (DM) today (Weinberg 1989).

A straightforward way to solve these problems is to consider models where the EOS evolves with time. Among the most-studied dynamical DE models are those involving scalar fields, for instance, quintessence (Caldwell et al. 1998; Peebles & Ratra 1988; Ratra & Peebles 1988; Wetterich 1988), phantom (Chiba et al. 2000; Caldwell 2002), quintom (Guo et al. 2005), and k-essence fields (Armendariz-Picon et al. 2000a, 2000b). In addition, there are many models in which the DE EOS is parameterized in terms of the scale factor or redshift (Magaña et al. 2014), for instance, the well-known Chevallier–Polarski–Linder ansatz (CPL, Chevallier & Polarski 2001; Linder 2003). The possibility of interactions between the DM and DE are also considered by several authors (Caldera-Cabral et al. 2009; Valiviita et al. 2010; Bolotin et al. 2015). These coupled models could alleviate both the coincidence problem and the tension among different cosmological data (He et al. 2011; Costa et al. 2014; Salvatelli et al. 2014; Valiviita & Palmgren 2015). Other interesting scenarios that have gained interest are the holographic dark energy (HDE) models, which are proposed in the context of a fundamental principle of quantum gravity, the holographic principle ('t Hooft 1993; Susskind 1995; Cárdenas & Perez 2010; Cárdenas et al. 2013). Although some HDE models could alleviate the coincidence problem and are in agreement with the cosmological data, they face many issues that must be solved (Zhang et al. 2010; Cárdenas et al. 2014; del Campo et al. 2014).

Thus, there are plenty of models with different theoretical motivations that are in agreement with some set of observational data and explain the accelerated expansion in the universe (Li et al. 2013). To discriminate among all these scenarios it is common to put constraints on their parameters using the distance modulus from SN Ia, the CMB anisotropies, and the BAO (Nesseris & Perivolaropoulos 2005, 2007; Shi et al. 2012). Many of the current data analyses are performed assuming a fiducial Λ cold DM model. Therefore, to improve the cosmological parameter estimation and to avoid biased constraints due to the assumption of a model, it is necessary to acquire high-precision data and to develop new complementary cosmological techniques, such as cosmography, which studies a set of observables of the universe's kinematics (see Gruber & Luongo 2014, and references therein).

Several authors have shown that the strong gravitational effect can be used as a powerful probe to test cosmological models (e.g., Jullo et al. 2010; Cao et al. 2012; Collett et al. 2012; Cárdenas et al. 2013; Chen et al. 2013; Lubini et al. 2014). Strong lensing (SL) occurs whenever the light rays of a source are strongly deflected by the lens, producing multiple images of the background source. The positions of these images depend on the properties of the lens mass distribution. Because the Einstein radii also depend on the cosmological model, the SL observations have been used to derive constraints on the DM density parameter, ${{\rm{\Omega }}}_{\mathrm{DM}},$ and the EOS for alternative DE models (see, for example, Biesiada 2006; Biesiada et al. 2010, 2011). In these previous works, the alternative cosmological models are tested by comparing (for the lens systems) the theoretical ratio of the angular diameter distances with an observable. This observable is typically estimated assuming a particular lens model along the standard cosmological paradigm. Nevertheless, the best way should be to test the cosmological model by reconstructing the lens model with that underlying the new cosmology. A pioneer work using a parametric reconstruction was performed by Jullo et al. (2010) to probe a flat constant wCDM model using the SL measurements in the Abell 1689 (A1689) galaxy cluster. They found that the DE EOS estimated with this technique is in agreement with those obtained using CMB and BAO. Recently, Lubini et al. (2014) investigated a novel nonparametric SL lens model to determine cosmological parameters. They applied this procedure using synthetic lenses and showed that it is possible to infer unbiased constraints from the assumed cosmological parameters. Therefore, SL modeling in galaxy clusters is a powerful and complementary method to put constraints on cosmological parameters (see also D'Aloisio & Natarajan 2011, and references therein).

In this paper we extend the previous analysis of Jullo et al. (2010) to test four alternative models using the SL measurements of the A1689 galaxy cluster. We investigate whether this technique is able to put narrow constraints on the dark parameters and the consistency of them with those provided by the SN Ia, BAO, and CMB data. The paper is organized as follows. In Section 2 we briefly describe the data used to constrain the cosmological parameters. In Section 3 we introduce the framework for a flat universe and the cosmological models to be tested. In Section 4 we define the method to obtain the constraints for each data set. We present the results in Section 5 and discuss them in Section 6. Finally, we give our conclusions in Section 7.

The following four data sets are used to test the alternative cosmological models: SL measurements in the A1689 galaxy cluster, SN Ia, BAO, and CMB.

SL in A1689. A1689 is among the richest clusters given the number density of galaxies in its core and one of the most luminous of galaxy clusters in X-ray wavelengths (Ebeling et al. 1996). It displays an incredibly large number of arc systems (see Limousin et al. 2007), and it has been studied using gravitational lensing by several authors (e.g., Limousin et al. 2007, 2013; Diego et al. 2015; Umetsu et al. 2015, and references therein). A$\;1689$ was previously used by Jullo et al. (2010) to simultaneously constrain the cluster mass distribution and DE EOS employing a SL parametric model. We refer the interested reader to that paper for a detailed description of the methodology to select the final catalog of multiple-image systems used to perform their analysis. In our present work, we are using the same catalog, which consists of 28 images derived from 12 families in the spectroscopic redshift range $1.15\lt {z}_{S}\lt 4.86.$

SN Ia. We use the sample presented by Ganeshalingam et al. (2013), consisting of 586 SN Ia in the redshift range 0.01–1.4, which considers 91 points of the Lick Observatory Supernova Search sample (Ganeshalingam et al. 2010).

BAO. The baryon acoustic oscillation signature is a useful standard ruler to constrain the expansion of the universe by the distance-redshift measurements from the clustering of galaxies with large-scale surveys (Blake & Glazebrook 2003; Seo & Eisenstein 2003). The BAO measurements considered in our analysis are obtained from the Six-degree-Field Galaxy Survey (6dFGS) BAO data (Beutler et al. 2011), the WiggleZ experiment (Blake et al. 2011), the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7), BAO distance measurements (Percival et al. 2010), the Baryon Oscillation Spectroscopic Survey (BOSS) SDSS Data Release 9 BAO distance measurements (SDSS DR9, Anderson et al. 2012), and the most recent BAO distance estimations from Data Release 11 (DR11) of the BOSS of SDSS (Delubac et al. 2015).

CMB. The CMB power spectrum is sensitive to the distance to the decoupling epoch, at redshift ${z}_{*},$ via the locations of peaks and the acoustic oscillations. The CMB measures two distance ratios related to the decoupling epoch: the acoustic scale ${l}_{A}({z}_{*}),$ and the shift parameter $R({z}_{*}).$ A quick way to confront a cosmological model with the CMB data without running a Bayesian global analysis of the power spectra is via the fitting of both distances (Wang & Mukherjee 2006; Wright 2007). The cosmological constraints estimated using this method are consistent with those obtained of the full analysis (Komatsu et al. 2009). Moreover, although these distance posteriors are computed assuming an underlying cosmology, Li et al. (2008) have demonstrated that these quantities are almost independent of the input DE models. We include CMB information by using the l_A, R, and ${z}_{*}$ posteriors from the WMAP nine-year measurements (Hinshaw et al. 2013).

A flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe with scale factor a and Hubble parameter $H(a)=\dot{a}/a$ is considered. For each cosmological model we use the following components: a source of cosmic acceleration, a cold DM, and a radiation fluid (γ, photons, and relativistic neutrinos). For this universe, the comoving distance from the observer to redshift z is given by

$$\begin{eqnarray}&&r(z)=\displaystyle \frac{c}{{H}_{0}}{\displaystyle \int }_{0}^{z}\displaystyle \frac{{{dz}}^{\prime }}{E({z}^{\prime })},\end{eqnarray} \tag{ 1 }$$

where $E(z)=H(z)/{H}_{0}$ and ${H}_{0}=H(0).$ The angular diameter distance for a source at redshift z is

$$\begin{eqnarray}&&{D}_{A}(z)=\displaystyle \frac{r(z)}{1+z}.\end{eqnarray} \tag{ 2 }$$

Since we are interested in the ability of SL measurements to constrain the parameters related to the cosmic acceleration and DE, during our analysis we set the Hubble parameter to ${H}_{0}=70\;\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{Mpc}}^{-1}$, and ${{\rm{\Omega }}}_{\mathrm{DM}0}=0.27.$ The current density parameter for radiation is ${{\rm{\Omega }}}_{\gamma 0}=2.469\times {10}^{-5}\;{h}^{-2}(1+0.2271{N}_{\mathrm{eff}}),$ where $h={H}_{0}/100\;\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{Mpc}}^{-1}$ and the number of relativistic species is set to ${N}_{\mathrm{eff}}=3.04$ (Komatsu et al. 2011). At low redshifts, ${{\rm{\Omega }}}_{\gamma }\;({\sim 10}^{-5})\ll {{\rm{\Omega }}}_{\mathrm{DM}},{{\rm{\Omega }}}_{\mathrm{DE}}\;(\sim 1),$ so we neglect this term when we use the A1689 SL measurements, but it is taken into account in the other data sets. The assumption of these fiducial values allows direct model comparisons because each model only has two free parameters related to DE. Bayliss et al. (2015) produced magnification maps for the Hubble Frontier Fields (HFF) galaxy clusters³ using priors in ${{\rm{\Omega }}}_{\mathrm{DM}0}$ and H₀. They obtain that varying the input cosmological parameters results in significant differences in the magnification maps. Nevertheless, the influence of the H₀ and its uncertainty in the lens reconstruction is subdominant because it cancels out when calculating the distance ratio (see Equations (1), (2) and (16)). On the other hand, the parameter estimation in the SL modeling could be slightly biased due to different choices of ${{\rm{\Omega }}}_{\mathrm{DM}0}.$ However, this bias is not statistically significant. Thus, for simplicity, we did not take into account the cosmological parameter (${{\rm{\Omega }}}_{\mathrm{DM}0},$ H₀) uncertainties in the SL lens modeling.

We choose the following alternative cosmological models: CPL, interacting dark energy (IDE), Ricci HDE, and modified polytropic Cardassian (MPC). In the following subsections we present the chosen models and the reasons for selecting them.

3.1. CPL Model

A natural extension of the ΛCDM scenario that could solve the coincidence problem is to allow the DE EOS to vary with time or redshift via some parameterization. One of the most popular functions is the CPL parameterization (Chevallier & Polarski 2001; Linder 2003) given by

$$\begin{eqnarray}&&w(z)={w}_{0}+{w}_{1}\displaystyle \frac{z}{1+z},\end{eqnarray} \tag{ 3 }$$

where ${w}_{0}=w(0),$ ${w}_{1}=w^{\prime} (0)$ are constants to be fitted by the data. The $E(z)$ function is for an FLRW universe where the DE EOS is parameterized and expressed as

$$\begin{eqnarray}{E}^{2}(z) & = & {{\rm{\Omega }}}_{\gamma 0}{(1+z)}^{4}+{{\rm{\Omega }}}_{\mathrm{DM}0}{(1+z)}^{3}\\ & & +\left(1-{{\rm{\Omega }}}_{\mathrm{DM}0}-{{\rm{\Omega }}}_{\gamma 0}\right)f(z),\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&f(z)=\mathrm{exp}\left(3{\displaystyle \int }_{0}^{z}\displaystyle \frac{1+w(z)}{1+z}{dz}\right).\end{eqnarray} \tag{ 5 }$$

The substitution of  Equation (3) into (5) results in

$$\begin{eqnarray}&&f(z)={(1+z)}^{3(1+{w}_{0}+{w}_{1})}\mathrm{exp}\left[-\displaystyle \frac{3{w}_{1}z}{1+z}\right].\end{eqnarray} \tag{ 6 }$$

Therefore, ${E}^{2}(z,{\rm{\Theta }})$ for the CPL parameterization reads as

$$\begin{eqnarray}{E}^{2}(z,{\rm{\Theta }}) & = & {{\rm{\Omega }}}_{\gamma 0}{(1+z)}^{4}+{{\rm{\Omega }}}_{\mathrm{DM}0}{(1+z)}^{3}\\ & & +\left(1-{{\rm{\Omega }}}_{\mathrm{DM}0}-{{\rm{\Omega }}}_{\gamma 0}\right){(1+z)}^{3(1+{w}_{0}+{w}_{1})}\\ & & \times \mathrm{exp}\left[-\displaystyle \frac{3{w}_{1}z}{1+z}\right],\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Theta }}=({w}_{0},{w}_{1})$ is the vector of the free parameters to be fitted by the data. The CPL parameterization is the fiducial model proposed by the Dark Energy Task Force (DETF) to study the cosmic acceleration (Albrecht et al. 2006). Therefore, Equation (7) has been widely used to put constraints on w₀ and $w1$ (see for example Su et al. 2011; Shi et al. 2012).

3.2. IDE Model

In IDE models there is a relation between the DE energy density (${\rho }_{\mathrm{DE}}$) and the DM energy density (${\rho }_{\mathrm{DM}}$) that could alleviate the cosmic coincidence problem. The general approach introduces a Q strength term in the right-hand side of the continuity equations for the dark components as follows (Amendola 2000; Dalal et al. 2001; Cai & Wang 2005; Guo et al. 2007; Caldera-Cabral et al. 2009; Valiviita et al. 2010):

$$\begin{eqnarray}&&\dot{{\rho }_{\mathrm{DM}}}+3H{\rho }_{\mathrm{DM}}=Q,\\ &&\dot{{\rho }_{\mathrm{DE}}}+3H\left(1+{w}_{x}\right){\rho }_{\mathrm{DE}}=-Q,\end{eqnarray} \tag{ 8 }$$

where w_x is the EOS of IDE. There are many choices for the phenomenological energy exchange term Q. One of them is to assume Q to be proportional to the Hubble rate, H, times either the energy densities or their sum or some other combination of the energy densities. We consider $Q=\delta H{\rho }_{\mathrm{DM}},$ δ being a constant to be fitted by the data (it is equivalent to the $Q=3\delta H{\rho }_{\mathrm{DM}}$ studied by He et al. 2011; Cao & Liang 2013; Costa et al. 2014). A positive δ describes an energy transfer or a decay of DM to DE, and a negative δ corresponds to an energy transfer from DE to DM. The ${E}^{2}(z)$ function (see its calculation in Guo et al. 2007; Bolotin et al. 2015) for this IDE reads as

