Arrows of Time and Chaotic Properties of the Cosmic Background Radiation

We advance a new viewpoint on the connection between the the thermodynamical and cosmological arrows of time, which can be traced via the properties of Cosmic Microwave Background (CMB) radiation. We show that in the Friedmann-Robertson-Walker Universe with negative curvature there is a necessary ingredient for the existence of the thermodynamical arrow of time. It is based on the dynamical instability of motion along null geodesics in a hyperbolic space. Together with special (de-correlated) initial conditions, this mechanism is sufficient for the thermodynamical arrow, whereas the special initial conditions alone are able to generate only a pre-arrow of time. Since the negatively curved space will expand forever, this provides a direct connection between the thermodynamical and cosmological arrows of time. The structural stability of the geodesic flows on hyperbolic spaces and hence the robustness of the proposed mechanism is especially stressed. We then point out that the main relations of equilibrium statistical thermodynamics (including the second law) do not necessarily depend on any arrow of time. Finally we formulate a {\it curvature anthropic principle}, which stipulates the negative curvature as a necessary condition for the time asymmetric Universe with an observer. CMB has to carry the signature of this principle as well.