pith. sign in

arxiv: 2301.09655 · v3 · submitted 2023-01-23 · 🌌 astro-ph.CO · astro-ph.IM· physics.comp-ph

Perturbation-theory informed integrators for cosmological simulations

Florian List , Oliver Hahn This is my paper

Pith reviewed 2026-05-24 10:13 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.IMphysics.comp-ph
keywords cosmological simulationsN-body integratorsLagrangian perturbation theorytime-stepping schemesZel'dovich approximationpower spectrumshell crossingFastPM
0
0 comments X

The pith

Integrators derived by matching leapfrog steps to Lagrangian perturbation theory trajectories exactly recover the Zel'dovich solution in 1D and need fewer timesteps to match power spectra in 2D and 3D cosmological simulations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs time-stepping schemes for N-body cosmological simulations by setting the particle displacements in a single drift-kick-drift leapfrog step equal to those predicted by Lagrangian perturbation theory. These schemes recover the exact analytic Zel'dovich solution for one-dimensional planar collapse before trajectories cross. In two and three dimensions the new integrators reproduce the power spectrum and bispectrum of the density field more accurately than both standard leapfrog and the FastPM scheme when only a small number of timesteps is used. The analysis further shows that any integrator's convergence order is limited to 3/2 after shell crossing because the acceleration field loses regularity there, and that symplecticity of the integrator has only minor influence on accuracy in the few-timestep regime.

Core claim

By equating the displacement produced by one Verlet drift-kick-drift step to the first-order Lagrangian perturbation theory displacement, a family of integrators is obtained that yields the exact Zel'dovich solution in one dimension before shell crossing. In higher dimensions these integrators require fewer timesteps than conventional or FastPM methods to reach a given accuracy in the power spectrum and bispectrum for fast approximate simulations with O(1-100) steps. Post-shell-crossing convergence is limited to order 3/2 for any integrator because the acceleration field is not sufficiently regular, and symplecticity plays only a secondary role when the number of timesteps is small.

What carries the argument

The LPT-matching condition that sets the coefficients of a single drift-kick-drift step so its displacement equals the Lagrangian perturbation theory prediction.

If this is right

  • The schemes exactly reproduce the analytic Zel'dovich solution in one-dimensional pre-shell-crossing collapse.
  • Fewer timesteps suffice to match power spectrum and bispectrum accuracy in two- and three-dimensional fast simulations.
  • Convergence order after shell crossing is capped at 3/2 for any integrator because the acceleration field lacks higher regularity.
  • Symplecticity has only minor effect on accuracy when the timestep count is small.
  • Timestep spacing and the presence of a decaying mode in the initial conditions both affect the achieved accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matching idea could be applied to second-order Lagrangian perturbation theory to extend the accurate regime deeper into the nonlinear regime.
  • Hybrid integrators that switch from LPT-matched steps to conventional steps after shell crossing could mitigate the 3/2-order limit.
  • The reduced importance of symplecticity suggests that energy conservation is secondary to local trajectory accuracy when only a handful of steps are affordable.
  • Adaptive choice of timestep spacing guided by the local LPT error estimate might further lower the step count needed for a target accuracy.

Load-bearing premise

That matching trajectories in one isolated step to first-order perturbation theory still improves global statistics such as power spectra once the simulation enters the mildly nonlinear regime in two or three dimensions.

What would settle it

A controlled 3D N-body run with exactly ten timesteps that directly compares the power-spectrum error of the new LPT-matched integrator against FastPM at identical step count and initial conditions.

Figures

Figures reproduced from arXiv: 2301.09655 by Florian List, Oliver Hahn.

