Mass scaling relations for dark halos from an analytic universal outer density profile

Context. The average matter density within the turnaround scale, which demarcates where galaxies shift from clustering around a structure to joining the expansion of the Universe, is an important cosmological probe. However, a measurement of the mass enclosed by the turnaround radius is difficult. Analyses of the turnaround scale in simulated galaxy clusters place the turnaround radius at about three times the virial radius in a Lambda CDM universe and at a (present-day) density contrast with the background matter density of the Universe of delta similar to 11. Assessing the mass at such extended distances from a cluster's center is a challenge for current mass measurement techniques. Consequently, there is a need to develop and validate new mass-scaling relations, to connect observable masses at cluster interiors with masses at greater distances. Aims. Our research aims to establish an analytical framework for the most probable mass profile of galaxy clusters, leading to novel mass scaling relations, allowing us to estimate masses at larger scales. We derive such analytical mass profiles and compare them with those from cosmological simulations. Methods. We used excursion set theory, which provides a statistical framework for the density and local environment of dark matter halos, and complement it with the spherical collapse model to follow the non-linear growth of these halos. Results. The profile we developed analytically showed good agreement (better than 30%, and dependent on halo mass) with the mass profiles of simulated galaxy clusters. Mass scaling relations were obtained from the analytical profile with offset better than 15% from the simulated ones. This level of precision highlights the potential of our model for probing structure formation dynamics at the outskirts of galaxy clusters.