LISA measurement of gravitational wave background anisotropy: Hexadecapole moment via a correlation analysis

We discuss spatial fluctuations in the gravitational wave background arising from unresolved Galactic binary sources, such as close white dwarf binaries, due to the fact the galactic binary source distribution is anisotropic. We introduce a correlation analysis of the two data streams of the Laser Interferometer Space Antenna (LISA) to extract spherical harmonic coefficients of the hexadecapole moment (l=4) related to the projected two-dimensional density distribution of the binary source population in an independent manner. In this analysis, we use a free parameter with which we can effectively control the frequency of the cartwheel rotation of LISA. The proposed technique complements and improves over previous suggestions in the literature to measure the gravitational wave background anisotropy based on the time modulation of data as LISA orbits around the Sun. Such techniques, however, are restricted only to certain combinations of spherical harmonic coefficients of the galaxy with no ability to separate them individually. With LISA, m=2,3 and 4 coefficients of the hexadecapole (l=4) can be measured with signal-to-noise ratios at the level of ten and above in a certain coordinate system. In addition to the hexadecapole coefficients, when combined with the time modulation analysis, the correlation study also can be used, in principle, to measure quadrupole coefficients of the binary distribution.