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We present a new statistics based on the turn-around radii of cluster halosto break the dark sector degeneracy between the $\Lambda$CDM model and thealternative ones with $f(R)$ gravity and massive neutrinos ($\nu$)characterized by the strength of the fifth force, $\vert f_{R0}\vert$, and thetotal neutrino mass, $M_{\nu}$. Analyzing the rockstar halo catalogs at thepresent epoch from the {\small DUSTGRAIN}-{pathfinder} $N$-body simulationsperformed for four different cosmologies, namely, $\Lambda$CDM ($\vertf_{R0}\vert=0$, $M_{\nu}=0.0$eV), fR6 ($\vert f_{R0}\vert=10^{-6}$,$M_{\nu}=0.0$eV), fR6+$0.06$eV ($\vert f_{R0}\vert=10^{-6}$, $M_{\nu}=0.06$eV)and fR5+$0.15$eV ($\vert f_{R0}\vert=10^{-5}$, $M_{\nu}=0.15$eV), which areknown to yield very similar conventional statistics to one another. For eachmodel, we select those cluster halos which do not neighbor any other largerhalos in their bound zones and construct their bound-zone peculiar velocityprofiles at $z=0$. Then, we determine the radial distance of each selected haloat which the bound-zone velocity becomes equal to the recession speed of theHubble flow as its turn around radius, and evaluate the cumulative probabilitydistribution of the ratios of the turn-around radii to the virial counterparts,$P(r_{t}/r_{v}\ge \alpha)$. The degeneracy between the fR6 and fR5+$0.15$eVmodels is found to be readily broken by the $10\sigma_{\Delta P}$ difference inthe value of $P(\alpha=4)$, while the $3.2\sigma_{\Delta P}$ difference betweenthe $\Lambda$CDM and fR6+$0.06$eV models is detected in the value of$P(\alpha=8.5)$. It is also found that the four models yield smallerdifferences in $P(\alpha)$ at higher redshifts.

Combined Effects of $f(R)$ Gravity and Massive Neutrinos on the Turn-Around Radii of Dark Matter Halos