Constraints On Dark Energy Models From Galaxy Clusters and Gravitational Lensing Data

The Sunyaev--Zel'dovich (SZ) effect is a global distortion of the Cosmic Microwave Background (CMB) spectrum as a result of its interaction with a hot electron plasma in the intracluster medium of large structures gravitationally viralized such as galaxy clusters (GC). Furthermore, this~hot gas of electrons emits X-rays due to its fall in the gravitational potential well of the GC. The~analysis of SZ and X-ray data provides a method for calculating distances to GC at high redshifts. On the other hand, many galaxies and GC produce a Strong Gravitational Lens (SGL) effect, which has become a useful astrophysical tool for cosmology. We use these cosmological tests in addition to more traditional ones to constrain some alternative dark energy (DE) models, including the study of the history of cosmological expansion through the cosmographic parameters. Using Akaike and Bayesian Information Criterion, we find that the $wCDM$ and $\Lambda CDM$ models are the most favoured by the observational data. In addition, we found at low redshift a peculiar behavior of slowdown of the universe, which occurs in dynamical DE models when we use data from GC.


II. GALAXY CLUSTERS
The GC are the biggest gravitational structures in the Universe. They are in the transition between the linear and nonlinear regimes of the structure formation. Gravitational lensing of background sources produced by these systems are used to infer the shape of matter distributions in the Universe. Nevertheless, some lensing results such as high Navarro-Frenk-White concentration parameters and the predictions of the Einstein radii distributions are in tension with the standard ΛCDM model [9]. Therefore, the study of GC is very important for cosmology because it offers information that can be used to develop cosmological tests that help to distinguish between different models of DE present in the literature. In what follows, we describe briefly three different data sets that will be used in the development of these cosmological tests: the GC (SZ/X-ray, f gas ) and SGL.
A. Angular diameter distance using SZ/X-Ray method The thermal SZ effect is a small distortion in cosmic microwave background (CMB) spectrum due to the inverse Compton scattering of the CMB photons when they pass through the hot gas of electrons in GC [10,11]. This small fluctuation in CMB temperature is characterized by ∆T sz /T cmb = f (ν, T e )y(n e , T e ), where y(n e , T e ) = los n e k B T e m e c 2 σ T dl, which is known as the Compton parameter, such that T cmb = 2.726 K, n e and T e are the temperature of CMB, electron number density and temperature of the hot gas, respectively. σ T is the Thomson cross section, k B is the Boltzmann constant, m e c 2 is the rest mass of the electron and the integration is along the line of sight (los). The dependence with the frequency of the thermal SZ effect is given through the term f (ν, T e ), which also introduces relativistic corrections (see [12] for more details and [13] for a more recent update).
On the other hand, gas in GC can reach temperatures of 10 7 −10 8 K and densities of the order of 10 −1 −10 −5 cm −3 , so they emit high amounts of energy in X-rays. The primary emission mechanisms of X-rays for a diffuse intra-cluster medium are collisional processes such as: free-free (Bremsstrahlung), free-bound (recombination) or bound-bound (mainly emission lines), with luminosities of the order of 10 44 erg/s or even higher and spatial extensions of several arcmin or larger, even at high redshift. X-rays' observations currently offer a powerful technique for building catalogs of galaxy clusters, which are very important for modern cosmology [14]. The X-ray GC emission is given by S x = 1 4π(1 + z) 4 n 2 e Λ eH (µ e /µ H )dl, (2) where Λ eH is the X-ray cooling function, µ is the molecular weight given by µ i = ρ/(n i m p ) and z is the cluster redshift [1,2]. Then, combining Equations (1) and (2) through n e , we can obtain experimental cosmological distance with triaxial symmetry, given by where ∆T SZ0 and S x0 are the central temperature decrement and the central surface brightness, respectively, which include all the physical constants and the terms resulting from the los integration, such that ∆T SZ0 ∝ d A (z), S x0 ∝ d A (x) and d A (z) = D c | ell exp h 3/4 (e proj /e 1 e 2 ) 1/2 , h is a function of GC shape and orientation, e proj is axial ratio of the major to minor axes of the observed projected isophotes and θ c,proj is the projection on the plane of the sky (pos) (see Appendix A for some useful relationships and Table VIII for some data used in these methods). The expression in Equation (3) is an observational quantity that depends basically on the physical and geometrical properties of the cluster (see [1] for more information about the astrophysical details). That method for measuring distances is completely independent of other techniques and is valid at any redshift. We use 25 measurements of angular diameter distances from GC obtained through SZ/X-ray method by De Filippis et al. (see Figure 5). In our analysis, we follow the standard procedure and minimize the χ 2 function where d A (z) is the angular diameter distance in a FLRW universe and σ 2 Dc are the errors associated with D c | ell exp (z i ) (see Table VIII in Appendix).

