Spatial Locality of Galaxy Correlation Function in Phase Space: Samples from the 2MASS Extended Source Catalog

We analyze the statistical properties and dynamical implications of galaxy distributions in phase space for samples selected from the 2MASS Extended Source Catalog. The galaxy distribution is decomposed into modes $\delta({\bf k, x})$ which describe the number density perturbations of galaxies in phase space cell given by scale band $\bf k$ to ${\bf k}+\Delta {\bf k}$ and spatial range $\bf x$ to ${\bf x}+\Delta {\bf x}$. In the nonlinear regime, $\delta({\bf k, x})$ is highly non-Gaussian. We find, however, that the correlations between $\delta({\bf k, x})$ and $\delta({\bf k', x'})$ are always very weak if the spatial ranges (${\bf x}$, ${\bf x}+\Delta {\bf x}$) and (${\bf x'}$, ${\bf x'}+\Delta {\bf x'}$) don't overlap. This feature is due to the fact that the spatial locality of the initial perturbations is memorized during hierarchical clustering. The highly spatial locality of the 2MASS galaxy correlations is a strong evidence for the initial perturbations of the cosmic mass field being spatially localized, and therefore, consistent with a Gaussian initial perturbations on scales as small as about 0.1 h$^{-1}$ Mpc. Moreover, the 2MASS galaxy spatial locality indicates that the relationship between density perturbations of galaxies and the underlying dark matter should be localized in phase space. That is, for a structure consisting of perturbations on scales from $k$ to $k+\Delta {k}$, the nonlocal range in the relation between galaxies and dark matter should {\it not} be larger than $|{\Delta {\bf x}}|=2\pi/|\Delta {\bf k}|$. The stochasticity and nonlocality of the bias relation between galaxies and dark matter fields should be no more than the allowed range given by the uncertainty relation $|{\Delta {\bf x}|| \Delta{\bf k}}|=2\pi$.