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 galaxydistributions in phase space for samples selected from the 2MASS ExtendedSource Catalog. The galaxy distribution is decomposed into modes $\delta({\bfk, x})$ which describe the number density perturbations of galaxies in phasespace cell given by scale band $\bf k$ to ${\bf k}+\Delta {\bf k}$ and spatialrange $\bf x$ to ${\bf x}+\Delta {\bf x}$. In the nonlinear regime,$\delta({\bf k, x})$ is highly non-Gaussian. We find, however, that thecorrelations between $\delta({\bf k, x})$ and $\delta({\bf k', x'})$ are alwaysvery 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 tothe fact that the spatial locality of the initial perturbations is memorizedduring hierarchical clustering. The highly spatial locality of the 2MASS galaxycorrelations is a strong evidence for the initial perturbations of the cosmicmass field being spatially localized, and therefore, consistent with a Gaussianinitial perturbations on scales as small as about 0.1 h$^{-1}$ Mpc. Moreover,the 2MASS galaxy spatial locality indicates that the relationship betweendensity perturbations of galaxies and the underlying dark matter should belocalized in phase space. That is, for a structure consisting of perturbationson scales from $k$ to $ k+\Delta {k}$, the nonlocal range in the relationbetween galaxies and dark matter should {\it not} be larger than $|{\Delta {\bfx}}|=2\pi/|\Delta {\bf k}|$. The stochasticity and nonlocality of the biasrelation between galaxies and dark matter fields should be no more than theallowed range given by the uncertainty relation $|{\Delta {\bf x}|| \Delta{\bfk}}|=2\pi$.Comment: 27 pages, 9 figures, accepted by Ap