B. The gas mass fraction fgas
Another independent cosmological technique is to derive d A using the gas mass fraction data from GC. In order to use f gas as a cosmological test, we need to assume that there is a proportion between the baryonic fraction of the GC and the global fraction of baryonic matter and DM. Moreover, it is necessary to assume that the baryonic fraction from clusters does not depend on the redshift [15]. This assumption is valid if one considers that these clusters are formed approximately by the same time. 1 (see [16] for more details). Thus, the gas mass fraction can be defined as f gas ≡ M gas /M tot , where M gas is the X-ray's gas mass and M tot is the total gravitational mass of GC respectively. To relate f gas with the parameters of a particular cosmological model, we can write M gas and M tot in terms of d A (z) as follows [17], where d A (z) is the angular diameter distance for a given cosmological model and d ΛCDM A (z) is the angular diameter distance for a reference model; in this case, let us assume the ΛCDM model. Here, Ω b and Ω 0m are the baryonic density parameter and the DM density parameter, respectively. The parameter b is the depletion factor that relates the baryonic fraction in clusters to the mean cosmic value. The constant α is the ratio between optically luminous baryonic mass in galaxies (stellar mass) to the baryonic X-ray gas mass in intracluster medium, and its value is given by α ≈ 0. 19 √ h [16]. The factor h is the normalized Hubble constant, that is, h = H 0 /100 km s −1 Mpc −1 . Let us use the f gas measurements from 42 GC obtained in [18]. The χ 2 is defined as where f gas is observational gas mass fraction data [18] and σ fgas are the systematic errors. In the analysis, we have considered b = 0.824 [16].

C. Gravitational lensing
The gravitational lens effect is one of the queen's tests of General Relativity. Strong gravitational lensing occurs when the light rays of a source are strongly deflected by the lens producing multiples images. The position of these images depend on the properties of the lens mass distribution [19]. Because the Einstein radii, θ E , also depends on a cosmological model, the SL observations can be used as an additional method to probe the nature of the DE [3,20]. In this work, we use the method that consists of comparing the ratio D of angular diameter distances between lens and source, d A (z l , z s ), and between observer and lens, d A (0, z s ), with its observable counterpart D obs given by where σ SIS is the Singular Isothermal Sphere (SIS) velocity dispersion and E(z, Θ) ≡ H(z, Θ)/H 0 , H(z, Θ) being the Hubble function. In order to put constraints on cosmological parameters through E(z, Θ), the Einstein radius θ E and the dispersion velocity σ SIS (exactly its central velocity dispersion σ 0 ) must be obtained by astrometric and spectroscopic means, respectively. In the first case, it depends on the lens modelling (either SIS, Singular Isothermal Ellipsoid (SIE) or Navarro-Frenk-White density profiles). In the second case, the velocity dispersion σ SIS of the mass distribution and the observed stellar velocity dispersion σ 0 need not be the same, since the halos of DM can have a greater speed of dispersion than the visible stars [21]. These effects can be taken into account through the following relationship σ SIS = f E σ 0 , where the parameter f E emulates the systematic errors in the RMS due to the difference between σ SIS and σ 0 ; the rms error caused by assuming the SIS model, since the observed image separation does not directly correspond to θ E and softened SIS potentials which tend to decrease the typical image separations [22]. In the present work we assume the best-fit reported in [20] (and references therein), where f E ≈ 1, which has been properly marginalized. On the other hand, GC can also act as sources to produce strong gravitational lensing showing giant arcs around GC. This phenomenon can be used to constrain the astrophysical properties of the cluster (projected mass) and cosmology [23]. If we assume the condition of hydrostatic equilibrium 2 and an approximation of spherical symmetry 3 [24], then a theoretical surface density can be described as where k B , m p , µ = 0.6 and β X are, respectively, the Boltzmann constant, the proton mass, the mean molecular weight and the slope of the β − model [25]. Although the hydrostatic equilibrium and isothermal hypotheses are very strong, the total mass density obtained under such assumption may lead to good estimates, even in dynamically active GC with irregular morphologies in X-rays. Then, combining this with the critical surface mass density for lensing Σ obs [26], We can get a Hubble constant independent ratio as where the parameters T X , β X and θ c can be obtained from X-ray observational data. The position of tangential critical curve θ t = θ arc , where θ arc is the observational arc position and = (1/ √ 1.2) ± 0.04 quantifies the slight difference with arc radius angle (See [27,28] for more details about the priors and 10 galaxy clusters used as sample).
In the present work we use a sample of 80 strong lensing systems by [20], which contains 70 data points from SLACS and LSD and 10 data points from GC. Again, the fit of the theoretical models to strong lensing observations can be found by the minimization of where the sum is over the sample and σ 2 D,i denotes the variance of D obs i . Additionally to these data sets defined in the Sections II A-II C, we will use 580 Supernovae data (SNIa) from Union 2.1 [29], the CMB shift parameter [30] (Planck 2013), as well as data from BAO (BOSS, WiggleZ, SDSS, 6dFGS) observations, adopting the three measurements of A(z) obtained from [31,32], and using the covariance among these data given in [33]. Each χ 2 function is constructed in a way analogous to the other tests considered above (see Appendix B).

