Photon Geodesics in FRW Cosmologies

Abstract: The Hubble radius is a particular manifestation of the Universe's gravitational horizon, R_h(t_0)=c/H_0, the distance beyond which physical processes remain unobservable to us at the present epoch. Based on recent observations of the cosmic microwave background (CMB) with WMAP, and ground-based and HST searches for Type Ia supernovae, we now know that R_h(t_0)~13.5 Glyr. This coincides with the maximum distance (ct_0~13.7 Glyr) light could have traveled since the big bang. However, the physical meaning of R_h is still not universally understood or accepted, though the minimalist view holds that it is merely the proper distance at which the rate of cosmic recession reaches the speed of light c. Even so, it is sometimes argued that we can see light from sources beyond R_h, the claim being that R_h lies at a redshift of only ~2, whereas the CMB was produced at a much greater redshift (~1100). In this paper, we build on recent developments with the gravitational radius by actually calculating null geodesics for a broad range of FRW cosmologies, to show---at least in the specific cases we consider here, including LCDM---that no photon trajectories reaching us today could have ever crossed R_h(t_0). We therefore confirm that the current Hubble radius, contrary to a commonly held misconception, is indeed the limit to our observability. We find that the size of the visible universe in LCDM, measured as a proper distance, is approximately 0.45ct_0.