Issue 
A&A
Volume 668, December 2022



Article Number  A135  
Number of page(s)  11  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202244366  
Published online  13 December 2022 
Constraining the dark energy models using baryon acoustic oscillations: An approach independent of H_{0} ⋅ r_{d}
^{1}
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria
email: dstaicova@inrne.bas.bg
^{2}
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
^{3}
Kavli Institute of Cosmology (KICC), University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
Received:
27
June
2022
Accepted:
12
September
2022
The H_{0} tension and the accompanying r_{d} tension are a hot topic in current cosmology. In order to remove the degeneracy between the Hubble parameter H_{0} and the sound horizon scale r_{d} from the baryon acoustic oscillation (BAO) datasets, we redefined the likelihood by marginalizing over the H_{0} ⋅ r_{d} parameter and then we performed a full Bayesian analysis for different models of dark energy (DE). We find that our datasets that are uncalibrated by early or late physics cannot constrain the DE models properly without further assumptions. By adding the type Ia supernova (SNIa) dataset, the models are constrained better with smaller errors on the DE parameters. The two BAO datasets we used – one with angular measurements and one with angular and radial ones, with their covariance – show statistical preferences for different models, with the Λ cold dark matter (ΛCDM) model being the best model for one of them. Adding the Pantheon SNIa dataset with its covariance matrix boosts the statistical preference for the ΛCDM model.
Key words: cosmological parameters / methods: data analysis / astronomical databases: miscellaneous
© The Authors 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
A turning point in modern cosmology is the measurement of the Hubble constant H_{0}, revealing the current accelerated expansion of the Universe (Riess et al. 1998; Freedman & Madore 2010). The estimation of H_{0} from the late Universe can be obtained from direct measurements such as distance ladders, strong lensing, and gravitational wave standard sirens (Freedman et al. 2001; Perlmutter et al. 1999; Riess et al. 2016, 2021). The latest SH0ES measurement based on the supernovae calibrated by Cepheids is H_{0} = 73.04 ± 1.04 km s^{−1} Mpc^{−1} at a confidence level of 68% (Riess et al. 2022b). Further improvement comes from a SH0ES measurement of the distance ladder calibrated by parallaxes of Cepheids in open clusters, which combined with all anchors yields H_{0} = 73.01 ± 0.99 km s^{−1} Mpc^{−1}Riess et al. (2022a).
Another type of measurement is provided by the Planck collaboration, which uses temperature and polarization anisotropies in the cosmic microwave background (CMB) to obtain H_{0} = 67.27 ± 0.6 km s^{−1} Mpc^{−1}. The discrepancy between local modelindependent measurements of H_{0} and the early Universe CMB values can reach 5.3σ and it is one of the fundamental problems in cosmology (Schöneberg et al. 2019; Di Valentino 2017; Di Valentino et al. 2021a,b; Perivolaropoulos & Skara 2022; Lucca 2021; Verde et al. 2019; Knox & Millea 2020; Jedamzik et al. 2021; Shah et al. 2021; Abdalla et al. 2022).
Baryon acoustic oscillations (BAOs) are sound waves in the baryonphoton plasma comprising the visible matter in the postinflationary Universe, which froze at a recombination epoch. Today, they are observed in the clustering of largescale structures by numerous galactic surveys (such as SDSS, DES, WiggleZ, BOSS). Due to rather simple physics of the plasma waves, BAOs can be considered as a standard ruler evolving with the Universe, thus providing another window into studying cosmological models (Dunkley et al. 2011; Addison et al. 2013; Aubourg & Bailey 2015; Cuesta et al. 2015; Ade et al. 2014a,b, 2016; Story et al. 2015; Alam et al. 2017; Troxel et al. 2018; Planck Collaboration VI 2020; Cuceu et al. 2019; Dainotti et al. 2021). A scale important for BAO measurements is set by the sound horizon at drag epoch. As it is known, at recombination epoch, the photons decouple from the baryons first, at z_{*} ≈ 1090, which gives rise to the CMB. The baryons stop feeling the drag of photons at the drag epoch, z_{d} ≈ 1059, which sets the standard ruler for the BAOs. The Planck Collaboration value of the sound horizon is Planck Collaboration VI (2020), and the latetime estimation for it is Arendse et al. (2020). Other estimations give numbers in this range, depending on the datasets in use, for example (e.g., Verde et al. 2017; Planck Collaboration VI 2020; Alam et al. 2021; Nunes & Bernui 2020; Nunes et al. 2020).
