Dark energy reconstruction based on the Padé approximation; an expansion around the \(\varLambda \)CDM
Abstract
We study the dynamical properties of dark energy based on a large family of Padé parameterizations for which the dark energy density evolves as the ratio between two polynomials in the scale factor of the universe. Using the latest cosmological data we perform a standard likelihood analysis in order to place constraints on the main cosmological parameters of different Padé models. We find that the basic cosmological parameters, namely \(({\varOmega _{m0}},h,{\sigma _{8}})\) are practically the same for all Padé parametrizations explored here. Concerning the free parameters which are related to dark energy we show that the bestfit values indicate that the equation of state parameter at the present time is in the phantom regime (\(w<1\)); however, we cannot exclude the possibility of \(w>1\) at \(1\sigma \) level. Finally, for the current family of Padé parametrizations we test their ability, via AIC, BIC and Jeffreys’ scale, to deviate from \(\varLambda \)CDM cosmology. Among the current Padé parametrizations, the model which contains two dark energy parameters is the one for which a small but nonzero deviation from \(\varLambda \)CDM cosmology is slightly allowed by the AIC test. Moreover, based on Jeffreys’ scale we show that a deviation from \(\varLambda \)CDM cosmology is also allowed and thus the possibility of having a dynamical dark energy in the form of Padé parametrization cannot be excluded.
1 Introduction
It is well known that the concept of dark energy (DE) was introduced in order to describe the accelerated expansion of the universe. Therefore, understanding the nature of DE is considered one of the most difficult and fundamental problems in cosmology. The introduction of a cosmological constant, \(\varLambda \) (\(\rho _{\varLambda }=\hbox {const}.\)), is perhaps the simplest form of DE which can be considered [1]. The outcome of this consideration is the concordance \(\varLambda \hbox {CDM}\) model, for which the \(\varLambda \) constant coexists with cold dark matter (CDM) and baryonic matter. In general, this model is a good description of the observed universe, since it is consistent with the cosmological data, namely Cosmic Microwave Background (CMB) [2, 3, 4, 5], Baryon Acoustic Oscillation (BAO) [6, 7, 8, 9, 10, 11] and Supernovae TypeIa (SnIa)[12, 13, 14, 15]. Despite the latter achievement \(\varLambda \hbox {CDM}\) suffers from the cosmological constant and the coincidence problems [16, 17, 18, 19, 20]. A third possible problem is related with the fact that the determination of the Hubble constant and the mass variance at \(8h^{1}\hbox {Mpc}\) have indicated a tension between the values resulting by the analysis of Planck data and the results obtained by the late time observational data [21, 22, 23].
An alternative avenue to overcoming the above problems is to introduce a dynamical DE, wherein the density of DE is allowed to evolve with cosmic time [24, 25, 26, 27, 28, 29, 30]. The first choice is to consider a DE fluid where the equation of state parameter varies with redshift, w(z). Usually, in these kinds of studies the EoS parameter can be written either as a firstorder Taylor expansion around \(a(z)=1\) [31, 32] or as a Padé parametrization [33, 34, 35, 36], where the corresponding free parameters are fitted by the cosmological data [37, 38, 39, 40, 41, 42, 43, 44]. Notice that for \(w>1\) we are in the quintessence regime [20, 45], namely the corresponding scalar field has a canonical Lagrangian form. In the case of \(w<1\) we are in the phantom region where the Lagrangian of the scalar field has a noncanonical form (Kessence) [45, 46, 47, 48, 49]. On the other hand, it is possible to reconstruct a DE model directly from observations. This approach is an excellent platform to study DE and indeed one may find several attempts in the literature. Specifically, one may use parametric criteria toward reconstructing directly the evolution of DE density \(\rho _\mathrm{de}(z)\) [50, 51, 52] and the potential of the scalar field [53].^{1} Comparing the two methods, namely w(z) and \(\rho _\mathrm{de}(z)\) for the same observational data sets, it has been found that the latter method leads to tighter constraints on the free parameters than the former [50, 51, 52].
In this work we have decided to reconstruct the evolution of the DE density, using the wellknown Padé approximation for which an unknown function [58, 59] is well approximated by the ratio of two polynomials. In contrast to the case where the Padé approximation is used to describe the DE EoS, our approach leads to an interesting parameterization which can be regarded as an expansion around the \(\varLambda \hbox {CDM}\). In Sect. 2 we introduce the concept of a Padé approximation in DE cosmologies. In Sect. 3 we briefly discuss the main features of the Bayesian analysis used in this work and we briefly present the observational data. In Sect. 4 we discuss the main results of our work; namely, we present the observational constraints on the fitted model parameters and we test whether a dynamical DE is allowed by the current data. Finally, in Sect. 5 we present our conclusions.
2 Reconstruction of dark energy using Padé approximation
2.1 Background evolution
Lastly, we would like to illustrate how extra terms of X(a) affect the Hubble parameter. As an example, we introduce the quantity \((1a)^{2}\) in Eq. (6), where the corresponding constants have been set either to \((b_{2},c_{2})=(0.1,0.1)\) or \((b_{2},c_{2})=(0.1,0.1)\). In Fig. 1 we present for the above set of \((b_{2},c_{2})\) parameters the evolution of \(\varDelta E\). It is obvious from the figure that the extra term \((1a)^{2}\) in the function X(a) does not really affect the cosmic expansion.