Figure 1
Figure 1. Figure 1: Comparison between the trajectory of an individual particle extracted from a two-dimensional simulation using the 𝛱-integrators PowerFrog, LPTFrog and FastPM, for 1, 2, and 4 integration steps to the end time. The initial particle position (marker in the lower left corner) and momentum are computed with 2LPT. For DKD schemes, hollow markers indicate the position of the particle after the first drift of eac… view at source ↗
Figure 2
Figure 2. Figure 2: Coefficient functions 𝑝(Δ𝑎, 𝑎) and 𝑞(Δ𝑎, 𝑎) defining the kick for the DKD integrators that we consider in this work for EdS cosmology, see Eq. (38b). By construction, 𝑝FastPM DKD = 𝑝Symplectic 2 and 𝑞TsafPM = 𝑞Symplectic 2. For non-integer values of 𝜖 < 3/2 (such as 𝜖 = 1.15 shown here), Δ𝑎 needs to remain small enough such that 𝑝𝜖 (Δ𝑎, 𝑎) ≥ 0 with the 𝜖 -integrator; the maximum value of Δ𝑎 for which this … view at source ↗
Figure 3
Figure 3. Figure 3: Initial conditions and reference solutions for our numerical experiments in 1D. Left: Initial displacement 𝛹 at 𝑎ini = 0.01 as a function of the Lagrangian coordinate 𝑞. ‘+’: Growing-mode-only case considered in Sections 5.1 and 5.2, ‘+−’: Mixed growing/decaying-mode case considered in Section 5.3. Centre: Displacement at the final time 𝑎end = 0.9 < 1.0 = 𝑎cross (pre-shell-crossing, for the examples in Sec… view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the numerically computed displacement field 𝛹 for the 1D growing-mode-only solution prior to shell-crossing at 𝑎end = 0.9 < 1.0 = 𝑎cross. Different colours correspond to different integrators, and different line styles indicate the time variable with respect to which the timesteps were taken to be uniform. To ensure a fair comparison between the methods, the 𝑥-axis shows the number of total … view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the numerically computed displacement field 𝛹 for the 1D growing-mode-only solution in the post-shell-crossing regime at 𝑎end = 2.0 > 1.0 = 𝑎cross. Now, the Zel’dovich-consistent integrators FastPM, LPTFrog, and PowerFrog no longer produce the exact solution and are on par with Symplectic 2 when suitable timesteps are chosen. Importantly, note that all methods converge at order 3/2, irrespec… view at source ↗
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Deviation of the energy in the 𝑁-body simulation from the cosmic energy (Layzer–Irvine) equation for a 1D growing-mode-only simulation that shell-crosses at 𝑎cross = 1.0, for uniform timesteps w.r.t. 𝑡˜, log 𝑎, and 𝑎. Clearly, the RK3 integrator and LPTFrog perform similarly w.r.t. this metric to the symplectic methods. The major source of energy errors is given by overly large timesteps. We will study ene… view at source ↗
Figure 8
Figure 8. Figure 8: Absolute difference between the 𝑁-body and LPT density contrast after a single timestep in two dimensions. Whereas the original FastPM KDK scheme is closest to the 1LPT (Zel’dovich) solution, it does not capture the corrections by higher LPT orders. On the other hand, the DKD variant of FastPM, LPTFrog, and TsafPM have a much smaller error w.r.t. 2LPT. PowerFrog achieves the best performance and produces a… view at source ↗
Figure 9
Figure 9. Figure 9: Kernel density estimates of the error distributions between the 𝑁-body simulation and different LPT orders after a single step in two dimensions, for the displacement (left) and the density field (right). As expected in view of its construction, the FastPM integrator produces the smallest residual w.r.t. to the 1LPT solution, but has a much larger error towards the 2LPT and 3LPT solutions. In contrast, the… view at source ↗
Figure 10
Figure 10. Figure 10: Quijote simulations: normalised cross-power spectra (top), and transfer functions of the power spectrum (middle) and equilateral bispectrum (bottom) between the fast simulations and our reference simulation that used 128 timesteps, for different integrators at 𝑧 = 0. The number of timesteps increases from left to right (uniformly spaced in 𝐷). Shaded regions indicate the standard deviation computed over 2… view at source ↗
Figure 11
Figure 11. Figure 11: Camels simulations: normalised cross-power spectra (top), and transfer functions of the power spectrum (middle) and equilateral bispectrum (bottom) between the fast simulations and our reference simulation that used 256 timesteps for different integrators at 𝑧 = 0. The number of timesteps increases from left to right (uniformly spaced in 𝐷). Shaded regions indicate the standard deviation computed over the… view at source ↗
Figure 12
Figure 12. Figure 12: Displacement errors (in ℎ −1 Mpc) for the Quijote (top) and Camels (bottom) experiments with different integrators as a function of the number of timesteps, w.r.t. the 𝐿 2 norm (left) and 𝐿 ∞ norm (right). Solid (dashed) lines correspond to the DKD (KDK) variant of the integrator. For PowerFrog, we only consider the DKD case here. The indicated convergence orders are for orientation only and do not necess… view at source ↗
read the original abstract

Large-scale cosmological simulations are an indispensable tool for modern cosmology. To enable model-space exploration, fast and accurate predictions are critical. In this paper, we show that the performance of such simulations can be further improved with time-stepping schemes that use input from cosmological perturbation theory. Specifically, we introduce a class of time-stepping schemes derived by matching the particle trajectories in a single leapfrog/Verlet drift-kick-drift step to those predicted by Lagrangian perturbation theory (LPT). As a corollary, these schemes exactly yield the analytic Zel'dovich solution in 1D in the pre-shell-crossing regime (i.e. before particle trajectories cross). One representative of this class is the popular FastPM scheme by Feng et al. 2016, which we take as our baseline. We then construct more powerful LPT-inspired integrators and show that they outperform FastPM and standard integrators in fast simulations in two and three dimensions with $\mathcal{O}(1 - 100)$ timesteps, requiring less steps to accurately reproduce the power spectrum and bispectrum of the density field. Furthermore, we demonstrate analytically and numerically that, for any integrator, convergence is limited in the post-shell-crossing regime (to order 3/2 for planar wave collapse), owing to the lacking regularity of the acceleration field, which makes the use of high-order integrators in this regime futile. Also, we study the impact of the timestep spacing and of a decaying mode present in the initial conditions. Importantly, we find that symplecticity of the integrator plays a minor role for fast approximate simulations with a small number of timesteps.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a class of time-stepping integrators for cosmological N-body simulations derived by matching the particle trajectories in a single drift-kick-drift Verlet step to those from Lagrangian perturbation theory (LPT). It shows that these schemes exactly reproduce the Zel'dovich solution in one dimension before shell-crossing. Building on FastPM as a baseline, the authors construct improved variants and demonstrate through 2D and 3D simulations that they require fewer timesteps to accurately match the power spectrum and bispectrum compared to standard leapfrog or FastPM. Additionally, they argue analytically and numerically that convergence is limited to order 3/2 post-shell-crossing due to the irregularity of the acceleration field, rendering high-order integrators ineffective in that regime, and find that symplecticity plays a minor role for simulations with small timestep counts.

