pith. machine review for the scientific record. sign in

arxiv: 2605.13948 · v1 · submitted 2026-05-13 · 🌌 astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Improved recipes for peculiar velocity power spectra using Evolution Mapping

Matteo Esposito , Ariel G. S\'anchez , Julien Bel , Andr\'es N. Ruiz
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:24 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords peculiar velocitiesvelocity divergence power spectrumevolution mappingredshift-space distortionsN-body simulationsσ12nonlinear clustering
0
0 comments X

The pith

New fitting functions for peculiar velocity power spectra reach 1-2 percent accuracy by parametrizing non-linear evolution with the clustering amplitude σ12.

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

The paper develops revised fitting formulas for the auto-power spectrum of velocity divergence Pθθ(k) and the density-velocity cross-spectrum Pδθ(k). It applies the Evolution Mapping framework to express these spectra in terms of σ12, the root-mean-square density fluctuation smoothed on 12 Mpc scales, rather than the conventional σ8. Calibration on multi-resolution gravity-only N-body simulations yields measurements converged to k approximately 0.56 per Mpc with percent-level precision across a wide range of σ12 values. Validation on independent simulations covering varied cosmologies shows the new fits outperform prior prescriptions while keeping errors from growth-history deviations subdominant for most practical models. This matters because accurate velocity statistics are essential for modeling redshift-space distortions and extracting cosmological parameters from galaxy redshift surveys.

Core claim

Fitting functions for Pθθ(k) and Pδθ(k) are constructed via the Evolution Mapping framework and expressed in terms of σ12; these functions reproduce simulation measurements to 1-2 percent on robust scales, systematically improve on existing recipes, and remain accurate when growth histories differ modestly from the mapped relation.

What carries the argument

The Evolution Mapping framework, which maps non-linear clustering evolution across cosmologies using σ12 as the controlling parameter instead of σ8.

Load-bearing premise

Neglecting small deviations from the exact evolution mapping relation caused by differing growth histories introduces only subdominant errors for most cosmologies of practical interest.

What would settle it

A measurement of Pθθ(k) or Pδθ(k) in a simulation whose linear growth factor evolves differently enough from the mapped case to push residuals above 2 percent on scales k less than 0.3 per Mpc would falsify the claim that such deviations remain subdominant.

Figures

Figures reproduced from arXiv: 2605.13948 by Andr\'es N. Ruiz, Ariel G. S\'anchez, Julien Bel, Matteo Esposito.