Consequently, our model can be considered as an expansion around the \(\varLambda \hbox {CDM}\) in the sense that adding extra terms changes the Hubble parameter slightly.
2.2 Growth of perturbations
An important quantity as regards testing the performance of the DE models at the perturbation level is \(f\sigma _8(a)\), where \(f=a\frac{\delta _m^{\prime }}{\delta _m}\) is the growth rate of clustering and \(\sigma _8(a)\) is the mass variance inside a sphere of radius 8\(h^{1}\)Mpc. The mass variance is written as \(\sigma _8(a)=\sigma _8\frac{\delta _m(a)}{\delta _m(a=1)}\), where \(\sigma _{8}\equiv \sigma _8(a=1)\) is the corresponding value at the present time. Notice that in our work we treat \(\sigma _8\) as a free parameter and thus it will be constrained by the available growth data.
For better comparison, the free parameters used in this figure are the same as those of Fig. 1, where we have set \(\varOmega _{m0}=0.3\), \(h=0.7\) and \(\sigma _8=0.8\). Overall, for phantom Padé cosmologies (\(b_1>c_{1}\); see the red line) we find that the expected differences are small at low redshifts, but they become larger for \(z\simeq 0.5\), reaching variations of up to \(\sim 3\%\), while they turn positive at high redshifts. Notice that the opposite behavior holds in the case of quintessence Padé cosmologies (\(b_1<c_{1}\); see the blue line).
3 Bayesian evidence and data processing
With the aid of the Padé parametrization which can be seen as an expansion around \(\varLambda \hbox {CDM}\) (\(w=1\)) our aim is to check whether the current observational data prefer a dynamical DE. First we consider a large body of Padé parametrizations (7) and then we test the statistical performance of each Padé model against the data.

We use the JLA SnIa data of (full likelihood version) [15].

The Baryon Acoustic Oscillations (BAO) data from 6dF [65], SDSS [66] and WiggleZ [67] surveys. Notice that details of the data concerned with processing and likelihoods can be found in [44, 68].

The Hubble parameter measurements as a function of redshift. We utilize the H(z) data set as provided by [69].

The CMB shift parameters as measured by the Planck team [70]. Notice that we use the covariance matrix which is introduced in Table 4 of [70].

The Hubble constant from [22].

For the growth rate data, in addition to the “Gold” growth data set \(f\sigma _8(z)\) provided by [71], we also use five new data points as collected by [72]. These new data points provide a growth rate at relatively higher redshifts and there is no overlap between these data and the gold sample. These new data points and their references are presented in Table 1.
New \(f\sigma _8\) data points which we use along with the Gold sample
Concerning the estimation of the sound horizon, needed when we compute the CMB and BAO likelihoods, we follow the procedure of [76]. Using the aforementioned data sets, we first perform a MCMC analysis to find the best value of the parameters and their uncertainties and then we quantify the statistical ability of each model to fit the observational data. To do this we use the MULTINEST sampling algorithm [77] and the python implementation pymultinest [78]. The latter technique was initially proposed in order to select the best model of AGN Xray spectra via a Bayesian approach.
4 Results and discussion
As we have already mentioned nowadays, testing the evolution of the DE EoS parameter is considered as one of the most fundamental problems in cosmology. We attempt to check such a possibility in the context of Padé parametrizations. Specifically, the family of Padé models and the corresponding free parameters used here are shown in Tables 2 and 3 respectively. Since, unlike the parameter estimation, the evidence strongly depends on the prior, we consider two different priors to show how the evidence changes due to different prior ranges. Here we select flat priors, which are often a standard choice. The upper panel in Table 3 shows a narrow range of priors while the lower panel presents a wider prior. The priors on the cosmological parameters (\(\varOmega _\mathrm{dm},\varOmega _\mathrm{ba},h,\sigma _8\)) are physically reasonable due to our understanding from observational data including SN Ia, CMB, BAO and growth rate of large scale structures. On the other hand, the range of priors in (\(\alpha ,\beta , M ,\varDelta M\)) comes from an analysis of the SN Ia data in [15, 79]. In contrast, we have no prior information regarding our free parameters in the Padé expansion (\(b_1,c_1,b_2,c_2,b_3,c_3,b_4,c_4\)). Therefore, we select priors on these parameters from the intuition that they should construct an expansion around the \(\varLambda { {CDM}}\) model. In this sense, we consider smaller prior ranges for parameters which are of higher order in the expansion. According to [61], one possibility in such a case is to consider a prior which maximizes the probability of the new model, given the data. In this case, if the evidence is not significantly larger than the simpler model, then we can say that the data does not support additional parameters. In addition to these two prior ranges, we have examined other prior ranges and our results did not change significantly.
