There is no evidence of axion dark matter in its most natural mass range, which implies that axion dark matter is not a plurality contributor to dark matter and probably makes up only a small percentage of it, if it exists at all.
The cosmic microwave background (CMB) places strong constraints on models of dark matter (DM) that deviate from standard cold DM (CDM), and on initial conditions beyond the scalar adiabatic mode. Here, the full Planck data set (including temperature, -mode polarisation, and lensing deflection) is used to test the possibility that some fraction of the DM is composed of ultralight axions (ULAs). This represents the first use of CMB lensing to test the ULA model. We find no evidence for a ULA component in the mass range . We put percent-level constraints on the ULA contribution to the DM, improving by up to a factor of two compared to the case with temperature anisotropies alone.
Axion DM also provides a low-energy window onto the high-energy physics of inflation through the interplay between the vacuum misalignment production of axions and isocurvature perturbations. We perform the first systematic investigation into the parameter space of ULA isocurvature, using an accurate isocurvature transfer function at all values. We precisely identify a "window of co-existence" for where the data allow, simultaneously, a contribution of ULAs to the DM, and contributions of isocurvature and tensors to the CMB power. ULAs in this window (and all lighter ULAs) are shown to be consistent with a large inflationary Hubble parameter, . The window of co-existence will be fully probed by proposed CMB-S4 observations with increased accuracy in the high- lensing power and low- and -mode polarisation. If ULAs in the window exist, this could allow for two independent measurements of in the CMB using the axion DM content and isocurvature, and the tensor contribution to -modes.Renée Hlozek, David J. E. Marsh, Daniel Grin "Using the Full Power of the Cosmic Microwave Background to Probe Axion Dark Matter" (August 18, 2017).
From the introduction in the body text:
A central role in the standard cosmological model is played by Cold Dark Matter (CDM). Using a combination of temperature (T), E-mode polarisation, and weak gravitational lensing power spectra of the CMB, the CDM density is measured to be
Ωch^2 = 0.1199 ± 0.0022 (Planck Collaboration et al. 2016b). CDM is the only form of Dark Matter (DM) required by the data, providing the backbone of the “cosmic web” of large scale structure on linear scales.
One well-motivated theoretical possibility for the CDM is the axion, a new hypothetical particle motivated by the charge-parity conjugation problem of quantum chromodynamics (QCD) (Peccei & Quinn 1977; Weinberg 1978; Wilczek 1978; Abbott & Sikivie 1983; Dine & Fischler 1983; Preskill et al. 1983). On scales of relevance to cosmology, QCD axions with mass ma < 10^−4 eV are produced non-thermally, have the correct cosmological relic density, and behave as CDM.
There are very few direct constraints on the QCD axion CDM parameter space (e.g. Asztalos et al. 2010). The lightest supersymmetric particle also provides a canonical CDM candidate (Jungman et al. 1996), though limits on “natural” regions of its parameter space are strong (e.g. Akerib et al. 2017).
The CMB places stringent limits on a variety of theoretical models for DM beyond CDM, including limits on the mass of standard model neutrinos (Planck Collaboration et al. 2016b), thermal axions (Archidiacono et al. 2013), DM interactions (Wilkinson et al. 2014b; Wilkinson et al. 2014a), and generalized models (Cyr-Racine et al. 2016; Thomas et al. 2016).
A particularly interesting bound comes from constraints on non-thermally produced ultralight axions (ULAs), as this establishes an absolute lower-bound on the DM particle mass from linear observables, ma > 10^−24 eV (Hlozek et al. 2015). The constraints of Hlozek et al. (2015) (hereafter, H15) not only bound the particle mass of the dominant component of DM, but also place stringent limits on the axion DM density over many orders of magnitude in particle mass (10^−33 eV < ma < 10^−24 eV) for models in which the DM is a mixture of “standard” CDM and ULAs.
Cosmology thus probes not only the abundance but also the composition of the dark sector. ULAs are expected to be ubiquitous in string theory (e.g. Svrcek & Witten 2006; Conlon 2006; Arvanitaki et al. 2010) and provide candidates for dark matter (if ma > 10^−27 eV) or dark energy (if ma > 10^−27 eV) components, depending on their mass.For fairly natural values of the axion initial field value in string models, φ¯ i ∼ fa < 10^17 GeV, where fa is the axion “decay constant”, ULAs contribute at the percent level to the cosmic critical density (Marsh 2016; Hui et al. 2017).
 The boundary between the two regimes is a matter of convention. Here it has been chosen as approximately the inverse Hubble parameter at matter radiation equality.
Within the range of particle mass (10^−33 eV < ma < 10^−24 eV) the axion ULA percentage of all mass-energy in the universe in the mid-range is most likely about one part per thousand and definitely not more than one part per hundred. It could potentially be almost 10% at the high end of 10^-24 eV with the high end of the error bars (less than half that at 10^25 eV), where it almost reaches a CDM threshold, and potentially even more at the extreme 10^-33 eV low end (almost totally absent at 10^-32 eV) where there is degeneracy between dark matter and dark energy contributions allow it to hit 20.3% of the total mass-energy of the universe.
The bottom line is that there are strict experimental constraints from the CMB on axion dark matter, and it is basically ruled out for many orders of magnitude of particle mass as the sole component of dark matter. Another recent study also disfavored axion dark matter. More constraints on axion dark matter from cosmology observations are here. QCD evidence suggests that if axions exist that they should be heavier than the ULA mass range (see also here).
This matters, of course, because experimental and observational evidence is increasingly ruling out dark matter particle candidates.