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On the Effective Spatial Separations in the Clustering of Faint Galaxies

Several recent measurements have been made of the angular correlation function $w(θ,m)$ of faint galaxies in deep surveys (e.g., Hubble Deep Field, HDF). Are the measured correlations indicative of gravitational growth of primordial perturbations or of the relationship between galaxies and (dark matter dominated) galaxy haloes? A first step in answering this question is to determine the typical spatial separations of galaxies whose spatial correlations, $ξ(r,z),$ 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 the perpendicular distance, $r_\perp,$ at the median redshift, z_{median}, of the galaxies. Over a wide range in $ξ$ growth rates $ε$ 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 well below 1 h^-1 Mpc. For $Ω_0=1$ and $λ_0=0$, the correlation signal at 4\arcsec predominant in the Villumsen et al. (1996) estimates of $w$ for faint HDF galaxies corresponds to r_eff(50%)(HDF) \approx 40 h^-1 kpc; other cosmologies 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 where halo and/or galaxy existence, let alone interactions, may modify the spatial correlation function. These measurements could maybe be used to probe the radial extent of haloes at high redshift.