STATISTICAL ANALYSIS OF NON-RANDOM PATTERNS

In [1] I have proposed an alternative method for analyzing the clustering morphology which is a generalization of the 2-point function $\xi(r)$ and is based on the spherical harmonic analysis. The method is drawn from molecular dynamics simulations, where it has been introduced for studying orientational order of supercooled liquids and metallic glasses [2]. Let us consider a system of N_p particles. The i-th particle has coordinates $\vec r_i$, in an arbitrary reference frame. For a specified cutoff radius R_c all the particles such that $\vert\vec r_i-\vec r_j\vert< R_c$ are neighbors of i. The line joining i to one of the j is termed a bond. The angular coordinates of the vector $\vec \Delta_{ji}\equiv \vec r_j - \vec r_i$ are $\theta_j, \phi_j$ and the quantity

$$Q_{lm}(\vec r_i)=\sum_{j \ne i} Y_{lm}(\theta_j,\phi_j),$$ (1)

is the coefficient of the spherical harmonic expansion of the angular density of the bonds associated with the particle i. In Eq. (1), and hereafter, summation is understood over all particles j of the distribution such that $\vert\vec r_i-\vec r_j\vert< R_c$.

The coefficients $Q_{lm}(\vec r_i)$ are defined as the bond-orientational order parameters and they can be drastically changed by a rotation of the reference systems. A natural quantity to consider , which is rotation invariant, is

$$Q_l(\vec r_i)=\sqrt {{{4\pi}\over{2l+1}} \sum_{m=-l}^{m=l} Q^{\star}_{lm}(\vec r_i) Q_{lm}(\vec r_i)}.$$ (2)

Using the addition theorem for the spherical harmonics, the expression simplifies to

$$Q_l(\vec r_i)=\sqrt { \sum_j \sum_k P_l(\gamma_{jk})},$$ (3)

where P_l is the Legendre polynomial,

$$\gamma_{jk}\equiv cos(\theta_{jk})=\vec \Delta_{ji}\cdot\vec\... ...\vert\vec \Delta_{ji}\vert\vert\vec \Delta_{ki}\vert)\nonumber$$

is the angle between two bonds and the summations in Eq. (3) are then independent of the chosen frame. An useful quantity is the auto-correlation G_l(r) of the coefficients $Q_l(\vec r_i)$. The function G_l is defined as follows: for all of the M_p pairs (i,k), such that $\vert\vec r_i-\vec r_k\vert=r \pm \Delta r$, where $\Delta r$ is the thickness of the radial bin, then G_l(r) is the sum over all of these pairs

$$G_l(r)={{1}\over{M_p}} \sum_i \sum_k{{4\pi}\over{2l+1}} \sum_{m=-l}^{m=l} Q^{\star}_{lm}(\vec r_i) Q_{lm}(\vec r_i+\vec r).$$ (4)

This equation can be greatly simplified: let j be the set of neighbors of the particle i and p that of the particle k, which satisfy $\vert\vec r_j-\vec r_i\vert< R_c$ and $\vert\vec r_p-\vec r_k\vert< R_c$. Then the summation becomes

$$G_l(r)={{1}\over{M_p}} \sum_i \sum_k \sum_j \sum_p P_l(\Gamma_{jp}),$$ (5)

with $\Gamma_{jp}$ being the angle between $\vec \Delta_{ji}$ and $\vec \Delta_{pk}$. The summation over the pairs is $\sum_i \sum_k$, with the sum over the particles k only for those particles with $\vert\vec r_i-\vec r_k\vert=r \pm \Delta r$.

The effectiveness of the statistical analysis in quantifying clustering morphology is studied [1] by applying the statistical estimator G_l to point distributions produced by an ensemble of cosmological N- body simulations with a CDM spectrum. The results shown that the statistical method defined by the function G_l(r) can be used to analyze the clustering morphology produced by gravitational clustering in a quantitative way. The function G_l(r) describes anisotropies in the clustering distribution by measuring the degree of correlation between the angular densities as seen from two different observers separated by r. G_l(r) can then be considered a statistical measure of clustering patterns, with different scales probed by varying the input parameters l and R_c. With large redshift surveys becoming available in the next few years, the proposed statistical method appears as a promising tool for analyzing patterns in the galaxy distribution.