Log-transforming the matter power spectrum
  We investigate whether non-linear effects on the large-scale power spectrum
of dark matter, namely the increase in small-scale power and the smearing of
baryon acoustic oscillations, can be decreased by a log-transformation or
emulated by an exponential transformation of the linear spectrum. To that end
we present a formalism to convert the power spectrum of a log-normal field to
the power spectrum of the logarithmic Gaussian field and vice versa. All
ingredients of our derivation can already be found in various publications in
cosmology and other fields. We follow a more pedagogical approach providing a
detailed derivation, application examples, and a discussion of implementation
subtleties in one text. We use the formalism to show that the non-linear
increase in small-scale power in the matter power spectrum is significantly
smaller for the log-transformed spectrum which fits the linear spectrum (with
less than 20% error) for redshifts down to 1 and $k\leq1.0\,h\,\mathrm{Mpc}$.
For lower redshifts the fit to the linear spectrum is not as good, but the
reduction of non-linear effects is still significant. Similarly, we show that
applying the linear growth factor to the logarithmic density leads to an
automatic increase in small-scale power for low redshifts fitting to
third-order perturbation spectra and Cosmic Emulator spectra with an error of
less than $20%$. Smearing of baryon acoustic oscillations is at least three
times weaker, but still present.