Many papers discuss the relation between H_{0} and the sound horizon scale r_{d} for different models (Aylor et al. 2019; Knox & Millea 2020; Pogosian et al. 2020; Aizpuru et al. 2021). Some claim that resolving the H_{0} tension is not enough, since one has to also take into account the model’s effect on the sound horizon. This means that one should rule out models that resolve the H_{0} tension without resolving the r_{d} tension simultaneously (Jedamzik et al. 2021; Aizpuru et al. 2021; de la Macorra et al. 2022). Since H_{0} and r_{d} are strongly connected, it seems hard to disentangle them without making any assumptions. In order to have an independent crosscheck on dark energy (DE) models’ constraints, we removed the dependence on H_{0} ⋅ r_{d} by marginalizing over it using a χ^{2} redefinition. Such an approach has already been used to different extents in the literature. In Lazkoz et al. (2005), it was performed on the type Ia supernovae (SNIa) Gold dataset to compare different parametrizations of H(z). Basilakos & Nesseris (2016) studied the growth index by comparing the Λ cold dark matter (ΛCDM) model to several DE models by marginalizing over M_{B} and σ_{8}. Anagnostopoulos & Basilakos (2018) studied different cosmological models by marginalizing over H_{0} and find that one cannot rule out nonflat models or dynamical DE. They observe that the timevarying equation of state parameter w(z) cannot be constrained by the current expansion data. Finally, Camarena & Marra (2021) used marginalization over H_{0} and M_{B} in different datasets to show that a hockeystick DE cannot solve the H_{0} tension.
One possible way to resolve the tension is by changing the DE model. The question whether the DE is a constant energy density or with a dynamical behavior has been studied in different works (Benisty et al. 2021; Capozziello & De Laurentis 2011; Bull et al. 2016; Di Valentino et al. 2021a; Yang et al. 2021). This motivates a host of DE parametrizations (Wang et al. 2018; Reyes & EscamillaRivera 2021; Colgáin et al. 2021; Liu et al. 2022) to be used in the search for deviations from the cosmological constant, Λ, in observational data. A justification for this can be found in numerous papers claiming that DE may resolve the Hubble tension, particularly for the early DE models (Gogoi et al. 2021; Poulin et al. 2019; Sakstein & Trodden 2020; Tian & Zhu 2021; Nojiri et al. 2021; Seto & Toda 2021; Hill et al. 2022).
In this work, we used two types of BAO datasets and we combined them with the Pantheon SNIa dataset. Then we marginalized over H_{0} ⋅ r_{d} and H_{0} and M_{B}, respectively. This allowed us to remove the need to take priors on these quantities, and thus it removed some of the implied assumptions on the models. Using this method, we studied ΛCDM, wCDM, the Chevallier–Polarski–Linder (CPL) parametrization of wwaCDM, and also two emergent DE models: pEDE and gEDE. We show that even with this more extensive marginalization, one can see differences in the predictions of the different models inferred from the different datasets. The latter is particularly interesting in view of the growing sensitivity toward the implied assumptions in processing the data. We then performed a statistical analysis on the obtained results using four wellestablished measures. We confirm that constraining w_{a} seems impossible from this method, while the errors on w_{0} improve significantly when we add SNIa. Surprisingly, the different BAO datasets show different preferences for the flatness of the universe.
The plan of the work is as follows: Section 2 formulates the relevant theory. Section 3 describes the method. Section 4 shows the results with a model comparison. Finally, Sect. 5 summarizes the results.
2. Theory
A Friedmann–Lemaître–Robertson–Walker metric with the scale parameter a = 1/(1 + z) is considered, where z is the redshift. The evolution of the Universe for it is governed by the Friedmann equation, which connects the equation of state for the ΛCDM background:
with the expansion of the Universe E(z)^{2} = H(z)/H_{0}, where H(z): = ȧ/a is the Hubble parameter at redshift z and H_{0} is the Hubble parameter today. Ω_{m}, Ω_{Λ}, and Ω_{K} are the fractional densities of matter, DE, and the spatial curvature at redshift z = 0. We ignored radiation, since we are looking at the late Universe. The spatial curvature is expected to be zero for a flat Universe, Ω_{K} = 0. We can expand this simple model by considering a DE component depending on z. This can be done with a generalization of the CPL parametrization (Chevallier & Polarski 2001; Linder 2003; Linder & Huterer 2005; Barger et al. 2006) of the wwaCDM model:
in which we considered three possible models:
which recover the ΛCDM for w_{0} = −1, w_{a} = 0.
