Proper size of the visible Universe in FRW metrics with a constant spacetime curvature

In this paper, we continue to examine the fundamental basis for the Friedmann-Robertson-Walker (FRW) metric and its application to cosmology, specifically addressing the question: What is the proper size of the visible universe? There are several ways of answering the question of size, though often with an incomplete understanding of how far light has actually traveled in reaching us today from the most remote sources. The difficulty usually arises from an inconsistent use of the coordinates or an over-interpretation of the physical meaning of quantities such as the so-called proper distance R(t) = a(t)r, written in terms of the (unchanging) co-moving radius r and the universal expansion factor a(t). In this paper, we prove for the five non-trivial FRW metrics with a constant spacetime curvature that, when the expansion began from an initial singularity, the visible universe today has a proper size equal to R_h(t_0/2), i.e., the gravitational horizon at half its current age. The exceptions are de Sitter and Lanczos, whose contents had pre-existing positions away from the origin. In so doing, we confirm earlier results showing the same phenomenon in a broad range of cosmologies, including ΛCDM, based on the numerical integration of null geodesic equations through an FRW metric.