Various models used in our analysis
\(M_1\)  \(\varLambda \hbox {CDM}\) 

\(M_2\)  \(b_1,c_1\ne 0 \) and all others equal to zero 
\(M_3\)  \(b_1,c_1,b_2,c_2\ne 0 \) and all others equal to zero 
\(M_4\)  \(b_1,c_1,b_2,c_2,b_3,c_3\ne 0 \) and all others equal to zero 
\(M_5\)  \(b_1,c_1,b_2,c_2,b_3,c_3,b_4,c_4\ne 0 \) and all others equal to zero 
Two ranges of the model parameters which we consider in this work. The upper panel (lower panel) indicates a narrow(broad) prior
Parameters  Prior (uniform)  Parameters  Prior (uniform) 

\(\varOmega _{dm}\)  [0.15, 0.30]  \(\alpha \)  [0.10, 0.16] 
\(\varOmega _{ba}\)  [0.02, 0.07]  \(\beta \)  [2.85, 3.25] 
h  [0.55,0.80]  M  [\({}19.6,{}19.2\)] 
\(\sigma _8\)  [0.50, 1.20]  \(\varDelta M\)  [\({}0.15,0.15\)] 
\(b_1\)  [\({}0.3,0.3\)]  \(c_1\)  [\({}0.3,0.3\)] 
\(b_2\)  [\({}0.2,0.2\)]  \(c_2\)  [\({}0.2,0.2\)] 
\(b_3\)  [\({}0.1,0.1\)]  \(c_3\)  [\({}0.1,0.1\)] 
\(b_4\)  [\({}0.05,0.05\)]  \(c_4\)  [\({}0.05,0.05\)] 
\(\varOmega _{dm}\)  [0.0, 0.60]  \(\alpha \)  [\({}0.3,0.3\)] 
\(\varOmega _{ba}\)  [0.0, 0.2]  \(\beta \)  [1.0, 4.0] 
h  [0.4, 1.]  M  [\({}21.,{}18.\)] 
\(\sigma _8\)  [0.3, 1.5]  \(\varDelta M\)  [\({}0.25,0.25\)] 
\(b_1\)  [\({}0.5,0.5\)]  \(c_1\)  [\({}0.5,0.5\)] 
\(b_2\)  [\({}0.3,0.3\)]  \(c_2\)  [\({}0.3,0.3\)] 
\(b_3\)  [\({}0.2,0.2\)]  \(c_3\)  [\({}0.2,0.2\)] 
\(b_4\)  [\({}0.1,0.1\)]  \(c_4\)  [\({}0.1,0.1\)] 
The bestfit values and the corresponding \(1\sigma \) uncertainties for the current Padé parametrizations. Notice that the \(\varLambda \hbox {CDM}\) model can be seen as a Padé \(M_{1}\) parametrization, namely \(b_{i}=c_{i}=0\)
Models/parameters  \(M_1\)  \(M_2\)  \(M_3\)  \(M_4\)  \(M_5\) 

\(\varOmega _m\)  \(0.2886\pm 0.0052\)  \(0.2836^{+0.0052}_{0.0059}\)  \(0.2830^{+0.0051}_{0.0058}\)  \(0.2846^{+0.0048}_{0.0054}\)  \(0.2843\pm 0.0056 \) 
h  \(0.6933\pm 0.0042\)  \(0.7026^{+0.0065}_{0.0050}\)  \(0.7042^{+0.0066}_{0.0059}\)  \(0.7023\pm 0.0058\)  \(0.7030\pm 0.0063\) 
\(\sigma _8\)  \(0.769\pm 0.023\)  \(0.766\pm 0.023 \)  \(0.765\pm 0.023\)  \(0.765\pm 0.023\)  \(0.765\pm 0.022\) 
\(\alpha \)  \(0.1413\pm 0.0050\)  \(0.1417\pm 0.0051\)  \(0.1417\pm 0.0050\)  \(0.1416\pm 0.0051\)  \(0.1417\pm 0.0051\) 
\(\beta \)  \(3.104\pm 0.057 \)  \(3.116\pm 0.058\)  \(3.117\pm 0.059\)  \(3.116\pm 0.058\)  \(3.114\pm 0.058\) 
M  \(19.074\pm 0.016\)  \(19.060\pm 0.018\)  \(19.057\pm 0.018\)  \(19.060\pm 0.017\)  \(19.059\pm 0.017\) 
\(\varDelta M\)  \(0.071\pm 0.017 \)  \(0.068^{+0.014}_{0.018}\)  \(0.067\pm 0.014\)  \(0.067^{+0.014}_{0.016}\)  \(0.067^{+0.014}_{0.016}\) 
\(b_1\)  –  \(0.03^{+0.30}_{0.17}\)  \(0.02^{+0.30}_{0.27}\)  \(0.00\pm 0.23\)  \(0.01\pm 0.22\) 
\(c_1\)  –  \(0.19^{+0.039}_{0.093}\)  \(0.14^{+0.13}_{0.21}\)  \(0.14^{+0.12}_{0.22}\)  \(0.14^{+0.11}_{0.20}\) 
\(b_2\)  –  –  \(0.03^{+0.18}_{0.13}\)  \(0.01\pm 0.16\)  \(0.01\pm 0.16\) 
\(c_2\)  –  –  \(0.126^{+0.056}_{0.16}\)  \(0.095^{+0.067}_{0.19}\)  \(0.092^{+0.088}_{0.18}\) 
\(b_3\)  –  –  –  \(0.039^{+0.15}_{0.061}\)  \(0.017^{+0.14}_{0.092}\) 
\(c_3\)  –  –  –  \(0.04^{+0.19}_{0.15}\)  \(0.03\pm 0.11\) 
\(b_4\)  –  –  –  –  \(0.092^{+0.11}_{0.075}\) 
\(c_4\)  –  –  –  –  \(0.005\pm 0.056\) 
