Fast Large Scale Structure Perturbation Theory using 1D FFTs
  The usual fluid equations describing the large-scale evolution of mass
density in the universe can be written as local in the density, velocity
divergence, and velocity potential fields. As a result, the perturbative
expansion in small density fluctuations, usually written in terms of
convolutions in Fourier space, can be written as a series of products of these
fields evaluated at the same location in configuration space. Based on this, we
establish a new method to numerically evaluate the 1-loop power spectrum (i.e.,
Fourier transform of the 2-point correlation function) with one-dimensional
Fast Fourier Transforms. This is exact and a few orders of magnitude faster
than previously used numerical approaches. Numerical results of the new method
are in excellent agreement with the standard quadrature integration method.
This fast model evaluation can in principle be extended to higher loop order
where existing codes become painfully slow. Our approach follows by writing
higher order corrections to the 2-point correlation function as, e.g., the
correlation between two second-order fields or the correlation between a linear
and a third-order field. These are then decomposed into products of
correlations of linear fields and derivatives of linear fields. The method can
also be viewed as evaluating three-dimensional Fourier space convolutions using
products in configuration space, which may also be useful in other contexts
where similar integrals appear.
