Bayesian Comparison of Interacting Scenarios

Bayesian Comparison of Interacting Scenarios

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September 12, 2019

We perform a Bayesian model selection analysis for different classes of phenomenological coupled scenarios of dark matter and dark energy with linear and non-linear interacting terms. We use a combination of some of the latest cosmological data such as type Ia supernovae (SNe Ia), cosmic chronometers (CC), cosmic microwave background (CMB) and two sets of baryon acoustic oscillations measurements, namely, 2-dimensional angular measurements (BAO2) and 3-dimensional angle-averaged measurements (BAO3). We find weak and moderate evidence against two-thirds of the interacting scenarios considered with respect to CDM when the full joint analysis is considered. About one-third of the models provide a description to the data as good as the one provided by the standard model. Our results also indicate that either SNe Ia, CC or BAO2 data by themselves are not able to distinguish among interacting models or CDM but the standard BAO3 measurements and the combination with the CMB data are indeed able to discriminate among them. We find that evidence disfavoring interacting models is weaker when we use BAO2 (data claimed to be almost model-independent) instead of the standard BAO3 measurements. These results help select classes of viable and non-viable interacting models in light of current data.

a,b]Antonella Cid a]Beethoven Santos c]Cassio Pigozzo d]Tassia Ferreira a,e]Jailson Alcaniz


[a]Observatório Nacional, 20921-400, Rio de Janeiro, RJ, Brasil. \affiliation[b]Departamento de Física, Grupo Cosmología y Partículas Elementales, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile. \affiliation[c]Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brasil. \affiliation[d]PPGCosmo, CCE, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brasil. \affiliation[e]Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, RN, Brasil

\emailAdd \ \ \ \


Interacting models, bayesian comparison, cosmological parameters

1 Introduction

In the coming decade cosmological data from a number of planned galaxy surveys and cosmic microwave background experiments will be used to tackle fundamental questions about the nature of the physical make-up of the Universe. Currently, the standard picture used to describe the available observations is the CDM cosmology, a model in which most of the clustered matter is effectively collisionless (dark matter) with the bulk of the energy density of the Universe behaving like the vacuum energy, (dark energy). With only half a dozen parameters, this remarkably simple model is able to explain most of the different sets of observations spanning over a large range of length scales (for a recent review, see [1]).

From the theoretical point of view, however, it is also well known that in order to provide a good description of the observed Universe the value of the vacuum energy density, Gev, leads to an unsettled situation in the interface between Cosmology and Particle Physics, since it differs from theoretical expectations by 60-120 orders of magnitude [2]. Moreover, although the evolution of these two dark components over the cosmic time is significantly different, their current energy densities are of the same order, which gives rise to the question whether this is only a coincidence or has a more fundamental reason. Such questions are known as the cosmological constant problems [3]. Thus, given the theoretical uncertainties on the nature and behavior of the dark energy a number of mechanisms of cosmic acceleration have been investigated, including modifications of gravity on large scales or a possible interaction between the components of the dark sector.

In particular, interacting models of dark energy and dark matter [4, 5, 6, 7, 8] are based on the premise that no known symmetry in Nature prevents or suppresses a non-minimal coupling between these components and, therefore, such possibility must be investigated in light of observational data111A usual critique to these scenarios is that, in the absence of a natural guidance from fundamental physics, one needs to specify a phenomenological interaction term between the dark components in order to establish a model and study their observational consequences. (for a recent review, see [9]). In some classes of these coupled models the coincidence problem above mentioned can be largely alleviated when compared with the standard cosmology. This was firstly discussed in reference [5], where the authors investigated asymptotic attractor behaviors for the ratio of the dark matter and dark energy densities. Since then, a number of interacting models with both numerical and analytical solutions have been proposed (see, e.g. [10, 6, 11, 12, 7, 13, 8, 14, 15, 16] and references therein).

In this paper, we study the observational viability of different classes of interacting scenarios, including both linear and non-linear interaction terms. In order to observationally distinguish between the different models we perform a Bayesian model selection analysis using current data of type Ia supernova (SN Ia) [17, 18], measurements of the baryon acoustic oscillation (BAO) from the 6dFGS [19], SDSS-MGS [20], BOSS-LOWZ [21], BOSS-CMASS [21], BOSS-DR12 [22], eBOSS [23, 24] and BOSS-Ly [25] surveys, along with BAO measurements obtained from SDSS-DR7 [26], SDSS-DR10 [27], SDSS-DR11 [28], SDSS-DR12Q [29]; measurements of the expansion rate from cosmic chronometers [30] and the estimate of the sound horizon scale at the last scattering reported by the Planck collaboration [31]. We find weak and moderate evidence against some of the interacting scenarios studied with respect to CDM when the full joint analysis is considered. We also discuss how such results can help select viable classes of interacting models.