The goodnessoffit statistics \(\chi ^{2}_{min}\), \(\varDelta \)AIC, \(\varDelta \)BIC, \(\varDelta \mathrm{ln}{\mathscr {E}}\) for the narrow and broad ranges of priors (N (B) stands for narrow(broad) prior ranges) and the Bayesian complexity for our models
Model  \(\chi ^2_\mathrm{min}\)  \(\varDelta \)AIC  \(\varDelta \)BIC  \(\varDelta \mathrm{\ln } {\mathscr {E}}_N \)  \(\varDelta \mathrm{\ln } {\mathscr {E}}_B \)  \(C_b(\hat{\theta }_b)\)  \(C_b(\hat{\theta }_m)\) 

\(M_1\)  733.20  0.0  0.0  0.00  0.00  9.09  8.89 
\(M_2\)  730.33  1.13  10.51  0.22  0.55  9.92  9.33 
\(M_3\)  729.99  4.79  23.55  0.65  0.82  9.85  9.62 
\(M_4\)  729.95  8.75  36.89  0.97  0.68  9.71  9.32 
\(M_5\)  729.85  12.65  50.17  1.02  0.95  9.77  9.6 
Another way of testing the ability of the models to fit the data is via the Bayesian evidence \({\mathscr {E}}\); namely, a model with the higher evidence is favored. In this context, in order to measure the significant difference between two models \(M_{i}\) and \(M_{j}\) we can use the Jeffreys scale [84], which is given by \(\varDelta {\ln \mathscr {E}}={\ln \mathscr {E}}_{M_{i}}{\ln \mathscr {E}}_{M_{j}}\). This model pair difference leads to the following situations: (1) \(0<\varDelta {\ln \mathscr {E}}<1.1\) suggests weak evidence against \(M_{j}\) model when compared with \(M_{i}\), (2) the restriction \(1.1<\varDelta \ln {\mathscr {E}}<3\) means that there is definite evidence against \(M_{j}\), while in the case of \(\varDelta \ln {\mathscr {E}}\ge 3\) such evidence becomes strong [85].
Our estimations of the Bayesian complexities are given in Table 5. Here we consider both cases for \(\hat{\theta }\), so \(\hat{\theta _b}\) (\(\hat{\theta _m}\)) indicates the best value (mean value) of the parameters.
Clearly, after considering the above statistical tests we find that the best model is the \(\varLambda \)CDM model, hence \(\mathrm{AIC}_\mathrm{min}\equiv \mathrm{AIC}_{M_{1}}\), \({\mathscr {E}}_{M_{j}}\equiv {\mathscr {E}}_{M_{1}}\). Using the model pair difference \(\varDelta \mathrm{AIC}\) we find strong evidence against models \(M_{4}\) and \(M_{5}\), namely \(\varDelta \mathrm{AIC} \gtrsim 10\). Also, in the case of the \(M_{3}\) model we have \(\varDelta \mathrm{AIC} \simeq 4.79\), which indicates positive evidence against that model, while for \(M_{2}\) model we obtain \(\varDelta \mathrm{AIC} \simeq 1.13\) and thus we cannot reject this model. In contrast to the AIC, BICs of our models indicate a “decisive” evidence against the \(M_2\), \(M_3\) \(M_4\) and \(M_5\) models. The main reason is that the BIC penalizes models with a high number of parameters more than AIC, specifically when there are large numbers of data points.
From the viewpoint of \(\varDelta {\ln \mathscr {E}}\)^{3} we argue that there is a weak evidence in favor of all dynamical models when compared with \(M_{1}\) (\(\varLambda \)CDM). Note that we consider two different prior ranges to check the possible dependency of the evidence on the prior (see Table 3). In fact, the evidence of each model is different considering different priors^{4} but \(\varDelta {\ln \mathscr {E}}\) (in our case) does not change significantly. In Table 5 the results are presented for both narrow (\(\varDelta {\ln \mathscr {E}}_{N}\) ) and broad (\(\varDelta {\ln \mathscr {E}}_B\)) priors.