Significance. If the reported performance improvements hold, this provides a principled route to more efficient approximate cosmological simulations for model-space exploration. The exact 1D pre-shell-crossing match to Zel'dovich and the analytic/numeric demonstration of the post-shell-crossing convergence limit (order 3/2 for planar collapse) are clear strengths that add both practical value and theoretical insight. The observation that symplecticity is secondary for low-timestep runs challenges standard assumptions in the integrator literature.

major comments (2)
  1. [sections presenting 2D/3D numerical experiments] The central performance claim—that the new LPT-informed integrators outperform FastPM and standard leapfrog for P(k) and B(k) in 2D/3D with O(1–100) timesteps—depends on the assumption that single-step first-order LPT trajectory matching remains advantageous once the simulation enters the mildly nonlinear regime. The numerical tests must include explicit convergence plots versus timestep number and direct comparisons against runs that incorporate higher-order LPT corrections to confirm the benefit survives.
  2. [analytic and numeric post-shell-crossing convergence discussion] The analytic argument that any integrator is limited to order-3/2 convergence post-shell-crossing (due to lacking regularity of the acceleration field) is load-bearing for the recommendation against high-order schemes in that regime; the manuscript should state the precise regularity assumption on the force field and confirm that the proposed integrators obey the same bound in the planar-wave test.
minor comments (2)
  1. The study of timestep spacing and the impact of a decaying mode in the initial conditions is mentioned but would benefit from dedicated quantitative figures or tables showing the sensitivity of the reported gains.
  2. [methods section] Notation for the drift-kick-drift operators and the specific LPT order used in each integrator variant should be made fully explicit in the methods section to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and constructive feedback. We address each major comment below. Where the suggestions strengthen the manuscript without altering its core claims, we will incorporate revisions.

read point-by-point responses

  1. Referee: [sections presenting 2D/3D numerical experiments] The central performance claim—that the new LPT-informed integrators outperform FastPM and standard leapfrog for P(k) and B(k) in 2D/3D with O(1–100) timesteps—depends on the assumption that single-step first-order LPT trajectory matching remains advantageous once the simulation enters the mildly nonlinear regime. The numerical tests must include explicit convergence plots versus timestep number and direct comparisons against runs that incorporate higher-order LPT corrections to confirm the benefit survives.

    Authors: We agree that explicit convergence plots versus timestep number would improve clarity. In the revised version we will add these for the 2D and 3D power-spectrum and bispectrum errors, comparing our LPT-matched integrators directly to FastPM and leapfrog across O(1–100) steps. On higher-order LPT corrections: our schemes are constructed by matching to first-order LPT within a single Verlet step; FastPM is likewise first-order. The reported gains are therefore relative to an equivalent baseline. Incorporating higher-order LPT into the force evaluation would change the underlying simulation rather than test the integrator. We will add a clarifying paragraph noting this scope and confirming that the advantage is demonstrated within the first-order LPT framework used throughout the paper. This constitutes a partial revision. revision: partial

  2. Referee: [analytic and numeric post-shell-crossing convergence discussion] The analytic argument that any integrator is limited to order-3/2 convergence post-shell-crossing (due to lacking regularity of the acceleration field) is load-bearing for the recommendation against high-order schemes in that regime; the manuscript should state the precise regularity assumption on the force field and confirm that the proposed integrators obey the same bound in the planar-wave test.

    Authors: The manuscript already derives the 3/2-order bound from the fact that the acceleration field is continuous but not differentiable across the shell-crossing surface in planar collapse. We will revise the analytic section to state the regularity assumption explicitly (acceleration belongs to C^0 but not C^1). We will also add a sentence confirming that the numerical convergence rate measured for our integrators in the planar-wave test saturates at the same 3/2 order, consistent with the bound applying to any integrator. This is a clarification of existing content. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivations anchored in external LPT and standard Verlet structure

full rationale

The paper constructs integrators by matching single D-K-D leapfrog trajectories to external Lagrangian perturbation theory (LPT) solutions, with FastPM (Feng et al. 2016) as an explicit baseline from other authors. The 1D pre-shell-crossing exact match to the Zel'dovich solution follows directly from using first-order LPT as the target, which is stated as a corollary rather than a novel prediction. Numerical tests of power/bispectrum accuracy in 2D/3D with O(1-100) steps are empirical validations against simulations, not reductions by construction. No parameters are fitted to data and then relabeled as predictions; no load-bearing self-citations appear; no uniqueness theorems or ansatzes are imported from the authors' prior work. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that LPT trajectories provide a useful matching target for single-step integrators and on the mathematical regularity analysis of the acceleration field after shell-crossing. No free parameters or invented entities are indicated.

axioms (2)
  • domain assumption Lagrangian perturbation theory supplies accurate particle trajectories in the pre-shell-crossing regime
    Invoked to derive the matching condition for the new integrators.
  • domain assumption The acceleration field loses sufficient regularity after shell-crossing to limit convergence order to 3/2 for planar collapse
    Used to conclude that high-order integrators are futile post-shell-crossing.

pith-pipeline@v0.9.0 · 5823 in / 1417 out tokens · 19599 ms · 2026-05-24T10:13:55.559790+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages · 36 internal anchors

  1. [1]

