Galaxy cluster number count has been proven to be a powerful cosmological probe. However, cosmological constraints established with galaxy cluster number count are highly dependent on the calibration of the mass-observable relations. Thanks to its nearly mass independence the specific mass accretion rate of galaxy clusters is nearly insensitive to the calibration of mass-observable relations. The study of galaxy cluster number count evolution allows to probe the galaxy cluster mass accretion history in the context of an homogenous Universe. In this paper, we use relative abundance matching to infer the galaxy cluster mass accretion rate (MAR) for $z in [0.0,0.6[$. Then, we use the MAR to set cosmological constraints. We found that this cosmological probe is sensitive to $sigma_8 Omega_{rm m}^{-0.3} H_0^{-0.2}$ whereas the galaxy cluster count is sensitive to $sigma_8 Omega_{rm m}^{0.3}$. We used the second $Planck$ catalog of Sunyaev-Zeldovich sources and we derive $sigma_8 Omega_{rm m}^{-0.3} H_0^{-0.2} = 0.75 pm 0.06$. This results is consistent with cosmological constraints derived from galaxy clusters number counts, angular power spectrum, and cosmic microwave background analyses. Therefore, the MAR is a key cosmological probe that can break the $sigma_8$-$Omega_{rm m}$ degeneracy and that is not sensitive to the calibration of the mass-observable relations and does not requires a parametric form for the galaxy cluster mass-function.
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