The paper is organized as follows: in section 2 we present the classes of interacting models studied in this work. In section 3 we present the datasets considered in the analysis. The Bayesian approach used to evaluate the performance of the different interacting scenarios is described in section 4. Section 5 is devoted to the results of the Bayesian comparison analysis. Finally, our results are summarized in section 6.

2 Interacting Cosmological Scenarios

Let us consider a homogeneous, isotropic and flat cosmological scenario described by the Friedmann-Lemaître-Robertson-Walker metric (FLRW) and assume that the total energy-momentum tensor for the matter content of the universe is conserved. In the most general case, the total matter density is composed of radiation (), baryons (), dark matter () and dark energy ().

Also, let us suppose that the dark components are allowed to interact between each other through a phenomenological interaction term :


where prime denotes a convenient derivative with respect to the function of the scale factor , is the barotropic index for dark energy and we adopt a pressureless dark matter component. Note that indicates an energy transfer from dark matter to dark energy and indicates the opposite. By adding equations (1) and taking we find:


and, by rearranging the terms, we obtain:


If we derivate any of equations (3) and replace (1) we can write a second order differential equation for a function  [6]:


where , , are constants, given in table 1, for the classes of interactions that are being considered in this work. The analytical solution of (4), describing the evolution of the dark sector [6], takes the following form:


where is the Hubble parameter and the constants , , and are given by:


Throughout this paper, we use the dimensionless density parameters with the subscript denoting their current values. It is worth emphasizing that the evolution of the dark sector is independent of that of radiation and baryons. Finally, the Hubble expansion rate in terms of the cosmological redshift can be written as:


with . Note that the radiation term includes the contribution of photons, , and neutrinos, .

Table 1: Definitions of the parameters , for interactions .

In reference [8] it was shown that the interacting models in table 1 can be described in a unified dark sector approach as a variable modified Chaplygin gas [32]. The authors point out that these scenarios can also be understood in terms of a varying barotropic index for the dark energy component. Furthermore, most of the interacting models shown in table 1 have been proposed in the literature in order to get a scaling solution for the coincidence problem, that is, in the context of these models the universe could approach a stationary stage at which the ratio between dark energy and dark matter densities becomes a constant. In particular, an interacting term of the type was first introduced in references [4, 5] from a study of a suitable coupling between a quintessence scalar field and a pressureless cold dark matter field. A generalization of this interaction was considered in reference [33] in order to overcome the coincidence problem near the transition time in a system that crosses the phantom divide line. On the other hand, the interaction was first presented in reference [6] as a convenient scenario for alleviating the coincidence problem, with an analytical solution for the dark sector. The interaction was firstly introduced in terms of a non-canonical scaling of the ratio of the dark matter and dark energy densities as an attempt to solve this same problem in reference [34]. This kind of interaction seems to be appealing since it is able to describe the large-scale evolution without instabilities or unphysical features [35]. Finally, interactions of the type and were studied in reference [12] in the realm of non-linear interactions alleviating the coincidence problem.

3 Data description

We focus on background data such as type Ia supernovae through the Joint Light-curve Analysis (JLA) compilation [17] and through the Pantheon sample [18]; baryon acoustic oscillation data from 6dFGS [19], SDSS-MGS [20], BOSS-LOWZ [21], BOSS-CMASS [21], BOSS-DR12 [22], eBOSS [23, 24] and BOSS-Ly [25]; the cosmic chronometers reported in reference [30] and the angular scale of the sound horizon at the last scattering [31]. Furthermore, we independently consider BAO measurements obtained by the angular separation between pairs of galaxies [27, 26, 28, 29]. In what follows, we briefly present each one of these datasets.

3.1 BAO data

In table 12 of Appendix A, the current available measurements of the BAO signal are shown. The isotropic BAO measurements are given in terms of the dimensionless ratio , where is a combination of the line-of-sight and transverse distance scales defined in reference [36],


is the speed of light, is the comoving angular diameter distance and the standard ruler length is usually interpreted as the comoving size of the sound horizon at the drag epoch where


with being the sound speed of the photon-baryon fluid,  [37] and the redshift at the drag epoch.