To this parametrization, we added another model, namely the phenomenologically Emergent Dark Energy (pEDE) model (Li & Shafieloo 2019, 2020) and its generalization (gEDE). gEDE is described by:
with pEDECDM recovered for , and ΛCDM for . The parameter z_{t} here is the transitional redshift, where Ω_{DE}(z_{t}) = Ω_{m}(1 + z_{t})^{3}. It should be noted that z_{t} is obtained as a solution of this equation, and thus it is not a free parameter, but a calculated one The analytical form of w(z) could then be obtained from the integral Eq. (2), see Li & Shafieloo (2020).
The BAO measurements provide different directions. The radial projection D_{H}(z) = c/H(z) gives:
which includes the parameter . The tangential BAO measurements are given in terms of the angular diameter distance D_{A}:
where sinn(x)≡sin(x), x, sinh(x) for Ω_{K} < 0, Ω_{K} = 0, Ω_{K} > 0 respectively. The Γ function is defined as:
where E(z) is related to the equation of state of the Universe as defined above. Thus, the measurement D_{A}/r_{d} can expressed as:
where:
A related quantity used in the radial BAO measurements is the comoving angular diameter distance D_{M} = D_{A}(1 + z).
Furthermore, we used the dataset featuring the BAO angular scale measurement θ_{BAO}(z). It gives the angular diameter distance D_{A} at the redshift z:
with:
We see that both D_{A}/r_{d} and θ_{BAO} and D_{H}/r_{d} depend on the quantity H_{0} ⋅ r_{d}, which can be eliminated from the corresponding χ^{2}, as we demonstrate in the next section.
Finally, we added the SNIa measurements, described by the luminosity distance μ(z). It is related to the Hubble parameter through the angular diameter distance as D_{A} = d_{L}(z)/(1 + z)^{2}. For the SNIa standard candles, the distance modulus μ(z) is related to the luminosity distance through
where d_{L} is measured in units of Mpc, and M_{B} is the absolute magnitude. There is a degeneracy between H_{0} and M_{B}, in such a way that total absolute magnitude reads: . This degeneracy, can also be used to remove the dependence on H_{0} and M_{B} in the χ^{2}.
3. Method
In order to infer the parameters of a certain model from the observations, one needs to define the appropriate χ^{2}. The goal of our analysis is to redefine the corresponding χ^{2} in all datasets, in a way that eliminates the dependence on degenerate parameters, such as H_{0} ⋅ r_{d} (or H_{0} and M_{B} for SNIa), but maintains the dependence on the equation of state that enters into Γ(z).
3.1. BAO redefinition
A DE model includes nfree parameters (i.e., Ω_{m}, Ω_{K}, w_{0}, w_{a}…), constrained by minimizing the χ^{2}:
where v_{obs} is a vector of the observed points at each z (i.e., D_{M}/r_{d}, D_{H}/r_{d}, D_{A}/r_{d} or θ_{BAO}) and v_{model} is the theoretical prediction of the model. It is possible to rewrite the vector as the dimensionless function multiplied by the parameter:
C_{ij} is the covariance matrix. For uncorrelated points the covariance matrix is a diagonal matrix, and its elements are the inverse errors . The statistics of the BAO are not fully Gaussian but we consider this as an approximation. Following the approach in Lazkoz et al. (2005), Basilakos & Nesseris (2016), Anagnostopoulos & Basilakos (2018), Camarena & Marra (2021), one can isolate in the χ^{2} by writing it as:
where:
Using Bayes’s theorem and marginalizing over c/(H_{0}r_{d}), we arrive at:
where D is the data we used, and the M is the model. Consequently, using , we get the marginalized χ^{2}:
This last equation is the final form of χ^{2}. Due to the marginalization procedure, this χ^{2} depends only on f(z) and h(z), which do not include H_{0} and r_{d} inside.
3.2. θ_{BAO} data
We used the same approach for the θ_{BAO}(z) measurements:
where and σ_{i} are the observational data and the corresponding uncertainties at the observed redshift z_{i}. The reconstructed , then, is the following:
where:
Using Bayes’s theorem and marginalizing over H_{0}r_{d}/c, we arrived at the marginalized χ^{2}, which is the same as in Eq. (17), only with A, B, and C now as functions of θ. This also depends only on h(z), without any dependence on H_{0} ⋅ r_{d}/c.