Figure 1
Figure 1. Figure 1: Velocity divergence auto- (left panel) and cross- (right panel) power spectra measured from the AletheiaMass simulations. The power spectra are multiplied by 𝑘 for a clearer display of the lines. Each colour indicates a different snapshot, with the corresponding values of 𝜎12 (𝑧) and 𝑧 indicated by the colour bar . The measurements are shown with (shaded) solid lines, while the dashed lines indicate the fi… view at source ↗
Figure 2
Figure 2. Figure 2: The dependence of the parameters for our 𝑃𝜃 𝜃 (𝑘) and 𝑃𝛿 𝜃 (𝑘) fitting prescriptions as a function of 𝜎12. The dashed lines indicate the fitted relationships of equation (10) and equation (11). The three colours categorise the parameters: in blue and orange, the parameters of the additive and mul￾tiplicative terms in equation (6), respectively; in green, the parameters of equation (7). AletheiaMass simulat… view at source ↗
Figure 3
Figure 3. Figure 3: Accuracy of the fitting functions of equation (6) (left panel) with parameters calibrated as in equation (10) (solid lines) compared to the 𝑃𝜃 𝜃 (𝑘) of the calibration set (AletheiaMass). The residuals are shown as the relative differences of the predicted power spectra, 𝑃𝜃 𝜃 (𝑘) fit, with respect to the measured ones, 𝑃𝜃 𝜃 (𝑘). The grey bands represent 1%, 3%, and 5% limits. The right panel shows the same… view at source ↗
Figure 4
Figure 4. Figure 4: Accuracy of the fitting functions of equation (7) (left panel) with parameters calibrated as in equation (11) (solid lines) compared to the 𝑃𝛿 𝜃 (𝑘) of the calibration set (AletheiaMass). The residuals are shown as the relative differences of the predicted power spectra, 𝑃𝛿 𝜃 (𝑘) fit, with respect to the measured ones, 𝑃𝛿 𝜃 (𝑘). The grey bands represent 1%, 3%, and 5% limits. The right panel shows the same… view at source ↗
Figure 5
Figure 5. Figure 5: Accuracy of the fitting functions of equation (6) (left panel) with parameters calibrated as in equation (10) (solid lines) compared to the 𝑃𝜃 𝜃 (𝑘) of the AletheiaEmu simulations. The residuals are shown as the relative differences of the predicted power spectra, 𝑃𝜃 𝜃 (𝑘) fit, with respect to the measured ones, 𝑃𝜃 𝜃 (𝑘). The grey bands represent 1%, 3%, and 5% limits. The right panel shows the same compar… view at source ↗
Figure 6
Figure 6. Figure 6: Accuracy of the fitting functions of equation (7) (left panel) with parameters calibrated as in equation (11) (solid lines) compared to the 𝑃𝛿 𝜃 (𝑘) of the AletheiaEmu simulations. The residuals are shown as the relative differences of the predicted power spectra, 𝑃𝛿 𝜃 (𝑘) fit, with respect to the measured ones, 𝑃𝛿 𝜃 (𝑘). The grey bands represent 1%, 3%, and 5% limits. The right panel shows the same compar… view at source ↗
Figure 7
Figure 7. Figure 7: Ratios of velocity divergence auto-power spectra (left panel) and cross-power spectra with the density field (right panel) measured from the Aletheia simulations relative to those of the reference cosmology (model 0). The different colours correspond to the various cosmologies detailed in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We present new fitting functions for the velocity divergence auto- and cross-power spectra, $P_{\theta\theta}(k)$ and $P_{\delta\theta}(k)$, calibrated on gravity-only $N$-body simulations. By applying the Evolution Mapping framework, we revise existing prescriptions to introduce a physically motivated parametrisation in terms of the clustering amplitude $\sigma_{12}$, the RMS density fluctuation smoothed at $12\,\text{Mpc}$. This approach improves robustness and extends the range of applicability beyond that of previous models. Our fits are calibrated using a suite of multi-resolution simulations, with numerical convergence carefully quantified and sampling artefacts mitigated through a conservative patching strategy. This yields converged measurements up to $k\simeq0.56\,\mathrm{Mpc}^{-1}$ and percent-level accuracy for both $P_{\theta\theta}(k)$ and $P_{\delta\theta}(k)$ over a wide range of $\sigma_{12}$. Validation against independent simulations spanning a broad range of cosmological models confirms an accuracy of $1$-$2$ per cent on scales where the measurements are robust, systematically outperforming existing prescriptions. We further assess the impact of deviations from the exact evolution mapping relation induced by differing growth histories. For most cosmologies of practical interest, we find that neglecting these effects introduces only subdominant errors. We show that expressing fitting functions in $h$-dependent units leads to spurious, unphysical dependencies on the Hubble parameter, even for models with identical linear clustering. This provides strong empirical support for parametrising non-linear evolution in terms of $\sigma_{12}$ rather than $\sigma_{8}$. Our fitting functions provide a robust description of velocity power spectra, with direct applications to redshift-space distortion modelling in galaxy redshift surveys.

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

1 major / 2 minor

Summary. The paper presents new fitting functions for the velocity divergence auto- and cross-power spectra P_θθ(k) and P_δθ(k), calibrated on gravity-only N-body simulations using the Evolution Mapping framework parametrized by the clustering amplitude σ12. The calibration employs multi-resolution simulations with quantified numerical convergence and a conservative patching strategy to achieve converged measurements up to k ≃ 0.56 Mpc^{-1} with percent-level accuracy. Validation against independent simulations across a broad range of cosmological models confirms 1-2% accuracy on robust scales, outperforming existing prescriptions, while assessing that deviations from exact evolution mapping due to growth histories are subdominant for most practical cosmologies. The work also demonstrates that h-dependent units introduce spurious dependencies, supporting the use of σ12 over σ8.

Significance. If the results hold, this manuscript provides improved, robust fitting functions for peculiar velocity power spectra with direct applications to redshift-space distortion modeling in galaxy redshift surveys. The strengths include the physically motivated σ12 parametrization that avoids unphysical Hubble dependencies, the careful multi-resolution calibration with convergence quantification, and validation on independent cosmologies showing systematic outperformance. These elements enhance the reliability and applicability of the recipes beyond previous models.