As we discuss in Appendix A, BAO measurements obtained through use a fiducial model in order to convert angles into distances. This fact motivates the use of a different set of BAO measurements (claimed to be almost model-independent), which considers only the transversal BAO signal through a geometrical feature such as the angular separation between pairs of galaxies,


As explained in reference [27], in order to estimate from (completely model-independent measurement of the BAO signal obtained from the 2-point angular correlation function) a shift factor is needed, given that the redshift shell-width, , is different from zero. In reference [27] the model-dependence of the shift factor is tested for several cosmologies and the overall conclusion is that the shift factor is almost model-independent, with the difference between and being for the considered .

On the other hand, the more recently available anisotropic BAO measurements (BOSS-DR12 and BOSS-Ly in table 12) consider observables in the transverse direction () as well as in the radial direction (), where both observables are correlated and together they provide complete information about the BAO signature at a given redshift. The corresponding covariance matrix for BOSS-DR12 and BOSS-Ly are given in references [22] and [25], respectively.

A relevant issue pointed out in the literature is the lack of informed correlations needed to use all the BAO data combined. In reference [20] the authors argue that SDDS-MGS [20] can be employed in combination with 6dFGS [19] and BOSS-LOWZ [21] because the overlapping volumes of the galaxy samples are small enough and consequently the correlations can be considered as negligible. Planck’s 2015 collaboration [38] uses these BAO data in addition to BOSS-CMASS [21] (uncorrelated) to constrain cosmological parameters. In this work we add to the four Planck’s 2015 BAO measurements the eBOSS measurement [23], which is uncorrelated with the others. From here on we refer to this set of isotropic BAO measurements as BAO3. Additionally we consider a second set of fourteen BAO measurements given by the transverse BAO signal through the angular feature (see table 2). From here on we refer to this set of angular BAO measurement as BAO2. For completeness, we also consider independently a third set of seven isotropic and anisotropic BAO measurements including updated data: 6dFGS [19], SDSS-MGS [20], BOSS-DR12 [22] replacing BOSS-LOWZ and BOSS-CMASS, eBOSS from reference [24] and BOSS-Ly [25], we show the results of this last analysis in the Appendix B.

Data from the Sloan Digital Sky Survey (SDSS) were employed in obtaining BAO2 measurements (see table 2) as well as some of the BAO3 data points, such as SDSS-MGS, BOSS-LOWZ, BOSS-CMASS, eBOSS. Furthermore, although WiggleZ is an independent survey, in reference [39] the authors pointed out that the volumes considered in the WiggleZ survey and the BOSS-CMASS partially overlap and, as such, they calculated the corresponding correlations. Due to this fact, in this work we assume a conservative approach and do not combine BAO3 and BAO2 measurements.

Reference Reference
0.235 [26] SDSS-DR7 0.550 [27] SDSS-DR10
0.365 [26] SDSS-DR7 0.570 [28] SDSS-DR11
0.450 [27] SDSS-DR10 0.590 [28] SDSS-DR11
0.470 [27] SDSS-DR10 0.610 [28] SDSS-DR11
0.490 [27] SDSS-DR10 0.630 [28] SDSS-DR11
0.510 [27] SDSS-DR10 0.650 [28] SDSS-DR11
0.530 [27] SDSS-DR10 2.225 [29] SDSS-DR12Q
Table 2: BAO measurements from angular separation of pairs of galaxies.

In the CDM scenario the standard ruler coincides with the sound horizon at the drag epoch, which can be determined in a model dependent way from CMB measurements. Nevertheless, in general the two quantities need not coincide [40] and some attempts in estimating a model independent low-redshift standard ruler has been made [41, 40]. On the other hand, it has been shown that the tension in could reflect a mismatch between the determination of the standard ruler for the acoustic scale and its standard value [42]. Given this, in this work we consider two different approaches in using the two different BAO datasets, the first one is to calculate the comoving size of the sound horizon (10) taking the redshift at the drag epoch given by , in accordance with Planck’s 2015 results [38]. The second approach is to consider the standard ruler for the acoustic scale as a free parameter, where we have chosen as prior the value reported in reference [40] (see table 5) instead of Planck’s results for the sound horizon since the former was obtained model-independently, just assuming the cosmological principle, a metric theory of gravity and a smooth expansion history without a fiducial cosmology at low redshift. This kind of methodology has been used before in different contexts, e.g., [43, 44].