    FastPM: a new scheme for fast simulations of dark matter and halos

    Y. Feng, M.-Y. Chu, U. Seljak, P. McDonald, FastPM: A new scheme for fast simulations of dark matter and haloes, Monthly Notices of the Royal Astronomical Society 463 (3) (2016) 2273–2286.arXiv:1603.00476, doi:10.1093/mnras/stw2123

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/stw2123 2016

  2. [2]

    R. E. Angulo, O. Hahn, Large-scale dark matter simulations, Living Reviews in Computational Astrophysics 8 (1) (2022) 1.arXiv: 2112.05165, doi:10.1007/s41115-021-00013-z

    work page doi:10.1007/s41115-021-00013-z 2022

  3. [3]

    Peebles, The Large-scale Structure of the Universe, Princeton Series in Physics, Princeton University Press, 1980

    P. Peebles, The Large-scale Structure of the Universe, Princeton Series in Physics, Princeton University Press, 1980

    work page 1980

  4. [4]

    Time stepping N-body simulations

    T. Quinn, N. Katz, J. Stadel, G. Lake, Time stepping N-body simulations, PreprintarXiv:astro-ph/9710043

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    The cosmological simulation code GADGET-2

    V.Springel,ThecosmologicalsimulationcodeGADGET-2,MonthlyNoticesoftheRoyalAstronomicalSociety364(4)(2005)1105–1134. arXiv:astro-ph/0505010, doi:10.1111/j.1365-2966.2005.09655.x

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1111/j.1365-2966.2005.09655.x 2005

  6. [6]

    R.Laureijs,J.Amiaux,S.Arduini,J.-L.Augueres,J.Brinchmann,R.Cole,M.Cropper,C.Dabin,L.Duvet,A.Ealet,etal.,Eucliddefinition study report, PreprintarXiv:1110.3193

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    LSST: from Science Drivers to Reference Design and Anticipated Data Products

    Ž. Ivezić, S. M. Kahn, J. A. Tyson, B. Abel, E. Acosta, R. Allsman, D. Alonso, Y. AlSayyad, S. F. Anderson, J. Andrew, et al., LSST: from science drivers to reference design and anticipated data products, The Astrophysical Journal 873 (2) (2019) 111.arXiv:0805.2366

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Wide-Field InfraRed Survey Telescope (WFIRST) Final Report

    J. Green, P. Schechter, C. Baltay, R. Bean, D. Bennett, R. Brown, C. Conselice, M. Donahue, X. Fan, B. Gaudi, et al., Wide-field infrared survey telescope (WFIRST) final report, PreprintarXiv:1208.4012

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    PKDGRAV3: Beyond Trillion Particle Cosmological Simulations for the Next Era of Galaxy Surveys

    D. Potter, J. Stadel, R. Teyssier, PKDGRAV3: beyond trillion particle cosmological simulations for the next era of galaxy surveys, Computational Astrophysics and Cosmology 4 (1) (2017) 2.arXiv:1609.08621, doi:10.1186/s40668-017-0021-1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1186/s40668-017-0021-1 2017

  10. [10]

    N. A. Maksimova, L. H. Garrison, D. J. Eisenstein, B. Hadzhiyska, S. Bose, T. P. Satterthwaite, AbacusSummit: a massive set of high- accuracy, high-resolution N-body simulations, Monthly Notices of the Royal Astronomical Society 508 (3) (2021) 4017–4037.arXiv: 2110.11398, doi:10.1093/mnras/stab2484

    work page doi:10.1093/mnras/stab2484 2021

  11. [11]

    HACC: Simulating Sky Surveys on State-of-the-Art Supercomputing Architectures

    S. Habib, A. Pope, H. Finkel, N. Frontiere, K. Heitmann, D. Daniel, P. Fasel, V. Morozov, G. Zagaris, T. Peterka, et al., HACC: Simulating sky surveys on state-of-the-art supercomputing architectures, New Astronomy 42 (2016) 49–65.arXiv:1410.2805

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    Heitmann, T

    K. Heitmann, T. D. Uram, H. Finkel, N. Frontiere, S. Habib, A. Pope, E. Rangel, J. Hollowed, D. Korytov, P. Larsen, et al., HACC Cosmological Simulations: First Data Release, The Astrophysical Journal Supplement Series 244 (1) (2019) 17.arXiv:1904.11966

    work page arXiv 2019

  13. [13]

    J. D. Emberson, H.-R. Yu, D. Inman, T.-J. Zhang, U.-L. Pen, J. Harnois-Déraps, S. Yuan, H.-Y. Teng, H.-M. Zhu, X. Chen, Z.-Z. Xing, Cosmological neutrino simulations at extreme scale, Research in Astronomy and Astrophysics 17 (8) (2017) 085.arXiv:1311.6130, doi:10.1088/1674-4527/17/8/85

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1674-4527/17/8/85 2017

  14. [14]

    T.Ishiyama,F.Prada,A.A.Klypin,M.Sinha,R.B.Metcalf,E.Jullo,B.Altieri,S.A.Cora,D.Croton,S.DeLaTorre,D.E.Millán-Calero, T. Oogi, J. Ruedas, C. A. Vega-Martínez, The Uchuu simulations: Data Release 1 and dark matter halo concentrations, Monthly Notices of the Royal Astronomical Society 506 (3) (2021) 4210–4231.arXiv:2007.14720, doi:10.1093/mnras/stab1755

    work page doi:10.1093/mnras/stab1755 2021

  15. [15]

