Capacitive Sensing of Test Mass Motion with Nanometer Precision over Millimeter-Wide Sensing Gaps for Space-Borne Gravitational Reference Sensors

We report on the performance of the capacitive gap-sensing system of the Gravitational Reference Sensor on board the LISA Pathfinder spacecraft. From in-flight measurements, the system has demonstrated a performance, down to 1 mHz, that is ranging between 0.7 and $1.8\text{ }\text{ }\mathrm{aF}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$. That translates into a sensing noise of the test mass motion within 1.2 and $2.4\text{ }\mathrm{nm}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ in displacement and within 83 and $170\text{ }\mathrm{nrad}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ in rotation. This matches the performance goals for LISA Pathfinder, and it allows the successful implementation of the gravitational waves observatory LISA. A $1/f$ tail has been observed for frequencies below 1 mHz, the tail has been investigated in detail with dedicated in-flight measurements, and a model is presented in the paper. A projection of such noise to frequencies below 0.1 mHz shows that an improvement of performance at those frequencies is desirable for the next generation of gravitational reference sensors for space-borne gravitational waves observation.