D. Statistic analysis
The procedure of finding a set of parameters for a given statistic is known as Maximum likelihood L max , that is, given a probability distribution this is maximum for the corresponding data set. The maximum likelihood estimate for the best fit parameters p m i is given by If L max (p m i ) has a Gaussian errors distribution, then χ 2 min (p m i ) = −2 ln L max (p m i ), which is our case [34]. In order to find the best values of the free parameters of the model, let us consider The Fisher matrix is used in the analysis of the constraint of cosmological models for different observational test [35,36]. It contains the Gaussian uncertainties σ 2 i of the different parameters p m i . Given the best fit χ 2 min (p m i , σ 2 i ) for a set of parameters p m i with uncertainties σ 2 i , the Fisher matrix is for each model m. The inverse of the Fisher matrix provides an estimate of the covariance matrix through [C cov ] = [F ] −1 . Its diagonal elements are the squares of the uncertainties in each parameter marginalizing over the others, while the off-diagonal terms yield the correlation coefficients between parameters. The uncertainties obtained in the propagation of errors are given by σ i = Diag [C cov ] ij . Notice that the marginalized uncertainty is always greater than (or at most equal to) the non-marginalized one: marginalization cant decrease the error, and only has no effect if all other parameters are uncorrelated with it 4 . Previously known uncertainties on the parameters, known as priors, can be trivially added to the calculated Fisher matrix. This is manifestly the case for us: a lot of standard cosmological datasets provide priors on our previously defined cosmological parameters. The analysis with the Fisher matrix is used to evaluate the errors on the best-fit parameters.
In our results, let us consider different cosmological models. Thus, a way to quantify which model best fit the data is consider a Bayesian comparison. We adopted the Akaike and Bayesian information criterion (AIC and BIC, respectively), which allows us to compare cosmological models with different degrees of freedom, with respect to the observational evidence and the set of parameters [37,38]. The AIC and BIC can be calculated as where L max is the maximum likelihood of the model under consideration (L max = exp − 1 2 χ 2 min ), d is the number of parameters and N the number of data points. The BIC imposes a strict penalty against extra parameters for any set with N data. The prefered model is that which minimizes the AIC and BIC. However, the absolute values of them are not of interest, only the relative values between the different models [39]. Therefore, the "strength of evidence" can be characterized in the form ∆AIC = AIC i − AIC min , ∆BIC = BIC i − BIC min , where the subindex i refers to value of AIC (BIC) for model i and AIC min (BIC min ) is the minimum value of AIC (BIC) among all the models [40]. We give the judgements for both critera as follows: (i) If ∆AIC(∆BIC) ≤ 2, then the concerned model has substantial support with respect to the reference model (i.e., it has evidence to be a good cosmological model), (ii) if 4 ≤ ∆AIC(∆BIC) ≤ 7, it is an indication for less support with respect to the reference model, and, finally, (iii) if ∆AIC(∆BIC) ≥ 10, then the model has no observational support. Thus, if we have a set of models of DE, first we should estimate the best fit χ 2 and then we can apply the AIC and BIC to identify which model is the preferred one by the observations. We also apply the reduced chi-square to see how well the model fit the data, which is defined as χ 2 red = χ 2 min /ν, where ν is the degrees of freedom usually given by N − d. Then, the total number of data points is: SNIa (580), CMB (3), BAO (7), d A (25), f gas (42), SGL (80), so N = 737. Priors used in the present analysis are standard and the most conservative possible and combining GC data with independent constraints from CMB, BAO and SNIa removes the need of priors for Ω b , and h leads to tighter constraints over Ω m , Ω k and the parameters that characterize the DE density for different cosmological models. On the other hand, SGL offers a great opportunity to constrain DE features without prior assumptions on the fiducial cosmology. In what follows, we present our main results.

