A pseudo-spectrum analysis of galaxy-galaxy lensing

We present the application of the pseudo-spectrum method to galaxy-galaxy lensing. We derive explicit expressions for the pseudo-spectrum analysis of the galaxy-shear cross spectrum, which is the Fourier space counterpart of the stacked galaxy-galaxy lensing profile. The pseudo-spectrum method corrects observational issues such as the survey geometry, masks of bright stars and their spikes, and inhomogeneous noise, which distort the spectrum and also mix the E-mode and the B-mode signals. Using ray-tracing simulations in N-body simulations including realistic masks, we confirm that the pseudo-spectrum method successfully recovers the input galaxy-shear cross spectrum. We also investigate the covariance of the galaxy-shear cross spectrum using the ray-tracing simulations to show that there is an excess covariance relative to the Gaussian covariance at small scales where the shot noise is dominated in the Gaussian approximation. We find that the excess of the covariance is consistent with the expectation from the halo sample variance (HSV), which originates from the matter fluctuations at scales larger than the survey area. We apply the pseudo-spectrum method to the observational data of Canada-France-Hawaii Telescope Lensing survey (CFHTLenS) shear catalogue and three different spectroscopic samples of Sloan Digital Sky Survey Luminous Red Galaxy (SDSS LRG), and Baryon Oscillation Spectroscopic Survey (BOSS) CMASS and LOWZ galaxies. The galaxy-shear cross spectra are significantly detected at the level of 7-10$\sigma$ using the analytic covariance with the HSV contribution included. We also confirm that the observed spectra are consistent with the halo model predictions with the halo occupation distribution parameters estimated from previous work. This work demonstrates the viability of galaxy-galaxy lensing analysis in the Fourier space.