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On the projected mass distribution around galaxy clusters : a Lagrangian theory of harmonic power spectra

Gravitational lensing allows to quantify the angular distribution of the convergence field around clusters of galaxies to constrain their connectivity to the cosmic web. We describe in this paper the corresponding theory in Lagrangian space where analytical results can be obtained by identifying clusters to peaks in the initial field. We derive the three-point Gaussian statistics of a two-dimensional field and its first and second derivatives. The formalism allows us to study the statistics of the field in a shell around a central peak, in particular its multipolar decomposition. The peak condition is shown to significantly remove power from the dipolar contribution and to modify the monopole and quadrupole. As expected, higher order multipoles are not significantly modified by the constraint. Analytical predictions are successfully checked against measurements in Gaussian random fields. The effect of substructures and radial weighting is shown to be small and does not change the qualitative picture. The non-linear evolution is shown to induce a non-linear bias of all multipoles proportional to the cluster mass.We predict the Gaussian and weakly non-Gaussian statistics of multipolar moments of a two-dimensional field around a peak as a proxy for the azimuthal distribution of the convergence field around a cluster of galaxies. A quantitative estimate of this multipolar decomposition of the convergence field around clusters in numerical simulations of structure formation and in observations will be presented in two forthcoming papers.