Ruling out light axions: the writing is on the wall

Concerning the further points raised by one of the Referees, we are happy that the EIC agrees that it is not necessary to further discuss the possibility that the prefactor g can be exponentially suppressed - as such models will not exhibit post-inflationary PQ symmetry breaking.

1. We have responded in detail to this Referee's question about the relationship between our results to refs.[55,57,59]. We believe we have provided a transparent, analytic understanding of the numerical results which is of value.

2. We had indeed misunderstood the question "In the case H_{inf} << f_{PQ}: why wouldn’t this automatically be the pre-inflationary axion scenario?” We agree that H_{inf} is the relevant parameter to compare with f_{PQ} - not V^{1/4}; for H_inf << f_PQ, the PQ symmetry is broken before inflation starts and in this sense this is the pre-inflationary scenario. What we are saying is that despite the fact that there are quantum fluctuations of order H_{inf}, those are too small to ever move the field far from its initial value and therefore will neither serve to generate a bias nor to cross from one vacuum to the other. This was pointed out by Linde & Lyth, Phys. Lett.B 246 (1990) 353 who showed that quantum fluctuations of the axion field during inflation can lead to formation of exponentially big axionic domain walls. We have now cited this paper and clarified the point.

We hope this addresses all the outstanding issues.

1) Added a statement in Footnote 2 (p.7): "Moreover recent lattice simulations show that taking super-horizon inflationary correlations into account can undermine the bias mechanism [39]".

[39] D. Gonzalez, N. Kitajima, F. Takahashi and W. Yin, Stability of domain wall network with initial inflationary fluctuations, and its implications for cosmic birefringence (2022), 2211.06849.

2) Clarified that our limit (48) on f_{PQ} and m_a is consistent with previous numerical results (p.12): "Our bound is consistent with previous numerical work [42, 58]."

3) Added clarification concerning the pre-inflationary axion scenario (p.7): "To generate a bias we must wait until the mean value of the field accumulates around the vacuum states. This can take a long time during which the causally connected patches at the time of PQ symmetry breaking inflate massively and *any domain walls that form are exponentially large [52]. Hence this is the same as pre-inflationary PQ symmetry breaking*."

[52] A. D. Linde and D. H. Lyth, Axionic domain wall production during inflation, Phys. Lett. B 246, 353 (1990), doi:10.1016/0370-2693(90)90613-B.