major comments (1)
  1. [Section discussing deviations from evolution mapping] The claim that neglecting deviations from the exact evolution mapping relation introduces only subdominant errors is load-bearing for the uniform 1-2% accuracy across cosmologies. While the paper performs an assessment, it should provide quantitative bounds on the residual errors (e.g., as a function of growth rate difference or redshift) to substantiate that these do not reach the quoted accuracy level in the validation set. Without such bounds, the range of applicability remains somewhat qualitative.
minor comments (2)
  1. [Abstract] The abstract provides limited detail on the exact functional form of the fitting functions and any post-hoc adjustments, which would help readers assess the implementation.
  2. [Calibration section] Clarify the exact patching strategy used to mitigate sampling artefacts and how it ensures the quoted convergence up to k ≃ 0.56 Mpc^{-1}.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review, positive assessment of the work, and recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Section discussing deviations from evolution mapping] The claim that neglecting deviations from the exact evolution mapping relation introduces only subdominant errors is load-bearing for the uniform 1-2% accuracy across cosmologies. While the paper performs an assessment, it should provide quantitative bounds on the residual errors (e.g., as a function of growth rate difference or redshift) to substantiate that these do not reach the quoted accuracy level in the validation set. Without such bounds, the range of applicability remains somewhat qualitative.

    Authors: We agree that providing explicit quantitative bounds would strengthen the manuscript and clarify the range of applicability. In the revised version, we will expand the relevant section (and add a supplementary figure if space permits) to report the residual errors explicitly as a function of growth-rate difference Δf and redshift for the validation cosmologies. This will show that, for the cosmologies of practical interest where |Δf| ≲ 0.05, the deviations remain below ~0.5% on the scales where our fits are calibrated, remaining well within the quoted 1–2% accuracy. We thank the referee for this suggestion. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical calibration and independent validation are self-contained

full rationale

The paper calibrates fitting functions for P_θθ(k) and P_δθ(k) directly on multi-resolution N-body simulations using the Evolution Mapping framework with σ12 parametrization, then validates accuracy at the 1-2% level on a separate suite of independent simulations spanning varied cosmologies. The assessment of subdominant errors from growth-history deviations is performed explicitly within the present work via direct comparison, and the claim of outperformance over prior prescriptions rests on these external measurements rather than any self-definitional reduction, fitted-input renaming, or load-bearing self-citation chain. All central results are therefore grounded in simulation data outside the fitted parameters themselves.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central results rest on empirical calibration to gravity-only N-body simulations rather than first-principles derivation; the fitting coefficients are free parameters tuned to data.

free parameters (1)
  • fitting function coefficients
    Numerical coefficients in the new fitting functions for P_θθ(k) and P_δθ(k) that are calibrated to simulation measurements.
axioms (1)
  • domain assumption Gravity-only N-body simulations sufficiently capture the non-linear evolution of peculiar velocities for the scales and cosmologies considered.
    Invoked when calibrating and validating the fitting functions on simulation outputs.

pith-pipeline@v0.9.0 · 5619 in / 1277 out tokens · 206010 ms · 2026-05-15T02:24:26.683336+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 11 internal anchors

  1. [1]

    , keywords =

    Evolution mapping: a new approach to describe matter clustering in the non-linear regime. , keywords =. doi:10.1093/mnras/stac1656 , archivePrefix =. 2108.12710 , primaryClass =

    work page doi:10.1093/mnras/stac1656

  2. [2]

    , journal =

    S\'anchez, Ariel G. , journal =. Arguments against using. 2020 , month =. doi:10.1103/PhysRevD.102.123511 , url =

    work page doi:10.1103/physrevd.102.123511 2020

  3. [3]

    Accelerating Universes with Scaling Dark Matter

    Accelerating Universes with Scaling Dark Matter. International Journal of Modern Physics D , keywords =. doi:10.1142/S0218271801000822 , archivePrefix =. gr-qc/0009008 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218271801000822

  4. [4]

    Exploring the Expansion History of the Universe , author =. Phys. Rev. Lett. , volume =. 2003 , month =. doi:10.1103/PhysRevLett.90.091301 , url =

    work page doi:10.1103/physrevlett.90.091301 2003

  5. [5]

    Galaxy Formation and Evolution

    work page

  6. [6]

    2024 , eprint=

    Phase-Space Delaunay Tessellation Field Estimator , author=. 2024 , eprint=

    work page 2024

  7. [7]

    , volume =

    Springel, Volker and Pakmor, Rüdiger and Zier, Oliver and Reinecke, Martin , title = ". , volume =. 2021 , month =. doi:10.1093/mnras/stab1855 , url =

    work page doi:10.1093/mnras/stab1855 2021

  8. [8]

    and Pontzen, Andrew , title = "

    Angulo, Raul E. and Pontzen, Andrew , title = ". , volume =. 2016 , month =. doi:10.1093/mnrasl/slw098 , url =

    work page doi:10.1093/mnrasl/slw098 2016

  9. [9]

