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The outer profile of dark matter halos: an analytical approach

A steepening feature in the outer density profiles of dark matter halos indicating the splashback radius has drawn much attention recently. Possible observational detections have even been made for galaxy clusters. Theoretically, Adhikari et al. have estimated the location of the splashback radius by computing the secondary infall trajectory of a dark matter shell through a growing dark matter halo with an NFW profile. However, since they imposed a shape of the halo profile rather than computing it consistently from the trajectories of the dark matter shells, they could not provide the full shape of the dark matter profile around the splashback radius. We improve on this by extending the self-similar spherical collapse model of Fillmore \& Goldreich to a $Λ$CDM universe. This allows us to compute the dark matter halo profile and the trajectories simultaneously from the mass accretion history. Our results on the splashback location agree qualitatively with Adhikari et al. but with small quantitative differences at large mass accretion rates. We present new fitting formulae for the splashback radius $R_{\rm sp}$ in various forms, including the ratios of $R_{\rm sp} / R_{\rm 200c}$ and $R_{\rm sp} / R_{\rm 200m}$. Numerical simulations have made the puzzling discovery that the splashback radius scales well with $R_{\rm 200m}$ but not with $R_{\rm 200c}$. We trace the origin of this to be the correlated increase of $Ω_{\rm m}$ and the average halo mass accretion rate with an increasing redshift.