DISCO-DJ II: a differentiable particle-mesh code for cosmology

The mildly non-linear regime of cosmic structure formation holds much of the information that upcoming large-scale structure surveys aim to exploit, making fast and accurate predictions on these scales essential. We present the $N$-body module of DISCO-DJ (DIfferentiable Simulations for COsmology - Done with Jax), designed to deliver high-fidelity, GPU-accelerated, and differentiable particle-mesh simulations tailored for cosmological inference. Theory-informed time integrators such as the recently introduced BullFrog method allow for accurate predictions already with few time steps (e.g. $6$ steps for per-cent-level accuracy in terms of the present-day power spectrum at $k \approx 0.2 \, h / \mathrm{Mpc}$ using $N = 512^3$ particles, which takes just a few seconds). To control discreteness effects and achieve high accuracy, the code incorporates a suite of advanced techniques, for example a custom non-uniform FFT implementation for force evaluation. Both forward- and reverse-mode differentiation are supported, with memory requirements independent of the number of time steps; in the reverse case, this is achieved through an adjoint formulation. We extensively study the effect of various numerical parameters on the accuracy. As an application of DISCO-DJ, we perform field-level inference by recovering $\sigma_8$ and the initial conditions from a noisy Gadget matter density field. Coupled with our recently introduced Einstein--Boltzmann solver, the DISCO-DJ ecosystem provides a self-consistent, fully differentiable pipeline for modelling the large-scale structure of the universe. The code is available at https://github.com/cosmo-sims/DISCO-DJ.