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.
