Galaxy Clustering at z ~ 2 and Halo Radii

The amplitude of the angular two-point galaxy correlation function w(\theta) for galaxies at z~2 is estimated for galaxies in the Hubble Deep Field by using a U < 27 complete sub-sample. (i) It is confirmed that the amplitude of the correlation can be corrected for the integral constraint 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(5'') = 0.10 \pm 0.09. (ii) If the biases introduced in faint galaxy selection due to obscuration by large objects are not corrected for by masking areas around them, then the estimate would be w(5'') =0.16\pm 0.07. (iii) The effective (3-D) galaxy pair separation at 5'' and this redshift range is ~ 25-250 /h kpc, so the correction to the spatial correlation function \xi(r) due to exclusion of overlapping galaxy dark matter haloes should be considered. For clustering stable in proper units in an \Omega=1,\lambda=0 universe, our w(5\arcs) estimate (a) implies a present-day correlation length of r_0 ~ 2.6^{+1.1}_{-1.7}/h Mpc if halo overlapping is ignored, but (b) for a present-day correlation length of r_0=5.5/h Mpc implies that a typical halo exclusion radius is r_halo=70^{+420}_{-30}/h kpc. (iv) The decreasing correlation period (DCP) of a high initial bias in the spatial correlation function is not detected at this redshift. For an \Omega=1,\lambda=0 universe and (proper) stable clustering, possible detections of the DCP in other work would imply that \xi at redshifts greater than z_t = 1.7\pm0.9 would be [(1+z)/(1+z_t)]^{2.1\pm3.6} times higher than at z_t, which is consistent with our lack of a detection at z ~ 2.Comment: 17 pages, 13 figures, accepted for MNRAS, additional FITS files with HDF images available at http://www.iap.fr/users/roukema/xi2

Homotopy symmetry in the multiply connected twin paradox of special relativity

In a multiply connected space, the two twins of the special relativity twin paradox move with constant relative speed and meet a second time without acceleration. The twins' situations appear to be symmetrical despite the need for one to be younger due to time dilation. Here, the suggestion that the apparent symmetry is broken by homotopy classes of the twins' worldlines is reexamined using space-time diagrams. It is found that each twin finds her own spatial path to have zero winding index and that of the other twin to have unity winding index, i.e. the twins' worldlines' relative homotopy classes are symmetrical. Although the twins' apparent symmetry is in fact broken by the need for the non-favoured twin to non-simultaneously identify spatial domain boundaries, the non-favoured twin {\em cannot} detect her disfavoured state if she only measures the homotopy classes of the two twins' projected worldlines, contrary to what was previously suggested. We also note that for the non-favoured twin, the fundamental domain can be chosen by identifying time boundaries (with a spatial offset) instead of space boundaries (with a temporal offset).Comment: 11 pages, 9 figures, v6: more elementary algebra, extra figure, accepted in MNRAS, the definitive version will be available at http://www.blackwell-synergy.co