We present a new approach to describe statistics of the non-linear matter density field that exploits a degeneracy in the impact of different cosmological parameters on the linear matter power spectrum, $P_{rm L}(k)$, when expressed in Mpc units. We classify all cosmological parameters into two groups, shape parameters, which determine the shape of $P_{rm L}(k)$, and evolution parameters, which only affect its amplitude at any given redshift. We show that the time evolution of $P_{rm L}(k)$ in models with identical shape parameters but different evolution parameters can be mapped from one to the other by relabelling the redshifts that correspond to the same values of $sigma_{12}(z)$, defined as the RMS linear variance in spheres of radius $12,{rm Mpc}$. We use N-body simulations to show that the same evolution mapping relation can be applied to the non-linear power spectrum, the halo mass function, or the full density field with high accuracy. The deviations from the exact degeneracy are the result of the different structure formation histories experienced by each model to reach the same value of $sigma_{12}(z)$. This relation can be used to drastically reduce the number of parameters required to describe the cosmology dependence of the power spectrum. We show how this degeneracy can be exploited to speed up the inference of parameter constraints from cosmological observations. We also present a new design of an emulator of the non-linear power spectrum whose predictions can be adapted to an arbitrary choice of evolution parameters and redshift.
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