The Complexity of our Curved Universe

The properties of the Cosmic Microwave Background (CMB) radiation must be different in flat, positively and negatively curved universes. This fact leads to a direct way of determining the geometry of the universe. The signature of the predicted effect of geodesic mixing, i.e., of the `chaotic' behavior of photon beams in negatively curved spaces peculiar to Anosov systems with strong statistical properties, has been detected while studying the COBE-DMR 4-year data \cite{GT}. This possible observation of the negative curvature of the universe suggests the need to search for more effective ways to analyze the CMB data expected from forthcoming high precision experiments. Information theory offers such a descriptor for the CMB sky maps - the Kolmogorov complexity - as well as provides novel insight into the problem of the loss of information and time asymmetry in a hyperbolic universe.