Measuring test mass acceleration noise in space-based gravitational wave astronomy

The basic constituent of interferometric gravitational wave detectors—the test-mass-to-test-mass interferometric link—behaves as a differential dynamometer measuring effective differential forces, comprising an integrated measure of gravity curvature, inertial effects, as well as nongravitational spurious forces. This last contribution is going to be characterized by the LISA Pathfinder mission, a technology precursor of future space-borne detectors like eLISA. Changing the perspective from displacement to acceleration can benefit the data analysis of LISA Pathfinder and future detectors. The response in differential acceleration to gravitational waves is derived for a space-based detector’s interferometric link. The acceleration formalism can also be integrated into time delay interferometry by building up the unequal-arm Michelson differential acceleration combination. The differential acceleration is nominally insensitive to the system’s free evolution dominating the slow displacement dynamics of low-frequency detectors. Working with acceleration also provides an effective way to subtract measured signals acting as systematics, including the actuation forces. Because of the strong similarity with the equations of motion, the optimal subtraction of systematic signals, known within some amplitude and time shift, with the focus on measuring the noise provides an effective way to solve the problem and marginalize over nuisance parameters. The $\mathcal{F}$ statistic, in widespread use throughout the gravitation waves community, is included in the method and suitably generalized to marginalize over linear parameters and noise at the same time. The method is applied to LPF simulator data and, thanks to its generality, can also be applied to the data reduction and analysis of future gravitational wave detectors.

Figure 1

TDI configuration in eLISA with one “mother” ${\mathrm{SC}}_1$ hosting two TMs and two “daughter” SCs. The SCs are labeled clockwise, while the arms are labeled counterclockwise; the light delays are labeled with primed delays clockwise and unprimed delays counterclockwise. The unequal arm interferometer ${\mathit{X}}_a$ [see Eq. (6)] is physically equivalent to the difference of a synthetic light beam originating from ${\mathrm{SC}}_1$ and bouncing first off ${\mathrm{SC}}_2$ and then off ${\mathrm{SC}}_3$ and another one bouncing first off ${\mathrm{SC}}_3$ and then off ${\mathrm{SC}}_2$.

Figure 2

Differential acceleration and displacement time-series for the nominal LPF measurement configuration. The dynamical range is dramatically reduced from displacement to acceleration. The measured acceleration is nominally insensitive to the free evolution of the system. Fitting directly acceleration turns out to be much easier than displacement.

Figure 3

Contributions to a one-day measurement of the residual differential acceleration, $a_{12,\mathrm{r}}$. In sequence, measured signals corresponding to the electrostatic suspension actuation and the two displacement outputs have been coherently subtracted from the total acceleration $a_{12}$. All signals injected in the main frequency band (the peaks between 1 and 50 mHz) are suppressed, thus allowing the accurate recovery of the noise floor, $a_{12,\mathrm{n}}$ down to a fraction of mHz.