Evolution of the real-space correlation function from next generation cluster surveys - Recovering the real-space correlation function from photometric redshifts

Context. The next generation of galaxy surveys will provide cluster catalogues probing an unprecedented range of scales, redshifts, and masses with large statistics. Their analysis should therefore enable us to probe the spatial distribution of clusters with high accuracy and derive tighter constraints on the cosmological parameters and the dark energy equation of state. However, for the majority of these surveys, redshifts of individual galaxies will be mostly estimated by multiband photometry which implies non-negligible errors in redshift resulting in potential difficulties in recovering the real-space clustering. Aims. We investigate to which accuracy it is possible to recover the real-space two-point correlation function of galaxy clusters from cluster catalogues based on photometric redshifts, and test our ability to detect and measure the redshift and mass evolution of the correlation length r(0) and of the bias parameter b(M,z) as a function of the uncertainty on the cluster redshift estimate. Methods. We calculate the correlation function for cluster sub-samples covering various mass and redshift bins selected from a 500 deg(2) light-cone limited to H < 24. In order to simulate the distribution of clusters in photometric redshift space, we assign to each cluster a redshift randomly extracted from a Gaussian distribution having a mean equal to the cluster cosmological redshift and a dispersion equal to sigma(z). The dispersion is varied in the range sigma((z=0)) = sigma(z)/1+z(c) = 0.005, 0.010, 0.030 and 0.050, in order to cover the typical values expected in forthcoming surveys. The correlation function in real-space is then computed through estimation and deprojection of w(p)(r(p)). Four mass ranges (from M-halo > 2 x 10(13) h(-1) M-circle dot to M-halo > 2 x 10(14) h(-1) M-circle dot) and six redshift slices covering the redshift range [ 0, 2] are investigated, first using cosmological redshifts and then for the four photometric redshift configurations. Results. From the analysis of the light-cone in cosmological redshifts we find a clear increase of the correlation amplitude as a function of redshift and mass. The evolution of the derived bias parameter b(M,z) is in fair agreement with theoretical expectations. We calculate the r(0) - d relation up to our highest mass, highest redshift sample tested (z = 2; Mhalo > 2 x 10(14) h(-1) M-circle dot). From our pilot sample limited to M-halo > 5 x 10(13) h(-1) M-circle dot (0.4 < z < 0.7), we find that the real-space correlation function can be recovered by deprojection of w(p)(r(p)) within an accuracy of 5% for sigma(z) = 0.001 x (1 + z(c)) and within 10% for sigma(z) = 0.03 x (1 + z(c)). For higher dispersions (besides sigma(z) > 0.05 x (1 + z(c))), the recovery becomes noisy and difficult. The evolution of the correlation in redshift and mass is clearly detected for all sigma(z) tested, but requires a large binning in redshift to be detected significantly between individual redshift slices when increasing sigma(z). The best-fit parameters (r(0) and gamma) as well as the bias obtained from the deprojection method for all sigma(z) are within the 1 sigma uncertainty of the z(c) sample.