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We show how $\varsigma$, the radial location of the minimum of thedifferential radial mass profile $M^\prime(r)$ of a galaxy cluster, can probethe theory of gravity. We derive $M^\prime(r)$ of the dark matter halos ofgalaxy clusters from N-body cosmological simulations that implement twodifferent theories of gravity: standard gravity in the $\Lambda$CDM model and$f(R)$. We extract 49169 dark matter halos in 11 redshift bins in the range$0\leq z\leq 1$ and in three different mass bins in the range$0.9<M_{200}/10^{14}h^{-1}$M$_\odot<11$. We investigate the correlation of$\varsigma$ with the redshift and the mass accretion rate (MAR) of the halos.We show that $\varsigma$ decreases from $\sim 3R_{200}$ to $\sim 2R_{200}$ when$z$ increases from 0 to $1$ in the $\Lambda$CDM model; at $z\sim 0.1$,$\varsigma$ decreases from $2.75R_{200}$ to $\sim 2.5R_{200}$ when the MARincreases from $\sim 10^4h^{-1}$M$_\odot$yr$^{-1}$ to $\sim 2\times10^5h^{-1}$M$_\odot$yr$^{-1}$; in the $f(R)$ model, $\varsigma$ is $\sim 15$%larger than in $\Lambda$CDM. We detect $\varsigma$ and its correlation withredshift and MAR in three samples of real clusters. We consider 129 clusters ofgalaxies from the Cluster Infall Regions in the Sloan Digital Sky Survey (CIRS)and the Hectospec Cluster Survey (HeCS), and 10 additional stacked clustersfrom the HectoMAP survey. The clusters span the redshift range $0.01 < z < 0.4$and the mass range $\sim 10^{14}-10^{15} h^{-1}$M$_{\odot}$. The values of$\varsigma$ of real clusters appear consistent with both the $\Lambda$CDM and$f(R)$ models, because the current uncertainties on the measured mass profilesare too large. We estimate that decreasing the uncertainties by a factor of tenand distinguishing between the two theories of gravity that we investigate hererequires $\gtrsim 400$ clusters, each with $\sim 200$ spectroscopic redshiftsof galaxies.

The mass distribution in the outskirts of clusters of galaxies as a probe of the theory of gravity