Relative information entropy in cosmology: The problem of information entanglement

The necessary information to distinguish a local inhomogeneous mass density field from its spatial average on a compact domain of the universe can be measured by relative information entropy. The Kullback-Leibler (KL) formula arises very naturally in this context, however, it provides a very complicated way to compute the mutual information between spatially separated but causally connected regions of the universe in a realistic, inhomogeneous model. To circumvent this issue, by considering a parametric extension of the KL measure, we develop a simple model to describe the mutual information which is entangled via the gravitational field equations. We show that the Tsallis relative entropy can be a good approximation in the case of small inhomogeneities, and for measuring the independent relative information inside the domain, we propose the R\'enyi relative entropy formula.

Received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under the grant agreement No. PCOFUND-GA-2009-246542; from CMAT through FEDER Funds COMPETE, and also from FCT projects Est-OE/MAT/UI0013/2014, SFRH/BCC/105835/2014 and CERN/FP/123609/2011