We present a numerical analysis supporting the evidence that the redshift evolution of the drifting coefficient of the field cluster mass function is capable of breaking several cosmic degeneracies. This evidence is based on the data from the CoDECS and DUSTGRAIN-pathfinder simulations performed separately for various non-standard cosmologies including coupled dark energy, $f(R)$ gravity and combinations of $f(R)$ gravity with massive neutrinos as well as for the standard $Lambda$CDM cosmology. We first numerically determine the field cluster mass functions at various redshifts in the range of $0le zle 1$ for each cosmology. Then, we compare the analytic formula developed in previous works with the numerically obtained field cluster mass functions by adjusting its drifting coefficient, $beta$, at each redshift. It is found that the analytic formula with the best-fit coefficient provides a good match to the numerical results at all redshifts for all of the cosmologies. The empirically determined redshift evolution of the drifting coefficient, $beta(z)$, turns out to significantly differ among different cosmologies. It is also shown that even without using any prior information on the background cosmology the drifting coefficient, $beta(z)$, can discriminate with high statistical significance the degenerate non-standard cosmologies not only from the $Lambda$CDM but also from one another. It is concluded that the evolution of the departure from the Einstein-de Sitter state and spherically symmetric collapse processes quantified by $beta(z)$ is a powerful probe of gravity and dark sector physics.
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