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REVIEW 2 major objections 208 references

GINKAKU reduces differences in nonlinear power spectra across N-body codes to below 1 percent by tuning TreePM accuracy parameters and linear-response neutrino terms.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.3

2026-06-29 10:51 UTC pith:C4YQ4JHG

load-bearing objection GINKAKU is a new TreePM code on FDPS for controlled DQ2 ensembles with neutrinos and clustering DE, showing ~1% code-to-code P(k) agreement after tuning. the 2 major comments →

arxiv 2605.28581 v1 pith:C4YQ4JHG submitted 2026-05-27 astro-ph.CO

GINKAKU: Scalable Cosmological Structure Formation Simulation Code and Post-processing Pipeline

Takahiro Nishimichi , Satoshi Tanaka , Kohji Yoshikawa This is my paper
classification astro-ph.CO
keywords N-body simulationscosmological structure formationmassive neutrinosdark energy clusteringnonlinear power spectrumhalo mass functionTreePM solver
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces GINKAKU, a new N-body code built on the FDPS framework for large-scale cosmological simulations that include massive neutrinos and clustering dark energy. It pairs a TreePM gravity solver with linear-response evolution of external components in the N-body gauge, avoiding full particle treatment for those species while preserving Newtonian dynamics on subhorizon scales. Internal convergence tests and direct comparisons against GADGET, PKDGRAV3, and RAMSES show that code-to-code scatter in the nonlinear power spectrum drops below the 1 percent level once internal accuracy settings are adjusted to a chosen fiducial configuration. The same setup is applied to initial Dark Quest II production runs of 3000 cubed particles in boxes up to 4 h inverse Gpc, followed by a post-processing pipeline that brings halo mass function scatter to roughly 1 percent. These runs recover the expected nonlinear signatures of scale-dependent growth from neutrinos and dark energy, supporting emulator construction for next-generation surveys.

Core claim

GINKAKU demonstrates that its TreePM solver plus linear-response treatment of massive-neutrino, radiation, and clustering-dark-energy perturbations in the N-body gauge produces nonlinear matter power spectra that agree with established codes to within about 1 percent once accuracy parameters are tuned, at modest extra cost, and that the same framework reproduces the expected scale-dependent growth signatures in production-scale runs.

What carries the argument

Linear-response treatment of external source terms (massive neutrinos, radiation, clustering dark energy) in the N-body gauge, coupled to a TreePM gravity solver on the FDPS framework.

Load-bearing premise

The linear-response treatment of external source terms remains accurate on subhorizon scales without requiring full nonlinear particle evolution for those components.

What would settle it

A side-by-side run on identical initial conditions in which the fiducial GINKAKU settings produce a nonlinear power spectrum differing by more than 1 percent from PKDGRAV3 or RAMSES across the relevant k-range would falsify the accuracy claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Ensemble production across multiple cosmological models with massive neutrinos and clustering dark energy becomes feasible while keeping power-spectrum errors below 1 percent.
  • The post-processing pipeline reduces inter-resolution scatter in the halo mass function to about 1 percent and supplies halo-shape data for intrinsic-alignment statistics.
  • A total matter power spectrum emulator can be constructed directly from the production runs.
  • Scale-dependent growth signatures from neutrinos and dark energy appear in the nonlinear regime as expected.

Where Pith is reading between the lines

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

  • If the linear-response approximation holds, other relativistic or early-universe components could be added at the linear level without full particle evolution.
  • The controlled accuracy opens the possibility of running larger ensembles or higher-resolution boxes while staying within survey requirements for systematic error budgets.
  • Direct comparison of the same initial conditions evolved in different gauges would test how much the N-body gauge choice affects the final 1 percent agreement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces GINKAKU, a new cosmological N-body code built on the FDPS framework that couples a TreePM gravity solver with linear-response treatment (in the N-body gauge) of external source terms for massive neutrinos, radiation, and clustering dark energy. It reports internal convergence studies and cross-comparisons with GADGET, PKDGRAV3, and RAMSES on shared initial conditions, claiming that tuning internal accuracy parameters reduces code-to-code differences in the nonlinear power spectrum below the ~1% level, with a production-grade fiducial setting identified at modest cost. The code is applied to an initial set of Dark Quest II production runs (eight cosmological models, 3000^3 particles in boxes up to 4 h^{-1} Gpc) processed by a renewed post-processing pipeline that reduces inter-resolution halo mass function spread to ~1% and includes halo-shape measurements; the runs reproduce expected nonlinear signatures of massive neutrinos and clustering dark energy, with a total matter power spectrum emulator presented in an accompanying paper.

