Positive vacuum energy and the N-bound

We argue that the total observable entropy is bounded by the inverse of the cosmological constant. This holds for all space-times with a positive cosmological constant, including cosmologies dominated by ordinary matter, and recollapsing universes. The argument involves intermediate steps which may be of interest in their own right. We note that entropy cannot be observed unless it lies both in the past and in the future of the observer's history. This truncates space-time to a diamond-shaped subset well-suited to the application of the covariant entropy bound. We further require, and derive, a novel Bekenstein-like bound on matter entropy in asymptotically de Sitter spaces. Our main result lends support to the proposal that universes with positive cosmological constant are described by a fundamental theory with only a finite number of degrees of freedom.