Galaxy clustering at z ∼2 and halo radii

The amplitude of the angular two‐point galaxy autocorrelation function w(θ) for galaxies at z ∼ 2 is estimated for galaxies in the Hubble Deep Field by using a U<27 complete subsample. The U ‐band selection ensures little contamination from z>2.5 galaxies, while photometric redshifts minimize the contribution from low redshift galaxies. (i) It is confirmed that the amplitude of the correlation can be corrected for the integral constraint (lack of large‐scale variance) without having to make assumptions about the shape of the correlation function and by avoiding the introduction of linear error terms. The estimate using this technique is w (θ≈5 arcsec)=0.10±0.09. Estimators that assume a power law of a given slope and include linear error terms would double this value. (ii) If the biases introduced in faint galaxy selection owing to obscuration by large objects are not corrected for by masking areas around them, then the estimate would be w (θ≈5 arcsec)=0.16±0.07. (iii) The effective (three‐dimensional) galaxy pair separation at 5 arcsec and this redshift range is ≈25‐‐250 h −1 kpc, so the correction to the spatial correlation function ξ( r ) owing to exclusion of overlapping galaxy dark matter haloes should be considered. For stable clustering in proper units in an Ω=1,λ=0 universe, our w (5 arcsec) estimate (a) implies a present‐day correlation length of r 0 ∼2.6 +1.1 −1.7 h −1 Mpc if halo overlapping is ignored, but (b) for a present‐day correlation length of r 0 =5.5 h −1 Mpc implies that a typical halo exclusion radius is r halo =70 +420 −30 h −1 kpc. For Ω 0 =0.1,λ 0 =0.9, the corresponding values are (a) r 0 ∼5.8 +2.4 −3.9 h −1 Mpc and (b) r halo <210 h −1 kpc (1σ upper limit). (iv) The decreasing correlation period (DCP) of a high initial bias in the spatial correlation function is not detected at this redshift. For an Ω=1,λ=0 universe and stable clustering in proper units, possible detections of the DCP in other work would imply that the values of ξ at redshifts greater than z t =1.7±0.9 would be [(1+ z )/(1+ z t )] 2.1±3.6 times their values at z t , which is consistent with our lack of detection at z ∼2.