    T., Mellema, G., Pen, U.-L., et al

    Crocce, Martín and Pueblas, Sebastián and Scoccimarro, Román , title = ". , volume =. 2006 , month =. doi:10.1111/j.1365-2966.2006.11040.x , url =

    work page doi:10.1111/j.1365-2966.2006.11040.x 2006

  10. [10]

    and Abel, Tom , year=

    Hahn, Oliver and Angulo, Raul E. and Abel, Tom , year=. The properties of cosmic velocity fields , volume=. , publisher=. doi:10.1093/mnras/stv2179 , number=

    work page doi:10.1093/mnras/stv2179

  11. [11]

    and Pezzotta, A

    Bel, J. and Pezzotta, A. and Carbone, C. and Sefusatti, E. and Guzzo, L. , year=. Accurate fitting functions for peculiar velocity spectra in standard and massive-neutrino cosmologies , volume=. doi:10.1051/0004-6361/201834513 , journal=

    work page doi:10.1051/0004-6361/201834513

  12. [12]

    DEMNUni: ISW, Rees-Sciama, and weak-lensing in the presence of massive neutrinos , volume=

    Carbone, Carmelita and Petkova, Margarita and Dolag, Klaus , year=. DEMNUni: ISW, Rees-Sciama, and weak-lensing in the presence of massive neutrinos , volume=. , publisher=. doi:10.1088/1475-7516/2016/07/034 , number=

    work page doi:10.1088/1475-7516/2016/07/034 2016

  13. [13]

    DEMNUni: the clustering of large-scale structures in the presence of massive neutrinos , volume=

    Castorina, Emanuele and Carbone, Carmelita and Bel, Julien and Sefusatti, Emiliano and Dolag, Klaus , year=. DEMNUni: the clustering of large-scale structures in the presence of massive neutrinos , volume=. , publisher=. doi:10.1088/1475-7516/2015/07/043 , number=

    work page doi:10.1088/1475-7516/2015/07/043 2015

  14. [14]

    Redshift-space distortions, pairwise velocities, and nonlinearities , volume=

    Scoccimarro, Román , year=. Redshift-space distortions, pairwise velocities, and nonlinearities , volume=. , publisher=. doi:10.1103/physrevd.70.083007 , number=

    work page doi:10.1103/physrevd.70.083007

  15. [15]

    and van de Weygaert, R

    Bernardeau, F. and van de Weygaert, R. , year=. A new method for accurate estimation of velocity field statistics , volume=. , publisher=. doi:10.1093/mnras/279.2.693 , number=

    work page doi:10.1093/mnras/279.2.693

  16. [16]

    doi:10.1093/mnras/290.3.566 , Eprint =

    , Keywords =. doi:10.1093/mnras/290.3.566 , Eprint =

    work page doi:10.1093/mnras/290.3.566

  17. [17]

    Generation of Vorticity and Velocity Dispersion by Orbit Crossing

    doi:10.1103/PhysRevD.80.043504 , Eid =. arXiv , Author =:0809.4606 , Journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.80.043504

  18. [18]

    2010, , 405, 1025, 10.1111/j.1365-2966.2010.16486.x

    , Keywords =. doi:10.1111/j.1365-2966.2010.17581.x , Eprint =

    work page doi:10.1111/j.1365-2966.2010.17581.x 2010

  19. [19]

    doi:10.1111/j.1745-3933.2012.01338.x , Eprint =

    , Keywords =. doi:10.1111/j.1745-3933.2012.01338.x , Eprint =

    work page doi:10.1111/j.1745-3933.2012.01338.x 2012

  20. [20]

    doi:10.1093/mnras/stu1610 , Eprint =

    , Keywords =. doi:10.1093/mnras/stu1610 , Eprint =

    work page doi:10.1093/mnras/stu1610

  21. [21]

    Sampling Artifact in Volume Weighted Velocity Measurement.--- I. Theoretical Modelling

    doi:10.1103/PhysRevD.91.043522 , Eid =. arXiv , Author =:1405.7125 , Journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.043522

  22. [22]

    Sampling Artifact in Volume Weighted Velocity Measurement.--- II. Detection in simulations and comparison with theoretical modelling

    doi:10.1103/PhysRevD.91.043523 , Eid =. arXiv , Author =:1409.6809 , Journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.043523

  23. [23]

    Kriging Interpolating Cosmic Velocity Field

    doi:10.1103/PhysRevD.92.083527 , Eid =. arXiv , Author =:1505.06827 , Journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.083527