Of course, such results disagree with Occam’s razor, which simply penalizes models with a large number of free parameters. Models \(M_5, M_4\) and \(M_3\) have eight, six and four free parameters more than the \(\varLambda \hbox {CDM}\), but the Bayesian evidence does not show any significant difference between them. Similar conclusions can be found in the work of [85] in which one proved that a linear model \(M_{a}\) with 14 free parameters provides the same value of Bayesian evidence as another model, \(M_{b}\), which contains four free parameters. According to these authors, the latter can be explained if the extra 10 parameters of \(M_{b}\) do not really improve the statistical performance of the model in fitting the data. In our case we confirm the results of [85] for Padé cosmologies; namely, the extra parameters of the \(M_3\), \(M_4\) and \(M_5\) parametrizations do not improve the corresponding DE models.
Moreover, the Bayesian complexity is a diagnostic tool to break the degeneracy when two competing models have almost the same evidence. Since, from the evidence alone, it is not clear that the extra parameters are unmeasured or improve the quality of the fit just enough to offset the Occam’s razor penalty term, the Bayesian complexity can be used to break this degeneracy (for more information see [61]). Our results indicate a slightly larger Bayesian complexity when the two competing models are \(M_1\) and \(M_2\), which is consistent with our conclusion that current data slightly prefer a dynamical dark energy model. The Bayesian complexities for other models are more and less the same, which indicates that current data are not good enough to measure the additional parameters and extra parameters are not needed.
Combining the aforementioned results we argue that, although the \(\varLambda \hbox {CDM}\) model reproduces very well the cosmological data, the possibility of a dynamical DE in the form of the \(M_{2}\) Padé model cannot be excluded by the data.
5 Conclusion
In this article we attempt to check whether a dynamical dark energy is allowed by the current cosmological data. The evolution of dark energy is treated within the context of a Padé parameterization, which can be seen as an expansion around the usual \(\varLambda \hbox {CDM}\) cosmology. Unlike most DE parameterizations (CPL and the like), in the case of the Padé parametrization the equation of state parameter does not diverge in the far future (\(a\gg 1\)) and thus its evolution is smooth in the range of \(a\in (0,+\infty )\).
Using the latest cosmological data we place observational constraints on the viable Padé dark energy models, by implementing a joint statistical analysis involving the latest observational data, SNIa (JLA), BAOs, direct measurements of H(z), and CMB shift parameters from Planck and growth rate data. In particular, we consider four Padé parametrizations, each with several independent parameters and we find that practically the examined Padé models are in very good agreement with observations. In all of them the main cosmological parameters, namely \((\varOmega _{m0},h,\sigma _{8})\), are practically the same. Regarding the free parameters of Padé parametrtization we show that, although the bestfit values indicate \(w<1\) at the present time, we cannot exclude the possibility of \(w>1\) at \(1\sigma \) level.
Finally, for all Padé models we quantify their deviation from \(\varLambda \hbox {CDM}\) cosmology through the AIC and Jeffreys scale. We find that the corresponding \(\chi ^{2}_\mathrm{min}\) values are very close to that of \(\varLambda \hbox {CDM}\), which implies that the chisquare estimator cannot distinguish Padé models from \(\varLambda \hbox {CDM}\). Among the family of current Padé parametrizations, the model which contains two dark energy parameters is the one for which a small but nonzero deviation from \(\varLambda \)CDM cosmology is slightly allowed by the AIC test. On the other hand, based on Jeffreys’ scale we show that a deviation from \(\varLambda \hbox {CDM}\) cosmology is also allowed, hence the possibility of a dynamical DE in the form of Padé parametrization cannot be excluded.
Furthermore, we estimate the socalled Bayesian complexity to realize whether the current data can constrain the extra parameters in our models or not. The Bayesian complexity is a measure of the effective number of parameters, which can be measured, given the data, and our results show a slightly larger Bayesian complexity for the \(M_2\) model, which is consistent with our conclusion as regards possible dynamical DE models. In contrast, the Bayesian complexity does not change significantly by adding extra parameters in our other models, which indicates that the current data is not good enough to measure the extra parameters.
