Optimal design of calibration signals in space-borne gravitational wave detectors

Future space-borne gravitational wave detectors will require a precise definition of calibration signals to ensure the achievement of their design sensitivity. The careful design of the test signals plays a key role in the correct understanding and characterization of these instruments. In that sense, methods achieving optimal experiment designs must be considered as complementary to the parameter estimation methods being used to determine the parameters describing the system. The relevance of experiment design is particularly significant for the LISA Pathfinder mission, which will spend most of its operation time performing experiments to characterize key technologies for future space-borne gravitational wave observatories. Here we propose a framework to derive the optimal signals—in terms of minimum parameter uncertainty—to be injected into these instruments during the calibration phase. We compare our results with an alternative numerical algorithm which achieves an optimal input signal by iteratively improving an initial guess. We show agreement of both approaches when applied to the LISA Pathfinder case.

Figure 1

Evolution of the algorithm to optimize the input signal for the harmonic oscillator case. The algorithm promotes the natural frequency of the oscillator ${\omega }_0=0.07$.

Figure 2

Comparison of the noise spectra for the two main interferometer channels for an analytical simple model (dashed line) and a noise data stream generated via a LPF state-space model (solid line).

Figure 3

Left: evaluation of the eigenvalue ${\lambda }_2$ of the $F_{2121}$ term for a single frequency injection. Right: Output of the numerical algorithm for the optimization of input signals based on the dispersion function applied to a LISA Pathfinder state-space model after 25 iterations. In this case, white noise was injected into the drag-free channel. In both cases (analytical and numerical), the analysis is repeated by rescaling by a factor of 25 and 50 the value of the second test mass stiffness, which is originally considered to be ${\omega }_0^2=-22\times {10}^{-6}{\mathrm{s}}^{-2}$.

Figure 4

Determinant of the $F_{2121}$ term for an injected signal with two independent frequencies. The determinant is evaluated for the case ${\omega }_2^2=-22\times {10}^{-7}{\mathrm{s}}^{-2}$ (left) and ${\omega }_2^2=50\times (-22\times {10}^{-7}){\mathrm{s}}^{-2}$ (right). The values on the legend are the $\log (\det [{\mathrm{F}}_{2121}])$.

Figure 5

Expected error on parameters for an injection in the drag-free channel considering ${\omega }_1^2$ and ${\omega }_2^2$ as the only relevant parameters. Black corresponds to the initial proposal of a white noise input, blue represents the expected error for the input signal as obtained with the proposed numerical algorithm after 25 iterations. Histograms are computed based on 5000 samples of a multivariate Gaussian distribution with the covariance matrix obtained by the numerical algorithm.

Figure 6

Normalized input spectrum for a Monte Carlo analysis running the experiment design algorithm, considering two input channels and five parameters. The solid line represents the mean, for each channel, of the 1000 runs while the grey shadow is the corresponding standard variation.