In the next few years Advanced LIGO (aLIGO) may see gravitational waves (GWs) from thousands of black hole (BH) mergers. This marks the beginning of a new precision tool for physics. Here we show how to search for new physics beyond the standard model using this tool, in particular the QCD axion in the mass range ma ~ 10^-14 to 10^-10 eV. Axions (or any bosons) in this mass range cause rapidly rotating BHs to shed their spin into a large cloud of axions in atomic Bohr orbits around the BH, through the effect of superradiance (SR). This results in a gap in the mass vs. spin distribution of BHs when the BH size is comparable to the axions Compton wavelength. By measuring the spin and mass of the merging objects observed at LIGO, we could verify the presence and shape of the gap in the BH distribution produced by the axion. The axion cloud can also be discovered through the GWs it radiates via axion annihilations or level transitions. A blind monochromatic GW search may reveal up to 10^5 BHs radiating through axion annihilations, at distinct frequencies within ~3% of $2 ma. Axion transitions probe heavier axions and may be observable in future GW observatories. The merger events are perfect candidates for a targeted GW search. If the final BH has high spin, a SR cloud may grow and emit monochromatic GWs from axion annihilations. We may observe the SR evolution in real time.
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