3.3. Supernova redefinition
Following the approach used in Di Pietro & Claeskens (2003), Nesseris & Perivolaropoulos (2004), Perivolaropoulos (2005), Lazkoz et al. (2005), we assumed no prior constraint on M_{B}, which is just some constant, and we integrated the probabilities over M_{B}. The integrated χ^{2} yields:
where:
Here μ^{i} is the observed luminosity, σ_{i} is its error, and the d_{L}(z) is the luminosity distance. The values of M and H_{0} don’t change the marginalized . In order to use the covariance matrix provided for the Pantheon dataset, one needs to transform D, E, F as follows:
where Δμ = μ^{i} − 5 log_{10}(d_{L}(z_{i})), E is the unit matrix, and is the inverse covariance matrix of the dataset. The total covariance matrix is given by C_{cov} = D_{stat} + C_{sys}, where comes from the measurement and C_{sys} is provided separately (Deng & Wei 2018). We notice that the forms of and are a bit different, since for the we removed the dependence of c/H_{0}r_{d}, which multiply the f(z) and, in the case of the parameter, is added to the value of μ.
In our analysis, we also consider the combined likelihood:
Here stands for the BAO or for the BAO_{θ} datasets independently. The distinction between the hyperparameters quantifying uncertainties in a dataset and the free parameters of the cosmological model is purely conceptual. It is important to note that the sodefined χ^{2} is not normalized, and thus its absolute value is not a useful measure of the quality of a given fit. Moreover, it is biased toward a larger number of parameters and is not very good for small datasets, such as the ones we used (Lazkoz et al. 2005). For this reason, we used it only to calculate the more balanced statistical measures, as is discussed in the following section.
3.4. Datasets and priors
In this work, we consider two different BAO datasets, to which we added the binned Pantheon supernovae dataset with its covariance matrix. The BAO datasets can be found summarized in Tables 1 and 2.
Compilation of BAO measurements from diverse releases of surveys such as SDSS, WiggleZ, and DES.
Compilation of angular BAO measurements from luminous red and blue galaxies, and quasars from diverse releases of the SDSS.
The first BAO dataset, shown on Table 1 and denoted as BAO, contains a combination of various angular measurements, to which we added points from the most recent to date eBOSS data release (DR16), which come as angular (D_{M}) and radial (D_{H}) measurements and their covariance. The points and the covariance matrices can be found in Cao & Ratra (2022). This choice of points allowed us to integrate the quantity H_{0} ⋅ r_{d} by summing the corresponding χ^{2} of the two types of measurements. While the covariance for some points is known and we include it in such cases, for the rest, we have to additionally test for possible correlations. To do so, we used the approach from Kazantzidis & Perivolaropoulos (2018), which we also used in Benisty & Staicova (2021). It consists of adding random correlation terms in the covariance matrix and testing the effect on the final result. Explicitly, we used
where σ_{i} is the 1σ error of the points. Applying the procedure shows that the points can be considered “effectively uncorrelated,” which allowed us to use them to infer the cosmological parameters. Even if there are small correlations, the procedure shows that the small correlations do not affect the final result considerately.
The second dataset shown in Table 2, denoted as BAO_{θ}, consists of 15 points, coming from transversal BAO measurements (Nunes et al. 2020). Importantly, the transversal BAO analysis does not need to assume a fiducial cosmology, particularly on the Ω_{K} parameter, which is included in the standard BAO analysis (Nunes et al. 2020). These points are claimed to be uncorrelated, but using this cosmologyindependent methodology means that their errors are larger than the errors obtained using the standard fiducial cosmology approach. One should note that using a fiducial cosmology is accounted for by the AlcockPaczynski distortion Lepori et al. (2017), so it does not compromise the integrity of the first dataset. However, we would like to investigate the overall effect of intrinsic assumptions in the final results and check if the two datasets are equivalent in this respect.
Finally, we added the Pantheon dataset, which contains 1048 supernovae luminosity measurements in the redshift range z ∈ (0.01, 2.3 Scolnic et al. 2018) binned into 40 points. To the statistical error, we also added the systematic errors as provided by the binned covariance matrix^{1}.
