The infall region as a complementary probe to cluster abundance

Galaxy cluster abundance measurements provide a classic test of cosmology. They are most sensitive to the evolved amplitude of fluctuations, usually expressed as $S_8 = \sigma_8\sqrt{\Omega_m/0.3}$. Thus, abundance constraints exhibit a strong degeneracy between $\sigma_8$ and $\Omega_{\rm m}$, as do other similar low-redshift tests such as cosmic shear. The mass distribution in the infall region around galaxy clusters, where material is being accreted from the surrounding field, also exhibits a cosmological dependence, but in this case it is nearly orthogonal to the $S_8$ direction in the $\Omega_m$--$\sigma_8$ plane, making it highly complementary to halo abundance or cosmic shear studies. We explore how weak lensing measurements of the infall region might be used to complement abundance studies, considering three different tests. The splashback radius is a prominent feature of the infall region; we show that detection of this feature in lensing data from the Euclid survey could independently constrain $\Omega_{\rm m}$ and $\sigma_8$ to $\pm 0.05$. Another feature, the depletion radius where the bias reaches a minimum, also shows cosmological dependence, though it is challenging to observe in practice. The strongest constraints come from direct measurements of the shear profile in the infall region at $2$--$4\,r_{200{\rm c}}$. Combining the latter with abundance constraints such as those reported from SRG$/$eROSITA should reduce the area of the error contours by an estimated factor of $1.2$ using a sample of clusters observed by the UNIONS survey, or a factor of $3$ using clusters observed by the Euclid Wide survey over a broader range of redshift.