  We show that the existence of clouds of ultralight particles surrounding
black holes during their cosmological history as members of a binary system can
leave a measurable imprint on the distribution of masses and orbital
eccentricities observable with future gravitational-wave detectors. Notably, we
find that for nonprecessing binaries with chirp masses ${\cal M} \lesssim
10\,M_\odot$, formed exclusively in isolation, larger-than-expected values of
the eccentricity, i.e. $e\gtrsim 10^{-2}$ at gravitational-wave frequencies
$f_{\rm GW} \simeq 10^{-2}\,$Hz, would provide tantalizing evidence for a new
particle of mass between $[0.5,2.5] \times 10^{-12}\,$eV in nature. The
predicted evolution of the eccentricity can also drastically affect the in-band
phase evolution and peak frequency. These results constitute unique signatures
of boson clouds of ultralight particles in the dynamics of binary black holes,
which will be readily accessible with the Laser Interferometer Space Antenna,
as well as future mid-band and Deci-hertz detectors.
