Using a fundamental discrete symmetry, ${\mathbf{Z}}_N$, we construct a two-axion model with the QCD axion solving the strong-$CP$ problem, and an ultralight axion (ULA) with $m_{\mathrm{ULA}}\approx 10^{-22}\mathrm{eV}$ providing the dominant form of dark matter (DM). The ULA is light enough to be detectable in cosmology from its imprints on structure formation, and may resolve the small-scale problems of cold DM. The necessary relative DM abundances occur without fine-tuning in constructions with decay constants $f_{\mathrm{ULA}}\sim {10}^{17}\mathrm{GeV}$, and $f_{\mathrm{QCD}}\sim {10}^{11}\mathrm{GeV}$. An example model achieving this has $N=24$, and we construct a range of other possibilities. We compute the ULA couplings to the standard model, and discuss prospects for direct detection. The QCD axion may be detectable in standard experiments through the $\overrightarrow{E}·\overrightarrow{B}$ and $G\tilde{G}$ couplings. In the simplest models, however, the ULA has identically zero coupling to both $G\tilde{G}$ of QCD and $\overrightarrow{E}·\overrightarrow{B}$ of electromagnetism due to vanishing electromagnetic and color anomalies. The ULA couples to fermions with strength $g\propto 1/f_{\mathrm{ULA}}$. This coupling causes spin precession of nucleons and electrons with respect to the DM wind with period $t\sim$ months. Current limits do not exclude the predicted coupling strength, and our model is within reach of the CASPEr-Wind experiment, using nuclear magnetic resonance.

• Received 10 November 2015

DOI:https://doi.org/10.1103/PhysRevD.93.025027