Photon geodesics in Friedmann-Robertson-Walker cosmologies

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 Wilkinson Microwave Anisotropy Probe, and ground-based and Hubble Space Telescope searches for Type Ia supernovae, we now know that R _h(t ₀) ∼13.5Glyr. This coincides with the maximum distance (ct ₀≈ 13.7Glyr) light could have travelled 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 Friedmann-Robertson-Walker cosmologies, to show - at least in the specific cases we consider here, including Λ cold dark matter (ΛCDM) - 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 ΛCDM, measured as a proper distance, is approximately 0.5ct ₀.