Significance. If the numerical control and post-processing claims hold, GINKAKU would provide a scalable platform for generating the large simulation ensembles needed for next-generation galaxy surveys, incorporating linear-response effects of massive neutrinos and clustering dark energy at controlled ~1% accuracy on power spectra and halo statistics.

major comments (2)
  1. [Abstract] Abstract: the central validation claim rests on cross-code agreement at <1% in nonlinear P(k) under a shared linear-response treatment for neutrinos/DE/radiation; this tests numerical consistency of the TreePM solver and parameter tuning across implementations but does not test the accuracy of the linear-response approximation itself on subhorizon scales (k ≳ 0.1 h Mpc^{-1}) where nonlinear evolution of those components could matter. No quantitative benchmark against a full particle-based neutrino reference is reported.
  2. [Abstract] Abstract and validation description: the statement that the DQ2 runs 'reproduce the expected nonlinear signatures' is presented as a qualitative consistency check rather than a quantitative test of the linear-response treatment's validity; this leaves the physical accuracy of the approximation unquantified for the production runs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on the scope of our validation. We respond point-by-point below and will revise the abstract and validation sections to clarify that the presented tests address numerical consistency of the TreePM implementation under the linear-response treatment, rather than the physical accuracy of the approximation itself.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central validation claim rests on cross-code agreement at <1% in nonlinear P(k) under a shared linear-response treatment for neutrinos/DE/radiation; this tests numerical consistency of the TreePM solver and parameter tuning across implementations but does not test the accuracy of the linear-response approximation itself on subhorizon scales (k ≳ 0.1 h Mpc^{-1}) where nonlinear evolution of those components could matter. No quantitative benchmark against a full particle-based neutrino reference is reported.

    Authors: We agree that the cross-code comparisons (with GADGET, PKDGRAV3, and RAMSES) under the shared linear-response treatment validate numerical consistency and parameter tuning of the TreePM solver, but do not test the physical accuracy of the linear-response approximation against a fully nonlinear treatment of neutrinos or clustering dark energy on subhorizon scales. No quantitative benchmark against a full particle-based neutrino reference is included, as this manuscript focuses on code implementation, internal convergence, and controlled production for the DQ2 campaign. We will revise the abstract to explicitly state the scope of the validation. revision: yes

  2. Referee: [Abstract] Abstract and validation description: the statement that the DQ2 runs 'reproduce the expected nonlinear signatures' is presented as a qualitative consistency check rather than a quantitative test of the linear-response treatment's validity; this leaves the physical accuracy of the approximation unquantified for the production runs.

    Authors: The statement that the DQ2 runs reproduce expected nonlinear signatures is intended as a qualitative consistency check demonstrating that the code produces physically plausible scale-dependent effects (e.g., neutrino suppression). It is not presented as a quantitative test of the linear-response approximation's validity. We will revise the abstract and validation description to make this distinction clearer and to note that the physical accuracy relies on the established linear-response formalism. revision: yes

Circularity Check

0 steps flagged

No significant circularity in validation or design claims

full rationale

The paper introduces GINKAKU as a new N-body code using TreePM solver plus linear-response treatment for neutrinos/DE/radiation in N-body gauge, then validates via direct cross-comparisons against independent external codes (GADGET, PKDGRAV3, RAMSES) on shared initial conditions. The central empirical claim—that tuning internal accuracy parameters reduces code-to-code nonlinear P(k) differences below ~1%—is a measured outcome of those external benchmarks, not a fitted parameter renamed as prediction or a derivation that reduces to its own inputs by construction. No self-citation load-bearing steps, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work appear in the abstract or described content. The linear-response approximation is presented as a deliberate design choice whose physical accuracy on subhorizon scales is not derived within the paper but taken as given for the production runs; the reported agreement tests numerical consistency across implementations rather than closing a self-referential loop. This is a standard code-validation manuscript with externally falsifiable benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; free parameters appear to be the internal accuracy settings tuned for 1% agreement. Standard cosmological N-body assumptions (Newtonian subhorizon dynamics, linear perturbation theory for non-particle components) are invoked without new derivation.

free parameters (1)
  • internal accuracy parameters
    Tuned to reduce code-to-code power-spectrum differences below 1%; specific values not stated in abstract.
axioms (1)
  • domain assumption Linear-response treatment of neutrinos, radiation, and clustering dark energy remains valid on subhorizon scales in N-body gauge
    Central design choice stated in abstract; no independent test of this approximation provided here.

