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Ruling out light axions: the writing is on the wall
by Konstantin A. Beyer, Subir Sarkar
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Authors (as registered SciPost users):  Konstantin Beyer · Subir Sarkar 
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Preprint Link:  https://arxiv.org/abs/2211.14635v3 (pdf) 
Date submitted:  20230208 17:37 
Submitted by:  Beyer, Konstantin 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We revisit the domain wall problem for QCD axion models with more than one quark charged under the PecceiQuinn symmetry. Symmetry breaking during or after inflation results in the formation of a domain wall network which would cause cosmic catastrophe if it comes to dominate the Universe. The network may be made unstable by invoking a 'tilt' in the axion potential due to Planck scale suppressed nonrenormalisable operators. Alternatively the random walk of the axion field during inflation can generate a 'bias' favouring one of the degenerate vacua, but we find that this mechanism is in practice irrelevant. Consideration of the axion abundance generated by the decay of the wall network then requires the PecceiQuinn scale to be rather low  thus ruling out e.g. the DFSZ axion with mass below 33~meV, where most experimental searches are in fact focussed.
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Anonymous Report 2 on 2023315 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2211.14635v3, delivered 20230315, doi: 10.21468/SciPost.Report.6907
Report
I think that the changes performed by the authors go in the right directions. However, I feel that they do not completely respond to the comments I made.
My point about the suppression of the explicit breaking is that the dimensionless coefficient could be extremely small. Indeed, if for instance the effect comes from nonperturbative semiclassical gravitational solutions with action $S$, then the coefficient is $g~\sim\exp(S)$ which can be much smaller than $10^{2}$, in which case Figure 1 would look different. If the coupling $g'$ that enters into $S$ is small, then $S$ would be enhanced by inverse powers of $g'$, e.g. as $S\sim 1/g'^2$, and $g$ can be even $e^{O(100)}$. What would be the lower bound on the allowed dimension of the operator that breaks PQ in this case?
I believe that the problem I mentioned about the `relaxation’ of the biased initial conditions during the evolution of the strings should be discussed in the text. It is a crucial issue that might prevent the bias solution of the domain wall problem to work, even if biased initial conditions were found. I think it would be more useful if the authors mentioned it at the start, so as to allow a reader to be aware of it. This would avoid a reader trying to construct a model with biased initial conditions that might not solve the domain wall problem in the first place, or to think of a possible way out.
The study of the dark matter abundance from domain wall decay has been carried out in the past by several collaborations. The authors often refer to the more detailed simulations of [55,56,58] to support their result, in particular that their estimate of the gravitational wave production differs of a factor of 5 from [55,56,58]. I would be grateful if the authors explained (at least to me) how the dark matter abundance and their bound on the axion mass ($\sim$60 meV) differs from the previous detailed literature ([55,56,58]) and what is the crucial new insight that leads to the new bound. In particular, do the domain walls reach a regime with a constant domain wall area parameter, as in some of the simulations the authors refer to?
As a minor comment, I believe that requiring min $V = 0$ is not precise, because the cosmological constant is small, but believed to be nonzero.
I feel that the authors should at least comment about the issues above before the paper can be published.
Report
The authors have addressed the point that I commented on, and based on my assessment, I can recommend that the paper be published.
Author: Subir Sarkar on 20230330 [id 3527]
(in reply to Report 1 on 20230305)We thank the Referee for acknowledging that we have addressed the point raised and for recommending publication.
Author: Subir Sarkar on 20230330 [id 3526]
(in reply to Report 2 on 20230315)Errors in usersupplied markup (flagged; corrections coming soon)
We are happy to comment on the issues raised by the Referee  see below:

The prefactor g can indeed be exponentially small  however such models do not have PQ symmetry breaking after inflation hence the issue is irrelevant (as the EditorinCharge also notes). We have kept g explicit in all equations and shown the effect of setting g << 1 in the figures  so the effect on the allowed dimension of the operator can easily be assessed.