We performed the H_{0} ⋅ r_{d}integration procedure, outlined in previous sections, first on the two different BAO datasets alone, and then on the combination of the appropriate BAO dataset plus the Pantheon dataset. The priors we used were: Ω_{m} ∈ (0.2, 0.4), w_{0} ∈ ( − 2, −0), w_{a} ∈ ( − 2, 1), and Ω_{K} ∈ ( − 0.3, 0.3). We set . For gEDE we used the redefinition , so that it could be plotted on the same plots as the other models. As mentioned before, z_{t} is not a free parameter, and thus it is not a parameter in the Markov chain Monte Carlo (MCMC), and it is found by solving the appropriate transcendental equation using the package sympy. Regarding the problem of likelihood maximization, we used an affineinvariant MCMC nested sampler, as it is implemented within the opensource package Polychord (Handley et al. 2015) with the GetDist package (Lewis 2019) to present the results. In Polychord, convergence is defined as when the posterior mass contained in the live points is p = 10^{−2} of the total calculated evidence. We checked that our chains were stable with respect to changes in the parameter p, and furthermore by checking the Geweke score and the GelmanRubin diagnostic with the package pymcmcstat.
4. Results
4.1. Posterior distributions
Figures 1–5, A.1 and A.2 in the appendix show the final values obtained by running the MCMC on the selected priors for the two different datasets, with the numerical values in the Tables II–VI. Since we integrated H_{0} and r_{d}, the only physically measured parameter that remained was Ω_{m}. We see that in all the cases, Ω_{m} is rather well constrained, even from the BAOonly datasets. The BAO_{θ}, as expected, gives larger errors that the inclusion of supernova data improves. The closest to the Planck measurement of Ω_{m} = 0.315 ± 0.007 (Planck Collaboration VI 2020) is the Log model for BAO_{θ} and the ΛLCDM model for BAO, and ΛCDM/OkCDM for BAO + SN and BAO_{θ} + SN, with the Log model being very close for the latter.
Fig. 1. Posterior distribution for for Ω_{m}, w_{0}, and w_{a} for different parametrizations of the wwaCDM model with the BAO and BAO_{θ} datasets to the left and to the right, respectively, and with the Pantheon data added to the bottom panels. 
Fig. 2. Posterior distribution for Ω_{m} and w_{0} for the wCDM model, with the BAO data in the upper panel and the BAO_{θ} data in the lower panel. 
Fig. 3. Posterior distribution for Ω_{m} and Ω_{k} for the Ω_{K}LCDM model with the BAO data in the upper panel and the BAO_{θ} data in the lower panel. 
Fig. 4. Posterior distribution for Ω_{m} and Δ in the gEDE model with the BAO data in the upper panel and the BAO_{θ} data in the lower panel, with the solid line corresponding to pEDE and the dashed line to ΛCDM. 
Fig. 5. The posterior distribution for Ω_{m} for the two 1parameter models: LCDM and pEDE, for the BAO and BAO_{θ} datasets. 
When we consider the other parameters, we see that the BAOonly datasets are not able to limit them properly. While the BAO_{θ} dataset values contain w_{0} = −1 within 1σ, the values for BAO infer w_{0} > −1. Adding the SNIa dataset (which we mark on the plots and tables as “SN”) improves the constraints significantly. With respect to the parameter w_{a}, the inferred values have very big errors. When it comes to Ω_{K}, BAO_{θ} gives values closer to a flat universe, while BAO points to Ω_{K} < 0 (a closed universe).
The two emergent DE models perform well in all the cases. The pEDE model has an error similar to ΛCDM, but at higher Ω_{m}. The gEDE model also prefers higher values for Ω_{m}.
As mentioned in the Theory section, Δ = 0 recovers ΛCDM, while Δ = −1 recovers pEDE. We see from Fig. 4 that ΛCDM is preferred only by BAO, while the other datasets prefer pEDE (i.e., Δ closer to −1) but with large error. On the other hand, z_{t} is consistent with the known results for z_{t} ∼ 0.2. It should be noted that in the tables and in the Appendix, we denote Δ → w_{0} and z_{t} → w_{a} for notation consistency with the other models.
The conclusion from our results is that the BAOalone datasets are useful mostly for constraining Ω_{m}, and to a lesser extent w_{0}, while they are much less sensitive to the other parameters, w_{a} and Ω_{k}. The BAO + SN datasets seem to give much better constraints on the DE parameters. Also, one can see that the BAO_{θ} dataset includes the Ω_{k} value of a flat universe, while the BAO dataset seems to exclude it at a 68% confidence level.
