3 Mark correlation functions

Mark correlation functions (MCFs) have been used for the characterization of marked point sets for quite some time (see Stoyan & Stoyan 1994 for an instructive introduction and overview). Beisbart & Kerscher (2000) introduced MCFs to cosmology and used them to quantify the luminosity and morphology dependent clustering in the observed galaxy distribution (see also Szapudi et al. 2000). Gottlöber et al. (2002) discussed the correlations of galaxy sized halos depending on the merging history with MCFs (see also Gottlöber et al. 2001). In a recent review Beisbart et al. (2002) show other physical applications of the MCFs and also how the MCFs can be calculated from stochastic models. To make this article self contained, we repeat briefly the basics of MCFs and then extend the formalism to allow for vector marks:

Consider the set of N points $\{{\vec{x}}_i\}_{i=1}^{N}$ and attach a mark m_i to each point ${\vec{x}}_i\in{\mathbb{R}}^3$ resulting in the marked point set $\{({\vec{x}}_i,m_i)\}_{i=1}^{N}$(Stoyan 1984; Stoyan & Stoyan 1994). In the following we use a variety of marks: the mass, the absolute value of the angular momentum, the spin parameter $\lambda$, and as vector valued marks, the principal axes of the second moment of the mass distribution and the angular momentum. Let $\rho$ be the mean number density of the points in space and $\rho^M(m)$ the probability density of the mark distribution. The mean mark is then $\overline{m}=\int{\rm d}m\ \rho^M(m)m$, and the variance is $V[m]=\int{\rm d}m\ \rho^M(m) (m-\overline{m})^2$. We assume that the joint probability $\rho^{SM}({\vec{x}},m)$ of finding a point at position ${\vec{x}}$ with mark M, splits into a space-independent mark probability and the constant mean density: $\rho^M(m)\times\rho$.
The spatial-mark product-density

$$\rho_2^{SM}(({\vec{x}}_1,m_1),({\vec{x}}_2,m_2))\ {\rm d}V_1 {\rm d}m_1\ {\rm d}V_2 {\rm d}m_2 ,$$ (2)

is the joint probability of finding a point at ${\vec{x}}_1$ with the mark m₁ and another point at ${\vec{x}}_2$ with the mark m₂. We obtain the spatial product density $\rho_2({\vec{x}}_1,{\vec{x}}_2)$ and the two-point correlation function $\xi(r)$ by marginalizing over the marks:

$$\rho^2\ (1+\xi(r)) \rho_2({\vec{x}}_1,{\vec{x}}_2)$$ =

$$\int{\rm d}m_1 \int{\rm d}m_2\ \rho_2^{SM}(({\vec{x}}_1,m_1),({\vec{x}}_2,m_2)) ,$$ = (3)

with $\xi(r)$ only depending on the distance $r=\vert{\vec{x}}_1-{\vec{x}}_2\vert$ of the points in a homogeneous and isotropic point set.
We define the conditional mark density:

$${\cal M}_2(m_1,m_2\vert{\vec{x}}_1,{\vec{x}}_2) =\left\{\begin{ar... ...rho_2({\vec{x}}_1,{\vec{x}}_2)\ne 0,\\ 0 & \textrm{ else}. \end{array} \right.$$ (4)

For a stationary and isotropic point distribution, ${\cal M}_2(m_1,m_2\vert r){\rm d}m_1{\rm d}m_2$ is the probability of finding the marks m₁ and m₂ of two galaxies located at ${\vec{x}}_1$ and ${\vec{x}}_2$, under the condition that they are separated by $r=\vert{\vec{x}}_1-{\vec{x}}_2\vert$. Now the full mark product-density can be written as

$$\rho_2^{SM}(({\vec{x}}_1,m_1),({\vec{x}}_2,m_2)) ={\cal M}_2(m_1,m_2\vert{\vec{x}}_1,{\vec{x}}_2)\ \rho_2({\vec{x}}_1,{\vec{x}}_2) .$$ (5)

If there is no mark-segregation ${\cal M}_2(m_1,m_2\vert r)$ is independent from r, and ${\cal M}_2(m_1,m_2\vert r)=\rho^M(m_1)\rho^M(m_2)$.

Starting from these definitions, especially using the conditional mark density ${\cal M}_2(m_1,m_2\vert r)$, one may construct several mark-correlation functions sensitive to different aspects of mark-segregation (Beisbart & Kerscher 2000). The basic idea was to consider weighted correlation functions conditional that two points can be found at a distance r. For a non-negative weighting function f(m₁,m₂) we define the average over pairs with separation r:

$$\left\langle f \right\rangle_{\rm P}(r) = \int{\rm d}m_1\int{\rm d}m_2\ f(m_1,m_2)\ {\cal M}_2(m_1,m_2\vert r).$$ (6)

$\left\langle f \right\rangle_{\rm P}(r)$ is the expectation value of the weighting function f (depending only on the marks), under the condition that we find a galaxy-pair with separation r. For a suitably defined integration measure, Eq. (6) is also applicable to discrete marks. The definition (6) is very flexible, and allows us to investigate the correlations both of scalar and vector valued marks.

