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Curvature and Flatness in a Brans-Dicke Universe

The evolution of a universe with Brans-Dicke gravity and nonzero curvature is investigated here. We present the equations of motion and their solutions during the radiation dominated era. In a Friedman-Robertson-Walker cosmology we show explicitly that the three possible values of curvature $κ=+1,0,-1$ divide the evolution of the Brans-Dicke universe into dynamically distinct classes just as for the standard model. Subsequently we discuss the flatness problem which exists in Brans-Dicke gravity as it does in the standard model. In addition, we demonstrate a flatness problem in MAD Brans-Dicke gravity. In general, in any model that addresses the horizon problem, including inflation, there are two components to the flatness issue: i) at the Planck epoch curvature gains importance, and ii) during accelerated expansion curvature becomes less important and the universe flattens. In many cases the universe must be very flat at the Planck scale in order for the accelerated epoch to be reached, thus there can be a residual flatness problem.