From the Gaussians we see that some DE models have multiple peaks, hinting at some degeneracy. The results do not seem to change with increasing numbers of live points, hinting that this is a property of the models themselves or of the selected datasets.
4.2. Model selection
To compare the different models, we used different wellknown statistical measures. We used the Akaike information criterion (AIC), the Bayesian information criterion (BIC), the deviance information criterion (DIC), and the Bayes factor (BF; Liddle 2007).
The AIC criterion is defined as
where ℒ_{max} is the maximum likelihood of the data under consideration, N_{tot} is the total number of data points, and k is the number of parameters. For large N_{tot}, this expression reduces to AIC ≃ −2ln(ℒ_{max})+2k, which is the standard form of the AIC criterion (Liddle 2007).
The BIC criterion is an estimator of the Bayesian evidence, (e.g., Liddle 2007), and is given as
The AIC and BIC criteria employ only the likelihood value at maximum. Since we evaluate this ℒ_{max} numerically, from the Bayesian analysis, one needs to use sufficiently long chains to ensure the accuracy of ℒ_{max} when evaluating AIC and BIC. The DIC (Liddle 2007) provides all the information obtained from the likelihood calls during the maximization procedure. The DIC estimator is defined as
where θ is the vector of parameters being varied in the model, the overline denotes the usual mean value, and D(θ) = − 2ln(ℒ(θ))+C, where C is a constant. We used these definitions to form the difference in the IC values of the default model (ΛCDM) and the other suggested models (namely, we calculated ΔIC_{model} = IC_{ΛCDM} − IC_{model}). The model with the minimal AIC is considered best, (Jeffreys 1939), so a positive ΔIC points to a preference toward the DE model, negative – toward ΛCDM with ΔIC≥2 signifying a possible tension, ΔIC≥6 – a medium tension, and ΔIC ≥ 10 – a strong tension. Finally, we used the Bayes factor, defined as
where p(dM_{i}) is the Bayesian evidence for model M_{i}. The evidence is difficult to calculate analytically, but in polychord, it is calculated numerically by the algorithm. In the tables below, we use the ln(B_{0i}), where “0” is ΛCDM, which we compare with all the other models (denoted by the index “i”). According to the Jeffry’s scale (Jeffreys 1939), ln(B_{ij}) < 1 is inconclusive for any of the models, 1–2.5 gives weak support for the model “i”, 2.5–5 is moderate and > 5 is strong evidence for the model “i”. A minus sign gives the same for model “j”.
The sodefined statistical measures for the two datasets are presented in Tables 3 and 4. In summary, the model comparison for the different datasets gives:

For the BAO dataset: the best model from the AIC, BIC, and DIC is ΛCDM, followed closely (within < 1 IC units) by pEDE. The BF agrees with that, with pEDE and gEDE being close to it. OkLCDM is comparable to LCDM.

For the BAO + SN dataset: the best model is ΛCDM from all IC measures. BF agrees with that for most models, with an inconclusive preference for wCDM (ln(BF) < − 1).

For the BAO_{θ} dataset, the best model for AIC and BIC is pEDE, followed by ΛCDM. For DIC the best model is CPL, with all wCDM and wwaCDM models being better than ΛCDM. The IC difference, however, is too small to signify any tension. The BF agrees with the DIC, with the CPL model being best and ΛCDM the worst. Again, this is inconclusive.

For the BAO_{θ} + SN dataset, with respect to the AIC and BIC, the best model is ΛCDM, but pEDE is very close to it. With respect to DIC, all the models give better results than ΛCDM, with CPL being best, but the statistical significance is extremely low. With respect to the BF, however, the three parametrizations of wwaCDM give the best results, with values representing a weak but nonnegligible support.
Constraints at 68% confidencelevel errors on the cosmological parameters for the different tested models for the two BAOonly datasets: BAO and BAO_{θ}.
Constraints at 68% confidencelevel errors on the cosmological parameters for the different tested models for the two BAO + SN datasets: BAO+SN and BAO_{θ} + SN.
From this comparison we see that first, the use of statistical measures does not give an entirely consistent view on selecting the best model. This can be due to a number of factors, such as slow convergence of some of the models, or priors not having the similar weight.
Second, the two BAO datasets have preferences for different models. This may be due to different intrinsic assumptions with which the measurements were made. The BAO_{θ} dataset, despite the larger errors, seems to give consistent results, with some weak support for DE models in the different measures. The more standard AIC and BIC, however, are always in favor of ΛCDM, with pEDE being close behind. The BAO dataset seems to always prefer ΛCDM in most measures.
