Information gain on reheating: The one bit milestone

We show that the Planck 2015 and BICEP2/KECK measurements of the cosmic microwave background (CMB) anisotropies provide together an information gain of $0.82\pm 0.13$ bits on the reheating history over all slow-roll single-field models of inflation. This corresponds to a 40% improvement compared to the Planck 2013 constraints on the reheating. Our method relies on an exhaustive CMB data analysis performed over nearly 200 models of inflation to derive the Kullback-Leibler entropy between the prior and the fully marginalized posterior of the reheating parameter. This number is a weighted average by the Bayesian evidence of each model to explain the data thereby ensuring its fairness and robustness.

Figure 1

One- and two-sigma confidence intervals associated wit the two-dimensional marginalized posterior distributions in the slow-roll parameter space $(P_{*},{\epsilon }_{1*},{\epsilon }_{2*},{\epsilon }_{3*})$ from combined Planck 2015 and BICEP2/KECK data sets.

Figure 2

Probability distributions (normalized to their maximum) for the rescaled reheating parameter $R_{\mathrm{reh}}$ associated with two of the 200 models analyzed: loop inflation on the left (LI) and supergravity brane inflation on the right (SBI), see Ref. [22]. The black curve corresponds to the prior (3) and is not exactly flat since the upper bound of Eq. (3) is slightly model dependent. The marginalized posterior obtained from Planck 2013 data is displayed in blue and is to be compared to the more constraining posterior obtained from the Planck 2015 data with BICEP2/KECK (red curve).

Figure 3

Information gain $D_{\mathrm{KL}}$ (in bits) given by Planck 2013 (left panel) and Planck 2015 with BICEP2/KECK (right panel) about the rescaled reheating parameter $\ln R_{\mathrm{reh}}$ as a function of the Bayesian evidence. Each circle represents one of the 200 models of the Encyclopaedia Inflationaris collection whose color traces the mean value of $\ln R_{\mathrm{reh}}$. The yellow band represents the one-sigma deviation around the mean value. For Planck 2015 and BICEP2/KECK, one gets $⟨D_{\mathrm{KL}}⟩=0.82\pm 0.13$. This corresponds to 40% improvement compared to Planck 2013.