Affine ANEC selects the closed FRW branch for geodesically complete cosmology

TL;DR

This paper investigates how spatial curvature influences the compatibility of Friedmann–Robertson–Walker cosmologies with the averaged null energy condition, revealing that closed models can be geodesically complete while flat and open models cannot.

Contribution

It demonstrates that positive spatial curvature allows for geodesically complete, ANEC-satisfying cosmologies, providing explicit constructions and contrasting them with flat models requiring NEC violation.

Findings

Flat and open non-static FRW models cannot be both null geodesically complete and ANEC-satisfying.

Closed FRW models with positive curvature can support nonsingular, geodesically complete cosmologies with NEC-respecting matter.

Explicit scalar-field constructions for closed FRW models are provided, including analytic and quadrature-based solutions.

Abstract

We study the relation between geodesic completeness, the averaged null energy condition (ANEC), and spatial curvature in Friedmann--Robertson--Walker (FRW) cosmology within classical general relativity. Using the affinely parameterized ANEC along radial null geodesics, we prove that non-static flat or open FRW spacetimes in the regular classes considered here cannot be both null geodesically complete and ANEC-satisfying. Bounded oscillatory or cyclic flat/open models do not circumvent the obstruction: the negative affine-ANEC bulk term accumulates over infinitely many cycles, giving \(I_{\rm ANEC}=-\infty\) for non-static periodic cases. Equivalently, within these classes, non-static ANEC-satisfying flat or open models are null incomplete. The sign obstruction is absent in closed \((k=+1)\) FRW geometry, where the positive curvature term enters the affine ANEC identity with the opposite sign and can support nonsingular, geodesically complete cosmologies with ordinary NEC-respecting matter. We give explicit closed-FRW scalar-field constructions, including a fully analytic quadratic reconstruction and a cubic-root reconstruction in closed quadrature, and contrast them with their flat realizations, which require NEC-violating support. Furthermore, we quantify how positive curvature can bias flat-model reconstructions toward an effective phantom equation of state, finding only a percent-level effect under current curvature bounds. The result is a curvature classification of ANEC-compatible eternal FRW cosmology: flat and open branches are obstructed, while the closed branch admits explicit complete realizations, with global de Sitter appearing as the vacuum limiting representative.