III. DARK ENERGY MODELS AND RESULTS
In order to put constraints on DE models using GC (d A , f gas ) and SGL, we need to compute the angular diameter distance of the model and compare it with observational data. In addition, to investigate whether a cosmological model can predict an accelerated expansion phase of the Universe, we must study the behavior of the deceleration parameter q(z). The angular diameter distance for a FLRW universe, from a source at redshift z, is given by where h is dimensionless Hubble parameter (H 0 = h 100 km s −1 Mpc −1 ) and the function sin ς(x) is defined such that it can be sinh(x) for Ω k > 0, sin(x) for Ω k < 0 and x for Ω k = 0 [41]. In the standard FLRW cosmology, the expansion rate as a function of the scale factor H(a) is given by the Friedmann equation as where H(a)/H 0 = E(a, Ω i ), H 0 is the current value of the expansion rate and the scale factor is related to redshift as 1 + z = a −1 , such that a 0 = 1 at present. In Equation (18), Ω i is a dimensionless energy densities relative to critical (ρ cri = 3H 2 0 /8πG) in the form of the i th component of the fluid density of: radiation (Ω r ), matter (Ω m ), curvature where Ω γ (h) = 2.469 × 10 −5 h −2 is the density of photons and N ef f = 3.046 is the effective number of neutrino species [42]. ω(a) = p(a)/ρ(a) is the EoS for DE, where p(a) is the fluid pressure. This EoS divides our models into two cases: when the energy density of the fluid is constant and the energy density of the fluid is dynamic. In all cosmological models, Ω k is a free parameter. A vector of parameters is considered for each DE model as Θ model Our analysis starts with standard cosmological model, where DE density is provided by the cosmological constant Λ. The expansion rate within ΛCDM context is given by where Ω r , Ω m and Ω Λ = 1 − Ω m − Ω k − Ω r , are the density parameters for radiation, matter and DE component, respectively. Here, the free parameter vector is Θ = {h, Ω b , Ω m , Ω k }. We find the best fit of parameters at 1σ confidence level (CL), whose results are shown in Table I.  In Table I, we can see the impact of adding the GC and SGL tests to the more traditional ones (CMB + BAO + SNIa), which evidently improves the constraints on the parameters of the model (see Figure 1).

B. wCDM model
The most simple extension of the ΛCDM model is to consider that the EoS remains constant but its value can be w = −1. In this case, the expansion rate for FLRW cosmology reads as where  Notice that in both cases the EoS has a phantom behavior and the standard model is excluded at least up to 2σ CL (see Figure 1). As the case of ΛCDM model, the curvature parameter changes from negative to positive (Table  II).

C. Chevalier-Polarski-Linder model
Another simple extension to the ΛCDM model is to allow for the EoS of the DE varies with the redshift. Several parameterizations have been considered in the literature. Here, let us consider the popular Chevallier-Polarski-Linder (CPL) model [43,44] where w 0 is the value of the DE state equation at the present and the parameter w 1 evaluates the dynamic character of DE. The FLRW E(z) for CPL parametrization is given by where Ω X = (1 − Ω k − Ω m − Ω r ) and The free parameters are Θ = {h, Ω b , Ω k , Ω m , w 0 , w 1 }. The best fit values at 1σ CL using CM B + BAO + SN Ia and full data set are summarized in Table III We can see that, for both the combined analyses, the CPL model allows a quintessential DE at the current time. The curvature parameter Ω k remains negative. The standard model remains within the 1σ and 2σ of CL for the present analysis (see Figure 1).

