Space-time curvature and the cosmic horizon: Derivations using the Newtonian world and the Friedmann-Robertson-Walker metric

Several relationships describing the distance versus time dependence of the cosmic horizon (R_h) for an expanding universe have been published within the past two decades. Some are based on the special conditions, including a flat universe geometry, and when applied for calculation return significantly different values. We present our derivation beginning with Newtonian world then following the Friedmann model from the viewpoint of an observer located at the origin of an expanding spherical, homogeneously matter-dominated universe; both geometrically flat and allowing space-time curvature. Our derivations for the cosmic horizon at the speed of light allow examination for the effects of matter density and space- time curvature. We also compare the fitness of several current models, including the recently proposed R_h = ct universe against the demands of the 580 Union 2.1 Type Ia supernovae distance and redshift data with uncommon, proper attention paid to data transformation and observational errors.