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Ridged Lagrangian Perturbation Theory (RLPT)

Francisco-Shu Kitaura, Francesco Sinigaglia

Abstract

Galaxy surveys demand fast large-scale structure forward models that preserve large-scale phases while providing realistic nonlinear morphology at fixed force resolution. Single-step Lagrangian Perturbation Theory (LPT) solvers are efficient, but they typically yield overly diffuse filaments and knots and underpredict small-scale clustering. We introduce Ridged Lagrangian Perturbation Theory (RLPT), a modular two-step scheme: a standard long-range LPT/ALPT transport is followed by a single post-processing Eulerian {ridging} update that reconstructs a short-range, curl-free displacement from the realised density field through a smooth scale separation and a Poisson inversion. This explicit completion layer is inexpensive, preserves the large-scale solution, and provides a small set of transparent parameters to tune the short-range response. We test RLPT against particle-mesh and $N$-body references and find that one additional ridging step systematically improves both nonlinear power and field-level agreement relative to 2LPT/ALPT baselines. Finally, we demonstrate that ridging can be repurposed as a deterministic subgrid relocation model: even when the underlying dark-matter field is only ``good enough'' on the mesh, ridging enables controlled tuning of tracer clustering beyond the nominal resolution, which is particularly relevant for mock-galaxy production and observational systematics sensitive to close pairs.

Ridged Lagrangian Perturbation Theory (RLPT)

Abstract

Galaxy surveys demand fast large-scale structure forward models that preserve large-scale phases while providing realistic nonlinear morphology at fixed force resolution. Single-step Lagrangian Perturbation Theory (LPT) solvers are efficient, but they typically yield overly diffuse filaments and knots and underpredict small-scale clustering. We introduce Ridged Lagrangian Perturbation Theory (RLPT), a modular two-step scheme: a standard long-range LPT/ALPT transport is followed by a single post-processing Eulerian {ridging} update that reconstructs a short-range, curl-free displacement from the realised density field through a smooth scale separation and a Poisson inversion. This explicit completion layer is inexpensive, preserves the large-scale solution, and provides a small set of transparent parameters to tune the short-range response. We test RLPT against particle-mesh and -body references and find that one additional ridging step systematically improves both nonlinear power and field-level agreement relative to 2LPT/ALPT baselines. Finally, we demonstrate that ridging can be repurposed as a deterministic subgrid relocation model: even when the underlying dark-matter field is only ``good enough'' on the mesh, ridging enables controlled tuning of tracer clustering beyond the nominal resolution, which is particularly relevant for mock-galaxy production and observational systematics sensitive to close pairs.
Paper Structure (27 sections, 20 equations, 9 figures)

This paper contains 27 sections, 20 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Method
  4. Step 1: long-range Lagrangian transport
  5. Step 2: Eulerian ridging (short-range potential completion)
  6. Spherical-collapse remapping on short-range modes
  7. Short-range potential displacement
  8. Optional extension: vortical ridging
  9. Helmholtz decomposition and update rule
  10. A minimal, calibratable vorticity-source model
  11. Smooth-particle ridging (SPRLPT): a SPH-inspired short-range completion
  12. Scheme
  13. Kernel density estimate
  14. Pressure-like driving (ridge contraction)
  15. Viscosity-like regularisation
  16. ...and 12 more sections

Figures (9)

  • Figure 1: Power spectra of the different LPT-based approximations compared with the FastPM particle-mesh solution with a volume of $L=100\,h^{-1}\,{\rm Mpc}$ side and $400^3$ particles. Upper panel:$z=0$; lower panel:$z=1$.
  • Figure 2: Cross power spectra (defined as $C(k)=\left\langle \hat{\delta}_{\rm A}(\mathbf{k})\,\hat{\delta}^{\,*}_{\rm B}(\mathbf{k})\right\rangle/\sqrt{P_{\rm A}(k)P_{\rm B}(k)}.$ for two fields $\mathrm{A}$ and $\mathrm{B}$) of the different LPT-based approximations compared with the FastPM particle-mesh solution with a volume of $L=100\,h^{-1}\,{\rm Mpc}$ side and $400^3$ particles. Upper panel:$z=0$; lower panel:$z=1$.
  • Figure 3: Power spectra of the different LPT-based approximations compared with the FastPMparticle-mesh solution with a volume of $L=200\,h^{-1}\,{\rm Mpc}$ side and $256^3$ particles. Upper panel:$z=0$; lower panel:$z=1$.
  • Figure 4: Cross power spectra of the different LPT-based approximations compared with the FastPMparticle-mesh solution with a volume of $L=200\,h^{-1}\,{\rm Mpc}$ side and $256^3$ particles. Upper panel:$z=0$; lower panel:$z=1$.
  • Figure 5: Power spectra of the different LPT-based approximations compared with the Abacus full $N$-body simulation with a volume of $L=2000\,h^{-1}\,{\rm Mpc}$ side and a mesh of $360^3$ cells. Upper panel:$z=0$; lower panel:$z=1$.
  • ...and 4 more figures