Constraining the dark energy equation of state using Bayes theorem and the Kullback-Leibler divergence

Data-driven model-independent reconstructions of the dark energy equation of state $w(z)$ are presented using Planck 2015 era CMB, BAO, SNIa and Lyman-$\alpha$ data. These reconstructions identify the $w(z)$ behaviour supported by the data and show a bifurcation of the equation of state posterior in the range $1.5{<}z{<}3$. Although the concordance $\Lambda$CDM model is consistent with the data at all redshifts in one of the bifurcated spaces, in the other a supernegative equation of state (also known as `phantom dark energy') is identified within the $1.5 \sigma$ confidence intervals of the posterior distribution. To identify the power of different datasets in constraining the dark energy equation of state, we use a novel formulation of the Kullback--Leibler divergence. This formalism quantifies the information the data add when moving from priors to posteriors for each possible dataset combination. The SNIa and BAO datasets are shown to provide much more constraining power in comparison to the Lyman-$\alpha$ datasets. Further, SNIa and BAO constrain most strongly around redshift range $0.1-0.5$, whilst the Lyman-$\alpha$ data constrains weakly over a broader range. We do not attribute the supernegative favouring to any particular dataset, and note that the $\Lambda$CDM model was favoured at more than $2$ log-units in Bayes factors over all the models tested despite the weakly preferred $w(z)$ structure in the data.