Quantifying the cosmic web - I. The large-scale halo ellipticity-ellipticity and ellipticity-direction correlations

The formation of dark matter haloes tends to occur anisotropically along the filaments of the cosmic web, which induces ellipticity-ellipticity (EE) correlations between the shapes of haloes, as well as ellipticity-direction (ED) cross-correlations between halo shapes and the directions to neighbouring haloes. We analyse the halo catalogue and the semi-analytic galaxy catalogue of the recent Millennium Run Simulation to measure the EE and ED correlations numerically at four different redshifts (z = 0, 0.5, 1 and 2). For the EE correlations, we find that (i) the major-axis correlation is strongest while the intermediate-axis correlation is weakest; (ii) the signal is significant at distances out to 10 h(-1) Mpc; (iii) the signal decreases as z decreases and (iv) its behaviour depends strongly on the halo mass scale, with larger masses showing stronger correlations at large distances. For the ED correlations, we find that (i) the correlations are much stronger than the EE correlations, and are significant even out to distances of 50 h(-1) Mpc; (ii) the signal also decreases as z decreases and (iii) it increases with halo mass at all distances. We also provide empirical fitting functions for the EE and ED correlations. The EE correlations are found to scale linearly with the linear density correlation function, xi(r), while the ED cross-correlation is found to scale as xi(1/2)(r) at large distances beyond 10 h(-1) Mpc. The best-fitting values of the fitting parameters for the EE and the ED correlations are all determined through chi(2)-statistics. Our results may be useful for quantifying the filamentary distribution of dark matter haloes over a wide range of scales.