$$\begin{eqnarray}&&{E}^{2}(z,{\rm{\Theta }})={{\rm{\Omega }}}_{\gamma 0}{(1+z)}^{4}\\ &&\qquad +(1-{{\rm{\Omega }}}_{\mathrm{DM}0}-{{\rm{\Omega }}}_{\gamma 0}){(1+z)}^{3(1+{w}_{x})}\\ &&\qquad +\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{DM}0}}{\delta +3{w}_{x}}\left[\delta {(1+z)}^{3(1+{w}_{x})}+3{w}_{x}{(1+z)}^{3-\delta }\right],\end{eqnarray} \tag{ 9 }$$

where the free parameters to be constrained by the data are ${\rm{\Theta }}=({w}_{x},\delta ).$ Equation (9) has been considered in flat (Guo et al. 2007; He et al. 2011; Cao & Liang 2013; Costa et al. 2014) and nonflat (Shi et al. 2012) models to put constraints on w_x and δ using several cosmological data points.

3.3. HDE with Ricci Scale and CPL Parameterization

Many DE models invoke the holographic principle (HP), which states that the number of degrees of freedom of a physical system should be finite and it should scale with its bounding area rather than with its volume ('t Hooft 1993; Susskind 1995; Fischler & Susskind 1998). In HDE it is required that the total energy in a region of size L should not exceed the mass of a black hole of the same size, so the HDE energy density satisfies ${L}^{3}{\rho }_{\mathrm{HDE}}\leqslant {M}_{{\rm{p}}}^{2}L$ (Cohen et al. 1999). This expression imposes a relationship between the ultraviolet (UV, related to the vacuum energy) and infrared (IR, related to the large scale of the universe) cutoffs. By saturating this inequality, we obtain the following DE energy density:

$$\begin{eqnarray}&&{\rho }_{\mathrm{HDE}}=\displaystyle \frac{3{c}_{H}^{2}{M}_{{\rm{p}}}^{2}}{{L}^{2}},\end{eqnarray} \tag{ 10 }$$

where the numerical constant c_H is related to the degree of saturation of the previous inequality. Therefore, the DE energy becomes dynamical, and the fine-tuning and coincidence problems can be solved. There are several ways to choose the IR cutoff, for example, the Hubble horizon or the event horizon (del Campo et al. 2011). Here, we consider ${L}^{2}=6/{\mathcal{R}},$ where ${\mathcal{R}}$ is the Ricci scalar defined as ${\mathcal{R}}=6(2{H}^{2}+\dot{H})$ (Gao et al. 2009; del Campo et al. 2013). We also consider that the DM and DE interact with each other, obeying Equation (8). Following the work by del Campo et al. (2011), we parameterize the EOS with the CPL ansatz (3). The $E(z,{\rm{\Theta }})$ parameter (see appendix) for this model is the following:

$$\begin{eqnarray}{E}^{2}(z,{\rm{\Theta }}) & = & \displaystyle \frac{{\left(1+z\right)}^{4}}{1+{x}_{0}+{y}_{0}}{\left(\displaystyle \frac{1+{x}_{0}}{f}\right)}^{2\alpha }\\ & & \times \left[f+{y}_{0}{\left(\displaystyle \frac{f}{1+{x}_{0}}\right)}^{2\alpha }\right],\end{eqnarray} \tag{ 11 }$$

where ${x}_{0}={{\rm{\Omega }}}_{\mathrm{DM}0}/{{\rm{\Omega }}}_{\mathrm{HDE}0},$ ${y}_{0}={{\rm{\Omega }}}_{\gamma 0}/{{\rm{\Omega }}}_{\mathrm{HDE}0},$ and ${{\rm{\Omega }}}_{\mathrm{HDE}0}=1-{{\rm{\Omega }}}_{\mathrm{DM}0}-{{\rm{\Omega }}}_{\gamma 0}.$ The function f and the exponent α are

$$\begin{eqnarray}&&f=1+{x}_{0}+z(1+3{w}_{1}+{x}_{0}),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\alpha =1-\displaystyle \frac{3({w}_{0}+{w}_{1})}{2(1+3{w}_{1}+{x}_{0})}.\end{eqnarray} \tag{ 13 }$$

The free parameter vector to be fitted by the data is ${\rm{\Theta }}=({w}_{0},{w}_{1}).$ A similar model without the radiation component was tested by Cárdenas et al. (2013). We present a new analytical solution for the Ricci HDE model with a CPL parameterization that includes a relativistic fluid.

3.4. MPC Model

The original Cardassian model was introduced by Freese & Lewis (2002) to explain the accelerated expansion of the universe without DE. Motivated by brane-world theory, this model modifies the Friedmann equation as ${H}^{2}=8\pi G{\rho }_{{\rm{m}}}/3+B{\rho }_{{\rm{m}}}^{n},$ where ${\rho }_{{\rm{m}}}$ is the total matter density. The second term in the right-hand side, known as the Cardassian term, drives the universe to an accelerated phase if the exponent n satisfies $n\lt 2/3.$ Gondolo & Freese (2002) introduced a simple generalization of the Cardassian model, the MPC, by introducing an additional exponent q (see also Wang et al. 2003). The modified Friedmann equation with this generalization can be written as

$$\begin{eqnarray}&&{H}^{2}=\displaystyle \frac{8\pi G}{3}{\rho }_{{\rm{m}}}{\left[1+{\left(\displaystyle \frac{{\rho }_{\mathrm{Card}}}{{\rho }_{{\rm{m}}}}\right)}^{q(1-n)}\right]}^{1/q},\end{eqnarray} \tag{ 14 }$$

where ${\rho }_{\mathrm{Card}}$ is the characteristic energy density with $n\lt 2/3$ and $q\gt 0.$ At early times, the universe is expanded according to the canonical Friedmann equation. However, at late times, the Cardassian term dominates, driving the universe to an accelerated expansion phase. Equation (14) reduces to the ΛCDM model for q = 1 and n = 0. Introducing a radiation term, the dimensionless ${E}^{2}(z,{\rm{\Theta }})$ parameter reads as

$$\begin{eqnarray}&&{E}^{2}(z,{\rm{\Theta }})={{\rm{\Omega }}}_{r}{(1+z)}^{4}+{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}\\ &&\qquad \times {\left[1+\left({\left(\displaystyle \frac{1-{{\rm{\Omega }}}_{r}}{{{\rm{\Omega }}}_{{\rm{m}}}}\right)}^{q}-1\right){(1+z)}^{3q(n-1)}\right]}^{1/q},\end{eqnarray} \tag{ 15 }$$

where the free parameter vector to be fitted by the data is ${\rm{\Theta }}=(q,n).$ The flat MPC model (Equation (15)) has been studied by several authors using different data without the radiation component (Feng & Li 2010) and also with a curvature term (Shi et al. 2012).

In this section we explain how the cosmological parameters are estimated for each different observational data set, and we also define the merit functions for each of them.

