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arxiv: 2604.17981 · v1 · submitted 2026-04-20 · 🌌 astro-ph.CO · astro-ph.GA

Recognition: unknown

Efficiently emulating distribution functions in gigaparsec volumes for varying cosmological parameters

Christopher C. Lovell , Max E. Lee , William J. Roper , Daniel Angl\'es-Alc\'azar , Shy Genel , Shivam Pandey , Francisco Villaescusa-Navarro
Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:16 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords halo mass functionoverdensity conditioningcosmological emulationN-body simulationslarge volume statisticscomputational efficiencyparameter variation
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The pith

Small regions with varying overdensities emulate the full halo mass function in gigaparsec volumes using 0.026 percent of the simulation volume.

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

The paper introduces a technique to reproduce the halo mass function across large cosmological volumes by selecting small simulation patches that span a range of local overdensities. An emulator is trained on these patches, taking as input both the overdensity and the global cosmological parameters, to predict the mass distribution of halos in each patch. By integrating the emulator's predictions over the known distribution of overdensities in the full volume, the global halo mass function is recovered accurately. This matters because it achieves the result with only a tiny fraction of the computational resources needed for direct large-volume runs, while also supporting changes to cosmological parameters and access to lower halo masses.

Core claim

We train a differentiable emulator on halo mass functions from small regions of varying overdensities extracted from large N-body simulations run with different Lambda-CDM parameters. The emulator is conditioned on the region's overdensity and the global parameters. Integrating the emulator outputs weighted by the overdensity distribution recovers the global halo mass function of the entire simulation box.

What carries the argument

The emulator conditioned on local overdensity and global cosmological parameters, which predicts the halo mass function in each small region so that the predictions can be integrated over the overdensity distribution.

If this is right

  • Targeted zoom simulations extracted from low-resolution parent volumes can emulate large-volume results at a fraction of the computational cost.
  • The dynamic range extends to lower halo masses than can be reached in standard periodic box simulations.
  • The same conditioning and integration approach applies to other dark matter and baryonic distribution functions.
  • Higher-order statistics can be emulated with direct implications for analyzing data from wide-field surveys.

Where Pith is reading between the lines

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

  • The differentiability of the emulator enables gradient-based methods for efficient cosmological parameter inference.
  • The approach could be paired with hydrodynamical simulations inside the small regions to capture baryonic effects without running full-volume hydrodynamical boxes.
  • If the overdensity dependence holds across different simulation resolutions and codes, the method could standardize emulation pipelines for existing datasets.

Load-bearing premise

The halo mass function within each small region depends only on the local overdensity value and the global cosmological parameters in a manner that one emulator can learn accurately enough to produce the correct global integral.

What would settle it

Measuring the halo mass function directly in a full large-volume simulation and comparing it to the one recovered by integrating the emulator trained on its small overdense subregions; significant mismatch would falsify the approach.

Figures

Figures reproduced from arXiv: 2604.17981 by Christopher C. Lovell, Daniel Angl\'es-Alc\'azar, Francisco Villaescusa-Navarro, Max E. Lee, Shivam Pandey, Shy Genel, William J. Roper.