pith-pipeline@v0.9.1-grok · 5852 in / 1259 out tokens · 29302 ms · 2026-06-29T10:51:56.423326+00:00 · methodology

0 comments
read the original abstract

We introduce GINKAKU, a new cosmological $N$-body code developed for the Dark Quest II (DQ2) simulation campaign and designed for controlled ensemble production across the cosmological model space required by next-generation galaxy surveys, including massive neutrinos and clustering dark energy. Built on the FDPS framework, GINKAKU couples a TreePM gravity solver with a linear-response treatment of external source terms for components not evolved as $N$-body particles, formulated in the $N$-body gauge. This design incorporates massive-neutrino perturbations, general-relativistic corrections, early-time radiation perturbations, and dark-energy clustering with non-unit effective sound speed at the linear level, while preserving Newtonian particle dynamics on subhorizon scales. The code is validated through internal convergence studies and cross-comparisons with GADGET, PKDGRAV3, and RAMSES on shared initial conditions: code-to-code differences in the nonlinear power spectrum can be reduced below $\sim1\%$ level by tuning internal accuracy parameters, and we identify a production-grade fiducial setting achieving this control at modest cost. We apply GINKAKU to an initial set of DQ2 production runs -- eight cosmological models with $3,000^3$ particles in boxes up to $4\,h^{-1}\mathrm{Gpc}$ -- processed by a renewed post-processing pipeline that reduces the inter-resolution spread of the halo mass function to $\sim 1\%$ and includes halo-shape measurements for intrinsic-alignment statistics. The scale-dependent-growth cosmologies reproduce the expected nonlinear signatures of massive neutrinos and clustering dark energy, demonstrating suitability for emulator-scale production. A total matter power spectrum emulator from these runs is presented in an accompanying paper. (abridged)

Figures

Figures reproduced from arXiv: 2605.28581 by Kohji Yoshikawa, Satoshi Tanaka, Takahiro Nishimichi.