4.1. Strong Lensing

In the SL regime, the light beams are deflected so strongly that they can result in the observation of several distorted images of a background source. The positions of the multiple images depend significantly on the characteristics of the lens mass distribution. Since the image positions are also related to the angular diameter distance ratios between the lens, source, and observer, they retain information about the underlying cosmology. In particular, this dependence of the lensing models on the geometry can be used to derive constraints on the DM density parameter and the DE EOS (see Jullo et al. 2010).

The cosmological models discussed in Section 3 were implemented in the LENSTOOL⁴ ray-tracing code, which uses a Bayesian Markov chain Monte Carlo method (Jullo et al. 2007). The model fitting is carried out taking into account the cosmological sensitivity of the angular size-redshift relation, when sources are at distinct redshifts (Link & Pierce 1998). Using this method, the angular diameter distance ratios for two images from different sources define the "family ratio" (see Jullo et al. 2010 for a detailed discussion), for which the constraints on cosmological parameters could be obtained:

$$\begin{eqnarray}&&{\rm{\Xi }}({z}_{1},{z}_{s1},{z}_{s2},{\rm{\Theta }})=\displaystyle \frac{D({z}_{1},{z}_{s1})}{D(0,{z}_{s1})}\displaystyle \frac{D(0,{z}_{s2})}{D({z}_{1},{z}_{s2})},\end{eqnarray} \tag{ 16 }$$

where Θ is the vector of cosmological parameters to be fitted, z₁ is the lens redshift, ${z}_{s1}$ and ${z}_{s2}$ are the two source redshifts, and $D({z}_{1},{z}_{2})$ is the angular diameter distance, calculated through Equations (1) and (2).

We computed the models performing the optimization in the source plane. We solved the lens equation in the source plane because it is computationally more efficient, and we checked with some models that source and image plane results were similar. Note that differences can appear for complex clusters with irregular shapes (e.g., MACS J0717.5+3745, Limousin et al. 2012), but this is not the case with Abell 1689. Every lensing mass model (regardless of the DE model) has a total of 21 free parameters, consists of two large-scale potentials and a galaxy-scale potential for the central brightest cluster galaxy, and includes the modeling of 58 of the brightest cluster galaxies.

For each of the image systems (12 families; see Section 2) with n images, we determine the goodness of fit for a particular set of model parameters defining a ${\chi }^{2}:$

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{n}\displaystyle \frac{{\left[M({{\boldsymbol{\beta }}}_{i}-\langle {\boldsymbol{\beta }}\rangle )\right]}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 17 }$$

where ${\beta }_{i}$ is the source plane position corresponding to image i, $\langle {\boldsymbol{\beta }}\rangle$ is the family barycenter, M is the magnification tensor, and ${\sigma }_{i}$ is the total error (see Jullo et al. 2007). The total ${\chi }^{2}$ is computed by summing over the whole set of families.

4.2. Type Ia Supernovae

The SN Ia samples give the distance modulus as a function of redshift ${\mu }_{\mathrm{obs}}(z)$ and its error ${\sigma }_{\mu }.$ Theoretically, the distance modulus is computed as

$$\begin{eqnarray}&&\mu (z)=5{\mathrm{log}}_{10}[{d}_{L}(z)/{\mathtt{Mpc}}]+{\mu }_{0},\end{eqnarray} \tag{ 18 }$$

where ${\mu }_{0}$ is a nuisance parameter that depends on the absolute magnitude of a fiducial SN Ia and the Hubble parameter. The $\mu (z)$ is a function of the cosmological model through the luminosity distance (measured in Mpc)

$$\begin{eqnarray}&&{d}_{L}(z)=(1+z)r(z),\end{eqnarray} \tag{ 19 }$$

where $r(z)$ is given by Equation (1). By marginalizing over ${\mu }_{0},$ we obtain ${\chi }_{\mathrm{SN}\;\mathrm{Ia}}^{2}=A-{B}^{2}/C,$ where

$$\begin{eqnarray}A & = & \displaystyle \sum _{i=1}^{586}\displaystyle \frac{{[\mu ({z}_{i})-{\mu }_{\mathrm{obs}}({z}_{i})]}^{2}}{{\sigma }_{{\mu }_{i}}^{2}},\\ B & = & \displaystyle \sum _{i=1}^{586}\displaystyle \frac{\mu ({z}_{i})-{\mu }_{\mathrm{obs}}({z}_{i})}{{\sigma }_{{\mu }_{i}}^{2}},\\ C & = & \displaystyle \sum _{i=1}^{586}\displaystyle \frac{1}{{\sigma }_{{\mu }_{i}}^{2}}.\end{eqnarray} \tag{ 20 }$$

The SN Ia constraints can be estimated by minimizing the ${\chi }_{\mathrm{SN}\;\mathrm{Ia}}^{2}.$

4.3. BAO Measurements

The 6dFGS BAO estimated the distance ratio ${d}_{z}=0.336\pm 0.015$ at $z=0.106$ (Beutler et al. 2011), where

$$\begin{eqnarray}&&{d}_{z}=\displaystyle \frac{{r}_{{\rm{s}}}({z}_{d})}{{D}_{V}(z)}.\end{eqnarray} \tag{ 21 }$$

The comoving sound horizon, ${r}_{{\rm{s}}}(z),$ is defined as

$$\begin{eqnarray}&&{r}_{{\rm{s}}}(z)=c{\displaystyle \int }_{z}^{\infty }\displaystyle \frac{{c}_{{\rm{s}}}({z}^{\prime })}{H({z}^{\prime })}{{dz}}^{\prime },\end{eqnarray} \tag{ 22 }$$

where the sound speed is ${c}_{{\rm{s}}}(z)=1/\sqrt{3(1+\bar{{R}_{b}}/(1+z)},$ with $\bar{{R}_{b}}=31500\;{{\rm{\Omega }}}_{b}{h}^{2}{({T}_{\mathrm{CMB}}/2.7\;{\rm{K}})}^{-4},$ ${{\rm{\Omega }}}_{b}$ is the baryonic density parameter, and T_CMB is the CMB temperature (2.726 K for WMAP nine-year; Hinshaw et al. 2013). The distance scale D_V is defined as

$$\begin{eqnarray}&&{D}_{V}(z)=\displaystyle \frac{1}{{H}_{0}}{\left[{(1+z)}^{2}{D}_{{\rm{A}}}{(z)}^{2}\displaystyle \frac{{cz}}{E(z)}\right]}^{1/3},\end{eqnarray} \tag{ 23 }$$

where ${D}_{{\rm{A}}}(z)$ is the angular diameter distance given by Equation (2).

