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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
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-12-15 15:46
Submitted by: Beyer, Konstantin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


We revisit the domain wall problem for QCD axion models with more than one quark charged under the Peccei-Quinn 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 non-renormalisable operators. Alternatively the random walk of the axion field during inflation can generate a 'bias' favouring one of the degenerate vacuua, 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 Peccei-Quinn scale to be rather low -- thus ruling out e.g. the DFSZ axion with mass below $\sim60$~meV, where most experimental searches are in fact focussed.

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Reports on this Submission

Anonymous Report 2 on 2023-1-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2211.14635v2, delivered 2023-01-23, doi: 10.21468/SciPost.Report.6591


The authors discussed the axion model with the domain wall (DW) number $N_{DW}>1$ with biased potential/initial conditions. By using the formula (4) of the DW decay rate to simulate the DW network decaying into axion+gravitaitonal waves evolution via the equations from number/energy conservations (35,36,37), they found that the resulting axion is too much to be consistent with the astrophysical bound.

The implicitly assumed philosophy is naturalness. Namely, the solution of the fine-tuning problem for the strong CP problem should not contain another fine-tuning problem in the forms of the Planck suppressed terms to let the DW decay. With this philosophy, with which I agree, a large decay rate of the DW network is in contradiction to the neutron electric-dipole moment experimental bound.

Of course, there can be some extensions to solve the problem (e.g., a relatively simple one may be an entropy dilution for the relativistic axions soon after the DW collapsing, i.e., DW network is formed and destroyed during the reheating),
I consider the paper interesting and important, showing the natural solution of the strong CP problem, with $N_{DW}>1$ and Peccei-Quinn symmetry breaking after the inflation, is strongly disfavored.

One comment is about their choice of $x_a$, the fraction of the DW induced axion abundance to the total dark matter abudance.
I consider it is more conservative to take $x_a=1$, i.e., it is the dominant dark matter. This is because, the axon produced with $K=100$ at, say, $T_{dec}\sim 1MeV$, becomes non-relativistic around $T=10$ keV. Naively, one can estimate that such axions have free-streaming length $\lesssim 0.01$ Mpc which is smaller than the Lyman-alpha constraint,$ \sim O(0.1)$ Mpc, and I consider that the component can be the dominant dark matter. In addition, there may be some thermalization during the redshift of the axion momentum. This is due to the self-quartic coupling, and Bose enhancement see, e.g., the discussions around Eq. 135 of the Axion cosmology by David Marsh, 1510.07633. Then the structure formation bound may be even weaker. Thus, I consider it should be conservative to take $x_a=1$. Since the authors claim the bound is robust, I would suggest the authors use $x_a=1$ for their sample value and conclusions.

Other than this, I am happy to recomend the paper be published.

Requested changes

1. use $x_a=1$ instead for the conservative bound.

  • validity: high
  • significance: top
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: good

Author:  Subir Sarkar  on 2023-02-07  [id 3318]

(in reply to Report 2 on 2023-01-23)
answer to question

We are grateful to Reviewer 2 for their thoughtful remarks and for finding our argument convincing.

Concerning the fraction x_a of the dark matter in axions created by the decay of the axion domain walls, it is true that our choice of x_a = 0.5 was rather simplistic. We did note that considerations of structure formation might impose a more stringent constraint. However we agree that taking x_a = 1 is most conservative. Accordingly we have done this and find our bound to now be reduced by a factor of ~2 to 33 meV. Our Figures 2 and 3 have been correspondingly redone with x_a = 1 and appropriate comments have been added in Section 3.3.

We had also missed citing a recent eprint by Notari et al (2211.03799) who carefully calculate the upper bound on the axion mass from considerations of structure formation to be 240 meV. This means that the axion window is reduced to 33-240 meV for DFSZ axions in the post-inflationary scenario. We have now said so in our Conclusions.

Anonymous Report 1 on 2023-1-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2211.14635v2, delivered 2023-01-22, doi: 10.21468/SciPost.Report.6583


The authors study an important open problem in axion physics, namely the dark matter abundance in the so-called post-inflationary scenario with domain wall number $N_W>1$. The authors consider an alternative to the usual mechanism to avoid domain wall overclosure, based on biased initial conditions, which unfortunately they find does not work. They also estimate the dark matter and gravitational wave abundance, providing a bound on the axion mass, if the U(1) Peccei--Quinn (PQ) symmetry has an explicit breaking (‘tilt’). The paper is sufficiently well written and the grammar is clear. I list below my main comments and concerns on the paper.

The authors assume the PQ violation takes the form of the operator in eq. (5), with a dimensionless coupling $g$ that parameterizes the breaking of the global U(1) symmetry assumed to be of order 1. However, this is not necessarily the case, as for instance these operators could be suppressed by $M_{\rm string}$. Additionally, although we expect gravity to break all global symmetries, we do not know to what amount these are broken; in particular, the size of the higher dimensional operators could be much smaller than order one in Planck units (this is the case in many instances of non-perturbative breaking via gravitational instantons, black holes and wormholes). This would change the conclusion about the allowed operator dimension and axion decay constant in Section 2.1. The authors should mention this and change the discussion appropriately.

