Numerically confirm inefficient parametric resonance during the inflationary transition

Confirm through a dedicated numerical simulation that parametric resonance remains inefficient when the Peccei–Quinn field transitions from the large-field minimum \(\rho_{\rm min}\) to the intermediate minimum \(\widetilde{\rho}_{\rm min}\) in an inflating background, so that the Peccei–Quinn symmetry is not restored.

Background

The proposed stepping-stone mechanism transfers the PQ field from a large-field minimum to an intermediate minimum during inflation. The authors estimate the resonance parameter and find it below the conventional threshold for efficient resonance, but emphasize that this estimate was derived under assumptions appropriate to a post-inflationary matter-dominated era rather than an approximately de Sitter background.

A dedicated simulation is therefore needed to evolve the transition and its fluctuations in the actual inflationary setting and to verify that resonant amplification does not restore the PQ symmetry.

References

A dedicated numerical simulation, which we leave for future work, is required to confirm that parametric resonance is indeed inefficient.

— A Stepping Stone Solution to the QCD Axion Isocurvature Problem  (2609.28840 - Lyu et al., 23 Sep 2026) in Section 4.1, Benchmark I

We note that $m_{\rho}(\rho_{\rm min}) \gtrsim H_{\rm inf}$ is a sufficient but not necessary condition. A lighter saxion may still be allowed if the resulting fluctuation amplitude always remains much smaller than the instantaneous PQ field value, which needs to be confirmed by numerical simulations.

— A Stepping Stone Solution to the QCD Axion Isocurvature Problem  (2609.28840 - Lyu et al., 23 Sep 2026) in Section 3.4, Avoiding PQ Symmetry Restoration