On the Effective Spatial Separations in the Clustering of Faint Galaxies

Several recent measurements have been made of the angular correlationfunction $w(\theta,m)$ of faint galaxies in deep surveys (e.g., Hubble DeepField, HDF). Are the measured correlations indicative of gravitational growthof primordial perturbations or of the relationship between galaxies and (darkmatter dominated) galaxy haloes? A first step in answering this question is todetermine the typical spatial separations of galaxies whose spatialcorrelations, $\xi(r,z),$ contribute most to w. The median spatial separation of galaxy pairs contributing to a fraction $p$of $w$ for a galaxy survey is denoted by r_{eff} and compared with theperpendicular distance, $r_\perp,$ at the median redshift, z_{median}, of thegalaxies. Over a wide range in $\xi$ growth rates $\epsilon$ and in$z_{median}, \r_eff(50%) \ltsim r_\perp$ and $\r_eff(90%) \sim 4 r_\perp.$ Values of r_eff for typical redshift distributions indicate that many $w$measurements correspond to spatial correlations at comoving length scales wellbelow 1 h^-1 Mpc. For $\Omega_0=1$ and $\lambda_0=0$, the correlation signal at4\arcsec predominant in the Villumsen et al. (1996) estimates of $w$ for faintHDF galaxies corresponds to r_eff(50%)(HDF) \approx 40 h^-1 kpc; othercosmologies and angles up to 10\arcsec can increase this to r_eff(50%)(HDF)$\ltsim 200 h^-1 kpc. The proper separations are (1+z) times smaller. These scales are small: the faint galaxy $w$ measurements are at scales wherehalo and/or galaxy existence, let alone interactions, may modify the spatialcorrelation function. These measurements could maybe be used to probe theradial extent of haloes at high redshift.Comment: 10 pages, 4 figures, \LaTeX files using aas2pp4.sty or aasms4.sty, accepted for publication in The Astrophysical Journa