A New Probe of $\mu$Hz Gravitational Waves with FRB Timing

We propose Fast Radio Burst (FRB) timing, which uses the precision measurements of the arrival time differences of repeated FRB signals along multiple sightlines, as a new probe of gravitational waves (GWs) around nHz to $\mu$Hz frequencies, with the highest frequency limited by FRB repeating period. The anticipated experiment requires a sightline separation of tens of AU, achieved by sending radio telescopes to space. We find the signal of arrival time difference induced by GWs depends only on the local GWs in the solar system and we can correlate the measurements from different FRB sources or the same source with different repeaters, which leads to a better sensitivity with a larger number of FRB repeaters detected. The projected sensitivity shows this method is a competitive probe in the nHz to $\mu$Hz frequency range. It can fill the '$\mu$Hz gap' between pulsar timing arrays and Laser Interferometer Space Antenna (LISA) and is complementary to other proposals of GW detection in this frequency band.