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

Abstract

We present 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 highresolution simulations.

What does the Marked Power Spectrum Measure? Insights from Perturbation Theory

“…which is the convolution of A and B with kernel X. 10 This is symmetric under A ↔ B if X is symmetric in its arguments. Furthermore, * F 2 2 [P L , P L ] (k) is equal to half the P 22 (k) term of the perturbative matter power spectrum.…”

On the fast random sampling and other properties of the three point correlation function in galaxy surveys

“…Our independently developed work extends on the latter using two different approaches. (See a similar discussion in Fourier space which also uses the multipole decomposition [36,37]).…”

Modeling the Marked Spectrum of Matter and Biased Tracers in Real- and Redshift-Space

“…Setting f = µ = 0 straightforwardly gives the real-space counterpart to this. 13 The functions A n (k) and B(k) depend on the second-order biases and have weak k dependence through W R (k); these are given in (B.6) & (B.10) in Appendix B. Furthermore, this uses the variance definitions…”

“…For the first option, the lowest-order extension lies in the three-point correlator, or bispectrum, though there exists information beyond even this [1][2][3]. The bispectrum has been considered in a number of works [e.g., [4][5][6][7][8], though its use for cosmological parameter inference still presents considerable challenges, with particular difficulties arising from its estimation [9][10][11][12][13] and high-dimensionality [14]. It has thus been little adopted, though Refs.…”