Concerning the possible 'relaxation’ of biased initial conditions, we have already noted as the Referee suggests at the start of 'Section 2.2 Bias’ (in footnote 2): "However, if the walls are connected by strings, the biased initial conditions mean that the initial string configuration is energetically disfavoured and would likely relax to an unbiased state by emitting axions before the formation of DWs. In any case we find here that bias does not solve the axion DW problem, hence further discussion of this issue is not warranted in the present paper. We have also added a citation to a eprint by Gonzalez et al, arXiv:2211.06849 that appeared after ours which argues that the bias solution does not work when inflationary correlations on superhorizon scales are taken into account.

Regarding how our bound on the axion mass differs from the previous detailed literature [55,56,58] ([56,57,59] in the current version) here are our comments on these papers:
 Hiramatsu et al. [56] focus 'On the estimation of gravitational wave spectrum from cosmic domain walls’; while their numerical code has been used in previous discussions of axion cosmology, they do not specifically consider axions.
 Saikawa [59] too provides ‘A review of gravitational waves from cosmic domain walls’  not specifically about axions.
 Kawasaki et al [56] discuss 'Axion dark matter from topological defects’ in detail. To answer the Referee, we do assume (our eq.20) that the domain wall network evolution reaches the regime with a constant domain wall area parameter before the wall network collapses, but Kawasaki et al [56] consider both the scaling solution, as well as a small deviation from scaling that is seen in their detailed numerical simulations (modelled as in their eq.3.14). However they conclude, as shown in their Fig.4, that "that there is no significant difference between the assumption of exact scaling and that of deviation from scaling”.
Concerning the dark matter abundance, they say "... axions can be responsible for the cold dark matter in the mass range ... m_a ≈ (10^−4 – 10^−2) eV with a mild tuning of parameters for the models with N_DW > 1”. We find however with *no tuning of parameters* a lower limit (eq.48) on the axion mass that is significantly higher: 3.3x10^−2 eV. As stated in the caption of our Fig.3: "If δ is finetuned to be 10^−2, then μ can increase to 10^−8 thus allowing the grey shaded region, i.e. lighter axions down to m_a ∼ 10−2 eV”. Kawasaki et al [56] show their constraints on m_a in their Fig.9 for N_DW = 6 and taking δ to be 10^−3 (=> m_a > 6x10^−3 eV), 10^−5 (=> m_a > 1.6x10^−3 eV) or 10^−8 (=> m_a > 8x10^−4 eV). Using the scaling with δ, these translate into a lower bound of m_a > 4x10−2 eV for δ=1 (taking N_DW=6), to be compared with our bound of m_a > 3.3x10^−2 eV (taking N_DW=2).
Another difference in our assumptions is that Kawasaki et al [56] interpret the neutron EDM bound to imply a more restrictive limit on the QCD theta parameter of 7x10^12 (their eq.4.9), whereas we adopt the, more standard, limit of 10^10 (our eq.10). Again since our results come from analytic solutions to the governing equations with clearly stated assumptions and choices of parameters (whose scaling is explicitly shown in eq.45) it can be seen that for the same adopted limit, the Kawaski et al [56] lower bound on m_a would change to m_a > 3x10−2 eV (δ=1, N_DW=6).
This is close enough to what we find  with a transparent, analytic treatment rather than relying on detailed numerical solutions. Note that some of these were done in previous work (Hiramatsu et al., Phys. Rev. D85 (2012) 105020) and have been improved on in subsequent work by the same authors e.g. Hiramatsu et al. [56]. So whereas the validity of the numerical simulations is established by our analytic result, there may be some uncertainty in the precise bound.

Concerning requiring min V=0, astronomers indeed say that the cosmological constant is not quite zero. However the value they obtain from interpreting observational data in the framework of the assumed maximally symmetric FriedmannLemaitreRobertsonWalker model is \Lambda ~ H_0^2 ~ (10^−42 GeV)^2. This corresponds to an energy density of \Lambda/8\pi G_N ~ (10^12 GeV)^4 … which is pretty much zero compared to the energy scales being considered here!

We hope our comments above (and in the text) satisfactorily address all the issues raised.