Footnotes
References
 1.P.J. Peebles, B. Ratra, Rev. Modern Phys. 75, 559 (2003)ADSMathSciNetCrossRefGoogle Scholar
 2.E. Komatsu, J. Dunkley, M.R. Nolta et al., ApJS 180, 330 (2009)ADSCrossRefGoogle Scholar
 3.N. Jarosik, C.L. Bennett, J. Dunkley, B. Gold, M.R. Greason, M. Halpern, R.S. Hill, G. Hinshaw, A. Kogut, E. Komatsu et al., ApJS 192, 14 (2011)ADSCrossRefGoogle Scholar
 4.E. Komatsu, K.M. Smith, J. Dunkley et al., ApJS 192, 18 (2011)ADSCrossRefGoogle Scholar
 5.X.I.V. Planck Collaboration, Astron. Astrophys. 594, A14 (2016)Google Scholar
 6.M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). https://doi.org/10.1103/PhysRevD.69.103501 ADSCrossRefGoogle Scholar
 7.S. Cole et al., MNRAS 362, 505 (2005). https://doi.org/10.1111/j.13652966.2005.09318.x ADSCrossRefGoogle Scholar
 8.D.J. Eisenstein et al., ApJ 633, 560 (2005). https://doi.org/10.1086/466512 ADSCrossRefGoogle Scholar
 9.W.J. Percival, B.A. Reid, D.J. Eisenstein et al., MNRAS 401, 2148 (2010)ADSCrossRefGoogle Scholar
 10.C. Blake, Mon. Not. R. Astron. Soc. 415, 2876 (2011). https://doi.org/10.1111/j.13652966.2011.18903.x ADSCrossRefGoogle Scholar
 11.B.A. Reid, L. Samushia, M. White, W.J. Percival, M. Manera et al., MNRAS 426, 2719 (2012). https://doi.org/10.1111/j.13652966.2012.21779.x ADSCrossRefGoogle Scholar
 12.A.G. Riess, A.V. Filippenko, P. Challis, et al., AJ 116, 1009 (1998)Google Scholar
 13.S. Perlmutter, G. Aldering, G. Goldhaber et al., ApJ 517, 565 (1999)ADSCrossRefGoogle Scholar
 14.M. Kowalski, D. Rubin, G. Aldering et al., ApJ 686, 749 (2008)ADSCrossRefGoogle Scholar
 15.M. Betoule et al., Astron. Astrophys. 568, A22 (2014). https://doi.org/10.1051/00046361/201423413 CrossRefGoogle Scholar
 16.S. Weinberg, Rev. Modern Phys. 61, 1 (1989)ADSMathSciNetCrossRefGoogle Scholar
 17.V. Sahni, A.A. Starobinsky, IJMPD 9, 373 (2000)ADSGoogle Scholar
 18.S.M. Carroll, Living Rev. Relat. 380, 1 (2001)ADSGoogle Scholar
 19.T. Padmanabhan, Phys. Rep. 380, 235 (2003)ADSMathSciNetCrossRefGoogle Scholar
 20.E.J. Copeland, M. Sami, S. Tsujikawa, IJMP D 15, 1753 (2006). https://doi.org/10.1142/S021827180600942X ADSCrossRefGoogle Scholar
 21.E.G.M. Ferreira, J. Quintin, A.A. Costa, E. Abdalla, B. Wang, Phys. Rev. D 95(4), 043520 (2017). https://doi.org/10.1103/PhysRevD.95.043520 ADSCrossRefGoogle Scholar
 22.A.G. Riess et al., Astrophys. J. 826(1), 56 (2016). https://doi.org/10.3847/0004637X/826/1/56 ADSMathSciNetCrossRefGoogle Scholar
 23.H. Hildebrandt, Mon. Not. R. Astron. Soc. 465, 1454 (2017). https://doi.org/10.1093/mnras/stw2805 ADSCrossRefGoogle Scholar
 24.R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998). https://doi.org/10.1103/PhysRevLett.80.1582 ADSCrossRefGoogle Scholar
 25.J.K. Erickson, R. Caldwell, P.J. Steinhardt, C. ArmendarizPicon, V.F. Mukhanov, Phys. Rev. Lett. 88, 121301 (2002). https://doi.org/10.1103/PhysRevLett.88.121301 ADSCrossRefGoogle Scholar
 26.C. ArmendarizPicon, V. Mukhanov, P.J. Steinhardt, Phys. Rev. D 63(10), 103510 (2001)ADSCrossRefGoogle Scholar
 27.R.R. Caldwell, Phys. Lett. B 545, 23 (2002)ADSCrossRefGoogle Scholar
 28.T. Padmanabhan, Phys. Rev. D 66, 021301 (2002)ADSCrossRefGoogle Scholar
 29.E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D 70, 043539 (2004). https://doi.org/10.1103/PhysRevD.70.043539 ADSCrossRefGoogle Scholar
 30.G.B. Zhao et al., Nat. Astron. 1, 627 (2017). https://doi.org/10.1038/s415500170216z ADSCrossRefGoogle Scholar
 31.M. Chevallier, D. Polarski, Int. J. Modern Phys. D 10, 213 (2001). https://doi.org/10.1142/S0218271801000822 ADSCrossRefGoogle Scholar
 32.E.V. Linder, Phys. Rev. Lett. 90, 091301 (2003)ADSCrossRefGoogle Scholar