    J. S. Bagla, TreePM: A Code for Cosmological N-Body Simulations, Journal of Astrophysics and Astronomy 23 (2002) 185–196.arXiv: astro-ph/9911025, doi:10.1007/BF02702282

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02702282 2002

  16. [16]

    Barnes, P

    J. Barnes, P. Hut, A hierarchical O(N log N) force-calculation algorithm, Nature 324 (6096) (1986) 446–449.doi:10.1038/324446a0

    work page doi:10.1038/324446a0 1986

  17. [17]

    A. W. Appel, An Efficient Program for Many-Body Simulation, SIAM Journal on Scientific and Statistical Computing 6 (1) (1985) 85–103

    work page 1985

  18. [18]

    Darden, D

    T. Darden, D. York, L. Pedersen, Particle mesh Ewald: An N log(N) method for Ewald sums in large systems, Journal of Computational Physics 98 (12) (1993) 10089–10092.doi:10.1063/1.464397

    work page doi:10.1063/1.464397 1993

  19. [19]

    Greengard, V

    L. Greengard, V. Rokhlin, A Fast Algorithm for Particle Simulations, Journal of Computational Physics 73 (2) (1987) 325–348.doi: 10.1016/0021-9991(87)90140-9

    work page doi:10.1016/0021-9991(87)90140-9 1987

  20. [20]

    Hockney, J

    R. Hockney, J. Eastwood, Computer Simulation Using Particles, Taylor & Francis, 1988

    work page 1988

  21. [21]

    O. Hahn, T. Abel, R. Kaehler, A new approach to simulating collisionless dark matter fluids, Monthly Notices of the Royal Astronomical Society 434 (2) (2013) 1171–1191.arXiv:1210.6652, doi:10.1093/mnras/stt1061

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/stt1061 2013

  22. [22]

    Michaux, O

    M. Michaux, O. Hahn, C. Rampf, R. E. Angulo, Accurate initial conditions for cosmological N-body simulations: minimizing truncation anddiscretenesserrors,MonthlyNoticesoftheRoyalAstronomicalSociety500(1)(2021)663–683. arXiv:2008.09588, doi:10.1093/ mnras/staa3149

    work page arXiv 2021

  23. [23]

    Colombi, Phase-space structure of protohalos: Vlasov versus particle-mesh, Astronomy and Astrophysics 647 (2021) A66.arXiv: 2012.04409, doi:10.1051/0004-6361/202039719

    S. Colombi, Phase-space structure of protohalos: Vlasov versus particle-mesh, Astronomy and Astrophysics 647 (2021) A66.arXiv: 2012.04409, doi:10.1051/0004-6361/202039719

    work page doi:10.1051/0004-6361/202039719 2021

  24. [24]

    Hayli, The method of the doubly individual step for N-body computations., in: D

    A. Hayli, The method of the doubly individual step for N-body computations., in: D. Bettis (Ed.), Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1974, pp. 304–312. doi:10.1007/BFb0066598

    work page doi:10.1007/bfb0066598 1974

  25. [25]

    S. L. W. McMillan, The Vectorization of Small-N Integrators, in: P. Hut, S. L. W. McMillan (Eds.), The Use of Supercomputers in Stellar Dynamics, Vol. 267, Springer, Berlin, Heidelberg, New York, 1986, p. 156.doi:10.1007/BFb0116406

    work page doi:10.1007/bfb0116406 1986

  26. [26]

    Y. B. Zel’dovich, Gravitational instability: An approximate theory for large density perturbations., Astronomy and Astrophysics 5 (1970) 84–89

    work page 1970

  27. [27]

    B.Marcos,T.Baertschiger,M.Joyce,A.Gabrielli,F.SylosLabini,LinearperturbativetheoryofthediscretecosmologicalN-bodyproblem, Physical Review D 73 (10) (2006) 103507.arXiv:astro-ph/0601479, doi:10.1103/PhysRevD.73.103507

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.73.103507 2006

  28. [28]

    Towards quantitative control on discreteness error in the non-linear regime of cosmological N body simulations

    M. Joyce, B. Marcos, T. Baertschiger, Towards quantitative control on discreteness error in the non-linear regime of cosmological N- body simulations, Monthly Notices of the Royal Astronomical Society 394 (2) (2009) 751–773.arXiv:0805.1357, doi:10.1111/j. 1365-2966.2008.14290.x. 5https://github.com/DifferentiableUniverseInitiative/jax_cosmo 6https://pypi....

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1111/j 2009

  29. [29]

    Physical Bayesian modelling of the non-linear matter distribution: new insights into the Nearby Universe

    J. Jasche, G. Lavaux, Physical Bayesian modelling of the non-linear matter distribution: New insights into the nearby universe, Astronomy and Astrophysics 625 (2019) A64.arXiv:1806.11117, doi:10.1051/0004-6361/201833710. URL https://www.aanda.org/10.1051/0004-6361/201833710

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361/201833710 2019

  30. [30]

    The frontier of simulation-based inference

    K. Cranmer, J. Brehmer, G. Louppe, The frontier ofsimulation-based inference, Proceedings of the NationalAcademy of Sciences 117 (48) (2020) 30055–30062.arXiv:1911.01429, doi:10.1073/pnas.1912789117. URL https://pnas.org/doi/full/10.1073/pnas.1912789117

    work page doi:10.1073/pnas.1912789117 2020

  31. [31]

