Covariant entropy bound and loop quantum cosmology

We examine Bousso’s covariant entropy bound conjecture in the context of radiation filled, spatially flat, Friedmann-Robertson-Walker models. The bound is violated near the big bang. However, the hope has been that quantum gravity effects would intervene and protect it. Loop quantum cosmology provides a near ideal setting for investigating this issue. For, on the one hand, quantum geometry effects resolve the singularity and, on the other hand, the wave function is sharply peaked at a quantum corrected but smooth geometry, which can supply the structure needed to test the bound. We find that the bound is respected. We suggest that the bound need not be an essential ingredient for a quantum gravity theory but may emerge from it under suitable circumstances.

Figure 1

The $({\tau }_i,{\tau }_f)$ region (in Planck units) where LQC corrections are significant. Since ${\tau }_i<{\tau }_f$, the striped portion is excluded. For $({\tau }_i,{\tau }_f)$ in the black region, surfaces $\mathcal{B}$ do not admit past light sheets. In the gray region, the expansion is negative near $\mathcal{B}$ but then becomes positive so that the light sheet $\mathcal{L}$ of $\mathcal{B}$ is incomplete. Points in the white region are the most interesting ones for testing the entropy bound. For $({\tau }_i,{\tau }_f)$ in this region, light sheets of surfaces $\mathcal{B}$ are complete.

Figure 2

$S/A$ is plotted in Planck units for points $({\tau }_i,{\tau }_f)$ in the white region of Fig. 1. (a) Shows the full plot and (b) zooms in on the portion of the plot where $S/A$ is near its maximum. The maximum value is $S/A\approx 0.244/{\ell }_{\mathrm{Pl}}^2$.