Inflationary resolution of the initial singularity
The inflationary paradigm has transformed our understanding of the early universe; yet most inflationary models are considered geodesically past-incomplete, suggesting a beginning of time or a primordial Big Bang singularity. The Borde-Guth-Vilenkin (BGV) theorem is often cited as demonstrating that all eternally inflating spacetimes must be past-incomplete. Utilizing a new theorem establishing geodesic completeness in generalized cosmologies, we present a simple, explicit class of inflationary solutions that are smooth, nonsingular, and geodesically complete for all time, including into the past. These models exhibit localized NEC violation but remain globally well-behaved in both temporal directions. The NEC violation is confined and controlled: the averaged null energy condition (ANEC) is satisfied in the strongest sense, while violations of smeared null energy conditions (SNEC) are uniformly bounded and become nonnegative under sufficiently wide smearings. Our results suggest that eternal inflation can arise from controlled NEC-violating dynamics, offering a new, nonsingular, and past-eternal picture of the universe.