Entropy of universe as entanglement entropy

We note that the observable part of universe at a certain time t_P is necessarily limited, when there is a beginning of universe. We argue that an appropriate spacetime region associated with an observer from t_I to t_P is the causal diamond which is the overlap of the past/future of the observer at t_P/t_I respectively. We also note that the overlap surface ∂D of the future and the past lightcones bisects the spatial section including ∂D into two regions D and D- where D is the region inside the causal diamond and D- the remaining part of the spatial section. We propose here that the entropy of universe associated with a causal diamond is given by an entanglement entropy where one is tracing over the Hilbert space associated with the region D- which is not accessible by the observer. We test our proposal for various examples of cosmological spacetimes, including flat or open FRW universes, by showing that the entropy as the area of ∂D divided by 4G is a non-decreasing function of time t_P as dictated by the generalized second law of thermodynamics. The closed, recollapsing universe corresponds to a finite system and there is no reason to expect the validity of the generalized second law for such a finite system.