Maximum likelihood map making with the Laser Interferometer Space Antenna

Given the recent advances in gravitational-wave detection technologies, the detection and characterization of gravitational-wave backgrounds (GWBs) with the Laser Interferometer Space Antenna (LISA) is a real possibility. To assess the abilities of the LISA satellite network to reconstruct anisotropies of different angular scales and in different directions on the sky, we develop a map-maker based on an optimal quadratic estimator. The resulting maps are maximum likelihood representations of the GWB intensity on the sky integrated over a broad range of frequencies. We test the algorithm by reconstructing known input maps with different input distributions and over different frequency ranges. We find that, in an optimal scenario of well understood noise and high frequency, high SNR signals, the maximum scales LISA may probe are ${\ell }_{\max }\lesssim 15$. The map-maker also allows to test the directional dependence of LISA noise, providing insight on the directional sky sensitivity we may expect.

Figure 1

Illustrations of the TDI 1 (left panel) and TDI 1.5 configuration $X$ (right panel). The three numbered grey circles represent the spacecrafts arranged in an equilateral triangle. The links are labeled ${\mathit{\ell }}_i$ in the left panel for clarity, and are the same as the ones in the right panel.

Figure 2

Normalized autocorrelated response of TDI channel X, ${\mathit{A}}_{XX}$, at time $t=0$ and in the Solar System barycenter (SSB) reference frame, at frequencies $f={10}^{-4}\mathrm{Hz}$, $f={10}^{-1}\mathrm{Hz}$, $f=5\times {10}^{-1}\mathrm{Hz}$ from left to right respectively.

Figure 3

Sky-integrated autocorrelated and cross-correlated responses, $A_{XX}$ and $A_{XY}$, across the frequency spectrum. Note that the slopes in the low frequency limit scale as $\propto f^4$.

Figure 4

Data power sample from the autocorrelated X channel, decomposed into injected signal (orange line) and simulated noise (red line) components. The Injected signal has and SNR of 0.6, and is evidently buried in the noise. The sample represents the data accumulated over a two-day period and then FFTed. The dashed grey lines are the noise model which informs the noise generation; the darker line is renormalized by a factor which accounts for the lower sampling rate, while the lighter line is the true instantaneous noise curve used to calculate the SNR.

Figure 5

An example input map from the simulations (left panel) to be compared to the final output maps obtained integrating with different frequency cutoffs, $f_{\max }=0.1\mathrm{Hz}$ (central panel) and $f_{\max }=0.01\mathrm{Hz}$ (right panel). These highlight the different resolutions the LISA channels have in different ranges of frequency.

Figure 6

Transfer functions $T_{\ell }$ for the average reconstructed $C_{\ell }\mathrm{s}$ obtained with different frequency cutoffs (left panel) and different spectral shapes, $\alpha =3$ and $\alpha =0$, both in the high frequency case $f_{\max }=0.1\mathrm{Hz}$ (right panel). Each simulation set consists of 50 maps, each a different realization of the same $C_{\ell }$ input. There appears to be a clear one-to-one relation between the resolution ${\ell }_{\max }$ of the instrument and the frequency cutoff. Conversely, there is an average difference of 5% between the transfer functions obtained with different spectral weightings, however this does not affect the resolution cutoff.

Figure 7

Input (left panel) and reconstructed output (central panel) maps in a low SNR case, integrating over a frequency range of $[{10}^{-4},{10}^{-1}]\mathrm{Hz}$. The input map is an $\ell C_{\ell }$ flat gaussian realization with ${\ell }_{\max }=10$, and in this case $\alpha =2/3$. The output map is a heavily smoothed version of the input, with some loss of power due to the strong conditioning required. The SNR of this particular reconstruction is shown in the right panel.

Figure 8

Sky distribution of the noise in the SSB frame. Note the imprinted 6-fold symmetry of the orbit, given by the three concentric orbits of the spacecrafts.

Figure 9

Sky inputs (left) and reconstructed outputs (right) for a Milky Way-like GWB distribution. The signal is injected in the range $[{10}^{-4},5\times {10}^{-3}]\mathrm{Hz}$. The top row is an extremely high SNR case, whereas the bottom row is an SNR 52, hence it emerges above the noise around ${10}^{-3}\mathrm{Hz}$. Both results are obtained over a full year of integration. These Mollweide projections are presented in log scale.