Figure 1
Figure 1. Figure 1: Scale dependence of the linear growth factor from z = 99 to z = 0. We plot the ratio of the CDM+baryon transfer function at these two redshifts for four models with different neutrino masses as indicated in the figure legend. The transfer function is computed in the N-body gauge as explained in the text. Alt text: Four overlapping curves of the linear growth ratio plotted against wavenumber. The curves sep… view at source ↗
Figure 2
Figure 2. Figure 2: Gauge dependence of the cb transfer function. We show the cb transfer function in N-body (solid), conformal Newtonian (dotted), and synchronous (dashed) gauge at z = 0 (thick) and z = 99 (thin). We show the fractional difference from N-body gauge in the bottom panel. Note that the results at z = 0 are vertically offset by +0.2 for clarity. We adopt here the fiducial cosmological model (see [PITH_FULL_IMAG… view at source ↗
Figure 3
Figure 3. Figure 3: Boost factor at different redshifts for models with massive (Mν = 0.24eV, an exaggerated value well above current observational bounds, adopted here for demonstration purposes; solid) and massless (dotted) neutrinos. Alt text: Plot of the boost factor against wavenumber for two cosmologies at several redshifts. The massive-neutrino curves rise above unity on large scales with redshift-dependent amplitude, … view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the cb power spectrum from simulations with different numbers of particles. We show the fractional difference relative to the highest resolution setting (HR: Lbox = 1h −1 Gpc, Np = 3,0003 ). The MR (Np = 1,5003 ; solid) and LR (Np = 7503 ; dashed) simulations are started from the same initial random field as HR, within the same cubic comoving volume, but with fewer particles. Two different s… view at source ↗
Figure 6
Figure 6. Figure 6: Convergence of the power spectrum from simulations with different softening length parameters ϵp and time step sizes controlled by the parameter ηglobal for different resolutions (Np = 5003 particles in Lbox = 250, 500 and 1000h −1Mpc at z = 0). We adopt the result with the smallest softening length and the most stringent time steps as the reference. The parameter ηglobal rescales both PM and tree time ste… view at source ↗
Figure 7
Figure 7. Figure 7: Dependence of the power spectrum on the softening parameter ϵp, as depicted in the color bar (in percent). The other accuracy parameters are fixed to the fiducial setting. We present the power spectrum normalized by the result of the “accurate” setting. The simulations employ Np = 5003 and Lbox = 500h −1Mpc, with results shown at z = 3, 2, 1 and 0. The vertical line marks the Nyquist wavenumber kNy for thi… view at source ↗
Figure 8
Figure 8. Figure 8: Dependence on the tree opening angle θ. We consider both the accurate and fiducial settings for the other accuracy parameters and show the fractional difference to the most stringent tree opening angle, θ = 0.1, respectively in the upper and lower panels. The vertical lines show the Nyquist wavenumber kNy for each resolution. Alt text: Two-row grid of power-spectrum ratio panels comparing tree opening angl… view at source ↗
Figure 9
Figure 9. Figure 9: Convergence of the matter power spectrum with respect to the number of PM grid points, Ng. The different line styles represent simulations with Ng = 5003 (solid), 1,0003 (dashed) and 1,5003 (dotted), each compared to the spectrum with Ng = 2,0003 , which serves as the reference with the highest PM force accuracy. The transition scale between the tree and PM forces is fixed to rs = 1.5Hp, corresponding to v… view at source ↗
Figure 10
Figure 10. Figure 10: Impact of refining the PM force time step on the matter power spectrum. The solid line represents the fiducial simulation setting with the PM force time step criterion parameter ηPM = 0.125, compared to a run with a more stringent value of ηPM = 0.05. Refining the PM time step demonstrates a negligible effect on the matter power spectrum, with fractional differences remaining well within the ±1% band acro… view at source ↗
Figure 11
Figure 11. Figure 11: Impact of reducing the number of PM grid points, Ng, to Np/2 3 from the fiducial setting of 2 3Np on the matter power spectrum. The upper (lower) panels display results for ϵg = 3 (4.5). For comparison, the fiducial case with Ng = 2 3Np is also shown as dot-dashed lines. We plot the fractional differences relative to the “accurate” case, which also has Ng = 2 3Np. With fewer grid points, power suppression… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the matter power spectra from simulations using different codes. We use the result from GINKAKU with the “accurate” setting as the reference and show the fractional differences relative to it. The results from L-GADGET2 (dashed; DQ1 default accuracy parameters) and PKDGRAV3 (dotted) are plotted. Additionally, we show the result from GINKAKU with the fiducial accuracy setting (solid), which i… view at source ↗
Figure 14
Figure 14. Figure 14: Our mass determination scheme based on the interior overdensity. We show the interior overdensity as a function of radius from the halo center, considering ∆ = 200 as the density threshold. We select three halos at different mass scales, as indicated in the figure legend (values from the HR simulation), matched among three simulations with different resolutions (HR: solid, MR: dashed, and LR: dotted; the … view at source ↗
Figure 16
Figure 16. Figure 16: Averaged radial density profile of halos. We randomly select 1, 10, 100 and 1000 halos with 988km/s < Vmax < 1014km/s from the HR simulation at z = 0. The vertical arrow indicates the softening length. Alt text: Plot of the averaged radial density profile of halos in a narrow circular-velocity range, for four stacking sample sizes. The profile becomes progressively smoother as more halos are stacked, with… view at source ↗
