Mock LISA data challenge for the Galactic white dwarf binaries

We present data analysis methods used in the detection and estimation of parameters of gravitational-wave signals from the white dwarf binaries in the mock LISA data challenge. Our main focus is on the analysis of challenge 3.1, where the gravitational-wave signals from more than $6\times {10}^7$ Galactic binaries were added to the simulated Gaussian instrumental noise. The majority of the signals at low frequencies are not resolved individually. The confusion between the signals is strongly reduced at frequencies above 5 mHz. Our basic data analysis procedure is the maximum likelihood detection method. We filter the data through the template bank at the first step of the search, then we refine parameters using the Nelder-Mead algorithm, we remove the strongest signal found and we repeat the procedure. We detect reliably and estimate parameters accurately of more than ten thousand signals from white dwarf binaries.

Figure 1

Comparison of the power spectra of the gravitational-wave signal response using the analytic formulas presented in this paper and Synthetic LISA [24].

Figure 2

Illustration of Algorithm  in two dimensions. Initial optimal lattice (top left) with the basis $\{{\overrightarrow{w}}_1,{\overrightarrow{w}}_2\}$, resolution vector ${\overrightarrow{v}}_0$ defining constraint surface and vector $\overrightarrow{l}$ is contracted along the vector $\overrightarrow{l}$ (top right) and rotated (bottom left) such that for the new lattice $\overrightarrow{l}={\overrightarrow{v}}_0$. Final suboptimal lattice has basis $\{{\overrightarrow{v}}_0,{\overrightarrow{w}}_1^{\prime }\}$ (bottom right).

Figure 3

Thickness of the lattices in 3 and 4 dimensions compared to the optimal thickness of $A_3^{*}\approx 1.4635$ and $A_4^{*}\approx 1.7655$. The upper diagram shows also the optimal lattice with covering radius $R(A_3^{*}{)}^2=5/48{\pi }^2$. The Voronoi cell, basis vectors, resolution vector, and part of the constraint surface are depicted; for this specific value of $R$ the head of the resolution vector lies on the constraint surface.

Figure 4

Estimation of the signals from white dwarf binaries in the challenge 3.1 data set for the band from 5 mHz to 5.1 mHz. Black color denotes the original data and the light (blue) color is the data after signals removed.

Figure 5

The zoom of Fig. 4 around the strongest signal identified.

Figure 6

Detection and correlations for the blind challenge 3.1 data set. The left panel shows the number of the signals detected by our search and selected by the procedure described in the text as a function of the frequency. The right panel displays the histogram of the correlation functions [Eq. (95)] between our estimated signals and the ones from the key.

Figure 7

Number of correlations of estimated signals with key signals as a function of the signal-to-noise ratio for signals with frequency below 3 mHz.

Figure 8

Errors of the parameters of the signals detected and verified in our search of the challenge 3.1 data set.

Figure 9

Errors of the estimation of parameters as in Fig. 8 but normalized by the variances obtained from the inverse of the Fisher information matrix.

Figure 10

Amplitude square of the Fourier transformed data. Upper signal is original challenge data (A-channel, includes full Galaxy and instrumental noise), middle is the data with selected removed signals, and lower is the data with all identified signals removed.

Figure 11

Power spectral density. Upper signal is the original challenge data, middle is reduced data (after removing all found signals), and lower is the instrumental noise.