Optimal non-linear transformations for large scale structure statistics
  Recently, several studies proposed non-linear transformations, such as a
logarithmic or Gaussianization transformation, as efficient tools to recapture
information about the (Gaussian) initial conditions. During non-linear
evolution, part of the cosmologically relevant information leaks out from the
second moment of the distribution. This information is accessible only through
complex higher order moments or, in the worst case, becomes inaccessible to the
hierarchy. The focus of this work is to investigate these transformations in
the framework of Fisher information using cosmological perturbation theory of
the matter field with Gaussian initial conditions. We show that at each order
in perturbation theory, there is a polynomial of corresponding order exhausting
the information on a given parameter. This polynomial can be interpreted as the
Taylor expansion of the maximally efficient "sufficient" observable in the
non-linear regime. We determine explicitly this maximally efficient observable
for local transformations. Remarkably, this optimal transform is essentially
the simple power transform with an exponent related to the slope of the power
spectrum; when this is -1, it is indistinguishable from the logarithmic
transform. This transform Gaussianizes the distribution, and recovers the
linear density contrast. Thus a direct connection is revealed between undoing
of the non-linear dynamics and the efficient capture of Fisher information. Our
analytical results were compared with measurements from the Millennium
Simulation density field. We found that our transforms remain very close to
optimal even in the deeply non-linear regime with \sigma^2 \sim 10.
