Planck data versus large scale structure: Methods to quantify discordance

Discordance in the Λ cold dark matter cosmological model can be seen by comparing parameters constrained by cosmic microwave background (CMB) measurements to those inferred by probes of large scale structure. Recent improvements in observations, including final data releases from both Planck and SDSS-III BOSS, as well as improved astrophysical uncertainty analysis of CFHTLenS, allows for an update in the quantification of any tension between large and small scales. This paper is intended, primarily, as a discussion on the quantifications of discordance when comparing the parameter constraints of a model when given two different data sets. We consider Kullback-Leibler divergence, comparison of Bayesian evidences and other statistics which are sensitive to the mean, variance and shape of the distributions. However, as a byproduct, we present an update to the similar analysis in [R. A. Battye, T. Charnock, and A. Moss, Phys. Rev. D 91, 103508 (2015)PRVDAQ1550-799810.1103/PhysRevD.91.103508], where we find that, considering new data and treatment of priors, the constraints from the CMB and from a combination of large scale structure (LSS) probes are in greater agreement and any tension only persists to a minor degree. In particular, we find the parameter constraints from the combination of LSS probes which are most discrepant with the Planck2015+Pol+BAO parameter distributions can be quantified at a ∼2.55σ tension using the method introduced in [R. A. Battye, T. Charnock, and A. Moss, Phys. Rev. D 91, 103508 (2015)PRVDAQ1550-799810.1103/PhysRevD.91.103508]. If instead we use the distributions constrained by the combination of LSS probes which are in greatest agreement with those from Planck2015+Pol+BAO this tension is only 0.76σ.