Fast large scale structure perturbation theory using one-dimensional fast Fourier transforms

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.

Figure 1

Loop corrections $P_{22}$ and $P_{13}$ to the SPT matter power spectrum computed with conventional quadrature integration of the convolution integrals (29) and (38), and with the new 1D FFT method of Eqs. (22), (25) and (41). The bottom panel shows the relative error between the two methods. Its size is dominated by the accuracy of the quadrature routine which was set to ${10}^{-5}$.