D. Interacting Dark Energy model
Cosmological models, where DM and DE are non minimally coupled throughout the evolution history of the universe, have been considered to solve the problem of the cosmic coincidence as well as the problem of the cosmological constant (models where DM interacts with vacuum energy or Interacting Dark Energy (IDE) Models-see [45,46] for general review). It has recently been shown that the current observational data can favor the late-time interaction in the dark sector [47][48][49][50][51][52]. In general, we assume that DM and DE interact via a coupling function Q given bẏ where ρ m and ρ x are the DM and DE density, respectively, with w x the EoS for DE. Here, Q = δH characterizes the strength of the interacting through the dimensionless coupling term δ, which establishes a transfer of energy from DE to DM for δ > 0, whereas for δ < 0 the energy transfer is the opposite. This model was originally introduced in [53], and then investigated in various contexts [54][55][56]. The expansion rate of the Universe for this model is given by where Ω X = (1 − Ω m − Ω k − Ω r ) and This model is characterized by the following set of parameters Θ = {h, Ω b , Ω k , Ω m , w x , δ}. We show the best fit values of these parameters in Table IV  Is interesting to note that, in both cases, EoS has a phantom behavior at present and the standard model is practically discarded at 1σ CL. The curvature parameter Ω k is positive. We can also notice that the case δ = 0 (absence of interaction) is excluded at least to 2σ CL for the present analysis, where we can appreciate that for both data sets the transfer of energy is from DM to DE (see Figure 1).

E. Early Dark Energy model
In early dark energy (EDE) scenarios, the DE density can be significant at high redshifts. This may be so if DE fluid tracks the dynamics of the background fluid density [57]. Here, we present the EDE model proposed by [58]. The FLRW equation for this model is where Ω X is given by and such that Ω X0 = 1 − Ω m − Ω k − Ω r is the current DE density, Ω e is the asymptotic early DE density and w 0 is the present DE EoE. Here, we have six free parameters Θ = {h, Ω b , Ω k , Ω m , Ω e , w 0 }. The best fit values of the model parameters are summarized in Table V  For this model, the EoS keeps a phantom behavior at the present time and the standard model is discarded at least to 2σ CL (see Figure 1). Ω k is positive in both cases.

F. Statistical discrimination models
In Table VI, we present the values for the analysis of the information criterion with respect to the five cosmological models presented above, for used data set, namely CMB + BAO + SNIa + d A + f gass + SGL. As we can see, ∆AIC  and ∆BIC are in favor of ωCDM and ΛCDM , respectively (approximately or less than two), and, hence, these models are in very good agreement with observations, which is also true for CPL, IDE and EDE models only with respect to ∆AIC. For models CPL, IDE and EDE, the value of ∆BIC is approximately equal to seven and therefore, according to this criterion, present less observational support.

