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

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 measure the redshift and mass evolution of the correlation length and the bias parameter as a function of the redshift uncertainty. We calculate the correlation function for cluster sub-samples covering various mass and redshift bins selected from a light-cone catalogue. To simulate the distribution of clusters in photometric redshift space, we assign to each cluster a redshift randomly extracted from a Gaussian distribution. The dispersion is varied in the range $\sigma_{(z=0)} = 0.001$ to $0.050$. The correlation function in real-space is computed through estimation and deprojection of $w_{p}(r_{p})$. Four mass ranges (from $M_{halo}> 2 \times 10^{13}$ to $M_{halo}> 2 \times 10^{14}$) and six redshift slices covering the redshift range [0,2] are investigated, using cosmological redshifts and photo-z configurations. We find a clear increase of the correlation amplitude as a function of redshift and mass for the $z_{c}$ samples. The evolution of the derived bias parameter is in agreement with theoretical expectations. From our pilot sample limited to $M_{halo}> 5 \times 10^{13} (0.4 < z < 0.7)$, we find that the real-space correlation function can be recovered by deprojection of $w_{p}(r_{p})$ within an accuracy of 5% for $\sigma_{z} = 0.001 \times (1 + z_{c})$ and within 10% for $\sigma_{z} = 0.03 \times (1 + z_{c})$. The evolution of the correlation in redshift and mass is clearly detected for all $\sigma_{z}$ tested. 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.