3.2 CMB data

To perform the full cosmic microwave background analysis building for the entire range of multipoles would require the study of all interacting models at a perturbative level, and also require the adaptation of Boltzmann codes such as CAMB or CLASS to obtain their anisotropy power spectrum. A simpler way to do that is to use the CMB compressed likelihood.

The compressed likelihood derived from Planck 2015 chains is composed by the set of parameters [31]. The shift parameter is, by construction, very dependent on the matter dominated epoch. This is not evident in Planck’s analysis since all models studied have approximately the same behavior in that period. However, when we are dealing with interacting models, the dynamics of the matter dominated epoch is affected, so that constraining to the values of reference [31] would include a bias on the analyses, and thereby artificially push the results to resemble the model with no interaction [31, 45]. In order to avoid this, we do not consider the shift parameter in our analysis. Besides, we do not use the spectral index since it does not appear explicitly in our analysis and we fix the physical baryon density to , as reported in [46].

Consequently, this means that the only contribution of CMB data we consider is the angular scale of the sound horizon at the last scattering:


where the comoving size of the sound horizon is evaluated at the redshift of last scattering . We compare the value obtained in our study to the one reported by the Planck collaboration in 2015, [31]. In order to elucidate if there is a noticeable change in our results in fixing or estimating we use two different methods, the first one considers , according with Planck’s 2015 results [38]. The second one contemplates the computation of the redshift at drag epoch from the following expression [47],


where we have considered and


This expression has been used before in studying interacting scenarios (see e.g. [8, 48]). Notice that (13) is weakly dependent on the dark matter contribution, the difference with Planck’s 2015 result for is lower than 1 for and .

3.3 Cosmic chronometers

All data from cosmic chronometers used here were obtained through the differential age method (see table 3). We were careful to use only these measurements, and not to include those obtained with BAO so as not to double count information in the joint analyses.

The procedure consists of taking the relative age of passively evolving galaxies, with respect to the redshift, as suggested by reference [49]. Most of the values are obtained from the BC03 catalogue [50], but values from older releases were also used [51]. With the ratio between the differential ages, and the respective difference in redshift, , obtaining becomes a simple task:


In our analysis, the theoretical value of is given by equation (8). Note that we only use data of redshift up to . This is motivated by a discussion in reference [52], where the authors argue that the expansion history data of the universe might not be necessarily smooth outside . Likewise, the authors of reference [53] have also shown that outside this range, the model of synthesis of stellar population adopted to derive the galaxy ages becomes relevant.

[km/s/Mpc] Ref. [km/s/Mpc] Ref. [km/s/Mpc] Ref.
[54] [54] [55]
[51] [53] [53]
[54] [30] [53]
[51] [51] [53]
[53] [30] [53]
[53] [30] [55]
[54] [30] [51]
[51] [30] [53]
Table 3: Estimated values of obtained using the differential age method.

Notice that the cosmic chronometer is the only method providing cosmology-independent, direct measurements of the expansion history of the universe [52].

3.4 Supernovae Ia

The best probe of the expansion history of the Universe on large-scales (up to ) is provided nowadays by observations of type Ia supernovae. The main reason for this is that SNe Ia are examples of "standardisable candles", due to the fact that their absolute magnitudes can be approximated by using light-curve templates to extract their stretch and color parameters. We use the JLA sample which contains a set of 740 spectroscopically confirmed SNe Ia [17] composed by several low-redshift () samples, the full three-year SDSS-II supernova survey sample [56] in the interval , the three-years data of the SNLS survey [57, 58] up to redshift and some high-redshift Hubble Space Telescope SNe [59] with redshift .

The predicted apparent magnitude of a SN Ia can be obtained from its light curve parameters through the linear relation:


where represents the set of parameters of the model, is the time stretching of the light curve and is the supernova color at its maximum brightness. In the expression above, is the theoretical distance modulus, given by:


where is the luminosity distance:


and .

In equation (16), the light-curve parameters and have different values for each supernova and are derived directly from the light-curves. However, the nuisance parameters , and are assumed to be constant for all the supernovae but differ for each cosmological model. Additionally, since the properties of the host galaxy can generate some effects on the intrinsic brightness of the SNe Ia, we follow reference [17] and model the relation between and the host galaxy stellar mass, , by assuming if . Thus, the nuisance parameters corresponding to the measurements are , , and .

