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

We present a new statistics based on the turn-around radii of cluster halos to break the dark sector degeneracy between the $\mathrm{\Lambda }$CDM model and the alternative ones with $f(R)$ gravity and massive neutrinos ($\nu$) characterized by the strength of the fifth force, $\mathrm{\mid }{f}_{R0}\mathrm{\mid }$, and the total neutrino mass, $M_{\nu }$. Analyzing the rockstar halo catalogs at the present epoch from the {\small DUSTGRAIN}-{pathfinder} $N$-body simulations performed for four different cosmologies, namely, $\mathrm{\Lambda }$CDM ($\mathrm{\mid }{f}_{R0}\mathrm{\mid }=0$, $M_{\nu }=0.0$eV), fR6 ($\mathrm{\mid }{f}_{R0}\mathrm{\mid }={10}^{-6}$, $M_{\nu }=0.0$eV), fR6+$0.06$eV ($\mathrm{\mid }{f}_{R0}\mathrm{\mid }={10}^{-6}$, $M_{\nu }=0.06$eV) and fR5+$0.15$eV ($\mathrm{\mid }{f}_{R0}\mathrm{\mid }={10}^{-5}$, $M_{\nu }=0.15$eV), which are known to yield very similar conventional statistics to one another. For each model, we select those cluster halos which do not neighbor any other larger halos in their bound zones and construct their bound-zone peculiar velocity profiles at $z=0$. Then, we determine the radial distance of each selected halo at which the bound-zone velocity becomes equal to the recession speed of the Hubble flow as its turn around radius, and evaluate the cumulative probability distribution of the ratios of the turn-around radii to the virial counterparts, $P(r_t\mathrm{/}{r}_v\ge \alpha )$. The degeneracy between the fR6 and fR5+$0.15$eV models is found to be readily broken by the $10{\sigma }_{\mathrm{\Delta }P}$ difference in the value of $P(\alpha =4)$, while the $3.2{\sigma }_{\mathrm{\Delta }P}$ difference between the $\mathrm{\Lambda }$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 smaller differences in $P(\alpha )$ at higher redshifts.