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We discuss how to efficiently and reliably estimate the level of agreement and disagreement on parameter determinations from different experiments, fully taking into account non-Gaussianities in the parameter posteriors. We develop two families of scalable algorithms that allow us to perform this type of calculations in increasing number of dimensions and for different levels of tensions. One family of algorithms rely on kernel density estimates of posterior distributions while the other relies on machine learning modeling of the posterior distribution with normalizing flows. We showcase their effectiveness and accuracy with a set of benchmark examples and find both methods agree with each other and the true tension within $0.5sigma$ in difficult cases and generally to $0.2sigma$ or better. This allows us to study the level of internal agreement between different measurements of the clustering of cosmological structures from the Dark Energy Survey and their agreement with measurements of the Cosmic Microwave Background from the Planck satellite.
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We provide a new interpretation for the Bayes factor combination used in the Dark Energy Survey (DES) first year analysis to quantify the tension between the DES and Planck datasets. The ratio quantifies a Bayesian confidence in our ability to combine the datasets. This interpretation is prior-dependent, with wider prior widths boosting the confidence. We therefore propose that if there are any reasonable priors which reduce the confidence to below unity, then we cannot assert that the datasets are compatible. Computing the evidence ratios for the DES first year analysis and Planck, given that narrower priors drop the confidence to below unity, we conclude that DES and Planck are, in a Bayesian sense, incompatible under LCDM. Additionally we compute ratios which confirm the consensus that measurements of the acoustic scale by the Baryon Oscillation Spectroscopic Survey (SDSS) are compatible with Planck, whilst direct measurements of the acceleration rate of the Universe by the SHOES collaboration are not. We propose a modification to the Bayes ratio which removes the prior dependency using Kullback-Leibler divergences, and using this statistical test find Planck in strong tension with SHOES, in moderate tension with DES, and in no tension with SDSS. We propose this statistic as the optimal way to compare datasets, ahead of the next DES data releases, as well as future surveys. Finally, as an element of these calculations, we introduce in a cosmological setting the Bayesian model dimensionality, which is a parameterisation-independent measure of the number of parameters that a given dataset constrains.
We investigate constraints on some key cosmological parameters by confronting metastable dark energy models with different combinations of the most recent cosmological observations. Along with the standard $Lambda$CDM model, two phenomenological metastable dark energy models are considered: (romannumeral1) DE decays exponentially, (romannumeral2) DE decays into dark matter. We find that: (1) when considering the most recent supernovae and BAO data, and assuming a fiducial $Lambda$CDM model, the inconsistency in the estimated value of the $Omega_{rm{m,0}}h^2$ parameter obtained by either including or excluding Planck CMB data becomes very much substantial and points to a clear tension~citep{sahni2014model,zhao2017dynamical}; (2) although the two metastable dark energy models that we study provide greater flexibility in fitting the data, and they indeed fit the SNe Ia+BAO data substantially better than $Lambda$CDM, they are not able to alleviate this tension significantly when CMB data are included; (3) while local measurements of the Hubble constant are significantly higher relative to the estimated value of $H_0$ in our models (obtained by fitting to SNe Ia and BAO data), the situation seems to be rather complicated with hints of inconsistency among different observational data sets (CMB, SNe Ia+BAO and local $H_0$ measurements). Our results indicate that we might not be able to remove the current tensions among different cosmological observations by considering simple modifications of the standard model or by introducing minimal dark energy models. A complicated form of expansion history, different systematics in different data and/or a non-conventional model of the early Universe might be responsible for these tensions.
The late-time modifications of the standard $Lambda$ Cold Dark Matter ($Lambda$CDM) cosmological model can be parameterized by three time-dependent functions describing the expansion history of the Universe and gravitational effects on light and matter in the Large Scale Structure. In this Letter, we present the first joint reconstruction of these three functions performed in a non-parametric way from a combination of recent cosmological observations. The reconstruction is performed with a theory-informed prior, built on the general Horndeski class of scalar-tensor theories. We find that current data can constrain 15 combined modes of these three functions with respect to the prior. Our methodology enables us to identify the phenomenological features that alternative theories would need to have in order to ease some of the tensions between datasets within $Lambda$CDM, and deduce important constraints on broad classes of modified gravity models.
The Phenomenologically Emergent Dark Energy model, a dark energy model with the same number of free parameters as the flat $Lambda$CDM, has been proposed as a working example of a minimal model which can avoid the current cosmological tensions. A straightforward question is whether or not the inclusion of massive neutrinos and extra relativistic species may spoil such an appealing phenomenological alternative. We present the bounds on $M_{ u}$ and $N_{rm eff}$ and comment on the long standing $H_0$ and $sigma_8$ tensions within this cosmological framework with a wealth of cosmological observations. Interestingly, we find, at $95%$ confidence level, and with the most complete set of cosmological observations, $M_{ u}sim 0.21^{+0.15}_{-0.14}$ eV and $N_{rm eff}= 3.03pm 0.32$ i.e. an indication for a non-zero neutrino mass with a significance above $2sigma$. The well known Hubble constant tension is considerably easened, with a significance always below the $2sigma$ level.
In this work, we present a new method to estimate cosmological parameters accurately based on the artificial neural network (ANN), and a code called ECoPANN (Estimating Cosmological Parameters with ANN) is developed to achieve parameter inference. We test the ANN method by estimating the basic parameters of the concordance cosmological model using the simulated temperature power spectrum of the cosmic microwave background (CMB). The results show that the ANN performs excellently on best-fit values and errors of parameters, as well as correlations between parameters when compared with that of the Markov Chain Monte Carlo (MCMC) method. Besides, for a well-trained ANN model, it is capable of estimating parameters for multiple experiments that have different precisions, which can greatly reduce the consumption of time and computing resources for parameter inference. Furthermore, we extend the ANN to a multibranch network to achieve a joint constraint on parameters. We test the multibranch network using the simulated temperature and polarization power spectra of the CMB, Type Ia supernovae, and baryon acoustic oscillations, and almost obtain the same results as the MCMC method. Therefore, we propose that the ANN can provide an alternative way to accurately and quickly estimate cosmological parameters, and ECoPANN can be applied to the research of cosmology and even other broader scientific fields.
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