 33.M. Adachi, M. Kasai, Prog. Theor. Phys. 127, 145 (2012). https://doi.org/10.1143/PTP.127.145 ADSCrossRefGoogle Scholar
 34.C. Gruber, O. Luongo, Phys. Rev. D 89(10), 103506 (2014). https://doi.org/10.1103/PhysRevD.89.103506 ADSCrossRefGoogle Scholar
 35.H. Wei, X.P. Yan, Y.N. Zhou, JCAP 1401, 045 (2014). https://doi.org/10.1088/14757516/2014/01/045 ADSCrossRefGoogle Scholar
 36.M. Rezaei, M. Malekjani, S. Basilakos, A. Mehrabi, D.F. Mota, ApJ 843, 65 (2017). https://doi.org/10.3847/15384357/aa7898 ADSCrossRefGoogle Scholar
 37.A.G. Riess, L.G. Strolger, J. Tonry, S. Casertano, H.C. Ferguson et al., ApJ 607, 665 (2004)ADSCrossRefGoogle Scholar
 38.U. Seljak et al., Phys. Rev. D 71, 103515 (2005). https://doi.org/10.1103/PhysRevD.71.103515 ADSCrossRefGoogle Scholar
 39.B.A. Bassett, M. Brownstone, A. Cardoso, M. Cortes, Y. Fantaye, R. Hlozek, J. Kotze, P. Okouma, JCAP 0807, 007 (2008). https://doi.org/10.1088/14757516/2008/07/007 ADSCrossRefGoogle Scholar
 40.J.B. Dent, S. Dutta, T.J. Weiler, Phys. Rev. D 79, 023502 (2009). https://doi.org/10.1103/PhysRevD.79.023502 ADSCrossRefGoogle Scholar
 41.P.H. Frampton, K.J. Ludwick, Eur. Phys. J. C 71, 1735 (2011). https://doi.org/10.1140/epjc/s100520111735x ADSCrossRefGoogle Scholar
 42.C.J. Feng, X.Y. Shen, P. Li, X.Z. Li, JCAP 1209, 023 (2012). https://doi.org/10.1088/14757516/2012/09/023 ADSCrossRefGoogle Scholar
 43.A. Mehrabi, M. Malekjani, F. Pace, Astrophys. Space Sci. 356(1), 129 (2015). https://doi.org/10.1007/s1050901421853 ADSCrossRefGoogle Scholar
 44.A. Mehrabi, S. Basilakos, F. Pace, MNRAS 452, 2930 (2015). https://doi.org/10.1093/mnras/stv1478 ADSCrossRefGoogle Scholar
 45.C. ArmendarizPicon, V.F. Mukhanov, P.J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000). https://doi.org/10.1103/PhysRevLett.85.4438 ADSCrossRefGoogle Scholar
 46.C. ArmendarizPicon, T. Damour, V.F. Mukhanov, Phys. Lett. B 458, 209 (1999). https://doi.org/10.1016/S03702693(99)006036 ADSMathSciNetCrossRefGoogle Scholar
 47.T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D 62, 023511 (2000). https://doi.org/10.1103/PhysRevD.62.023511 ADSCrossRefGoogle Scholar
 48.T. Chiba, S. Dutta, R.J. Scherrer, Phys. Rev. D 80, 043517 (2009). https://doi.org/10.1103/PhysRevD.80.043517 ADSCrossRefGoogle Scholar
 49.L. Amendola, S. Tsujikawa, Dark Energy: Theory and Observations (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
 50.Y. Wang, P.M. Garnavich, Astrophys. J. 552, 445 (2001). https://doi.org/10.1086/320552 ADSCrossRefGoogle Scholar
 51.I. Maor, R. Brustein, J. McMahon, P.J. Steinhardt, Phys. Rev. D 65, 123003 (2002). https://doi.org/10.1103/PhysRevD.65.123003 ADSCrossRefGoogle Scholar
 52.Y. Wang, K. Freese, Phys. Lett. B 632, 449 (2006). https://doi.org/10.1016/j.physletb.2005.10.083 ADSCrossRefGoogle Scholar
 53.A.A. Mamon, K. Bamba, S. Das, Eur. Phys. J. C 77(1), 29 (2017). https://doi.org/10.1140/epjc/s100520164590y ADSCrossRefGoogle Scholar
 54.T. Holsclaw, U. Alam, B. Sanso, H. Lee, K. Heitmann, S. Habib, D. Higdon, Phys. Rev. D 84, 083501 (2011). https://doi.org/10.1103/PhysRevD.84.083501 ADSCrossRefGoogle Scholar
 55.M. Sahlen, A.R. Liddle, D. Parkinson, Phys. Rev. D 72, 083511 (2005). https://doi.org/10.1103/PhysRevD.72.083511 ADSCrossRefGoogle Scholar
 56.M. Sahlen, A.R. Liddle, D. Parkinson, Phys. Rev. D 75, 023502 (2007). https://doi.org/10.1103/PhysRevD.75.023502 ADSCrossRefGoogle Scholar
 57.R.G. Crittenden, G.B. Zhao, L. Pogosian, L. Samushia, X. Zhang, JCAP 1202, 048 (2012). https://doi.org/10.1088/14757516/2012/02/048 ADSCrossRefGoogle Scholar
 58.A. Baker, P. GravesMorris, Pade Approximants (1996)Google Scholar
 59.M. Adachi, M. Kasai, Progr. Theor. Phys. 127, 145 (2012). https://doi.org/10.1143/PTP.127.145 ADSCrossRefGoogle Scholar
