BGV Theorem: Do expanding-on-average universes require infinite speeds?

In 2003, the so-called BGV theorem was published in an updated paper. (The theorem is also called the Borde-Guth-Vilenkin theorem, after the papers's authors: Alan Guth, Alvin Borde, and Alexander Vilenkin). The BGV theorem is a straightforward proof that any universe which expands on average has to connect in a finite amount of time to a past boundary.^{1 , 2} The proof moves by way of demonstrating that any such eternally inflating spacetime would require that faster-than-light travel be possible. (In fact, the required speed asymtotes to infinity.) Is this theorem correct? Do expanding-on-average spacetimes require such speeds?

1. In their own words:
   • Alan Guth, Alvin Borde, and Alexander Vilenkin: “Our argument shows that null and time-like geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics.” [“Inflationary spacetimes are not past-complete” at arXiv:gr-qc/0110012 (2003) 3.] To define some terms:
   • Geodesic (in General Relativity) = def. The path that space takes a free floating particle through, as it expands or contracts. The particle ignores all influences other than gravity.
   • Geodesically incomplete = def. A geodesic whose path is incomplete (finite): it terminates at some point. Geodesic incompleteness is symptomatic of a beginning/ending to classical spacetime.
   You can think of geodesics as the time-directioned lines in canonical Big Bang cone picture. The lines may be infinite towards the future, but they terminate in the past at the supposed singularity.
2. Notably, the boundary is most likely a singularity. Geodesic incompleteness is symptomatic of singularities.

   • Stephen Hawking & Roger Penrose: “One can then recognize the occurrence of singularities by the existence of incomplete geodesics that cannot be extended to infinite values of the affine parameter. Definition of Singularity: A spacetime is singular if it is timelike or null geodesically incomplete but cannot be embedded in a larger spacetime. This definition reflects the most objectionable feature of singularities, that there can be particles whose history has a beginning or end at a finite time” [The Nature of Space and Time (Princeton, 1996), 15.]
   • Aron Wall: “If we assume that the universe was always expanding, so that the BGV theorem applies, then presumably there must have been some type of initial singularity.” [“Did the Universe Begin? III: BGV Theorem. http://www.wall.org/~aron/blog/did-the-universe-begin-iii-bgv-theorem/ ]

• Gary Gibbons: “These examples also underscore the need for a theory of initial conditions in order to understand cosmology and the initial singularity or big bang. As emphasized by Penrose among others, elementary thermodynamic arguments indicate that the Universe began in a very special state and even proponents of eternal inflation have had to concede, following Borde, Guth, and Vilenkin, that eternity is past incomplete.” [“Singularities” in The Quantum Structure of Space and Time eds Gross, Henneaux, Sevrin (World Scientific Publishing, 2005), 60.]
• Gregory Vereshchagin: “However, inflation is insufficient when we face the challenge of cosmological singularity. In fact, it is shown that inflationary spacetimes are geodesically incomplete and therefore initial cosmological singularity is present even if inflation is of the eternal time and final singularity never occurs. [“Gauge Theories of Gravity with the Scalar Field Cosmology,” in Frontiers in Field Theory ed. Kovras (Nova Science, 2005), 215.]
• Mary-Jane Rubenstein: “In 2003, however, Arvind Borde, Alan Guth, and Alexander Vilenkin published a highly influential paper arguing that inflationary models cannot be ‘past eternal’” [Worlds Without End: The Many Lives of the Multiverse (Colombia, 2014), 284.]