Quantum of volume in de Sitter space

We apply the nonstandard loop quantum cosmology method to quantize a flat Friedmann-Robertson-Walker cosmological model with a free scalar field and the cosmological constant $\Lambda >0$. Modification of the Hamiltonian in terms of loop geometry parametrized by a length $\lambda$ introduces a scale dependence of the model. The spectrum of the volume operator is discrete and depends on $\Lambda$. Relating quantum of the volume with an elementary lattice cell leads to an explicit dependence of $\Lambda$ on $\lambda$. Based on this assumption, we investigate the possibility of interpreting $\Lambda$ as a running constant.

Figure 1

Evolution of volume $v$ as a function of the parameter $u$. The function is periodic in variable $u$ with the period equal to $2K$. The solutions in different periods are separated by vertical asymptotes.

Figure 2

Ratio ${\Delta }_{\delta }/\Delta$ as a function of $\delta$ (black line). We see that $\underset{\delta \to 0}{lim}{\Delta }_{\delta }/\Delta =1$ as well as $\underset{\delta \to 1}{lim}{\Delta }_{\delta }/\Delta =\infty$. The (red) dot represents the case with $\Lambda =0$.

Figure 3

Scale dependence of $\Lambda$. The arrows indicate flow from UV ($\lambda ={\lambda }_0$) to IR limit ($\lambda \to \infty$). The dotted part of the curve denotes the region of large $\lambda$ ($\lambda \gg {\lambda }_1$) where considerations become speculative. In the plot ${\lambda }_0=2.4l_{\mathrm{Pl}}$ and ${\lambda }_1=3.8l_{\mathrm{Pl}}$.