Homogeneous Cosmological Models with Bounded Potential

Under the assumption that there exists a finite global upper bound to the gravitational potential, 2Gm/c2r, it is shown that simply connected homogeneous models based on the Walker-Robertson line element must have zero mean density if the curvature parameter le is 0 or -1. For homogeneous models with k = +1, all global potential bounds less than or equal to the Schwarzschild limit imply a positive cosmological constant. If the potential is globally less than 0.912 (spherical space) or 0 75 (elliptical space), and if the physically realizable pressures are at all future epochs bounded below by zero and above by photon gas pressure, pc2/3, k = + 1 models will expand monotonically without limit-for all future time Asymptotic values for large times are in the usual notations: H = c ~f (X/3), q = -1, p = 0, and p = 0. Under the same conditions, the present value of the deceleration parameter is less than -1, so that the asymptotic value is approached from below. The cosmological constant is bounded below by 3H02/c2, where H0 is the present value of the Hubble parameter