Bayesian parameter estimation in the second LISA Pathfinder mock data challenge

A main scientific output of the LISA Pathfinder mission is to provide a noise model that can be extended to the future gravitational wave observatory, LISA. The success of the mission depends thus upon a deep understanding of the instrument, especially the ability to correctly determine the parameters of the underlying noise model. In this work we estimate the parameters of a simplified model of the LISA Technology Package instrument. We describe the LISA Technology Package by means of a closed-loop model that is used to generate the data, both injected signals and noise. Then, parameters are estimated using a Bayesian framework, and it is shown that this method reaches the optimal attainable error, the Cramér-Rao bound. We also address an important issue for the mission: how to efficiently combine the results of different experiments to obtain a unique set of parameters describing the instrument.

Figure 1

The LTP MDC2 model. Left: Simplified scheme of the LTP instrument. Only two out of the four heterodyne interferometers are represented here: the one measuring spacecraft to first test mass distance, $o_{x1}$, and the one measuring test mass to test mass distance, $o_{x\Delta }$. See text for a description of terms appearing in the picture. Right: The previous is described as a control loop: the boxes describe the interferometer (IFO), controllers and dynamics of the test masses. The circles represent noise contributions, diamonds are signal injection points and the triangles denote cross-couplings between the first ($o_{x1}$) and second channel ($o_{x\Delta }$).

Figure 2

Amplitude spectral density of a noise realization of the LTP MDC2 noise model compared to analytical curves. We compare the noise of the first channel ($S_{\mathrm{x}1}$), the second channel ($S_{\mathrm{x}\Delta }$) and the absolute value of the cross-spectra between both ($S_{\mathrm{x}1:\mathrm{x}\Delta }$).

Figure 3

The three MDC2 experiments. From left to right: scheme of injected signal, input signal and output signal. From top to bottom: experiment 1, 2 and 3. Only $x_{12}$ output is shown for experiments 2 and 3, the response of the first channel to the injected signal is similar to the one shown in experiment 1.

Figure 4

Histograms of the MCMC samples illustrating the individual parameters’ marginal posterior probability distributions as computed with the last 3500 samples of the chain. All histograms are plotted with the same y axis range, up to 250 counts. Black vertical lines illustrate the true parameter values. Parameters $G_{\mathrm{df}}$, $G_{\mathrm{sus}}$ and ${\delta }_{21}$ are dimensionless; dimensions for stiffness parameters are $[{\omega }_1^2]=[{\omega }_2^2]={\mathrm{s}}^{-2}$.