To calculate such mark correlation functions from the cluster data we use an estimator taking into account the periodic boundaries of the simulation box. We obtain virtually identical results for the estimator without boundary corrections (Beisbart & Kerscher 2000).

   
3.1 Correlations of scalar marks

For scalar marks the following mark correlation functions have proven to be useful (Beisbart & Kerscher 2000):

• The simplest weight to be used is the mean mark,
  

  $$k_{\rm m}(r) \equiv \frac{\left\langle m_1+m_2 \right\rangle_{\rm P}(r)}{2\ \overline{m}},$$ (7)

  
  
  that quantifies the deviation of the mean mark of pairs with separation rfrom the overall mean mark $\overline{m}$. A $k_{\rm m}>1$ indicates mark segregation for point pairs with a separation r, specifically their mean mark is larger than the overall mark average.
  
• Accordingly, higher moments of the marks may be used to quantify mark segregation. The mark variance ${\rm var}(r)$
  

  $${\rm var}(r) \equiv \left\langle \left(m_1-\left\langle m_1 \right\rangle_{\rm P}(r)\right)^2 \right\rangle_{\rm P}(r),$$ (8)

  
  
  gives information about the fluctuations of the marks of points which are separated by a distance r to other members of the set. A ${\rm var}(r)/V[m]$ larger than one indicates a substantially increased scattering of the marks compared to the overall variance.
• The mark covariance (Cressie 1991) is
  
   

  $${\rm cov}(r) \equiv \left\langle \big(m_1-\left\langle m_1 \right\rangle_{\rm P}(r)\b... ...big(m_2-\left\langle m_2 \right\rangle_{\rm P}(r)\big) \right\rangle_{\rm P}(r)$$

  $$\left\langle m_1 m_2 \right\rangle_{\rm P}(r) - \left\langle m_1 \right\rangle_{\rm P}(r)\left\langle m_2 \right\rangle_{\rm P}(r).$$ (9)

  
  
  Mark segregation can be detected by looking whether ${\rm cov}(r)$ differs from zero. A ${\rm cov}(r)$ larger than zero, e.g., indicates that points with separation r tend to have similar marks.

3.2 Correlations of vector marks

To study the alignment effects between the clusters we attach to each cluster the orientation as a vector mark ${\vec{l}}$ with $\vert{\vec{l}}\vert=1$. The orientation is either given by the direction of the major half-axis of the mass ellipsoid, or the direction of the angular momentum. The distance vector between two clusters is ${\vec{r}}$, and the normalized direction is $\hat{\vec{r}}={\vec{r}}/r$. We consider the following MCFs:

 

$${\cal A}(r) \left\langle \vert{\vec{l}}_1\cdot{\vec{l}}_2\vert \right\rangle_{\rm P}(r),$$

$${\cal F}(r) \frac{1}{2} \left\langle \vert{\vec{l}}_1\cdot{\hat{\vec{r}}}\vert + \vert{\vec{l}}_2\cdot{\hat{\vec{r}}}\vert \right\rangle_{\rm P}(r),$$ (10)

where "$\cdot$'' denotes the scalar product. Due to the spatial symmetry of the mass ellipsoids we use the absolute values of these scalar products.

• ${\cal A}(r)$ quantifies the direct ${\cal A}$lignment of the vectors ${\vec{l}}_1$and  ${\vec{l}}_2$ (see also Stoyan & Stoyan 1994 and their $k_{\rm d}$). ${\cal A}(r)$ is proportional to the cosine of the angle between ${\vec{l}}_1$and ${\vec{l}}_2$;
• ${\cal F}(r)$ quantifies the ${\cal F}$ilamentary alignment of the vectors ${\vec{l}}_1$ and ${\vec{l}}_2$ with the line connecting both clusters. ${\cal F}(r)$is proportional to the cosine of the angle between ${\vec{l}}_1$ and the direction vector $\hat{\vec{r}}$ connecting the points.

In three dimensions no mark segregation implies ${\cal A}(r)={\cal F}(r)=0.5$, in two dimensions ${\cal A}(r)={\cal F}(r)=2/\pi$. Beisbart et al. (2002) provide some further explanations and discuss in which sense ${\cal A}(r)$, ${\cal F}(r)$ provide a complete characterization of the vector correlations.