    C. Modi, F. Lanusse, U. Seljak, FlowPM: Distributed TensorFlow implementation of the FastPM cosmological N-body solver, Astronomy and Computing 37 (2021) 100505.arXiv:2010.11847, doi:10.1016/j.ascom.2021.100505. URL https://linkinghub.elsevier.com/retrieve/pii/S2213133721000597

    work page doi:10.1016/j.ascom.2021.100505 2021

  32. [32]

    Y. Li, C. Modi, D. Jamieson, Y. Zhang, L. Lu, Y. Feng, F. Lanusse, L. Greengard, Differentiable Cosmological Simulation with Adjoint Method, PreprintarXiv:2211.09815

    work page arXiv

  33. [33]

    Villaescusa-Navarro, C

    F. Villaescusa-Navarro, C. Hahn, E. Massara, A. Banerjee, A. M. Delgado, D. K. Ramanah, T. Charnock, E. Giusarma, Y. Li, E. Allys, A. Brochard, C. Uhlemann, C.-T. Chiang, S. He, A. Pisani, A. Obuljen, Y. Feng, E. Castorina, G. Contardo, C. D. Kreisch, A. Nicola, J. Alsing, R. Scoccimarro, L. Verde, M. Viel, S. Ho, S. Mallat, B. Wandelt, D. N. Spergel, The...

    work page doi:10.3847/1538-4365/ab9d82 2020

  34. [34]

    Villaescusa-Navarro, D

    F. Villaescusa-Navarro, D. Anglés-Alcázar, S. Genel, D. N. Spergel, R. S. Somerville, R. Dave, A. Pillepich, L. Hernquist, D. Nelson, P. Torrey, D. Narayanan, Y. Li, O. Philcox, V. La Torre, A. M. Delgado, S. Ho, S. Hassan, B. Burkhart, D. Wadekar, N. Battaglia, G. Contardo, The CAMELS project: Cosmology and Astrophysics with MachinE Learning Simulations,...

    work page arXiv 2021

  35. [35]

    C.Rampf,U.Frisch,O.Hahn,Unveilingthesingulardynamicsinthecosmiclarge-scalestructure,MonthlyNoticesoftheRoyalAstronomical Society 505 (1) (2021) L90–L94.arXiv:1912.00868, doi:10.1093/mnrasl/slab053

    work page doi:10.1093/mnrasl/slab053 2021

  36. [36]

    Arnold, Mathematical methods of classical mechanics, Vol

    V. Arnold, Mathematical methods of classical mechanics, Vol. 60, Springer, 1989

    work page 1989

  37. [37]

    T. Abel, O. Hahn, R. Kaehler, Tracing the dark matter sheet in phase space, Monthly Notices of the Royal Astronomical Society 427 (1) (2012) 61–76.arXiv:1111.3944, doi:10.1111/j.1365-2966.2012.21754.x

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1111/j.1365-2966.2012.21754.x 2012

  38. [38]

    A. G. Doroshkevich, V. S. Ryaben’kii, S. F. Shandarin, Nonlinear theory of the development of potential perturbations, Astrophysics 9 (2) (1973) 144–153.doi:10.1007/BF01011421

    work page doi:10.1007/bf01011421 1973

  39. [39]

    A Convenient Set of Comoving Cosmological Variables and Their Application

    H. Martel, P. R. Shapiro, A convenient set of comoving cosmological variables and their application, Monthly Notices of the Royal Astronomical Society 297 (2) (1998) 467–485.arXiv:astro-ph/9710119, doi:10.1046/j.1365-8711.1998.01497.x

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1046/j.1365-8711.1998.01497.x 1998

  40. [40]

    T.Buchert,J.Ehlers,LagrangiantheoryofgravitationalinstabilityofFriedman-Lemaitrecosmologies–second-orderapproach: animproved model for non-linear clustering, Monthly Notices of the Royal Astronomical Society 264 (1993) 375–387.doi:10.1093/mnras/264.2. 375

    work page doi:10.1093/mnras/264.2 1993

  41. [41]

    F. R. Bouchet, S. Colombi, E. Hivon, R. Juszkiewicz, Perturbative Lagrangian approach to gravitational instability., Astronomy and Astrophysics 296 (1995) 575.arXiv:astro-ph/9406013

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  42. [42]

    The recursion relation in Lagrangian perturbation theory

    C.Rampf,TherecursionrelationinLagrangianperturbationtheory,JournalofCosmologyandAstroparticlePhysics2012(12)(2012)004. arXiv:1205.5274, doi:10.1088/1475-7516/2012/12/004

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2012/12/004 2012

  43. [43]

    Rampf, Cosmological Vlasov-Poisson equations for dark matter, Reviews of Modern Plasma Physics 5 (1) (2021) 10.doi:10.1007/ s41614-021-00055-z

    C. Rampf, Cosmological Vlasov-Poisson equations for dark matter, Reviews of Modern Plasma Physics 5 (1) (2021) 10.doi:10.1007/ s41614-021-00055-z

    work page 2021

  44. [44]

    Rampf, S

    C. Rampf, S. O. Schobesberger, O. Hahn, Analytical growth functions for cosmic structures in aΛCDM Universe, Monthly Notices of the Royal Astronomical Society 516 (2) (2022) 2840–2850.arXiv:2205.11347, doi:10.1093/mnras/stac2406. URL http://arxiv.org/abs/2205.11347

    work page doi:10.1093/mnras/stac2406 2022

  45. [45]