The redshift z_d at the baryon drag epoch is fitted with the formula proposed by Eisenstein & Hu (1998):

$$\begin{eqnarray}&&{z}_{d}=\displaystyle \frac{1291{({{\rm{\Omega }}}_{{\rm{m}}}{h}^{2})}^{0.251}}{1+0.659\;{({{\rm{\Omega }}}_{{\rm{m}}}{h}^{2})}^{0.828}}[1+{b}_{1}{({{\rm{\Omega }}}_{b}{h}^{2})}^{{b}_{2}}],\end{eqnarray} \tag{ 24 }$$

where

$$\begin{eqnarray}&&{b}_{1}=0.313{\left({{\rm{\Omega }}}_{{\rm{m}}}\;{h}^{2}\right)}^{-0.419}\left[1+0.607{\left({{\rm{\Omega }}}_{{\rm{m}}}\;{h}^{2}\right)}^{0.674}\right],\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{b}_{2}=0.238{\left({{\rm{\Omega }}}_{{\rm{m}}}\;{h}^{2}\right)}^{0.223}.\end{eqnarray} \tag{ 26 }$$

Therefore, the ${\chi }^{2}$ for the BAO data point from 6dFGS is

$$\begin{eqnarray}&&{\chi }_{6\mathrm{dFGS}}^{2}={\left(\displaystyle \frac{{d}_{z}-0.336}{0.015}\right)}^{2}.\end{eqnarray} \tag{ 27 }$$

The WiggleZ BAO estimated three points for the acoustic parameter $A(z)$ (Eisenstein et al. 2005):

$$\begin{eqnarray}&&A(z)=\displaystyle \frac{{D}_{V}(z)\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{H}_{0}^{2}}}{{cz}}.\end{eqnarray} \tag{ 28 }$$

The observational data are ${\bar{A}}_{\mathrm{obs}}=(0.474,0.442,0.424)$ for the effective redshifts $z=0.44,0.6$, and 0.73, respectively.

Thus, the ${\chi }^{2}$ for the WiggleZ BAO data is given by

$$\begin{eqnarray}&&{\chi }_{\mathrm{WiggleZ}}^{2}=({\bar{A}}_{\mathrm{obs}}-{\bar{A}}_{\mathrm{th}}){C}_{\mathrm{WiggleZ}}^{-1}{({\bar{A}}_{\mathrm{obs}}-{\bar{A}}_{\mathrm{th}})}^{T},\end{eqnarray} \tag{ 29 }$$

where ${\bar{A}}_{\mathrm{th}}$ denotes the theoretical value for the acoustic parameter $A(z)$ and ${\bar{A}}_{\mathrm{obs}}$ is the observed one. The inverse covariance ${C}_{\mathrm{WiggleZ}}^{-1}$ is given by

$$\begin{eqnarray}{C}_{\mathrm{WiggleZ}}^{-1}=\left(\begin{array}{ccc}1040.3 & -807.5 & 336.8\\ -807.5 & 3720.3 & -1551.9\\ 336.8 & -1551.9 & 2914.9\end{array}\right).\end{eqnarray} \tag{ 30 }$$

Similarly, for the SDSS DR7 BAO distance measurements, the ${\chi }^{2}$ can be expressed as

$$\begin{eqnarray}&&{\chi }_{\mathrm{DR}7}^{2}=({\bar{{d}_{z}}}_{\mathrm{obs}}-{\bar{{d}_{z}}}_{\mathrm{th}}){C}_{\mathrm{DR}7}^{-1}{({\bar{{d}_{z}}}_{\mathrm{obs}}-{\bar{{d}_{z}}}_{\mathrm{th}})}^{T},\end{eqnarray} \tag{ 31 }$$

where ${\bar{{d}_{z}}}_{\mathrm{obs}}=(0.190195,0.1097)$ are the data at $z=0.2$ and 0.35, respectively (Percival et al. 2010). Here ${\bar{{d}_{z}}}_{\mathrm{th}}$ denotes the theoretical distance ratio given by Equation (21). The inverse covariance matrix ${C}_{\mathrm{DR}7}^{-1}$ reads as

$$\begin{eqnarray}{C}_{\mathrm{DR}7}^{-1}=\left(\begin{array}{cc}30124 & -17227\\ -17227 & 86977\end{array}\right).\end{eqnarray} \tag{ 32 }$$

The SDSS DR9 estimated the distance ratio ${d}_{z}=0.0732\pm 0.0012$ at $z=0.57$ (Anderson et al. 2012). For this BAO data point, the ${\chi }^{2}$ function is given by

$$\begin{eqnarray}&&{\chi }_{\mathrm{DR}9}^{2}={\left(\displaystyle \frac{{d}_{z}-0.0732}{0.0012}\right)}^{2}.\end{eqnarray} \tag{ 33 }$$

The most recent measured position of the BAO peak from SDSS DR11 determines ${D}_{H}/{r}_{d}=9.18\pm 0.28$ at $z=2.34,$ where ${D}_{H}=c/H$ and ${r}_{d}={r}_{s}({z}_{d})$ (Delubac et al. 2015). We compute the ${\chi }^{2}$ for this point as

$$\begin{eqnarray}&&{\chi }_{\mathrm{DR}11}^{2}={\left(\displaystyle \frac{{D}_{H}/{r}_{d}-9.18}{0.28}\right)}^{2}.\end{eqnarray} \tag{ 34 }$$

Therefore, the total ${\chi }^{2}$ function for the BAO measurements is

$$\begin{eqnarray}&&{\chi }_{\mathrm{BAO}}^{2}={\chi }_{6\mathrm{dFGS}}^{2}+{\chi }_{\mathrm{WiggleZ}}^{2}+{\chi }_{\mathrm{DR}7}^{2}+{\chi }_{\mathrm{DR}9}^{2}+{\chi }_{\mathrm{DR}11}^{2}.\end{eqnarray} \tag{ 35 }$$

The BAO constraints can be estimated by minimizing Equation (35).

4.4. CMB

We use the following WMAP nine-year distance posterior (Hinshaw et al. 2013) for a flat ΛCDM universe: ${l}_{{A}^{\mathrm{obs}}}=302.40,$ ${R}^{\mathrm{obs}}=1.7246$, ${z}_{*}^{\mathrm{obs}}=1090.88,$ and the inverse covariance matrix

$$\begin{eqnarray}{C}_{\mathrm{WMAP}9}^{-1}=\left(\begin{array}{ccc}3.182 & 18.253 & -1.429\\ 18.253 & 11887.879 & -193.808\\ -1.429 & -193.808 & 4.556\end{array}\right).\end{eqnarray} \tag{ 36 }$$

The acoustic scale is defined as

$$\begin{eqnarray}&&{l}_{{\rm{A}}}=\displaystyle \frac{\pi r({z}_{*})}{{r}_{{\rm{s}}}({z}_{*})},\end{eqnarray} \tag{ 37 }$$

and the redshift of decoupling ${z}_{*}$ is given by (Hu & Sugiyama 1996)

$$\begin{eqnarray}&&{z}_{*}=1048[1+0.00124{({{\rm{\Omega }}}_{b}{h}^{2})}^{-0.738}][1+{g}_{1}{({{\rm{\Omega }}}_{{\rm{m}}}{h}^{2})}^{{g}_{2}}],\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{g}_{1}=\displaystyle \frac{0.0783{({{\rm{\Omega }}}_{b}{h}^{2})}^{-0.238}}{1+39.5{({{\rm{\Omega }}}_{b}{h}^{2})}^{0.763}},{g}_{2}=\displaystyle \frac{0.560}{1+21.1{({{\rm{\Omega }}}_{b}{h}^{2})}^{1.81}}.\end{eqnarray} \tag{ 39 }$$

The shift parameter is defined as (Bond et al. 1997)

$$\begin{eqnarray}&&R=\displaystyle \frac{\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{H}_{0}^{2}}}{c}r({z}_{*}).\end{eqnarray} \tag{ 40 }$$

Thus, the CMB constraints can be estimated by minimizing

$$\begin{eqnarray}&&{\chi }_{\mathrm{CMB}}^{2}={X}^{T}{C}_{\mathrm{CMB}}^{-1}X,\end{eqnarray} \tag{ 41 }$$

where

$$\begin{eqnarray}&&X=\left(\begin{array}{c}{l}_{{\rm{A}}}^{\mathrm{th}}-{l}_{{\rm{A}}}^{\mathrm{obs}}\\ {R}^{\mathrm{th}}-{R}^{\mathrm{obs}}\\ {z}_{*}^{\mathrm{th}}-{z}_{*}^{\mathrm{obs}}\end{array}\right),\end{eqnarray} \tag{ 42 }$$

and the superscripts "th" and "obs" refer to the theoretical and observational values, respectively.

