We apply the caustic technique to samples of galaxy clusters stacked in
redshift space to estimate the gravitational potential in the cluster’s outer
region and test modifications to the standard theory of gravity. We separate
122 galaxy clusters from the HeCS-SZ, HeCS-redMapper, and HeCS samples into
four samples with increasing mass; we estimate four robust, highly constraining
caustic profiles for these samples. The caustic masses of the four stacked
clusters agree within $ 10\%$ with the corresponding median values of each
cluster sample. By adopting the NFW density profile to model the gravitational
potential, we recover the caustic profile $\mathcal{A}(r)$ up to radius $r_{\rm
p} \sim 4.0\, {\rm Mpc}$. This comparison is a first-order validation of the
mass-concentration relation for galaxy clusters expected in the $\Lambda$CDM
model. We thus impose this correlation as a prior in our analysis. Based on our
stacked clusters, we estimate the value of the filling factor, which enters the
caustic technique, $\mathcal{F}_{\beta} = 0.59\pm 0.05$; we derive this value
using real data alone and find it consistent with the value usually adopted in
the literature. We then use the caustic profiles $\mathcal{A}(r)$ of the
stacked clusters to constrain the chameleon gravity model. We find that the
caustic profiles provide a stringent upper limit of $|f_{\rm R0}| \lesssim 4
\times 10^{-6}$ at $95\%$ C.L. limits in the $f(\mathcal{R})$ scenario. Der
formalism developed here shall be further refined to test modifications to
gravity in the extended outer weak gravitational regions of galaxy clusters.

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