Testing template-fitting models for the multipoles of the two-point clustering of galaxy clusters

The \textit{Euclid} satellite will deliver a catalogue of optically-selected galaxy clusters spanning from around $2000$ deg$^2$ in Data Release (DR) 1 to around $14\,000$ deg$^2$ in DR3. We assess the validity of cluster clustering (CC) models for template-fitting, which complements the full-shape methodology in providing cosmological information from the anisotropy of the redshift-space two-point correlation function (2PCF). Both will be used to analyse the cluster 2PCF multipoles in \textit{Euclid}. We analyse the multipoles of the two-point redshift-space clustering of galaxy clusters simulated with the semi-analytic \code{PINOCCHIO} code using third-order Lagrangian perturbation theory, assuming a \textit{Euclid} DR1-like footprint of 500 deg$^2$ in the Northern Hemisphere and 1400 deg$^2$ in the Southern Hemisphere. We estimate the first three even multipoles of the 2PCF and associated covariance matrix from 1000 DR1-like synthetic catalogues. We study the impact of the modelling of nonlinearities, halo bias, and photometric redshift uncertainties on the 2PCF. We apply three clustering models to the mock catalogues at $0<z<2$ and virial mass $M_{\rm vir}>10^{14}\;h^{-1}\,M_\odot$ under realistic and optimistic photometric redshift uncertainty scenarios. We formulate a set of permissive and conservative criteria that should be fulfilled by the multipole cut-off scales and validate against 100 mock catalogues via inference of the growth rate times the matter power spectrum normalisation parameter, $f\sigma_8$. We test the dispersion, Scoccimarro, and Taruya--Nishimichi--Saito models. We find that the simplest of the three -- the dispersion model -- yields unbiased inferences on $f\sigma_8$ from CC down to $10$ $h^{-1}$ Mpc in a DR1-like setting. All clustering models provide very similar goodness-of-fit metrics in the presence of DR1-like cluster redshift uncertainties.