Accurate Estimators of Correlation Functions in Fourier Space
  Efficient estimators of Fourier-space statistics for large number of objects
rely on Fast Fourier Transforms (FFTs), which are affected by aliasing from
unresolved small scale modes due to the finite FFT grid. Aliasing takes the
form of a sum over images, each of them corresponding to the Fourier content
displaced by increasing multiples of the sampling frequency of the grid. These
spurious contributions limit the accuracy in the estimation of Fourier-space
statistics, and are typically ameliorated by simultaneously increasing grid
size and discarding high-frequency modes. This results in inefficient estimates
for e.g. the power spectrum when desired systematic biases are well under
per-cent level. We show that using interlaced grids removes odd images, which
include the dominant contribution to aliasing. In addition, we discuss the
choice of interpolation kernel used to define density perturbations on the FFT
grid and demonstrate that using higher-order interpolation kernels than the
standard Cloud in Cell algorithm results in significant reduction of the
remaining images. We show that combining fourth-order interpolation with
interlacing gives very accurate Fourier amplitudes and phases of density
perturbations. This results in power spectrum and bispectrum estimates that
have systematic biases below 0.01% all the way to the Nyquist frequency of the
grid, thus maximizing the use of unbiased Fourier coefficients for a given grid
size and greatly reducing systematics for applications to large cosmological
datasets.