The parameters of the alternative models are determined by minimizing the ${\chi }^{2}$ function for each data set. For all models, first we calculate the minimum values using the SL data with the priors described by Jullo et al. (2010). Then, we estimate the constraints using the SN Ia, BAO, and CMB, comparing the different data sets. Furthermore, we could use a refined ${\chi }_{\mathrm{min}}^{2}$ criteria, for instance, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), to discern which model is preferred by the data. However, since all of the tested models have only two free parameters, the information provided by the ${\chi }_{\mathrm{min}}^{2}$ values is sufficient, and the AIC and BIC criteria do not provide further information.

CPL. The best fits on the EOS parameters w₀ and w₁ for the CPL model and the estimated ${\chi }^{2}$ using each data set are listed in Table 1. Note that the limits derived for the A1689 SL data are in tension with those obtained with the SN Ia, BAO, and nine-year WMAP data. Actually, the A1689 constraint on w₀ is positive, implying no cosmic acceleration. Figure 1 shows the marginalized contours at $1\sigma ,$ $2\sigma$, and $3\sigma$ for the CPL parameters. The inset shows the region where the different contours overlap.

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Table 1.  CPL Model

Data Set	${\chi }_{\mathrm{min}}^{2}$	FoMa	w0	w1
A1689	264.9	8.20	0.43 ± 0.48	$-{6.45}_{-0.36}^{+3.60}$
SN Ia	574.13	24.41	−0.82 ± 0.14	−1.51 ± 0.91
BAO	3.77	7.89	−0.94 ± 0.26	−1.55 ± 1.72
CMB	0.363	21.54	−0.59 ± 0.58	−1.38 ± 2.36

Note. Best fits for the w₀ and w₁ CPL parameters estimated from the SL measurements in A1689, SN Ia, BAO, and CMB.

^aWe define the FoM in Section 6.5.

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IDE. We have carried out two different analyses using the SL data: the first one, A1689M1, with the (astrometric) errors in the image positions of the systems previously used by Jullo et al. (2010) (also used in the other three models of this paper) and the other, A1689M2, in which we have set the errors as five times the values of the fiducial model, that is, five times the errors in model A1689M1, so that we obtain a reduced ${\chi }^{2}\simeq 1.$ These large errors take into account other possible sources of uncertainties in the SL measurements, such as systematic errors due to the complexities in the mass distribution and the line-of-sight structures (e.g., Jullo et al. 2010; D'Aloisio & Natarajan 2011). In Section 6.2 we will resume the discussion.

The best fits on the w_x and δ parameters for both runs of the IDE model and the estimated ${\chi }^{2}$ using each data set are listed in Table 2. Note that A1689M1 constraints on w_x and δ are in disagreement with the estimations of the other cosmological tests. In the second analysis, considering larger errors on the SL data, we obtain that the best fit on w_x is in agreement at $1\sigma$ with the other data. Figure 2 shows the marginalized $1\sigma ,$ $2\sigma$, and $3\sigma$ confidence contours in the plane ${w}_{x}-\delta$ for each data set. The inset shows the region where the A1689M2 contours overlap the SN Ia, BAO, and CMB contours.

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Table 2.  IDE Model

Data Set	${\chi }_{\mathrm{min}}^{2}$	FoM	wx	δ
A $1689{\rm{M}}1$	256.7	127.063	−0.32 ± 0.07	$-{2.0}_{-0.0}^{+0.30}$
A $1689{\rm{M}}2$	25.9	4.55	−1.53 ± 0.42	−0.21 ± 0.80
SN Ia	574.95	38.76	−0.95 ± 0.08	0.77 ± 0.69
BAO	4.61	1060.52	−1.10 ± 0.13	−0.0093 ± 0.014
CMB	0.081	18488.1	−0.97 ± 0.02	−0.0017 ± 0.003

Note. Best fits for the w_x and δ IDE parameters estimated from the SL measurements in A1689, SN Ia, BAO, and CMB.

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Ricci HDE. The best fits on the w₀ and w₁ parameters and the ${\chi }^{2}$ obtained using each data set are shown in Table 3. Note that the A1689 best fits are consistent with those of the CMB data. Nevertheless, there is a tension between these values and the BAO and SN Ia constraints. Figure 3 shows the marginalized confidence contours at $1\sigma ,$ $2\sigma ,$ and $3\sigma$ in the parameter space w₀–w₁. The inset shows the region where the contours overlap.

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Table 3.  HDE Model

Data Set	${\chi }_{\mathrm{min}}^{2}$	FoM	w0	w1
A1689	279.82	24.85	$-{1.60}_{-0.0}^{+0.13}$	${1.97}_{-0.66}^{+0.01}$
SN Ia	575.135	153.89	−0.96 ± 0.10	0.21 ± 0.22
BAO	5.79	241.52	−2.03 ± 0.21	2.10 ± 0.21
CMB	0.081	14725.7	−1.48 ± 0.01	1.51 ± 0.01

Note. Best fits for the w₀ and w₁ Ricci HDE parameters estimated from the SL measurements in A1689, SN Ia, BAO, and CMB.

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MPC. The best fits on the q and n parameters and the estimated ${\chi }^{2}$ using each data set are listed in Table 4. Note that the A1689 constraints on q and n are consistent with the limits given by the CMB data. Although these SL and CMB best fits are in tension with the estimations obtained with SN Ia and BAO observations, the errors for the q parameter are statistically larger, so the constraints for each data point are consistent at $1\sigma .$ Figure 4 shows the marginalized confidence contours at $1\sigma ,$ $2\sigma ,$ and $3\sigma$ in the parameter space q–n. An interesting aspect of the SL contours is that the A1689 data provided two $1\sigma$ regions. The inset shows a region where the different contours overlap.

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Table 4.  MPC Model

Data Set	${\chi }_{\mathrm{min}}^{2}$	FoM	q	n
A1689	266.7	2.54	5.2 ± 2.25	0.41 ± 0.25
SN Ia	574.52	18.69	3.20 ± 2.19	0.32 ± 0.08
BAO	3.59	7.97	3.29 ± 3.30	0.26 ± 0.13
CMB	0.363	37.63	4.52 ± 3.27	0.49 ± 0.05

Note. Best fits for the q and n Cardassian parameters estimated from the SL measurements in A1689, SN Ia, BAO, and CMB.

