We study a model for a nonsingular cosmic bounce in $\mathcal{N}=1$ supergravity, based on supergravity versions of the ghost condensate and cubic Galileon scalar field theories. The bounce is preceded by an ekpyrotic contracting phase, which prevents the growth of anisotropies in the approach to the bounce and allows for the generation of scale-invariant density perturbations that carry over into the expanding phase of the Universe. We present the conditions required for the bounce to be free of ghost excitations, as well as the tunings that are necessary in order for the model to be in agreement with cosmological observations. All of these conditions can be met. Our model thus provides a proof-of-principle that nonsingular bounces are viable in supergravity, despite the fact that during the bounce the null energy condition is violated.

We propose a scenario to stabilize all geometric moduli—that is, the complex structure, Kähler moduli, and the dilaton—in smooth heterotic Calabi-Yau compactifications without Neveu-Schwarz three-form flux. This is accomplished using the gauge bundle required in any heterotic compactification, whose perturbative effects on the moduli are combined with nonperturbative corrections. We argue that, for appropriate gauge bundles, all complex structure and a large number of other moduli can be perturbatively stabilized—in the most restrictive case, leaving only one combination of Kähler moduli and the dilaton as a flat direction. At this stage, the remaining moduli space consists of Minkowski vacua. That is, the perturbative superpotential vanishes in the vacuum without the necessity to fine-tune flux. Finally, we incorporate nonperturbative effects such as gaugino condensation and/or instantons. These are strongly constrained by the anomalous $U(1)$ symmetries, which arise from the required bundle constructions. We present a specific example, with a consistent choice of nonperturbative effects, where all remaining flat directions are stabilized in an anti-de Sitter vacuum.

We explicitly confirm that spatially flat nonsingular bouncing cosmologies make sense as effective theories. The presence of a nonsingular bounce in a spatially flat universe implies a temporary violation of the null energy condition, which can be achieved through a phase of ghost condensation. We calculate the scale of strong coupling and demonstrate that the ghost–condensate bounce remains trustworthy throughout, and that all perturbation modes within the regime of validity of the effective description remain under control. For this purpose we require the perturbed action up to third order in perturbations, which we calculate in both flat and co-moving gauge—since these two gauges allow us to highlight different physical aspects. Our conclusion is that there exist healthy descriptions of nonsingular bouncing cosmologies providing a viable resolution of the big-bang singularities in cosmological models. Our results also suggest a variant of ekpyrotic cosmology, in which entropy perturbations are generated during the contracting phase, but are only converted into curvature perturbations after the bounce.