We can conclude that from the two datasets of BAO points, only the BAO dataset has a strong preference for ΛCDM. Adding the Pantheon dataset to it boosts this preference to statistical significance. The fact that ΛCDM is not the best model statistically in all of the cases for the BAOonly datasets may be due to the big uncertainty related to the BAO measurements or the specifics of the chosen dataset. While including the Pantheon dataset decreases the deviation in general, it does not eliminate it entirely for BAO_{θ}. This could be due to the different redshift distributions of BAO and Pantheon affecting the model fit: the maximum redshift for the binned Pantheon is vs. for BAO, and the median redshifts are accordingly vs. . Taking into consideration the big errors of the DE parameters for the different models and that all the evidence against ΛCDM is weak, we see that one needs much better BAO data to get a statistically strong preference, if indeed it exists.
5. Discussion
In order to avoid the problem of the degeneracy between H_{0} and r_{d} in the BAO measurements, and the assumptions on the data it imposes, this paper removes the combination H_{0} ⋅ r_{d} entirely by marginalizing over it in the χ^{2}. We used two different BAO datasets to test our approach. The first one, named BAO, comes from different measurements provided by surveys such as SDSS, WiggleZ, and DES, in addition to radial measurements coming from the eBOSS data release DR16 with their covariances. The other dataset is the BAO_{θ} compilation that measures θ(z), which is based on angular BAO measurements obtained from analyses of luminous red galaxies, blue galaxies, and quasars. These transversal BAO data have the advantage of being weakly dependent on the cosmological model. Both D_{A}/r_{d}, D_{M}/r_{d} and D_{H}/r_{d} provided from the first dataset, and θ(z) provided from the second one, depend only on the combination H_{0} ⋅ r_{d}, which we integrate out. In a similar way, one can integrate out the dependence on H_{0} and M_{B} in the Pantheon SNIa dataset, leaving all the likelihoods depending purely on the equation of state, namely Ω_{m} and the DE parameters Ω_{Λ}, w_{0}, and w_{a}, which allows us to use these datasets to infer the corresponding cosmological parameters.
We find that the BAOonly datasets infer Ω_{m} very well, close to the expected values and with a small error, but they are not sufficient to significantly constrain the parameters of the DE models. The errors on w_{0} and particularly on w_{a} are significant within the rather wide priors that we use. The errors of the BAO_{θ} dataset are larger than the errors of the BAO dataset, as expected.
Adding the SNIa dataset reduces the errors, especially for the w_{0} parameter. For the BAO + SN dataset, we find w = −0.986 ± 0.045. For w w_{a}CDM, we find w_{0} = −1.18 ± 0.139, w_{a} = −0.376 ± 0.672. From the BAO_{θ} + SN dataset, we find w = −1.08 ± 0.14 for the wCDM model. For w w_{a}CDM, we find w_{0} = −1.09 ± 0.09, w_{a} = −0.31 ± 0.74. As for the curvature, the BAO + SN dataset prefers a closed, almost flat, universe (Ω_{k} = −0.21 ± 0.07, while the BAO_{θ} + SN dataset prefers a flat one (Ω_{k} = −0.09 ± 0.15). In both cases, the gEDE model is closer to pEDE than to ΛCDM.
Comparing to the SDSSIV results (Alam et al. 2021), we see that they predict w_{0} = −0.939 ± 0.073, w_{a} = −0.31 ± 0.3 when one considers BAO+SN+CMB, but w_{0} = −0.69 ± 0.15 when only the BAO dataset is used. Thus, our results are consistent in both cases, with the BAO+SN value for w_{0} a little lower and the BAOonly value, very close to theirs. The mean value for w_{a} is close, but with a much larger error. However, we see that in the SDSSIV results, the error on w_{a} is also rather large. Our results also predict a negative Ω_{k}, with a larger error. One should note, however, that while we include some of the most recent BAO measurements, we include only the angular part of SDSSIII DR12, due to its interredshift covariance. Also, the BAO_{θ} dataset has larger inherent errors, and thus it is be expected to lead to larger errors in the inferred parameters. Finally, under the procedure we applied, some precision was lost due to the marginalization itself. Taking into account all this, we see that the procedure we employed still gives results close to those expected.