I believe that there may also be another problem with postulating initial conditions after inflation containing a bias on the axion field (i.e., effectively, more points where $a=0$). Thus, even if such initial conditions were obtained, they may not solve the domain wall problem. This is because the boundary of domain walls are axion strings, and biased initial conditions mean that the initial string configuration has the fundamental domain $[-\pi v_a, \pi v_a]$ wrapped in a non-uniform way in space. This configuration is energetically unfavorable, and the string system will relax to an unbiased configuration before domain walls dominate, radiating the surplus of energy into axion waves. The authors should take this into account and change the discussion appropriately. It would be interesting to understand if a theory without strings could have the domain walls destroyed before they dominate (but perhaps this is ruled out by the authors’ work).

The dyanamics of the domain walls and string might also be significantly more complex than the authors discuss and, in particular, it might not be the case that the assumptions leading to eqs. (40) and (41) are valid. The authors should discuss how their assumptions (i.e. the domain-wall oscillating bubbles) used in their estimate in eqs. (40) and (41) compare with the other assumptions discussed in the literature, tested for instance in the numerical simulations of [55,56,58], and how they change the bound on the axion mass. Note that the dependence on the symmetry breaking parameter $\mu$ in eqs. (40) and (41) is hidden, at least in part, in the exponential integral functions, which is hard to understand at a first sight and should be clarified.

I believe that changes addressing these concerns are important before the article could be published in Scipost.

Minor comments and typos:
- why is it important that ${\rm min}V = 0$ below eq. (5)?
- from the discussion at the end of pg. 7, it is not immediately obvious what could concretely be the additional source of the potential that leads to relevant biased initial conditions
- in eq. (21) there is a double round bracket
- vacuua-> vacua, throughout the paper
- at the end of pg. 2: “a much stronger bound obtains” -> “a much stronger bound is obtained” ?
- on pg. 3 “the DFSZ axion..” -> “the DFSZ axion.”
- $z$ and $x$ in the definition of $E_n$ below eq. (41) do not match

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Subir Sarkar  on 2023-02-07  [id 3317]

(in reply to Report 1 on 2023-01-22)
answer to question

We thank Reviewer 1 for a careful reading of our paper and for pointing out several typos (which we have now corrected - thanks!)

Concerning the substansive point made, viz. that the scale at which global symmetries are violated by quantum gravity effects (call it M_QG) need not be M_Pl but can e.g. be M_string - this is indeed true. However this will only make our constraints stronger; since we wish to be conservative, we take M_QG to be M_Pl. We also agree that the size of the higher dimensional operators (parameterised by g) can be much smaller than order one for non-perturbative symmetry breaking. However this does not affect the two opposing constraints on f_PQ which we discuss, viz. from above by requiring that the neutron does not acquire an observable EDM, and from below by requiring the domain wall network to disappear sufficiently fast. It affects only their interpretation, as in our eqs.(10) and (14). We can thus address the Reviewer's concern by simply replacing M_Pl by M_QG in these equations. In Fig.1, we have now explicitly said in the caption that the constraint shown on f_PQ is obtained taking M_QG to be M_Pl; we had already shown the effect of taking |g| to be 10^-2 rather than 1. Since the constraint shown can easily be scaled for any other choice of M_QG, we hope this is satisfactory.

The Reviewer may also be right that 'bias' need not solve the domain wall problem when the walls end on strings, as in the case of axions. Indeed if biased initial conditions result in energetically unfavorable configurations, then the string system will relax to an unbiased configuration before the walls come to dominate. However since we have shown that *in any case* bias is not relevant for the axion domain wall problem (because of the large separation of scales between f_PQ and \Lambda_QCD when the walls form), we do not discuss this here further - it does however merit further study. We have added footnote 2 to this effect in Section 2.2.

The Reviewer wishes to understand better whether our assumption that the wall network decays exponentially is justfied. Indeed the dynamics is quite complex; as seen even in our early simulation [31] (Fig.3) the co-moving domain wall energy density is not a smooth exponential with conformal time (as in eq.3) but exhibits 'bounces’ as bubbles of the disfavored vacuum collapse and radiate away the energy contained in the walls. This affects the total energy density very little however, and subsequent more detailed simulations [55,56,58] confirm this. However our estimate of the radiated gravitational waves uses the quadrupole formula which strictly applies only to a bubble oscillating at its natural frequency set by its size. The more detailed calculation [56] suggests the value can be higher by up to a factor of ~5, but given that energy density of the gravitational waves generated is sub-dominant by a factor of ~10^6, this does not affect our bound on the axion mass. We have made appropriate comments to this effect in Section 3.3. We also agree that the dependence on the symmetry breaking parameter μ in eq.(42), which follows from eqs.(40) and (41), is hidden in the exponential integral functions - this is why we have shown explicitly the dependence of the derived constraint on μ in Fig.3.

As for the Reviewer's question "why is it important that minV=0 below eq. (5)?", this is of course just the notorious Cosmological Constant problem - we are simply highlighting what is usually brushed under the carpet!

Finally the Reviewer notes that "from the discussion at the end of pg. 7, it is not immediately obvious what could concretely be the additional source of the potential that leads to relevant biased initial conditions". We agree - however this issue is beyond the scope of the present work. We have cited interesting ideas by Ferrer et al. [51] and Caputo & Reig [52] in this context. However as we have already remarked the necessary alignment of f_PQ and H_infl is quite challenging.

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