Discrete fuzzy de Sitter cosmology

We analyze the spectrum of time observable in noncommutative cosmological model introduced in [5], defined by $(\rho, s=\frac 12)\,$ 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, $\,t_n \sim \ell\, \log\, n$+const, where $\ell$ characterizes noncommutativity and the usual assumption is $\,\ell=\ell_{Planck}$. 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.