Figure 15
Figure 15. Figure 15: Halo masses for three different density threshold values (∆ = 200,400 and 800 from top to bottom, as indicated by the y-axis labels) as a function of the maximum circular velocity Vmax in the HR simulation at z = 0. The left panels display all the structures found by ROCKSTAR, while the right panels show only central halos, determined based on the procedure explained in the main text. Alt text: Six-panel … view at source ↗
Figure 17
Figure 17. Figure 17: Logarithmic slope of the averaged radial density profile of central halos in the HR simulation at z = 0 for different density thresholds. We consider the 100,000 halos with the largest Vmax before the central-satellite split (solid), and after discarding satellites with three values of ∆ (dashed: 200, dotted: 400 and dot dashed: 800). The inset zooms in to show how the location of the steepest slope chang… view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of the halo mass function measured from simulations with different resolutions. We plot the ratio of the halo mass function measured from simulations started with exactly the same initial conditions in a (1h −1 Gpc)3 box, but traced by different numbers of particles. We adopt the highest resolution simulation with 3,0003 particles as the reference, and plot the results of lower resolution simul… view at source ↗
Figure 19
Figure 19. Figure 19: presents a two-dimensional projection of halo shapes as illustrated by elliptical boundaries, using the three different weighting schemes. The upper panels show ellipses at various radii for each scheme. The unweighted case, depicted in the left panel, appears the most stretched, particularly at the largest radius corresponding to Mvir, as it is influenced signif￾icantly by contributions from the outer re… view at source ↗
Figure 20
Figure 20. Figure 20: Distribution of the angle between the major axis of subhalo shapes and the direction toward the host halo center. We stack subhalos contained within the top 20 most massive host halos in the simulation box (Mvir > 2 × 1015 h −1 M⊙). The increasing trend indicates that the major axes of subhalos are preferentially oriented toward the center of the host halo. Each column corresponds to a different weighting… view at source ↗
Figure 21
Figure 21. Figure 21: Distribution of ellipticities at Mvir for all halos more massive than 1011 h −1 M⊙ found by ROCKSTAR in a simulation box (without removing subhalos). Columns correspond to the unweighted I (equation 35), the inverse-square weighted ˆI (equation 36), and the ellipsoidal-distance weighted ˜I (equation 37), as indicated in the figure legend. Alt text: Three-panel histogram of halo ellipticity at the virial m… view at source ↗
Figure 24
Figure 24. Figure 24: Matter power spectrum measured from a single simulation box with a comoving volume (1h −1 Gpc)3 traced by Np = 3,0003 particles. The solid line represents the final estimate obtained using our method, while the dashed line corresponds to the standard FFT-based approach. The inset highlights a zoomed-in view of the BAO wiggles, illustrating the improved smoothness and precision of our method. Alt text: Plo… view at source ↗
Figure 25
Figure 25. Figure 25: Correlation functions measured from five realizations randomly chosen from the 100 at z = 0. We plot the halo auto, halo-cb cross and cb auto correlation functions from top to bottom. Also shown by the lines starting at around 10h −1 Mpc are the linear theory estimates with random noise corresponding to the same random realizations. We consider a halo sample with the number density 3.16 × 10−4 (h −1 Mpc)−… view at source ↗
Figure 26
Figure 26. Figure 26: Propagators for the matter (cb) and halo samples with different number densities as indicated in the figure legend. The top panel compares the measurements from the 100 realizations (dashed) and fits to each of them (solid). The lower 3 panels show the residual of the fit normalized by the low-k limit of the fit (i.e., the linear bias parameter). Alt text: Four-panel plot. The top panel shows the propagat… view at source ↗
Figure 27
Figure 27. Figure 27: Our cleaning treatment on the cb correlation function. Various different estimates of the correlation function, multiplied by the separation squared, are plotted (see the legend for details) with the switching scales, x1 and x2, as well as xpeak and xtrough characterizing the BAO bump. The raw measurement from the simulation ξˆ sim is performed with the direct pair counting method below the scale x1, and … view at source ↗
Figure 29
Figure 29. Figure 29: Matter (cb) power spectra measured with our new method (solid), as compared to the standard FFT-based method (dashed). We show the Nyquist wavenumber of the latter by the vertical arrow in the top panel. The middle panel compares the statistical error of the two methods. The bottom panel shows the fractional difference between the two sets of measurements. The FFT-based measurement starts to blow up near … view at source ↗
Figure 30
Figure 30. Figure 30: Same as figures 28 and 29, but for the halo-cb cross correlation and power spectrum. Alt text: Two side-by-side three-panel groups: the left for the halo-matter cross correlation function and the right for the corresponding power spectrum. In both, the cleaning procedure reduces the scatter among realizations and the statistical error without introducing systematic bias. 0 50 100 x 2 ξh(x)[(h −1Mpc) 2 ] n… view at source ↗
Figure 31