IV. HISTORY OF THE EXPANSION AND COSMOGRAPHY
The kinematics of the universe can be described through the Hubble parameter H(t) and its dependence on time, i.e., the deceleration parameter q(t) [59] . Following [60], the scale factor a(t) can be expanded in Taylor series around the current time (t 0 ) as: where in general we can have a kinematic description of the cosmic expansion through the set of parameters: where the last term is know as jerk parameter j (t). The great advantage of this method is that we can investigate the cosmic acceleration without assuming any modification of gravity theory or DE model, due mainly to its geometric approximation. Although more terms of the series can be analyzed, we are only interested in the first three terms for the present work. The deceleration and jerk parameter are obtained as and The history of expansion is fit through deceleration parameter, which characterize whether the universe is currently accelerated or decelerated If q(z) > 0,ä(z) < 0, then the expansion decelerates as expected due to gravity produced by DM, baryonic matter or radiation. The discovery that the universe today presented an accelerated expansion already has about one decade and a half old [61,62]. A simple explanation for this phenomenon is the cosmological constant Λ, which, however, does not offer a consistent theoretical explanation based on physical foreground. The information about the dynamics of the expansion can be obtained through Equations (32) and (33), which directly depends on the cosmological model. In general, if Ω X = 0 is sufficiently large (i.e., Ω X > Ω m ), then q(z) < 0 andä(z) > 0, which translates into an accelerated expansion as it is shown by the observations. If the accelerated expansion is driven by a new type of fluid, then is important to identify if fluid energy density is constant or dynamic.
Model χ 2 red Parameters Λ CDM 1.11 h = 0.722 ± 0.012, Ωm = 0.2640 ± 0.0093, Ω k = −0.13 ± 0.16, Ω b = 0.0410 ± 0.0014 ω CDM 1.11 h = 0.722 ± 0.012, Ωm = 0.2685 ± 0.0093, Ω k = −0.14 ± 0.88, ω = −0.99 ± 0.73, Ω b = 0.0409 ± 0.0015 CPL 1.14 h = 0.721 ± 0.011, Ωm = 0.274 ± 0.013, Ω k = −0.5 ± 1.8, ωa = −1.5 ± 2.2, ω0 = −0.60 ± 0.50, Ω b = 0.0411 ± 0.0014 IDE 1.14 h = 0.721 ± 0.011, Ωm = 0.274 ± 0.012, Ω k = 0.2 ± 2.5, ωx = −1.1 ± 3.1, δ = 3.9 ± 13.0, Ω b = 0.0411 ± 0.0015 EDE 1.14 h = 0.720 ± 0.012, Ωm = 0.276 ± 0.014, Ω k = 0.3 ± 1.7, ω0 = −0.8 ± 1.7, Ωe = −1.1 ± 1.7, Ω b = 0.0412 ± 0.0015 In the present cosmographic analysis, we use of data from GC (d A + f gas ), where we can see that these do not provide a tight constraint on curvature and DE parameters, mainly due to the degeneracy presented between these parameters and the large systematic errors of the samples (see Table VII), which can lead to large discrepancies with respect to the standard model. Despite this, we are more interested in the analysis of the behavior of low redshift of each cosmological model with respect to these data sets. Figure 2 shows the plot of the deceleration parameter q(z) and, as expected, the models studied give q(z) < 0 at late times and q(z) > 0 at earlier epoch. All cosmological models present a redshift of transition (z t ) between the two periods; however, all models of dynamical DE present an interesting behavior of slowing down of acceleration at low redshift (late times), which can be characterized through the change of sign of the parameter j(z) (CPL: j(z low ) → 0, when z low ∼ 0.50; IDE: j(z low ) → 0, when z low ∼ 0.41; EDE: j(z low ) → 0, when z low ∼ 0.23). We can interpret j(z) as the slope at each point of q(z), which indicates a change in acceleration. This result is consistent with the one presented by Barrow, Bean and Magueijo [63], in which arises the possibility of a scenario with accelerated expansion of the universe and that does not imply an eternal accelerated expansion. In [60], an extensive analysis of this possibility is made (see also [64]), which includes a cosmographic analysis like the one presented in the current work. This transient accelerating phase can be also a clear behavior of dynamical DE at low redshift for models with variation of the density of DE over time.

V. SUMMARY AND DISCUSSION
In the present work, we compared alternative cosmological models of DE using data obtained from GC and SGL in addition to more traditional ones, getting the best-fit value of parameters for each one. On the other hand, applying the Akaike and Bayesian information criteria, we determine which of these models is the most favored by current observational data. Our analysis shows that ωCDM and ΛCDM DE models are preferred by ∆AIC and ∆BIC, respectively. For the first time, we report that the ωCDM model is favored by observational data at least with ∆AIC; however, the ΛCDM model remains the best fit for ∆BIC. In Figure 1, we can see that ΛCDM model is excluded at least 2σ CL for ωCDM , IDE and EDE models, combining all data sets (see also Tables II, IV and V). Models such as CPL, IDE and EDE, although they are penalized given their large number of free parameters, have a good fit with the observational data.
On the other hand, we carried out the study of the history of cosmic expansion through the H(z), q(z) and j(z) parameters with data from GC (d A,clusters + f gas ). We find new evidence showing anomalous behavior of the deceleration parameter q(z) in later times (z low < 0.5), suggesting that the expansion of the universe could decelerate in the near future (Figure 2), which was pointed out in previous works with SNIa (for CPL [65,66]), f gas (for CPL and different parameterizations of w(z) [67,68]) and BAO (for CPL, IDE and EDE [69]). Other types of mechanisms were also taken into account to explain this phenomenon (see, for example, [70]). This perspective raises the possibility that an accelerated expansion does not imply the eternal expansion, even in the presence of DE [71]. This cosmic slowing down of acceleration only appears in dynamic models of DE (CPL, IDE and EDE), which in principle can be an indication of the need for a scalar field such as quintessence or phantom (see, for example, [72]). Finally, in Figure 3, we show the results for jerk parameter j(z) obtained from our kinematic analysis, where we can appreciate a considerable deviation from ΛCDM (black curve) in late times (z < 0.5) for CPL, IDE and EDE models. A more careful study might give insight into this anomalous behavior, which may also represent a challenge for alternative models to DE including modified gravity models.
As we can see, the fit of observational data acquires slightly larger values of χ 2 min with respect to ΛCDM , when GC and SGL data are added to the more traditional ones as CMB + BAO + SNIa, which may be mainly due to their large systematic errors (GC + SGL) (see Tables I-V). However, the potential of these data sets as cosmological tests is very high, since, for example, the increase in the number of data points and the reduction of systematic errors leads to better constraints in parameters such as DE, which is of fundamental interest for modern cosmology.
Then the electron density distribution is described by five parameters in a ellipsoidal triaxial β-model: n e0 , β, e 1 , e 2 and r c3 .
The projection along the los of the electron density distribution, to a generic power m in the observer coordinate system is given by where d A is the angular diameter distance in a FRW universe, θ i ≡ x i,obs /d A e proj is the projected angular position on the plane of the sky (pos) of the intrinsic orthogonal coordinate system x i,obs and h is a function of the GC shape and orientation: h = e 2 1 sin 2 θ Eu sin 2 ϕ Eu + e 2 2 sin 2 θ Eu cos 2 ϕ Eu + cos 2 θ Eu , such that θ Eu and ϕ Eu are the Euler angles in the GC coordinate system (see Fig. 4) and θ c,proj ≡ θ c3 e proj e 1 e 2 If we assume that the intracluster medium is described by an isothermal triaxial β-model distribution with m=1 we obtain where ∆T sz0 is the central temperature decrement of SZ effect, which is given by e proj is the axial ratio of the major to minor axes of the observed projected isophotes and θ c,proj is the projection on the (pos).
On the other hand, the X-Ray surface brightness for intracluster medium with m=2, is given by where the central surface brightness S x0 is with µ i ≡ ρ/(n i m p ) the molecular weight.