The analyses involving the JLA dataset were carried out by comparing the predicted magnitude from equation (16) against the observed ones of the JLA sample (table F.3 of reference [17]), which are denoted by and represent the observed peak magnitude in rest-frame band. The Monte Carlo analyses for the JLA SNe Ia sample were performed by assuming a multivariate Gaussian likelihood of the form




where corresponds to the covariance matrix of the measurements, estimated accounting for various statistical and systematic uncertainties (we refer the reader to reference [17] for more information about these uncertainties).

Since the Pantheon supernovae sample [18] came to light during the development of this work, for completeness we include in Appendix B a brief description of this dataset and the results when replacing the JLA compilation with the Pantheon compilation.

3.5 Joint analysis

By using the same methodology as in the case of the JLA SNe compilation, we consider a multivariate Gaussian likelihood for BAO2, BAO3, CC and CMB data. The chi-square function for the measurement of a generic function is defined as follows:


where represents the measured value for at redshift , whereas is computed assuming a model with parameters . The function stands for the functions , , and for BAO2, BAO3, CC and CMB data, respectively. The sum in equation (21) runs over the data in table 2 for BAO2 and table 3 for CC. For BAO3 we consider 6dFGS, SDSS-MGS, BOSS-LOWZ, BOSS-CMASS and eBOSS as described in section 3.1. Independently, in the Appendix B we consider the following set of updated BAO measurements: 6dFGS, SDSS-MGS, BOSS-DR12, eBOSS and BOSS-Ly. For CMB, we take just a single data at . In the case of the joint analysis, the total likelihood is obtained as the product of individual likelihoods associated to each data as in equation (19) by using the chi-square definition in equation (21). For example, the full joint analysis considering BAO2 data is given by: .

4 Methodology: Bayesian model selection

The Bayesian inference method constitutes a robust statistical technique for parameter estimation and model selection, and over the last years has been widely used in the study of cosmological scenarios [48, 60, 61, 62, 63]. Bayesian inference is based on the Bayes’ theorem, which updates our knowledge of a given model (or hypothesis) in light of new available data (or information). Mathematically, the Bayes’ theorem gives us the posterior probability for a set of parameters , given the data , described by a model ,


where , and stand for the likelihood, prior and evidence, respectively.

The evidence in equation (22) constitutes just a normalization constant in the Bayesian parameter estimation approach, however, it becomes a key ingredient in the Bayesian model comparison approach. In order to compare the performance of different models given a set of data, we use the Bayes’ factor defined in terms of the evidence of models and as:


where the evidence corresponds to the average value of the likelihood over the entire model parameter space allowed, before we observe the new data [64], that is:


If the models and have the same prior probability, then the Bayes’ factor gives the posterior odds of the two models.

Monte Carlo sampling techniques are widely applied nowadays to construct the posterior distribution in equation (22), since it is very difficult to compute the posterior numerically (see references [65, 66] for applications of some Monte Carlo algorithms in cosmology). In this sense, we performed the analyses involving the data described in section 3 by applying the nested sampling (NS) Monte Carlo algorithm [67], which is well known for its efficiency in the evidence computation since it is designed to directly estimate the relation between the likelihood function and the prior mass, thus obtaining the evidence (and its uncertainty) immediately by summation, while also computing the samples from the posterior distribution as an optional by-product. To compute the evidence values and generate the posterior distributions we used the MultiNest222 [68, 69, 70] algorithm, requiring a global log-evidence tolerance of 0.01 as a convergence criterion and working with a set of 1000 live points to improve the accuracy in the estimate of the evidence. With this number of live points, the number of samples for all posterior distributions was of order .

The Jeffreys’ scale [71] gives us an empirical measure for interpreting the strength of the evidence in comparing two competing models. In order to perform model comparison in this work we use a conservative version of the Jeffreys’ scale defined in reference [72] (see table 4). Usually for the evidence in favor/against model relative to model is interpreted as inconclusive. On the other hand, the thresholds shown in table 4 for weak, moderate and strong evidences in favor/against the tested model correspond to posterior odds of about 3:1, 12:1 and 150:1, respectively [72]. Here, we take CDM as the reference model , as such, the subscripts in the Bayes’ factor (23) will be omitted hereafter. Note that from now on, means support in favor of the CDM model.