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6.1. CPL

6.2. IDE

6.3. Ricci HDE

6.4. MPC

6.5. Merit of the SL Method

As we showed and discussed above, the SL technique provides complementary constraints to the standard cosmological probes. It is important to stress that the determination of which cosmological model is favored by the data, mainly by the SL measurements, is far from the scope of the present work (the current data do not allow us to undertake such a detailed analysis). Nevertheless, by considering standard errors in the SL data, the IDE model gives the lowest value of the SL ${\chi }_{\mathrm{min}}^{2},$ so it is favored by the A1668 SL measurements. However, as discussed in Section 6.2, the SL constraints for this IDE model are in disagreement with those of SN Ia, BAO, and CMB. The CPL model is the second one preferred by the SL data.

Another useful tool to quantify the ability of each observational data set to constrain the cosmological parameters is considering the figure of merit (FoM, Albrecht et al. 2006; Wang 2008; Su et al. 2011). The DETF defined the FoM for the CPL model as the inverse of the area enclosed by the 95% confidence level contour of $(w0,{wa})$ (Albrecht et al. 2006). Wang (2008) introduced a more general definition given by

$$\begin{eqnarray}&&{\rm{FoM}}=\displaystyle \frac{1}{\sqrt{\mathrm{det}{\rm{Cov}}(f1,f2,f3,\ldots )}},\end{eqnarray} \tag{ 43 }$$

where ${\rm{Cov}}(f1,f2,f3,\ldots )$ is the covariance matrix of the cosmological parameters f_i. A larger FoM means stronger constraints on the parameters because it corresponds to a smaller error ellipse. We have computed the FoM of the cosmological models for each data set. The results are shown in the third column of Tables 1–4. For a more intuitive comparison, we show in Figure 5 the values of the FoM for each model using each data set. Note that for the CPL model, the SL probe gives slightly more stringent constraints than the BAO test. In the case of A1689M1, although the SL FoM is ∼3.3 times greater than that for SN Ia, the constraints obtained from this lens model are inconsistent with the other tests. For the A1689M2 analysis with large errors on the SL measurements, the SL FoM has the lowest value, indicating that this technique provides weak constraints on the IDE parameters. The FoMs for the HDE and MPC models from the A1689 data are also the lowest when compared with the other cosmological probes. Although it is very difficult to compare the SL FoM for different models, our computations suggest that the SL technique provides statistically significant constraints on the CPL parameters but a weak one for the other models.

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In spite of the general low FoM for the SL probe compared with the other data sets, it is important to remark that we only use the measurements of one galaxy cluster, namely, A1689. To improve the SL constraints on the parameters of the alternative cosmological models, we need to consider the SL measurements in other galaxy clusters. This work is a first test of the capability of this technique to constrain unusual cosmological models. D'Aloisio & Natarajan (2011) showed that by using the data from 10 simulated galaxy clusters, each one with 20 multiply imaged families, the estimated constraints on the parameters of the wCDM and CPL models are improved. We plan to extend this analysis using the data that will come from future surveys, such as the Frontier Fields (FF) program of the Hubble Space Telescope. In addition, it is crucial that future efforts also take into account additional uncertainties in the lens modeling due to the line-of-sight structure and other systematic errors (Host 2012; Bayliss et al. 2014; Jaroszynski & Kostrzewa-Rutkowska 2014; McCully et al. 2014). Finally, it is important to point out that the cosmological constraints obtained in this work could be affected by other unknown systematics (Bayliss et al. 2014, 2015; Zitrin et al. 2015), such as the SL modeling technique.

One of the main goals of observational cosmology is to elucidate what gives origin to the late cosmic acceleration in the universe. A wide set of theoretical models have been proposed to explain this cosmic feature (Li et al. 2013), and they need to be tested with observational data (Nesseris & Perivolaropoulos 2005, 2007; Albrecht et al. 2006; Lazkoz et al. 2008). In this paper, we put constraints on four alternative cosmological models: CPL parameterization, IDE, Ricci HDE, and MPC. We mainly focus on a powerful and not fully exploited technique that uses the SL measurements in the ${\rm{A}}1689$ galaxy cluster (Jullo et al. 2010). The advantage of the method presented here is that the cosmological parameters are estimated by modeling the SL features for each underlying cosmology. Additionally, we use the SN Ia, BAO, and CMB signals as complementary probes. We have shown that for the CPL model the SL method provides constraints in agreement with those estimated with the other probes. We performed two analyses for the IDE model, one with standard errors in the SL measurements and the other with larger errors to take into account other sources of uncertainties (D'Aloisio & Natarajan 2011; Host 2012; Bayliss et al. 2014; Jaroszynski & Kostrzewa-Rutkowska 2014; McCully et al. 2014; Zitrin et al. 2015). We found that the limits on the IDE parameters derived from the standard error analysis are in disagreement with the standard tests. Moreover, the confidence contours do not overlap with those of SN Ia, BAO, and CMB. Nevertheless, if larger errors in the SL measurements are considered, the SL estimations are consistent with the constraints obtained from other probes. Therefore, underestimating the total error can lead to erroneous constraints on the parameters of the IDE model. For the Ricci HDE, the SL data give weak constraints on the DE EOS parameters. In addition, we also found a tension between the bounds obtained from the SN Ia, BAO, and CMB data. Finally, the estimations for the MPC parameters using the SL test are similar to the SN Ia, BAO, and CMB constraints. We also calculate the FoMs to quantify the goodness of fit using the different data. We found that in general the SL constraints are weak when compared with other tests. Also, the contours do not always overlap each other, suggesting some systematic errors in the models of the observables that remain to be investigated. Nevertheless, it is worthy to note that we use only data from one galaxy cluster. The cosmological constraints could be improved if more SL data are used (D'Aloisio & Natarajan 2011). Our results show that this is a powerful technique that will be used in the future when more data are available, in particular those for the HFF clusters.

We thank the anonymous referee for thoughtful suggestions. J. M. acknowledges the support from ESO Comité Mixto, Gemini 32130024, ECOS-CONICYT C12U02, and the hospitality of the Laboratoire d'Astrophysique de Marseille (LAM), where part of this work was done. V. M. acknowledges support from FONDECYT 1120741, ECOS-CONICYT C12U02, and Centro de Astrofísica de Valparaíso. V. C. acknowledges support from FONDECYT Grant 1110230 and DIUV 13/2009. T. V. thanks Dr. V. Motta for the kind invitation to work in Valparaíso, as well as the staff of the Instituto de Física y Astronomía of the Universidad de Valparaíso. E. J. acknowledges the support of Centre National d'Etudes Spatiales (CNES). This work was granted access to the High Performance Computing (HPC) resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program 'Investissements d'Avenir' supervised by the Agence Nationale pour la Recherche.

