Sub-Femto-$g$ Free Fall for Space-Based Gravitational Wave Observatories: LISA Pathfinder Results

We report the first results of the LISA Pathfinder in-flight experiment. The results demonstrate that two free-falling reference test masses, such as those needed for a space-based gravitational wave observatory like LISA, can be put in free fall with a relative acceleration noise with a square root of the power spectral density of $5.2\pm 0.1\mathrm{fm}{\mathrm{s}}^{-2}/\sqrt{\mathrm{Hz}}$, or $(0.54\pm 0.01)\times {10}^{-15}$g$/\sqrt{\mathrm{Hz}}$, with g the standard gravity, for frequencies between 0.7 and 20 mHz. This value is lower than the LISA Pathfinder requirement by more than a factor 5 and within a factor 1.25 of the requirement for the LISA mission, and is compatible with Brownian noise from viscous damping due to the residual gas surrounding the test masses. Above 60 mHz the acceleration noise is dominated by interferometer displacement readout noise at a level of $(34.8\pm 0.3)\mathrm{fm}/\sqrt{\mathrm{Hz}}$, about 2 orders of magnitude better than requirements. At $f\le 0.5\mathrm{mHz}$ we observe a low-frequency tail that stays below $12\mathrm{fm}{\mathrm{s}}^{-2}/\sqrt{\mathrm{Hz}}$ down to 0.1 mHz. This performance would allow for a space-based gravitational wave observatory with a sensitivity close to what was originally foreseen for LISA.

Figure 1

Gray: ASD of $\mathrm{\Delta }g$, $S_{\mathrm{\Delta }g}^{1/2}(f)$, measured for 6.5 days starting 127 days after launch. The ASD is the result of averaging 26 periodograms of 40 000 s each, which results in a relative error ($1\sigma$) of 10% in $S_{\mathrm{\Delta }g}^{1/2}$. The effective spectral resolution, set by the spectral window, is $\mathrm{\Delta }f\simeq \pm 50\mu \mathrm{Hz}$. The absolute calibration of the measurement is better than 5%. Red: ASD of the same time series after correction for the centrifugal force (visible at the lowest frequencies). Light blue: ASD after correction for the pickup of spacecraft motion by the interferometer (IFO), visible in the 20–200 mHz range. Dashed smooth black line: $S_{\mathrm{\Delta }g}(f)=S_0+S_{\mathrm{IFO}}{(2\pi f)}^4$ with $S_0^{1/2}=(5.57\pm 0.04)\mathrm{fm}{\mathrm{s}}^{-2}/\sqrt{\mathrm{Hz}}$ and $S_{\mathrm{IFO}}^{1/2}=(34.8\pm 0.3)\mathrm{fm}/\sqrt{\mathrm{Hz}}$. Note that the level of $S_0$ has decreased further in subsequent measurements, as quoted in the abstract and shown in Fig. 3. Shaded areas: LISA and LISA Pathfinder requirements for $\mathrm{\Delta }g$. The LISA single test-mass acceleration requirement [2] has been multiplied by $\sqrt{2}$ to be presented here as a differential acceleration.

Figure 2

A schematic of LPF. The figure shows TM1, TM2, and the optical bench beam paths for measuring $\mathrm{\Delta }x$ and $x_1$. The measurement of $\mathrm{\Delta }x$ drives the electrostatic suspension of TM2, which applies the necessary electrostatic forces by means of the electrodes represented by the four gold plates facing TM2. All other electrodes surrounding the TMs are not shown. The measurement of $x_1$ drives the drag-free control loop that uses the micronewton thrusters to exert forces on the spacecraft. The figure depicts the $x$ and $y$ axes we use in this Letter, while $z$ is normal to the figure.

Figure 3

(a) Square root of the averaged power spectral density of $\mathrm{\Delta }g$ in the 3–8 mHz and 0.1–0.4 mHz frequency bands as a function of time. This is calculated from a numerical integration over the relevant band, $\overline{S}\equiv \frac{1}{f_2-f_1}{\int }_{f_1}^{f_2}S(f)df$. (b) Quasistatic value of $\overline{\mathrm{\Delta }g}$ as a function of time, both for the raw time series and, to highlight the unexplained portion of this, the same data corrected for the differential gravitation calculated from the depletion of propellant mass. This correction is large and negative, resulting in consistently higher $\overline{\mathrm{\Delta }g}$ values for the corrected data series compared to the native one. All time series are based on 100 000 s groups of 50% overlapped windows from the 11 noise runs, with 10 000 s windows used for the 3–8 mHz band and 40 000 s for the other three data series. In all cases the errors are assigned based on the scatter between the averaged windows. Gaps in the data series shown here are due to the presence of both station-keeping maneuvers and other experiments.