Table 4: The Jeffreys’ scale, empirical measure for interpreting the evidence in comparing two models and as presented in reference [72]. The left column indicates the threshold for the logarithm of the Bayes factor and the right column the interpretation for the strength of the evidence above the corresponding threshold.

In this work, we consider two different approaches in using the data, where the goal is to elucidate if there is a noticeable impact in our final results by considering different numbers of free parameters and different kinds of priors for some specific parameters. In our first approach, we use the Planck’s values [38] for the redshift at the drag epoch and last scattering, and , respectively. The priors considered here are shown in table 5. We have chosen a uniform prior for the parameters appearing in all the studied models such as , , , , and , and a Gaussian prior for the parameters defining only some of the models, i.e., and the interacting parameters and . For the parameter we choose a conservative uniform prior between 0 and 1, and for the dimensionless Hubble parameter we adopt a range 10 times wider than the 1 value reported by Riess et al. in reference [73]; we emphasize that Riess’ result is considered only through this prior and not used as an independent data in the analysis. For the JLA parameters, , , and , we use a range 20 times wider than the values reported by Betoule et al. in reference [17]. The prior for the parameter corresponds to the 1 value informed by Planck [38] and the prior for the interacting parameters is the same for all the studied models, of order with a negative mean (see reference [8]), in such a way that most of the models in table 1, , , favor a transfer from dark energy to dark matter provided that and are positive defined. This kind of transfer is expected for some interacting models based on thermodynamical arguments [74].

In the second approach we change the prior for the parameter to a Gaussian prior (or equivalently we consider the value reported by Riess et al. in reference [73] as a data), we compute from (13) instead of fixing it and we consider a free parameter with a Gaussian prior as in reference [40].

In a nutshell, we study the following scenarios as two independent sets of priors (see table 5) and definitions,


Both scenarios consider uniform priors for the remaining parameters defined in table 5.

Given that we expect interacting models not to affect the physics of the primordial universe but only to modify the evolution of the dark sector recently, we fix the following parameters: [46], [75], , [76].

Parameter Status Prior Ref.
Global Parameter Uniform: -
Global Parameter Uniform: [73]
Gaussian: [73]
Variable state parameter Gaussian: [38]
Interacting models Gaussian: [8]
Interacting models Gaussian: [8]
Global, JLA parameter Uniform: [17]
Global, JLA parameter Uniform: [17]
Global, JLA parameter Uniform: [17]
Global, JLA parameter Uniform: [17]
Global Parameter Gaussian: [40]
Table 5: Priors on the free parameters of the studied models. For a Gaussian prior we inform and for a Uniform prior represent .

5 Analysis and Results

We perform a Bayesian comparison analysis of the interacting models presented in table 1 in terms of the strength of the evidence according to Jeffreys’ scale (see table 4). In this study we used the priors shown in table 5 and considered different combinations of background data such as, type Ia supernovae, cosmic chronometers, baryon acoustic oscillations and cosmic microwave background. Our main results are summarized in tables 611. We labeled different realizations of models in table 1 with numerical subscripts as follows: 0, 1, 2, 3, 4 meaning , , , , , respectively. We do not include the analysis for the interacting model because it reduces to the CDM scenario.

In tables 69, the results were obtained using the priors given by (25), while tables 1011 the priors used were those of (26). In tables 811 we have not reported the nuisance parameters because the variation in the best fit estimation among different models is negligible (see Appendix C).

In table 6 we observe that by considering JLA, BAO2 or CC by themselves or even the joint analysis with BAO2 + CC, we get inconclusive evidence for the interacting models and for the CDM model when compared to CDM. We remark that the BAO2 and CC data that were used are almost model-independent and model-independent, respectively, and these alone or jointly are not able to rule out any of the models considered in this work compared to CDM. We also observe that by analyzing BAO2 + CC + CMB we find weak or moderate evidence disfavoring some of the interacting models.

In table 7 we show the logarithm of the Bayesian evidence and the interpretation of the strength of the evidence for the analysis with BAO3, BAO3 + CC and BAO3 + CC + CMB. We note that some of the interacting models are disfavored with a weak evidence in the studies with BAO3 and BAO3 + CC, and these evidences become moderate (in most of the cases) when we add CMB data to the analysis. Moreover, we notice that the models presenting inconclusive evidence in this analysis are the same as in the study of BAO2 + CC + CMB, which seems to indicate that the addition of the CMB data to the analysis, even when it is a single data point, contributes significantly to the evidence disfavoring most of the interacting models. A different example of the importance in considering CMB measurements is studied in reference [77], where the authors show that a power law scenario is supported by SNe Ia alone or combined with BAO data, nevertheless the addition of a single CMB measurement to the joint analysis rules out this scenario.