In this paper we have revisited the HDE with the Ricci scale model presented by del Campo et al. (2013) and Cárdenas et al. (2013), but we have taken into account the radiation component. We consider a flat FLRW universe with DM, HDE, and radiation. By assuming the possibility of interaction between the dark components, the dynamics of this universe are governed by the system

$$\begin{eqnarray}&&{H}^{2}=\displaystyle \frac{8\pi G}{3}\left({\rho }_{\mathrm{DM}}+{\rho }_{\mathrm{HDE}}+{\rho }_{r}\right),\end{eqnarray} \tag{ 44a }$$

$$\begin{eqnarray}&&\dot{{\rho }_{\mathrm{DM}}}+3H{\rho }_{\mathrm{DM}}=Q,\end{eqnarray} \tag{ 44b }$$

$$\begin{eqnarray}&&\dot{{\rho }_{\mathrm{HDE}}}+3H(1+\omega ){\rho }_{\mathrm{HDE}}=-Q,\end{eqnarray} \tag{ 44c }$$

$$\begin{eqnarray}&&\dot{{\rho }_{\gamma }}+4H{\rho }_{\gamma }=0,\end{eqnarray} \tag{ 44d }$$

such that the total energy, $\rho ={\rho }_{\mathrm{DM}}+{\rho }_{\mathrm{HDE}}+{\rho }_{\gamma },$ is conserved. Here, $\omega \equiv {p}_{\mathrm{HDE}}/{\rho }_{\mathrm{HDE}}$ is the EOS parameter of the HDE, and p_HDE is the pressure associated with the holographic component. We may write the holographic energy density as

$$\begin{eqnarray}&&{\rho }_{\mathrm{HDE}}=\displaystyle \frac{3{c}_{H}^{2}{M}_{{\rm{p}}}^{2}}{{L}^{2}},\end{eqnarray} \tag{ 45 }$$

where L represents the IR cutoff scale and M_p is the reduced Planck mass. In the HDE model, it is assumed that the energy in a given box should not exceed the energy of a black hole of the same size. This means that ${L}^{3}{\rho }_{\mathrm{HDE}}\leqslant {M}_{{\rm{p}}}^{2}L,$ and in this context the numerical constant c_H in Equation (45) is related to the degree of saturation of the previous expression. Here, we consider the Ricci scalar, ${\mathcal{R}},$ as the IR cutoff, i.e., ${L}^{2}=6/{\mathcal{R}},$ where ${\mathcal{R}}\equiv 6(2{H}^{2}+\dot{H})$ (Gao et al. 2009; del Campo et al. 2013); then

$$\begin{eqnarray}&&{\rho }_{H}=3\;{c}_{H}^{2}\;{M}_{p}^{2}\;\displaystyle \frac{{\mathcal{R}}}{6}=\alpha \left(2{H}^{2}+\dot{H}\right),\end{eqnarray} \tag{ 46 }$$

where $\alpha =3{c}_{H}^{2}/8\pi G.$ By defining $x\equiv {\rho }_{\mathrm{DM}}/{\rho }_{\mathrm{HDE}}$ and $y\equiv {\rho }_{\gamma }/{\rho }_{\mathrm{HDE}},$ the Friedmann and Raychauduri equations can be rewritten as

$$\begin{eqnarray}&&{H}^{2}=\displaystyle \frac{{c}_{H}^{2}}{\alpha }{\rho }_{\mathrm{HDE}}(1+x+y),\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&\dot{H}=-\displaystyle \frac{3}{2}{H}^{2}\left(1+\displaystyle \frac{y}{3(1+x+y)}+\displaystyle \frac{\omega }{1+x+y}\right).\end{eqnarray} \tag{ 48 }$$

The substitution of Equations (47) and (48) into (46) leads to the condition

$$\begin{eqnarray}&&\displaystyle \frac{2}{{c}_{H}^{2}}=1+x-3\omega .\end{eqnarray} \tag{ 49 }$$

This expression can be evaluated at a = 1 to obtain ${C}_{1}=1+{x}_{0}-3{\omega }_{0},$ where ${x}_{0}={{\rm{\Omega }}}_{\mathrm{DM}0}/{{\rm{\Omega }}}_{\mathrm{HDE}0},$ ${{\rm{\Omega }}}_{\mathrm{HDE}0}=1-{{\rm{\Omega }}}_{\mathrm{DM}0}-{{\rm{\Omega }}}_{\gamma 0},$ ${{\rm{\Omega }}}_{\mathrm{DM}0}$, and ${{\rm{\Omega }}}_{\mathrm{HDE}0}$ are the current DM and HDE density parameters, respectively. Thus, $x={x}_{0}+3(\omega -{\omega }_{0}).$

On the other hand, the differentiation of y with respect to the cosmological time t yields

$$\begin{eqnarray}&&\dot{y}=\left(\displaystyle \frac{\dot{{\rho }_{\gamma }}}{{\rho }_{\gamma }}-\displaystyle \frac{\dot{{\rho }_{\mathrm{HDE}}}}{{\rho }_{\mathrm{HDE}}}\right).\end{eqnarray} \tag{ 50 }$$

We take the time derivative of Equation (46) to obtain

$$\begin{eqnarray}&&\dot{{\rho }_{\mathrm{HDE}}}=\alpha \left(4H\dot{H}+\ddot{H}\right).\end{eqnarray} \tag{ 51 }$$

Similarly, the differentiation of Equation (48) results in

$$\begin{eqnarray}\displaystyle \frac{\ddot{H}}{\dot{H}} & = & -3H\displaystyle \frac{{C}_{1}+4\omega +4/3\;y}{{C}_{1}+3\omega +y}+\displaystyle \frac{4\dot{\omega }+4/3\;\dot{y}}{{C}_{1}+4\omega +4/3\;y}\\ & & -\displaystyle \frac{3\dot{\omega }+\dot{y}}{{C}_{1}+3\omega +y}.\end{eqnarray} \tag{ 52 }$$

By combining Equations (44d) and (50)–(52), we obtain the following differential equation for y:

$$\begin{eqnarray}&&{y}^{\prime }=y\displaystyle \frac{3\omega ^{\prime} -{C}_{1}}{3\omega +{C}_{1}},\end{eqnarray} \tag{ 53 }$$

where ' stands for the derivative with respect to e-foldings $N=\mathrm{ln}a,$ i.e., ${}^{\prime }=d/{dN}.$ In the N space, Equation (48) reads as

$$\begin{eqnarray}&&{H}^{\prime }=-\displaystyle \frac{3}{2}H\left(\displaystyle \frac{{C}_{1}+4\omega +4/3\;y}{{C}_{1}+3\omega +y}\right).\end{eqnarray} \tag{ 54 }$$

For the EOS CPL parameterization (Equation (3)), the system of Equations (53)–(54) has the following analytical solution:

$$\begin{eqnarray}E(z) & = & \displaystyle \frac{1}{\sqrt{1+{x}_{0}+{y}_{0}}}{\left(1+z\right)}^{2}{\left(\displaystyle \frac{1+{x}_{0}}{f}\right)}^{\alpha }\\ & & \times \sqrt{f+{y}_{0}{\left(\displaystyle \frac{f}{1+{x}_{0}}\right)}^{2\alpha }},\end{eqnarray} \tag{ 55 }$$

where ${y}_{0}={{\rm{\Omega }}}_{\mathrm{DM}0}/{{\rm{\Omega }}}_{\mathrm{HDE}0},$ $f=1+{x}_{0}+z(1+3{w}_{1}+{x}_{0}),$ and the exponent α is given by

$$\begin{eqnarray}&&\alpha =1-\displaystyle \frac{3({w}_{0}+{w}_{1})}{2(1+3{w}_{1}+{x}_{0})}.\end{eqnarray} \tag{ 56 }$$

Thus, we present a new analytical solution of $E(z)$ for the Ricci HDE model with CPL parameterization including the radiation fluid.