On the past-completeness of inflationary spacetimes

Understand

We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past.

• Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation.
• It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate $H_{avg}>0$) must be incomplete in null and timelike past directions.
• This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic.