4 Correlations in the orientation of clusters' shape

As described in Sect. 2 we determine the mass ellipsoid for each of the cluster sized halos. The distribution of the intrinsic shapes of clusters in our sample is illustrated in Fig. 2. There we plot the ratios between the lengths of the three principal axes $a\ge b\ge c$: the upper panel shows the ratios a/b versus a/c, whereas the lower panel shows b/c versus a/c. For prolate rotational ellipsoids (a>b, b = c) all points would coincide with the dashed diagonal on the upper panel. In the case of oblate rotational ellipsoids (a=b, a>c) all data points would coincide with the dashed diagonal plotted in the lower panel. Clearly the majority of the data points tend to be concentrated rather along the upper panels' diagonal than along the lower panels' diagonal. Therefore, the clusters in our simulations typically have prolate shape, in good agreement with observations (e.g. Basilakos et al. 2000; Cooray 2000) and other CDM simulations, cp. (Cole & Lacey 1996).

Figure 2: Ratios of the length of the clusters principal axes ( $a\ge b\ge c$). Upper panel: a/b versus a/c, for prolate rotational ellipsoids (a>b, b = c) all points would coincide with the dashed diagonal; lower panel: b/c versus a/c, for oblate rotational ellipsoids (a=b, a>c) all data points would coincide with the dashed diagonal.

In the following we will use the direction of the major axis of the mass ellipsoid (the "a-axis'') as a vector mark ${\vec{l}}$. Figure 3 shows the mark correlation functions ${\cal A}(r)$ and ${\cal F}(r)$. The increased ${\cal A}(r)$ towards smaller scales indicates that pairs of clusters with a distance smaller than 30 h^-1 Mpc prefer a parallel orientation of their orientation axes ${\vec{l}}_1$ and ${\vec{l}}_2$. This deviation of $\sim$$10\%$ from the signal of a purely random orientation is clearly outside the random fluctuations shown by the shaded region. We determine these random fluctuations by repeatedly and randomly redistributing the orientation axes among the clusters. The positions of the clusters remain fixed. The filamentary alignment of the orientation of the clusters ${\vec{l}}_1$and ${\vec{l}}_2$ towards the connecting vector $\hat{\vec{r}}$ as quantified by ${\cal F}(r)$ is significantly stronger (up to 20% deviation from the signal of purely random orientation on small scales). Remarkably this signal extends out to 100 h^-1 Mpc. In a qualitative picture this may be explained by a large number of clusters elongated in the direction of the filaments of the large scale structures. Filaments are prominent features found in the observed galaxy distribution (Huchra et al. 1990) as well as in N-body simulations (Melott & Shandarin 1990), often with an extent up to 100 h^-1 Mpc. The inspection of the density field of the simulation utilized here confirms this view (see Fig. 5).

Figure 3: Mark correlations using the 3D orientation of the dark matter halo, specified by the direction of the major axis ${\vec{l}}$ of the mass ellipsoid, as vector mark. The shaded area is obtained by randomizing the orientation among the clusters.

Up to now we used the orientation of the mass ellipsoid in three dimensional space. Most observations, however, only provide the projected galaxy number density or X-ray intensity. In the following we will investigate whether the clear signal found in Fig. 3 will be reduced by projection. In close analogy to our three dimensional analysis we determine the orientation of the mass-ellipse of a halo from the mass-density orthogonally projected onto a side of the simulation box. We repeat our analysis with the major axis of the mass ellipse in two-dimensions as a two-dimensional vector mark. Also the normalized direction $\hat{\vec{r}}$is given in the plane. However, for the radial distance r we use the three dimensional separation of the clusters. With this setting, we mimic the observational constraints, e.g. in the REFLEX cluster survey. In this projected sample the signal in ${\cal A}(r)$ is only visible on scales smaller than 10 h^-1 Mpc (Fig. 4). The amplitude of the deviation from random orientation (in two dimension $2/\pi$) is reduced, as well. But still, the filamentary alignment quantified with ${\cal F}(r)$ indicates a strong alignment of the orientation of the clusters with the connecting vector (Fig. 4). Although the amplitude is slightly reduced, this alignment is still visible out to scales of 100 h^-1 Mpc, as in the three dimensional analysis.

Figure 4: Mark correlations with the normalized major axis $\vert{\vec{l}}\vert=1$ of the projected cluster-density (2D orientation) as vector mark. The shaded area is obtained by randomizing the orientation among the clusters.

The question arises, why do we find two different regimes of the vector correlations, using either ${\cal A}(r)$ or ${\cal F}(r)$. The direct alignment, quantified with ${\cal A}(r)$ extends out to 30 h^-1 Mpc, whereas the filamentary alignment ( ${\cal F}(r)$) is visible out to 100 h^-1 Mpc. A glimpse on the real density distribution in Fig. 5 sheds light on this topic. Clearly one can see filamentary structures with sizes extending up to 100 h^-1 Mpc, for example the two pronounced filaments which end up in the knot at $x\approx275~$h^-1 Mpc and $y\approx75~$h^-1 Mpc. Generically, such filaments are not straight but crinkled and show a lot of small branches. This implies that the coherence of the angles between the major axes of clusters harbored in these filamentary structures is lost soon. Consequently the positive correlation seen with ${\cal A}(r)$ is confined to comparatively small scales $\lesssim$30 h^-1 Mpc (in the case of projected data even to $\lesssim$10 h^-1 Mpc). On the other hand, the signal exploring the angle between orientation and connecting vector is not affected by the small scale disorder. Thus ${\cal F}(r)$ is tracing the filamentary alignment out to scales of 100 h^-1 Mpc.

Figure 5: Projected density field of a $\sim$7.5 h-1 Mpc slice through the simulation box. The density is calculated on a 3003 grid which corresponds to a smoothing length of $\sim$1.7 h-1 Mpc. The maximum value of projected density is $\approx$ $1.5\times 10^4 \rho /\rho _{\rm mean}$. Gray scales are chosen according to $\log$ of density.