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tokens: that gas outside halos, not necessarily following any given profile, also contributes to the signal. Furthermore, we would like to conduct an analysis that is as model independent as possible, and thus blind to any specific model for the gas spatial distribution. The correction by a redshift-averaged quantity introduced in the last equation above was justified by Hand et al. (2012) through the need to correct for possible redshift evolution of the tSZ signal in the sources. In practice, this correction allows us to measure relative changes in temperature anisotropies between nearby galaxy pairs after minimizing other noise sources, such as CMB residuals. Fig. 1 Absolute value of versus distance to the observer after obtaining AP kSZ temperature estimates on the real positions of CGs (black line) and on a rotated CG configuration (red line). The green line provides the theoretical expectation for the rms of , and the blue histogram displays the radial distribution of the CGs (see text for further details). We next conduct an analysis into the motivation behind this -dependent monopole correction. In Fig. 1 we display the amplitude of for versus the comoving distance to the observer . The value of the aperture chosen in this exercise is 8 arcmin. The black line refers to the real position of the CGs, while the red line refers to a rotated-on-the-sky configuration of the CGs. Since we are plotting absolute values of , the solid (dashed) parts of these lines refer to positive (negative) values of . The amplitude of these curves is close to the solid green line, which corresponds to the theoretical prediction for the rms of under the assumption that the AP temperature estimates are dominated by CMB residuals. That is, the amplitude of the green curve equals , where K is the rms of the 8 arcmin AP kSZ estimates and is the number of CGs effectively falling under the redshift Gaussian window given in Eq. ( 5 ). While K is computed from real data, its amplitude is very close to the theoretical predictions of Fig. 6 in Hernández-Monteagudo et al. (2006b) , for an angular aperture of 8 arcmin after considering the CMB exclusively. We can see that the amplitude of is lowest for those distances where the number density of CGs is highest (as displayed, in arbitrary units, by the blue histogram in Fig. 1 ). Therefore Fig. 1 shows not only that AP kSZ estimates are dominated by CMB residuals, but also that the difference in for CG pair members lying 50–100 Mpc away in radial distance typically amounts to few times K. We shall show below that this amplitude is not completely negligible when compared to the typical kSZ pairwise momentum amplitude between CGs at large distances, and thus subtracting becomes necessary. However, this should not be the case when cross-correlating AP kSZ measurements with estimates of radial peculiar velocities (see Sect. 3.2 below), since in this case the residual will not be correlated with those velocities and should not