$30$ particles. The subsample with the most massive haloes starts at $1000$ particle members corresponding to $M\geq 7.2\times 10^{13}\,h^{-1}\,M_{\odot}$.

=0.5

The left panels of Figure \[fig:3D\] show the cross-correlation function between low mass DM haloes and DM particles (top-left) in the directions parallel and perpendicular to their angular momenta, and in all directions (diamonds, triangles and squares, respectively). The lower panel shows the ratios between these three estimates and the correlation function considering all neighbors; their relative amplitudes help identify alignments in either direction. As can be seen, the relative amplitudes of the correlation function when using neighbors in the directions parallel and perpendicular to the angular momentum of the DM halo are different; for small separations $R<1$h$^{-1}$Mpc, there is an excess of correlation in the direction perpendicular to the angular momentum of up to a $20\%$ effect (between correlations parallel and perpendicular to the angular momentum, lower panel), whereas for larger separations the alignment diminishes and shows a tendency to reverse (although, as can be seen in the figure, errors are larger at these separations); we come back to this later in this section where we use a higher signal estimator for alignments.

When using higher mass DM haloes, 