Detecting dark matter oscillations with gravitational waveforms

We consider the phase shift in the gravitational wave signal induced by fast oscillations of scalar dark matter surrounding binary systems, which could be probed by the future experiments LISA and DECIGO. This effect depends on the local matter density and the mass of the dark matter particle. We compare it to the phase shift due to a standard dynamical friction term, which should generically be present. We find that the effect associated with the oscillations only dominates over the dynamical friction for dark matter masses below $10^{-21}$ eV, with masses below $10^{-23}$ eV implying cloud sizes that are too large to be realistic. Moreover, for masses of the order of $10^{-21}$ eV, LISA and DECIGO would only detect this effect for dark matter densities greater than that in the solar system by a factor $10^5$ or $10^4$ respectively. We conclude that this signal can be ignored for most dark matter scenarios unless very dense clouds of very light dark matter are created early in the Universe at a redshift $z\sim 10^4$.