We performed a number of statistical tests for model comparison. The two BAO datasets show small statistical preferences for different models: ΛCDM for the BAO dataset and DE (wwaCDM, but also pEDE/gEDE) for the BAO_{θ} dataset. When we added the SN dataset, ΛCDM remained the best model for the BAO+SN dataset, but the BAO_{θ} + SN dataset shows a weak but nonnegligible preference for DE models.
Our conclusion is that one cannot sufficiently constrain the DE models from the chosen uncalibrated, mostly angular, BAO datasets alone. Adding the SNIa dataset to further reduce the errors and to remove some possible degeneracy helps, but it only helps to constrain w_{0} and not so much w_{a}. However, the results on Ω_{m} and w_{0} seem constrained enough to confirm the usefulness of this new approach. A downside is that, for the moment, it is not possible to include all correlated D_{M} − D_{H} measurements, since it is not possible to integrate out H_{0} ⋅ r_{d} for a covariance matrix over different z. For this reason we did not use all known correlations in the BAO data, which will improve on the errors and, thus, could lead to better constraints. We predict that future measurements of the BAO would increase the efficiency of the approach, as long as the correlation between some redshifts was not large. In any case, the marginalization approach offers a new perspective on the degeneracy H_{0} − r_{d} − Ω_{m} since, in this case, the only varying parameter is Ω_{m} and it could be a tool for an independent crosscheck on DE models.
Acknowledgments
We thank Eleonora DiValentino and Sunny Vagnozzi for useful comments and discussions. We would like to also thank the anonymous referee for their helpful comments regarding the manuscript. D.B. thanks to the Grants Committee of the Rothschild and the Blavatnik Cambridge Fellowships for generous supports. D.B. acknowledges a Postdoctoral Research Associateship at the Queens’ College, University of Cambridge. D.S. is thankful to Bulgarian National Science Fund for support via research grant KP06N58/5. We have received partial support from European COST actions CA15117 and CA18108.
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Appendix A: Corner plots for the different datasets
Fig. A.1. Posterior distribution for Ω_{m} and w_{0}, w_{a} for different parametrizations of DE, with the BAO data only in the upper panel and the combined BAO + Pantheon data in the lower panel. 
Fig. A.2. Posterior distribution for for Ω_{m},w_{0}, and w_{a} for different parametrizations of DE, with the BAO_{θ} data only in the upper and the combined BAO + Pantheon data in the lower panel. 
All Tables
Compilation of BAO measurements from diverse releases of surveys such as SDSS, WiggleZ, and DES.
Compilation of angular BAO measurements from luminous red and blue galaxies, and quasars from diverse releases of the SDSS.
Constraints at 68% confidencelevel errors on the cosmological parameters for the different tested models for the two BAOonly datasets: BAO and BAO_{θ}.
Constraints at 68% confidencelevel errors on the cosmological parameters for the different tested models for the two BAO + SN datasets: BAO+SN and BAO_{θ} + SN.
All Figures
Fig. 1. Posterior distribution for for Ω_{m}, w_{0}, and w_{a} for different parametrizations of the wwaCDM model with the BAO and BAO_{θ} datasets to the left and to the right, respectively, and with the Pantheon data added to the bottom panels. 

In the text 
Fig. 2. Posterior distribution for Ω_{m} and w_{0} for the wCDM model, with the BAO data in the upper panel and the BAO_{θ} data in the lower panel. 

In the text 
Fig. 3. Posterior distribution for Ω_{m} and Ω_{k} for the Ω_{K}LCDM model with the BAO data in the upper panel and the BAO_{θ} data in the lower panel. 

In the text 
Fig. 4. Posterior distribution for Ω_{m} and Δ in the gEDE model with the BAO data in the upper panel and the BAO_{θ} data in the lower panel, with the solid line corresponding to pEDE and the dashed line to ΛCDM. 

In the text 
Fig. 5. The posterior distribution for Ω_{m} for the two 1parameter models: LCDM and pEDE, for the BAO and BAO_{θ} datasets. 

In the text 
Fig. A.1. Posterior distribution for Ω_{m} and w_{0}, w_{a} for different parametrizations of DE, with the BAO data only in the upper panel and the combined BAO + Pantheon data in the lower panel. 

In the text 
Fig. A.2. Posterior distribution for for Ω_{m},w_{0}, and w_{a} for different parametrizations of DE, with the BAO_{θ} data only in the upper and the combined BAO + Pantheon data in the lower panel. 

In the text 
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