    A. D. Chernin, D. I. Nagirner, S. V. Starikova, Growth rate of cosmological perturbations in standard model: Explicit analytical solution, Astronomy and Astrophysics 399 (2003) 19–21.arXiv:astro-ph/0110107, doi:10.1051/0004-6361:20021763

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361:20021763 2003

  46. [46]

    Evolution of density perturbations in a realistic universe

    M. Demianski, Z. A. Golda, A. Woszczyna, Evolution of density perturbations in a realistic universe, General Relativity and Gravitation 37 (12) (2005) 2063–2082.arXiv:gr-qc/0504089, doi:10.1007/s10714-005-0180-2

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10714-005-0180-2 2005

  47. [47]

    R. D. Skeel, D. J. Hardy, Practical construction of modified hamiltonians, SIAM J. Sci. Comput. 23 (4) (2001) 1172–1188.doi: 10.1137/S106482750138318X

    work page doi:10.1137/s106482750138318x 2001

  48. [48]

    Hairer, M

    E. Hairer, M. Hochbruck, A. Iserles, C. Lubich, Geometric numerical integration, Oberwolfach Reports 3 (1) (2006) 805–882.doi: 10.14760/OWR-2006-14

    work page doi:10.14760/owr-2006-14 2006

  49. [49]

    Yoshida, Construction of higher order symplectic integrators, Physics Letters A 150 (5-7) (1990) 262–268

    H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A 150 (5-7) (1990) 262–268. doi:10.1016/ 0375-9601(90)90092-3

    work page 1990

  50. [50]

    R. I. McLachlan, M. Perlmutter, G. Quispel, On the nonlinear stability of symplectic integrators, BIT Numer. Math. 44 (2004) 99–117. doi:10.1023/B:BITN.0000025088.13092.7f

    work page doi:10.1023/b:bitn.0000025088.13092.7f 2004

  51. [51]

    Shang, Kam theorem of symplectic algorithms for hamiltonian systems, Numer

    Z.-j. Shang, Kam theorem of symplectic algorithms for hamiltonian systems, Numer. Math. 83 (1999) 477–496. doi:10.1007/ s002110050460

    work page 1999

  52. [52]

    A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in hamilton’s function, in: Dokl. Akad. Nauk SSSR, Vol. 98, 1954, pp. 527–530

    work page 1954

  53. [53]

    Moser, On invariant curves of area-preserving mappings of an annulus, Nachr

    J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, II (1962) 1–20

    work page 1962

  54. [54]

    V.I.Arnol’d,Proofofatheoremofa.n.kolmogorovontheinvarianceofquasi-periodicmotionsundersmallperturbationsofthehamiltonian, Russ. Math. Surv. 18 (5) (1963) 9.doi:10.1070/RM1963v018n05ABEH004130. URL https://dx.doi.org/10.1070/RM1963v018n05ABEH004130

    work page doi:10.1070/rm1963v018n05abeh004130 1963

  55. [55]

    Magnus, On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (4) (1954) 649–673.doi:10.1002/cpa.3160070404

    W. Magnus, On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (4) (1954) 649–673.doi:10.1002/cpa.3160070404

    work page doi:10.1002/cpa.3160070404 1954

  56. [56]

    The Magnus expansion and some of its applications

    S. Blanes, F. Casas, J. A. Oteo, J. Ros, The Magnus expansion and some of its applications, Physics Reports 470 (5-6) (2009) 151–238. arXiv:0810.5488, doi:10.1016/j.physrep.2008.11.001

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physrep.2008.11.001 2009

  57. [57]

    Kolmogorov, On the notion of mean, Selected works of AN Kolmogorov 25 (1991) 144–146

    A. Kolmogorov, On the notion of mean, Selected works of AN Kolmogorov 25 (1991) 144–146

    work page 1991

  58. [58]

    R. Teyssier, Cosmological hydrodynamics with adaptive mesh refinement: A new high resolution code called RAMSES, Astronomy and Florian List & Oliver Hahn/Journal of Computational Physics (2024) 39 Astrophysics 385 (1) (2002) 337–364.arXiv:0111367, doi:10.1051/0004-6361:20011817

    work page doi:10.1051/0004-6361:20011817 2024

  59. [59]

    J. E. Campbell, On a law of combination of operators, Proc. London Math. Soc. 29 (1897) 14–32

    work page

  60. [60]

    H. F. Baker, Alternant and continuous groups, Proc. London Math. Soc. 3 (1905) 24–47

    work page 1905

  61. [61]

    Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipziger Ber

    F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipziger Ber. 58 (1906) 19–48

    work page 1906

  62. [62]

    S. A. Chin, C.-R. Chen, Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation, The Journal of Chemical Physics 114 (17) (2001) 7338–7341

    work page 2001

  63. [63]

    Forest, R

    E. Forest, R. D. Ruth, Fourth-order symplectic integration, Physica D Nonlinear Phenomena 43 (1) (1990) 105–117.doi:10.1016/ 0167-2789(90)90019-L

    work page 1990

  64. [64]

    Candy, W

    J. Candy, W. Rozmus, A Symplectic Integration Algorithm for Separable Hamiltonian Functions, Journal of Computational Physics 92 (1) (1991) 230–256.doi:10.1016/0021-9991(91)90299-Z

    work page doi:10.1016/0021-9991(91)90299-z 1991

  65. [65]

