A Faster Fourier Transform? Computing Small-Scale Power Spectra and Bispectra for Cosmological Simulations in $\mathcal{O}(N^2)$ Time

We present $\mathcal{O}(N^2)$ estimators for the small-scale power spectrum and bispectrum in cosmological simulations. In combination with traditional methods, these allow spectra to be efficiently computed across a vast range of scales, requiring orders of magnitude less computation time than Fast Fourier Transform based approaches alone. These methods are applicable to any tracer; simulation particles, halos or galaxies, and take advantage of the simple geometry of the box and periodicity to remove almost all dependence on large random particle catalogs. By working in configuration-space, both power spectra and bispectra can be computed via a weighted sum of particle pairs up to some radius, which can be reduced at larger $k$, leading to algorithms with decreasing complexity on small scales. These do not suffer from aliasing or shot-noise, allowing spectra to be computed to arbitrarily large wavenumbers. The estimators are rigorously derived and tested against simulations, and their covariances discussed. The accompanying code, HIPSTER, has been publicly released, incorporating these algorithms. Such estimators will be of great use in the analysis of large sets of high-resolution simulations.