 60.L.R. Abramo, R.C. Batista, L. Liberato, R. Rosenfeld, Phys. Rev. D 79, 023516 (2009). https://doi.org/10.1103/PhysRevD.79.023516 ADSCrossRefGoogle Scholar
 61.R. Trotta, Contemp. Phys. 49, 71 (2008). https://doi.org/10.1080/00107510802066753 ADSCrossRefGoogle Scholar
 62.J. Martin, C. Ringeval, R. Trotta, Phys. Rev. D 83, 063524 (2011). https://doi.org/10.1103/PhysRevD.83.063524 ADSCrossRefGoogle Scholar
 63.T.D. Saini, J. Weller, S.L. Bridle, Mon. Not. R. Astron. Soc. 348, 603 (2004). https://doi.org/10.1111/j.13652966.2004.07391.x ADSCrossRefGoogle Scholar
 64.A.I. Lonappan, S Kumar Ruchika, B.R. Dinda, A.A. Sen, Phys. Rev. D 97(4), 043524 (2018). https://doi.org/10.1103/PhysRevD.97.043524 ADSCrossRefGoogle Scholar
 65.F. Beutler, C. Blake, M. Colless, D.H. Jones, L. StaveleySmith et al., MNRAS 423, 3430 (2012). https://doi.org/10.1111/j.13652966.2012.21136.x ADSCrossRefGoogle Scholar
 66.L. Anderson, E. Aubourg, S. Bailey, D. Bizyaev, M. Blanton et al., MNRAS 427(4), 3435 (2013). https://doi.org/10.1111/j.13652966.2012.22066.x ADSCrossRefGoogle Scholar
 67.C. Blake, E. Kazin, F. Beutler, T. Davis, D. Parkinson et al., MNRAS 418, 1707 (2011). https://doi.org/10.1111/j.13652966.2011.19592.x ADSCrossRefGoogle Scholar
 68.G. Hinshaw et al., ApJS 208, 19 (2013). https://doi.org/10.1088/00670049/208/2/19 ADSCrossRefGoogle Scholar
 69.O. Farooq, F.R. Madiyar, S. Crandall, B. Ratra, Astrophys. J. 835(1), 26 (2017). https://doi.org/10.3847/15384357/835/1/26 ADSCrossRefGoogle Scholar
 70.P.A.R. Ade et al., Astron. Astrophys. 594, A14 (2016). https://doi.org/10.1051/00046361/201525814 CrossRefGoogle Scholar
 71.S. Nesseris, G. Pantazis, L. Perivolaropoulos, Phys. Rev. D 96(2), 023542 (2017). https://doi.org/10.1103/PhysRevD.96.023542 ADSCrossRefGoogle Scholar
 72.L. Kazantzidis, L. Perivolaropoulos, Phys. Rev. D 97(10), 103503 (2018). https://doi.org/10.1103/PhysRevD.97.103503 ADSCrossRefGoogle Scholar
 73.Y. Wang, G.B. Zhao, C.H. Chuang, M. PellejeroIbanez, C. Zhao, F.S. Kitaura, S. RodriguezTorres (2017)Google Scholar
 74.H. GilMarn, et al., (2018). https://doi.org/10.1093/mnras/sty453
 75.G.B. Zhao, et al., (2018)Google Scholar
 76.Y. Wang, S. Wang, Phys. Rev. D 88(4), 043522 (2013). https://doi.org/10.1103/PhysRevD.88.043522. https://doi.org/10.1103/PhysRevD.88.069903. [Erratum: Phys. Rev. D88, no.6,069903(2013)]
 77.F. Feroz, M.P. Hobson, M. Bridges, Mon. Not. R. Astron. Soc. 398, 1601 (2009). https://doi.org/10.1111/j.13652966.2009.14548.x ADSCrossRefGoogle Scholar
 78.J. Buchner, A. Georgakakis, K. Nandra, L. Hsu, C. Rangel, M. Brightman, A. Merloni, M. Salvato, J. Donley, D. Kocevski, Astron. Astrophys. 564, A125 (2014). https://doi.org/10.1051/00046361/201322971 ADSCrossRefGoogle Scholar
 79.S. Wang, S. Wen, M. Li, JCAP 1703(03), 037 (2017). https://doi.org/10.1088/14757516/2017/03/037 ADSCrossRefGoogle Scholar
 80.H. Akaike, IEEE Trans. Autom. Control 19, 716 (1974)ADSCrossRefGoogle Scholar
 81.G. Schwarz, Ann. Statist. 6, 461 (1978)ADSMathSciNetCrossRefGoogle Scholar
 82.K.P. Burnham, D.R. Anderson, Model selection and multimodel inference:a practical informationtheoretic approach, 2nd edn. (Springer, New York, 2002)zbMATHGoogle Scholar
 83.K.P. Burnhama, D.R. Anderson, Sociol. Meth. Res. 33, 261 (2004)CrossRefGoogle Scholar
 84.H. Jeffreys, Theory of Probability (1961)Google Scholar
 85.S. Nesseris, J. GarciaBellido, JCAP 1308, 036 (2013). https://doi.org/10.1088/14757516/2013/08/036 ADSCrossRefGoogle Scholar
 86.D.J. Spiegelhalter, N.G. Best, B.P. Carlin, A. van der Linde, J. R. Statist. Soc. B 64(4), 583 (2002). https://doi.org/10.1111/14679868.00353 CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}