    Taruya, S

    A. Taruya, S. Colombi, Post-collapse perturbation theory in 1D cosmology - beyond shell-crossing, Monthly Notices of the Royal Astro- nomical Society 470 (4) (2017) 4858–4884.doi:10.1093/MNRAS/STX1501

    work page doi:10.1093/mnras/stx1501 2017

  66. [66]

    doi:10.1086/167681

    A.L.Melott,S.F.Shandarin,Gravitationalinstabilitywithhighresolution,TheAstrophysicalJournal343(1989)26. doi:10.1086/167681. URL http://adsabs.harvard.edu/doi/10.1086/167681

    work page doi:10.1086/167681 1989

  67. [67]

    S.Tassev,M.Zaldarriaga,D.J.Eisenstein,SolvinglargescalestructureinteneasystepswithCOLA,JournalofCosmologyandAstroparticle Physics 2013 (6) (2013) 036.arXiv:1301.0322, doi:10.1088/1475-7516/2013/06/036

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2013/06/036 2013

  68. [68]

    L-PICOLA: A parallel code for fast dark matter simulation

    C. Howlett, M. Manera, W. J. Percival, L-PICOLA: A parallel code for fast dark matter simulation, Astronomy and Computing 12 (2015) 109–126.arXiv:1506.03737

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  69. [69]

    sCOLA: The N-body COLA Method Extended to the Spatial Domain

    S. Tassev, D. J. Eisenstein, B. D. Wandelt, M. Zaldarriaga, sCOLA: The N-body COLA Method Extended to the Spatial Domain, Preprint arXiv:1502.07751

    work page internal anchor Pith review Pith/arXiv arXiv

  70. [70]

    ICE-COLA: Towards fast and accurate synthetic galaxy catalogues optimizing a quasi $N$-body method

    A. Izard, M. Crocce, P. Fosalba, ICE-COLA: Towards fast and accurate synthetic galaxy catalogues optimizing a quasi-N-body method, Monthly Notices of the Royal Astronomical Society 459 (3) (2016) 2327–2341.arXiv:1509.04685, doi:10.1093/mnras/stw797

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/stw797 2016

  71. [71]

    J. Koda, C. Blake, F. Beutler, E. Kazin, F. Marin, Fast and accurate mock catalogue generation for low-mass galaxies, Monthly Notices of the Royal Astronomical Society 459 (2) (2016) 2118–2129.arXiv:1507.05329, doi:10.1093/mnras/stw763

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/stw763 2016

  72. [72]

    Rampf, O

    C. Rampf, O. Hahn, Shell-crossing in aΛcDM Universe, Monthly Notices of the Royal Astronomical Society Letters 501 (1) (2021) L71–L75.doi:10.1093/mnrasl/slaa198

    work page doi:10.1093/mnrasl/slaa198 2021

  73. [73]

    A. E. Bayer, A. Banerjee, Y. Feng, A fast particle-mesh simulation of non-linear cosmological structure formation with massive neutrinos, Journal of Cosmology and Astroparticle Physics 2021 (01) (2021) 016.doi:10.1088/1475-7516/2021/01/016

    work page doi:10.1088/1475-7516/2021/01/016 2021

  74. [74]

    M., Tremonti, C., et al

    Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, A. Sobolevskiˇı, Reconstruction of the early Universe as a convex optimization problem, Monthly Notices of the Royal Astronomical Society 346 (2) (2003) 501–524.arXiv:0304214, doi: 10.1046/j.1365-2966.2003.07106.x

    work page doi:10.1046/j.1365-2966.2003.07106.x 2003

  75. [75]

    McQuinn, M

    M. McQuinn, M. White, Cosmological perturbation theory in 1+ 1 dimensions, Journal of Cosmology and Astroparticle Physics 2016 (01) (2016) 043. doi:10.1088/1475-7516/2016/01/043

    work page doi:10.1088/1475-7516/2016/01/043 2016

  76. [76]

    G. F. Corliss, Integrating ODEs in the complex plane—pole vaulting, Mathematics of Computation 35 (152) (1980) 1181–1189

    work page 1980

  77. [77]

    J. E. Chambers, Symplectic Integrators with Complex Time Steps, The Astronomical Journal 126 (2) (2003) 1119–1126.doi:10.1086/ 376844

    work page 2003

  78. [78]

    A critical look at cosmological perturbation theory techniques

    J. Carlson, M. White, N. Padmanabhan, Critical look at cosmological perturbation theory techniques, Physical Review D 80 (4) (2009) 043531. arXiv:0905.0479, doi:10.1103/PhysRevD.80.043531

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.80.043531 2009

  79. [79]

    A.Klypin,F.Prada,Darkmatterstatisticsforlargegalaxycatalogues: Powerspectraandcovariancematrices,MonthlyNoticesoftheRoyal Astronomical Society 478 (4) (2018) 4602–4621.arXiv:1701.05690, doi:10.1093/mnras/sty1340

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/sty1340 2018

  80. [80]

    M. Pietroni, Structure formation beyond shell-crossing: Nonperturbative expansions and late-time attractors, Journal of Cosmology and Astroparticle Physics 2018 (6).arXiv:1804.09140, doi:10.1088/1475-7516/2018/06/028

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2018/06/028 2018

Showing first 80 references.