BullFrog: Multi-step perturbation theory as a time integrator for cosmological simulations

TL;DR

BullFrog introduces a multi-step perturbation time integrator for cosmological N-body simulations that converges rapidly to the exact solution, improving upon existing LPT-inspired methods without extra computational cost.

Contribution

The paper presents BullFrog, a second-order accurate integrator that automatically performs 2LPT steps, enhancing simulation accuracy without additional computational overhead.

Findings

BullFrog converges rapidly to the exact solution with increasing time steps.

It outperforms other LPT-inspired integrators like FastPM and COLA.

Implementation in existing N-body codes is straightforward.

Abstract

Modelling the cosmic large-scale structure can be done through numerical N-body simulations or by using perturbation theory. Here, we present an N-body approach that effectively implements a multi-step forward model based on Lagrangian Perturbation Theory (LPT) in a $\Lambda$CDM Universe. This is achieved by introducing the second-order accurate BullFrog integrator, which automatically performs 2LPT time steps to second order without requiring the explicit computation of 2LPT displacements. Importantly, we show that BullFrog trajectories rapidly converge to the exact solution as the number of time steps increases, at any moment in time, even though 2LPT becomes invalid after shell-crossing. As a validation test, we compare BullFrog against other N-body integrators and high-order LPT, both for a realistic $\Lambda$CDM cosmology and for simulations with a sharp UV cutoff in the initial conditions. The latter scenario enables controlled experiments against LPT and, in practice, is particularly relevant for modelling coarse-grained fluids arising in the context of effective field theory. We demonstrate that BullFrog significantly improves upon other LPT-inspired integrators, such as FastPM and COLA, without incurring any computational overhead compared to standard N-body integrators. Implementing BullFrog in any existing N-body code is straightforward, particularly if FastPM is already integrated.