Discrete fuzzy de Sitter cosmology

We analyze the spectrum of time observable in noncommutative cosmological model introduced in [5], defined by $(\rho ,s=\frac{1}{2})$ representation of the de Sitter group. We find that time has peculiar property: it is not self-adjoint, but appropriate restrictions to the space of physical states give self-adjoint extensions. Extensions have discrete spectrum with logarithmic distribution of eigenvalues, $t_n\sim \mathrm{\ell }\log n$+const, where $\mathrm{\ell }$ characterizes noncommutativity and the usual assumption is $\mathrm{\ell }={\mathrm{\ell }}_{Planck}$. When calculated on physical states, radius of the universe is bounded below by $, \ell, \sqrt{\frac 34, \left( \frac 14 +\rho^2\right)}, $, which resolves the big bang singularity. An immediate consequence of the model is a specific breaking of the original symmetry at the Planck scale.