In tables 89 the results for the full joint analysis, including SNe Ia from JLA compilation, are shown considering independently the measurements of BAO2 and BAO3, respectively. We present the best fit parameters for the studied models, along with the logarithm of the Bayesian evidence, the logarithm of the Bayes’ factor and the interpretation for the strength of the evidence. It is interesting to note that, for the results in table 8 as well as the results in table 9, the evidence remains inconclusive for the models CDM, , , , , , and when compared to CDM. These models are the same as those presenting inconclusive evidence in the analyses BAO2 + CC + CMB and BAO3 + CC + CMB. Observe that the inconclusive models correspond to or and the energy transfer for these models turns out to be from dark energy to dark matter. In this context, in the recent work [16] interacting scenarios and were studied, performing a joint analysis with the full CMB temperature anisotropy spectrum, JLA and BAO3. The authors found, using theoretical arguments and observational results, that while models with are virtually discarded, there is still room for models with .

From tables 89 we notice that the moderated evidence found in the analysis BAO2 + CC + CMB becomes weak when we consider the full analysis including JLA. On the other hand, the weak evidences obtained in the analysis with JLA + BAO2 + CC + CMB become all moderate when we take BAO3, instead of BAO2 in the full joint analysis. This result is expected since BAO3 data implicitly seem to favor the CDM scenario. It is worth pointing out here that the value of the Hubble parameter turns out to be higher for all the studied scenarios when we consider BAO2 instead of BAO3 in the full joint analysis (see tables 8 and 9), this seems to indicate that the BAO data play an important role in the tension [78], we also observe that the values of the fitted barotropic index and the interacting parameters are approximately one order of magnitude lower in the full joint analysis including BAO3 compared to the analysis with BAO2, which reinforce the indication for a preference of CDM in BAO3 data.

In tables 1011 the analogous results, to those in tables 89, for the full joint analysis with the JLA compilation are shown considering independently the measurements of BAO2 and BAO3, respectively. The differences between tables 810 and 911 rely on considering scenario (25) instead of scenario (26) for the priors. We observe that in tables 1011, as in tables 89, the evidence remains inconclusive for the models CDM, , , , , , and when compared to CDM. However, from tables 1011 it is not so clear that BAO3 data favor the CDM scenario, a similar behavior is observed in tables 1314 in Appendix B where, instead of the JLA, we use the most up to date SNe Ia sample (Pantheon).

In tables 10 and 11 we notice that in the best fit estimation for the parameter, the error is in general of order 1-2 and the dispersion (comparing higher to lower estimation for all the models) is always lower than 1. The parameter presents a similar behavior, the error is around 1 and the dispersion around . Nevertheless, when comparing the best fit for between tables 10 and 11 the difference is around 1, while for the best fit is close to . From this, we conclude that in comparing CDM with interacting models there is no prominent difference in the best fit estimation for the parameters or and while there is no distinctive variation in the estimation in using the full analysis with BAO2 or BAO3, the standard ruler becomes closer to the prior (and also closer to the Planck’s 2015 estimation [38]) when using BAO3 data. An analogous behavior is observed from tables 1314 in the Appendix B. Also, there is no noticeable difference in the order of magnitude in estimating parameters , or (see tables 1011 and 1314).

For completeness, in table 15 of Appendix B we have included the full joint analysis with the set of priors (26) and an updated set of BAO measurements, including data from: 6dFGS [19], SDSS-MGS [20], BOSS-DR12 [22], eBOSS [24] and BOSS-Ly [25]. The differences in parameter estimation between tables 14 and 15 are negligible, the major change is presented in the interpretation of the strength of the evidence, in those cases where the evidence is weak in table 14 it becomes moderate in table 15, in both cases the evidence is supporting CDM.