Figure 1
Figure 1. Figure 1: Figure demonstrating the region selection procedure. Bottom panel shows a density slice of one of the Quijote BSQ simulations, with depth 50 ℎ −1Mpc. The two inset panels show the density distribution in two selected regions, one with high overdensity and one low, each (50 ℎ −1Mpc) 3 in volume. Haloes in each region are shown with circles whose radius is proportional to their mass. The inset plot to the bo… view at source ↗
Figure 2
Figure 2. Figure 2: Evaluation of the region level conditional HMF emulator. Top two rows: six held-out examples selected at the 50th, 95th, and 99th percentiles of the Poisson deviance distribution, 𝐸𝑆, shown at the bottom left of each sub-panel. Each panel also show the global cosmological parameters, 𝜽, and the overdensity percentile of the region, 𝛿𝑆. Bottom-left: distribution of 𝐸𝑆 across all test regions. Bottom-middle:… view at source ↗
Figure 3
Figure 3. Figure 3: Fractional error in the global HMF predictions across the test set. The Grey shaded regions show the median and 1𝜎 distribution of the shot noise across the test set. The solid orange line shows the median fractional error, and the shaded region the 1𝜎 distribution. 4 RESULTS 4.1 Conditional HMF region-level predictions In [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example global HMF predictions from held out test simulations in the Quijote BSQ suite. Each panel shows the cosmological parameters of the held out simulation, the true HMF (blue) with poisson confidence intervals, the baseline from the training regions (grey dashed), and the predicted HMF in orange with 68% uncertainties. Each panel also shows the fractional error in the counts in each bin, alongside the… view at source ↗
Figure 5
Figure 5. Figure 5: The predicted global HMF as a function of the cosmological parameters Ω𝑚, Ω𝑏, ℎ, 𝜎8 and 𝑛𝑠. Each panel shows the impact of varying a single parameter whilst keeping the other parameters fixed to the fiducial values stated in the bottom right. competitive with other emulators available to date trained on suites of periodic simulations (McClintock et al. 2019; Bocquet et al. 2020; Sáez-Casares et al. 2024; S… view at source ↗
Figure 7
Figure 7. Figure 7: Median absolute fractional error against halo mass bin for all simulations in the test set. Coloured lines show different numbers of training simulations 𝑁sim, and solid and dashed lines show either 1 or 10 samples from each simulation, respectively. The grey shaded region shows the median contribution from shot noise across the test set. simulation changes shape and volume over cosmic time, it is more cha… view at source ↗
Figure 6
Figure 6. Figure 6: Median absolute fractional error across all mass bins as a function of (top panel) number of simulations 𝑁sim, and (bottom panel) total number of regions (𝑁sim × 𝑁samp). Coloured lines show the number of regions selected within each simulation. Numbers printed by each point in the top panel state the total number of regions used. The orange circle denotes our fiducial simulation and sample selection shown … view at source ↗
Figure 9
Figure 9. Figure 9: As for [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Impact of varying the volume of each selected region, from (27.3 ℎ −1Mpc) 3 to (102 ℎ −1Mpc) 3 . Top: Median absolute fractional er￾ror as a function of number of simulations across the test set (where a single region is selected from each simulation). Bottom: Median absolute fractional error against total simulated volume selected across all regions. techniques have been presented in recent years that pro… view at source ↗
read the original abstract

We present a new method for emulating the halo mass function (HMF) and other distribution functions in large effective volumes, down to low halo masses, whilst simultaneously modifying large ranges of parameters, for a fraction of the cost of traditional periodic cosmological simulations. We demonstrate the method by selecting small regions, $V \sim (50 \,h^{-1}{\rm Mpc})^3$, with a range of overdensities from the Quijote suite, consisting of tens of thousands of $(1 \,h^{-1}{\rm Gpc})^3$ $N$-body simulation volumes run with varying $\Lambda$CDM parameters. We train a differentiable emulator, conditioned on the overdensity of the region and these global parameters, to reproduce the halo mass function in these regions. We then successfully recover the global distribution of halo masses of the entire box by integrating over the overdensity distribution. Our approach uses just $\sim\,$0.026% of the original simulation volume, and suggests that suites of targeted `zoom' simulations, extracted from low resolution parent volumes, can be used to emulate large volume simulations at a fraction of the computational cost, whilst simultaneously pushing the dynamic range to much lower masses than can be achieved in periodic simulations. We discuss emulation of other key dark matter and baryonic distribution functions, as well as higher order statistics, with implications for the interpretation of upcoming wide field surveys on observatories such as Euclid, Roman and Rubin.

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 paper presents a method to emulate the halo mass function (HMF) in gigaparsec volumes by extracting and training a neural network emulator on small sub-volumes ((50 h^{-1} Mpc)^3) selected from the Quijote suite of N-body simulations. The emulator is conditioned on local overdensity and varying cosmological parameters; the global HMF is then recovered by integrating the emulator output over the known overdensity distribution of the full box, using only ~0.026% of the original volume. The approach is positioned as a cost-effective alternative to full periodic simulations for distribution functions and higher-order statistics relevant to surveys such as Euclid, Roman, and Rubin.

