Seismic gravity-gradient noise in interferometric gravitational-wave detectors

When ambient seismic waves pass near an interferometric gravitational-wave detector, they induce density perturbations in the earth which produce fluctuating gravitational forces on the interferometer's test masses. These forces mimic a stochastic background of gravitational waves and thus constitute noise. We compute this noise using the theory of multimode Rayleigh and Love waves propagating in a layered medium that approximates the geological strata at the LIGO sites. We characterize the noise by a transfer function $T(f) \equiv \tilde x(f)/\tilde W(f)$ from the spectrum of direction averaged ground motion $\tilde W(f)$ to the spectrum of test mass motion $\tilde x(f) = L\tilde h(f)$ (where $L$ is the length of the interferometer's arms, and $\tilde h(f)$ is the spectrum of gravitational-wave noise). This paper's primary foci are (i) a study of how $T(f)$ depends on the various seismic modes; (ii) an attempt to estimate which modes are excited at the LIGO sites at quiet and noisy times; and (iii) a corresponding estimate of the seismic gravity-gradient noise level. At quiet times the noise is below the benchmark noise level of ``advanced LIGO interferometers'' (although not by much near 10 Hz); it may significantly exceed this level at noisy times. The lower edge of our quiet-time noise is a limit beyond which there is little gain from further improvements in vibration isolation and thermal noise, unless one also reduces seismic gravity-gradient noise. Two methods of reduction are discussed: monitoring the earth's density perturbations, computing their gravitational forces, and correcting the data for those forces; and constructing narrow moats around the interferometers' test masses to shield out the fundamental-mode Rayleigh waves, which we suspect dominate at quiet times.