The axion is a hypothetical massive scalar boson with a zero electric charge whose existence would (1) suppress CP violation in strong force interactions despite the fact that there is naturally a term in the strong force equations that could introduce and quantify CP violation in these interactions, and (2) provide a non-thermal dark matter particle.
The axion was originally proposed by Peccei and Quinn in a 1977 paper. The only experiment claiming to have observed an axion did so in 1986, but was subsequently discredited with the conclusion was retracted by the group that did the original experiment in 2007. Other experimental efforts to observe an axion over the last three and a half decades have produced null results.
There are several other ways that the strong CP problem could be resolved (e.g. (i) the up quark could actually have a near zero mass or (ii) the physical constant related to CP violation by the strong force could be zero simply because that is the arbitrary value of that experimentally determined physical constant (an "unnatural" choice but not one that violates any fundamental principle of the Standard Model), or (iii) it is my own conjecture that the fact that this is related to the fact that the gluon has a zero rest mass).
There are also many other possible dark matter candidates.
Thus, neither of these considerations demand the axion as a solution, although the axion be a solution to one or both of these problems if indeed it does exist.
The new constraints from Big Bang Nucleosynthesis on Axion properties.
A new pre-print by Blum, et al., examines observational limits on the axion mass and axion decay constant due to Big Bang Nucleosynthesis, because the role that the axion plays in strong force interactions would impact the proportions of light atoms of different types created in the early universe.
The study concludes that (1) the product of the axion mass and axion decay constant must be approximately 1.8*10^-9 GeV^2, and (2) that in order to solve the strong CP problem and be consistent with astronomy observations, that axion mass must be between 10^-16 eV and 1 eV in mass (with a 10^-12 eV limitation likely due to the hypothesis that the decay constant is less than the Planck mass). The future CASPEr2 experiment could place a lower bound on axion mass of 10^-12 eV to 10^-10 eV and would leave the 1 eV upper bound unchanged.
Other studies argue that the axion decay constant must be less than 10^9 GeV (due to constraints from observations of supernovae SN1987A) and propose an axion mass on the order of 6 meV (quite close to the muon neutrino mass if one assumes a normal hierarchy and a small electron neutrino mass relative to the muon neutrino-electron neutrino mass difference) or less. Estimates of the axion mass in the case of non-thermal production of axions, which are favored if it is a dark matter particle, are on the order of 10^-4 to 10^-5 eV. There are also order of magnitude estimates of the slight predicted coupling of axions to photons.
Other studies placing observational limitations on massive bosons as dark matter candidates apply only to bosons much heavier than the axion.
Cosmology implications of axion dark matter
The observational constraints on axion mass put it in the same vicinity as that of heavy neutrinos, which would be considered "hot dark matter," But, the mass-velocity dispersion relation used to distinguish "cold", "warm" and "hot" dark matter which refers to the velocity dispersion of a dark matter candidate, does not apply to dark matter candidates that are not thermal relics like axions. Thus, axion dark matter could have a velocity dispersion consistent with "cold" or "warm" dark matter despite having a mass that would make it "hot dark matter" (which is ruled out by observational evidence) if it were a thermal relic.
However, since axion dark matter is not a thermal relic, it cannot be assumed to produce a cosmology consistent with the empirically validated six parameter lamda CDM cosmology model whose parameters were recently refined by the Planck satellite's cosmic background radiation observations. This model assumes thermal relic dark matter, although it is not very specific regarding its properties. The new paper does not address the cosmology implications of a non-thermal relic dark matter candidate, beyond its impact on Big Bang Nucleosynthesis if the axion mass and decay constant are outside a specified range.
Footnote on the Neutron EDM
The new paper states claims that there is tension between experimental measurement of the neutron electric dipole moment and the Standard Model expectation for its value at page 1 which it states that:
The QCD theta term . . . induces a neutron electric dipole moment (EDM) approximately equal to 2.4*10^-16 theta*e*cm [5 - Pospelov and Ritz, 83 Phys. Rev. Lett. 2526 (1999)] that is in tension with experiment for theta greater than 10^-10 [6 - Baker et al., 97 Phys.Rev.Lett. 131801 (2005)][7 - Harris, et al., 82 Phys.Rev.Lett. 904 (1999)].In contrast, Wikipedia cites a current limit of the neutron EDM of less than 10^-24 e*cm, and a Standard Model expectation of 10^-32 e*cm. Wikipedia also states that the neutron EDM constrains the strong CP violation theta angle to be less than 10^-10 radians, which is not a tension between experiment and the Standard Model expectation.
Harris (1999) states that the neutron EDM is less than 6.3*10^-26 e*cm at the 90% confidence interval and does not set forth a minimum value for the neutron EDM as the Blum, et al. (2014) preprint claims.
Pospelov and Ritz (1999) does contain a 2.4*10^-16 theta*e*cm result to a 40% precision. Their paper also states that this translates into a limit on the strong CP violation theta angle to be less than 3*10^-10, but does not support an experimental minimum for theta of 10^-10 as the Blum, et al. (2014) preprint claims.
Pospelov, Ritz and Huber, in a 2007 preprint, cite at endnote 28 to Harris and Baker papers above, just as the new paper by Blum, et al. does, for the proposition that the neutron EDM is less than 3*10^-26 e*cm, which again, fails to support that claim that there is experimental tension with the Standard Model prediction. They also note that experiments are underway which could bound the neutron EDM experimentally to less than 10^-28 e*cm, closing two of the six orders of magnitude between the Standard Model prediction and the experimental constraint (and placing a considerably more strict limitation on the strong CP violating angle theta); of course, the new experiments could instead actually measured the neutron EDM and discover new physics.
Baker et al., 97 Phys.Rev.Lett. 131801 (2006) which is cited by both Wikipedia (with the correct year) and by the preprint (with the wrong year but the same journal page number reference) states that the 90% confidence interval upper limit on the neutron EDM is 2.9*10^-26 e*cm and does not undertake to convert that figure into a constraint on the strong CP angle theta or suggest any floor on either the neutron EDM or theta. Baker, et al. responded to a comment on their paper and stood by their results in a Response of 2007. Therefore, it appears that the new paper's assertion is simply incorrect.
Thus, Blum, et al. (2014) simply do not understand their source or the state of the research when they claim that there is experimental support for the existence of an axion based upon tension between the Standard Model prediction of the neutron EDM and experimental data showing that theta is not less than 10^-10 based upon Pospelov and Ritz (1999), Harris (1999) and Baker (sic. 2005), or alternately did not write what they meant to say in the sentences quoted above.
I would very much hope that this will be corrected between the preprint prior to publication and I have corresponded with the authors on this point asking them to correct the apparent error. I will update this post if there are new developments on this point.
UPDATE January 29, 2014: After exchanging e-mail with the authors, it has become clear that the issue is one of clarity of phrasing, rather than intended meaning. They state in a prompt reply to my e-mail endorsed by the authors collectively that:
There seems to be a slight misunderstanding here: what we say in our preprint is that the QCD theta term would violate existing experimental upper bounds on the neutron EDM, if theta was larger than ~10^-10. We do not suggest that theta is larger than 10^-10. Thus, we seem to be in agreement with your point of view. We're sorry if this point was not sufficiently clear in the paper.