Figure 31. Figure 31: Same as figures 28 and 29, but for the halo auto correlation and power spectrum. We adopt a smaller FFT grid (1603 ) for our new method here (vertical arrow in the left panel); the “FFT-based” reference curve in the right panel keeps the original grid. Direct pair counting is quicker for halos, whose number is much smaller than matter particles, so we can extend it up to 10dg in reasonable time. Alt text:… view at source ↗
Figure 32
Figure 32. Figure 32: Comparison of the total matter power spectrum from the DQ2 production runs (symbols with error bars) against four published predictions, with the NgenHalofit prediction (Smith & Angulo 2019) adopted as the reference (zero ratio). The other three models are shown by the dot-dashed (Mira-Titan Universe), solid (EuclidEmulator2), and dashed (BACCO) lines. The light (dark) shaded region marks the 2% (1%) devi… view at source ↗
Figure 33
Figure 33. Figure 33: Comparison of the halo mass function at z = 0 from simulations with three different box sizes (upward triangles, circles, and downward triangles for Lbox = 1, 2, and 4h −1 Gpc, respectively) and with two different criteria for the central/satellite split (DQ1 pipeline: open symbols; DQ2 pipeline: filled symbols). The upper panel shows the absolute mass function; the solid line is the DarkEmulator (Nishimi… view at source ↗
Figure 34
Figure 34. Figure 34: Same as figure 33, but with the respective mass correction factors applied (equation 31 for DQ1, equation 32 for DQ2). The bold dash-dotted lines in the middle panels show the ratio of the Tinker et al. (2008) fitting formula with the exponential-damping parameter adjusted from c = 1.19 to c = 1.175, which captures the high-mass upturn relative to DarkEmulator seen in the larger simulation boxes (see the … view at source ↗
Figure 35
Figure 35. Figure 35: Ratio of the total matter power spectra for different cosmological models, measured at three representative redshifts (z = 3, 1, and 0 from top to bottom; see the figure legend). Left: models with different neutrino masses, Mν, normalized to the massless case. Right: models with different dark energy sound speeds, cs , normalized to the model with cs = 1. In both panels, symbols denote the simulation meas… view at source ↗
Figure 36
Figure 36. Figure 36: The wall-clock time of simulations from z = 49 to 0 as a function of number of CPU cores. Dashed lines show the ideal scaling for each size of the problem. Note that we show three, one, and zero measurements for the largest problem size of 3,0723 particles (diamonds) for GINKAKU, L-GADGET2 and PKDGRAV3, respectively. Alt text: Plot of total wall-clock time against the number of central-processing-unit cor… view at source ↗
Figure 39
Figure 39. Figure 39: Convergence of the cb power spectrum from L-GADGET2 (with the DQ1 default setting) toward GINKAKU as the time-step parameter is tightened, shown for three resolutions and several time-step settings. Alt text: Plot of the fractional power-spectrum difference between L-Gadget2 and Ginkaku against wavenumber, for three resolutions and several time-step settings. The differences shrink as the time-step parame… view at source ↗
Figure 38
Figure 38. Figure 38: Impact of neglecting the scale-dependent growth due to massive neutrinos on the total matter power spectrum. We show the ratio of the spectrum with our linear response method to that with the backscaling method adopted in our previous study. The linear theory predictions and the simulation results are shown by the dotted and solid lines, respectively, at six redshifts (z = 3, 2.03, 1.30, 0.74, 0.32, and 0… view at source ↗
Figure 40
Figure 40. Figure 40: Code-to-code comparison between GINKAKU and PKDGRAV3. Each panel plots the fractional difference PG INKAKU/PPKDGRAV3 − 1 as a function of wavenumber. Rows correspond to z = 3, 1, and 0 from top to bottom, and columns to Lbox = 1,000, 500, and 250h −1 Mpc from left to right. The four line styles in each panel show GINKAKU runs at progressively tighter accuracy settings, while the PKDGRAV3 reference always … view at source ↗
Figure 41
Figure 41. Figure 41: Same as figure 18, but with ∆ = 400. Alt text: Two-panel halo mass function ratio comparison for three resolutions, in the same format as the main-text figure for an overdensity threshold of two hundred but here for a threshold of four hundred. The empirical correction in the right panel brings all resolutions close to the five-percent band [PITH_FULL_IMAGE:figures/full_fig_p042_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: Same as figure 18, but with ∆ = 800. Alt text: Two-panel halo mass function ratio comparison for three resolutions, in the same format as for a threshold of two hundred but here for a threshold of eight hundred. Residuals exceed the five-percent band only at high redshift and the lowest mass bins. modestly at ∆ =800 when one combines a high overdensity def￾inition, a marginal particle count, and a high re… view at source ↗
Figure 43
Figure 43. Figure 43: Halo mass function for different mass definitions, measured from the highest-resolution simulation after the mass correction in equation (32) is applied. Circles, triangles, and squares correspond to ∆ = 200, 400, and 800, respectively. The upper panel shows the mass function, and the lower panel shows the ratio to the ∆ = 200 case. The Tinker08 fitting formula (Tinker et al. 2008) is overplotted using th… view at source ↗
Figure 44
Figure 44. Figure 44: First three growth-only response functions Gn, scaled by 1/n!, defined as the n-th derivative of the matter power spectrum with respect to the background linear density contrast, measured from SU simulations performed with our code. The five panels correspond to z = 3, 2, 1, 0.5, and 0 from top to bottom; within each panel, Gn/n! for n = 1, 2, and 3 are overplotted. The horizontal lines indicate the corre… view at source ↗

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