Galaxy clusters data
Table VIII shows us the experimental cosmological distance with triaxial symmetry from De Filippis et al. obtained by the method S-Z/X-Ray [1]. Column 1 shows the cluster identification name, column 2 give the correspond redshift, column 3 is gas temperature, column 4 is central temperature decrement, column 5 is the term of dependence with frequency with relativistic corrections and column 6 show us the experimental cosmological distance. Fig 5. show us the angular diameter distance vs reshift and the data sample from De Filippis et al.  We use the Union 2.1 compilation, which contains a sample of 580 data points. We can get the luminosity distance through the relation d L (z) = (1 + z) 2 d A (z), then to fit cosmological model by minimizing the χ 2 value defined by where µ(z) ≡ 5 log 10 [d L (z)/Mpc] + 25 is the theoretical value of the distance modulus, and we have marginalized over the nuisance parameter µ 0 and µ obs .

CMB
A standar observational test is the angular scale of sound horizon (r s ) at time of decoupling (z cmb ∼ 1090), which is encoded in the location of the first peak of the CMB power spectrum l T T 1 . We include CMB information of Planck 13 data [? ], whose minimization is given by such that where ω b = Ω b h 2 . Here l A is the "acoustic scale" defined as where d A (z cmb ) is the angular diameter distance and z cmb is the redshift of decoupling given by [73], The "shift parameter" R is defined as [74] R = √ Ω m c d A (z cmb )(1 + z cmb ). (B8) C −1 cmb in Eq. (B3) is inverse covariance matrix for (R, l A , ω b ), which to Planck 13 data is: where σ i = (0.18, 0.0094, 0.00030) and normalized covariance matrix is: This test contributes with 3 data points to the statistical analysis.

BAO
The large scale correlation function measured from SDSS, includes a peak which was identified with the expanding spherical wave of baryonic perturbations from acoustic oscillations at recombination, whose current comoving scale corresponds to 150M pc. The expected BAO scale depends on the scale of the sound horizon at recombination and on transverse and radial scales at the mean redshift of galaxies in the survey. To obtain constraints on cosmological model we begin with χ 2 for WiggleZ BAO data [32], which is given by whereĀ obs = (0.447, 0.442, 0.424) is data vector at z = (0.44, 0.60, 0.73) andĀ th (z, p i ) is given by [75] A th = D V (z) Ω m H 2 0 cz , in which D V (z) is the distance scale defined as Here, d A (z) is the angular diameter distance. Additionally, C −1 W iggleZ is the inverse covariance matrix for the WiggleZ data set given by Similarly, for the SDSS DR7 BAO distance measurements, χ 2 can be expressed as [76]