Evolution mapping: a new approach to describe matter clustering in the
  non-linear regime
  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.
