Reconstructing Positions and Peculiar Velocities of Galaxy Clusters within 20000 km/sec. I: The Cluster 3D Dipole

Starting from the observed distribution of galaxy clusters in redshift space we use a two--step procedure to recover their distances and peculiar velocities. After statistically correcting for the unobserved cluster distribution in the zone of avoidance ($|b|\le 20^{\circ}$) and also for a smooth absorption at higher $|b|$'s, we use a dynamical iterative algorithm to recover the real--space cluster positions by minimizing the redshift space distortions. The whole procedure assumes that clusters trace the mass, that peculiar velocities are caused by gravity and that linear perturbation theory applies. The amplitude of the cluster dipole measured in the 3D space turns out to be $\sim 23\%$ less than that measured in redshift space. In both cases the dipole direction is aligned with the Cosmic Microwave Background dipole within $\sim 10^{\circ}$, taking into account the Virgocentric infall component of the Local Group motion. Observational errors, limitations in the reconstruction procedure and the intrinsic cosmological variance, which is the dominant source of uncertainty, render a stringent determination of the $\beta$ parameter whose central value turns out to be $\beta \approx 0.2$ while its total uncertainty is $\pm 0.1$. This implies that for a cluster-mass bias parameter of $\sim 5$, a flat Universe is not excluded, contrary to previous cluster-dipole $z$-space analysis.