In reference [8] some of the models studied in this work were analyzed in a joint analysis considering Union2.1 + BAO3 + H(z) + CMB, and a comparison among the models was performed using the Akaike and Bayesian information criteria (AIC and BIC, respectively). At this point, it is worth mentioning that, differently from these methods, the Bayesian model comparison applied in this analysis selects the best-fit model by comparing the compromise between quality of the fit and predictability, and by evaluating if the extra degrees of freedom of a given model are indeed required by the data, preferring the model that describes the data well over a large fraction of their prior volume. The results of [8] in terms of BIC indicate that the interacting models have "strong evidence against" when compared to CDM and this strength of evidence changes to "evidence against" (or moderate) when compared to CDM but using binned JLA data instead of Union2.1. Besides, the BAO measurements used included WiggleZ data, but the full JLA dataset was not considered.

The authors also compared the interacting models with the CDM model obtaining "not enough evidence against" or inconclusive evidence for the interacting models with three free parameters analyzed (, , , , , and in this work). In our study, the evidence interpretation remains the same in most of the models when we compare them to CDM instead of CDM. Finally, the ordering of the evidence in terms of the number of free parameters for the interacting models reported in reference [8] with the BIC approach is not observed in our work using a full Bayesian comparison approach.

In reference [48] the model was analyzed in a model comparison approach with JLA data and a different set of BAO3 measurements. The authors found moderate evidence in favor of this model compared to CDM. Instead, in this work we found moderate evidence disfavoring the model. The main differences between our datasets and the ones used in reference [48] are that (i) we are considering CMB data and high redshift values of BAO2 and BAO3 data that have become available only recently. Furthermore, the authors of reference [48] used WiggleZ data, which is not considered in the set of BAO3 data used in this work. Besides, the authors use the Eisenstein’s approximation for the redshift at the drag epoch, whereas here we use the Planck’s value in scenario (25) or as a free parameter in scenario (26).

In figure 1 we summarize our results in terms of the interpretation of the Bayes’ factor considering the Jeffreys’ scale. We show the Bayes’ factor for the full joint analysis of the several realizations of interacting models in table 1 compared to CDM. We consider independently the full joint analysis using BAO2 and BAO3 data. We see that using BAO3 instead of BAO2 shift the Bayes factor towards a better support for the CDM model in most of the cases. In figures 2 and 3 we show examples of contour plots and PDFs in our analysis (models and ), for scenarios (25) and (26) as priors, respectively. The figures show the differences in parameter estimation considering BAO2 or BAO3 in the full joint analysis. In figure 2 we notice a tension in the parameter estimation using BAO2 or BAO3, this tension is slightly released when we enlarge the parameter space by adding the barotropic index to the analysis. In figure 3 we find that the tension between the parameters estimated is transfered to the parameter. The behavior observed in figures 2 and 3 is analogous for all the studied scenarios, and therefore other contour plots are not shown for brevity.

In short, our results indicate that the inconclusive evidences obtained for scenarios , , , , , and in comparison to CDM are maintained in the full joint analysis, this is independent of the chosen priors (scenarios (25) or (26)) or the JLA-Pantheon sample interchange. Nonetheless, in considering as a free parameter and applying a Gaussian prior to ( in scenario (26) is not relevant in this statement, see Appendix B ) it is not so clear that BAO3 data is favoring the CDM scenario or that BAO2 data is alleviating the tension as it was when considering the scenario (25) as priors. Our results seem to indicate that the BAO3 model-dependency is partially contained in the estimation of as the sound horizon at the drag epoch through (10).

Model Interpretation
CDM Inconclusive
Table 6: Bayesian evidence () for the models studied in this work. For the analysis with BAO2, CC, JLA and BAO2+CC the evidence is inconclusive for all cases. As for the joint analysis BAO2+CC+CMB, note that favors CDM. These results consider as priors scenario (25).
Dataset BAO3 BAO3 + CC BAO3 + CC + CMB
Model Interpretation Interpretation Interpretation
CDM - - -
CDM Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Weak Weak Moderate
Weak Weak Moderate
Weak Weak Moderate
Weak Weak Weak
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Weak Weak Moderate
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Weak Weak Moderate
Inconclusive Inconclusive Inconclusive
Table 7: Bayesian evidence () and interpretation for the models considered in this work. Note that favors the CDM model. These results consider as priors scenario (25).
Model Interpretation
CDM - - - 0 -
CDM - - Inconclusive
- - Weak
- - Inconclusive
- Weak
- Weak
- - Weak
- Weak
- Weak
- - Inconclusive
- - Weak
- - Inconclusive
- Weak
- Inconclusive
- Weak
- Inconclusive