Significance. If the central recovery claim holds with quantified accuracy, the method would enable efficient emulation of distribution functions across large parameter spaces and dynamic ranges by leveraging existing large-volume suites and targeted sub-volumes. This could reduce computational costs while extending to lower masses and other statistics, with direct relevance to interpreting wide-field survey data.

major comments (2)
  1. [Abstract] Abstract: the claim that the global HMF is 'successfully recover[ed]' by integrating the emulator over the overdensity distribution is presented without reported error bars, quantitative validation metrics (e.g., fractional residuals or Kolmogorov-Smirnov statistics), or details on how overdensity bins are chosen and the integration is performed numerically. This prevents assessment of whether the integrated result matches the full-box truth within expected uncertainties.
  2. [Method] Method and results sections: the workflow assumes the HMF inside each (50 h^{-1} Mpc)^3 patch is fully determined by its scalar mean overdensity plus global cosmological parameters, with no residual dependence on larger-scale environment (tidal shear, neighboring structures, or variance on scales >50 h^{-1} Mpc). No test is described that isolates or quantifies such residual correlations, which would propagate directly into systematic bias in the integrated global HMF even if the emulator fits the training patches well.
minor comments (2)
  1. [Abstract] The abstract mentions emulation of 'other key dark matter and baryonic distribution functions' and 'higher order statistics,' but the presented results focus exclusively on the HMF; clarify whether these extensions are demonstrated or left for future work.
  2. [Methods] Notation for the overdensity variable and the precise definition of the integration measure over the overdensity distribution should be introduced explicitly with an equation in the methods section for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that have helped improve the clarity and robustness of our work. We address each of the major comments below and have made corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the global HMF is 'successfully recover[ed]' by integrating the emulator over the overdensity distribution is presented without reported error bars, quantitative validation metrics (e.g., fractional residuals or Kolmogorov-Smirnov statistics), or details on how overdensity bins are chosen and the integration is performed numerically. This prevents assessment of whether the integrated result matches the full-box truth within expected uncertainties.

    Authors: We agree that the abstract should include quantitative support for the recovery claim to allow immediate assessment. The detailed validation, including fractional residuals, error bars on the integrated HMF, and the numerical procedure for binning and integrating over the overdensity distribution, is presented in the results section of the manuscript. We have revised the abstract to incorporate a concise statement of the achieved accuracy (sub-percent level agreement with the full-box HMF) along with a brief description of the integration method, while preserving the abstract's length and focus. revision: yes

  2. Referee: [Method] Method and results sections: the workflow assumes the HMF inside each (50 h^{-1} Mpc)^3 patch is fully determined by its scalar mean overdensity plus global cosmological parameters, with no residual dependence on larger-scale environment (tidal shear, neighboring structures, or variance on scales >50 h^{-1} Mpc). No test is described that isolates or quantifies such residual correlations, which would propagate directly into systematic bias in the integrated global HMF even if the emulator fits the training patches well.

    Authors: This is a substantive point about the completeness of the environmental modeling. Our method is predicated on the local mean overdensity (on the chosen 50 h^{-1} Mpc scale) being the dominant driver of the HMF, with larger-scale variations accounted for via the measured overdensity distribution of the parent volume. The end-to-end success in recovering the global HMF to high accuracy provides indirect validation that residual correlations do not dominate. We have revised the methods and discussion sections to explicitly articulate this assumption, to discuss its physical basis and potential limitations with reference to the literature on environmental effects, and to note that any unaccounted bias would have appeared in the integrated comparison. A direct isolation test for larger-scale residuals was not included in the original submission; we view the addition of the explicit discussion as a partial but appropriate response that strengthens the paper without requiring new simulations. revision: partial

Circularity Check

0 steps flagged

No circularity: training on conditional sub-volume HMFs and integration over overdensity PDF are independent steps

full rationale

The paper selects small sub-regions from external Quijote N-body simulations, trains a differentiable emulator to reproduce the HMF conditioned only on local overdensity and global cosmological parameters, then integrates the emulator output against the separately measured overdensity distribution to recover the global HMF. This integration is not equivalent to any training input by construction; the global result functions as an empirical validation rather than a mathematical identity. No equations, self-citations, or ansatzes reduce the claimed recovery to a fitted quantity or prior result. The method is self-contained against the full-volume benchmark and does not invoke uniqueness theorems or rename known patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that local HMF is a function of overdensity and global parameters only, plus standard integration over a known PDF. No new physical entities are introduced.

free parameters (1)
  • emulator neural network weights
    Fitted during training on the selected subregions; their values are not reported.
axioms (2)
  • domain assumption The halo mass function in a subvolume is fully determined by its overdensity and the global cosmological parameters.
    Invoked when conditioning the emulator and when integrating to recover the global distribution.
  • domain assumption The overdensity distribution of the full volume is known or can be measured independently.
    Required for the integration step to produce the global HMF.

pith-pipeline@v0.9.0 · 5597 in / 1324 out tokens · 38353 ms · 2026-05-10T04:16:05.784949+00:00 · methodology

discussion (0)

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