  24. [24]

    doi:10.1093/mnras/stu2043 , Eprint =

    , Keywords =. doi:10.1093/mnras/stu2043 , Eprint =

    work page doi:10.1093/mnras/stu2043

  25. [25]

    Villaescusa-Navarro, Francisco and Naess, Sigurd and Genel, Shy and Pontzen, Andrew and Wandelt, Benjamin and Anderson, Lauren and Font-Ribera, Andreu and Battaglia, Nicholas and Spergel, David N. , year=. Statistical Properties of Paired Fixed Fields , volume=. , publisher=. doi:10.3847/1538-4357/aae52b , number=

    work page doi:10.3847/1538-4357/aae52b

  26. [26]

    Smith, R. E. and Peacock, J. A. and Jenkins, A. and White, S. D. M. and Frenk, C. S. and Pearce, F. R. and Thomas, P. A. and Efstathiou, G. and Couchman, H. M. P. , year=. Stable clustering, the halo model and non-linear cosmological power spectra , volume=. , publisher=. doi:10.1046/j.1365-8711.2003.06503.x , number=

    work page doi:10.1046/j.1365-8711.2003.06503.x 2003

  27. [27]

    2012, ApJ, 761, 152, doi: 10.1088/0004-637X/761/2/152

    Takahashi, Ryuichi and Sato, Masanori and Nishimichi, Takahiro and Taruya, Atsushi and Oguri, Masamune , year=. REVISING THE HALOFIT MODEL FOR THE NONLINEAR MATTER POWER SPECTRUM , volume=. , publisher=. doi:10.1088/0004-637x/761/2/152 , number=

    work page doi:10.1088/0004-637x/761/2/152

  28. [28]

    Fast, accurate and perturbative forward modeling of galaxy clustering. Part II. Redshift space. , keywords =. doi:10.1088/1475-7516/2025/11/055 , archivePrefix =. 2411.04513 , primaryClass =

    work page doi:10.1088/1475-7516/2025/11/055 2025

  29. [29]

    Renormalized cosmological perturbation theory , author =. Phys. Rev. D , volume =. 2006 , month =. doi:10.1103/PhysRevD.73.063519 , url =

    work page doi:10.1103/physrevd.73.063519 2006

  30. [30]

    Peculiar Velocity Decomposition, Redshift Space Distortion and Velocity Reconstruction in Redshift Surveys. II. Dark Matter Velocity Statistics

    doi:10.1103/PhysRevD.88.103510 , Eid =. arXiv , Author =:1308.0886 , Journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.88.103510

  31. [31]

    doi:10.1111/j.1365-2966.2007.12190.x , Journal =

    work page doi:10.1111/j.1365-2966.2007.12190.x 2007

  32. [32]

    Velocity Field Statistics and Tessellation Techniques: Unbiased Estimators of $\Omega$

    Large Scale Structure: Tracks and Traces , Editor =. astro-ph/9803143 , Pages =

    work page internal anchor Pith review Pith/arXiv arXiv

  33. [33]

    Angulo, R. E. and White, S. D. M. , year=. One simulation to fit them all - changing the background parameters of a cosmologicalN-body simulation , ISSN=. doi:10.1111/j.1365-2966.2010.16459.x , journal=

    work page doi:10.1111/j.1365-2966.2010.16459.x 2010

  34. [34]

    , keywords =

    3 per cent-accurate predictions for the clustering of dark matter, haloes, and subhaloes, over a wide range of cosmologies and scales. , keywords =. doi:10.1093/mnras/staa3117 , archivePrefix =. 2001.03176 , primaryClass =

    work page doi:10.1093/mnras/staa3117 2001

  35. [35]

    Weakly Non-Linear Gaussian Fluctuations and the Edgeworth Expansion

    Weakly Nonlinear Gaussian Fluctuations and the Edgeworth Expansion. , keywords =. doi:10.1086/175420 , archivePrefix =. astro-ph/9308012 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1086/175420

  36. [36]

    Continuous Fields and Discrete Samples: Reconstruction through Delaunay Tessellations

    Continuous fields and discrete samples: reconstruction through Delaunay tessellations. , keywords =. doi:10.48550/arXiv.astro-ph/0011007 , archivePrefix =. astro-ph/0011007 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.astro-ph/0011007

  37. [37]

    , keywords =

    Improved two-point correlation function estimates using glass-like distributions as a reference sample. , keywords =. doi:10.1093/mnras/stab1622 , archivePrefix =. 2011.05747 , primaryClass =

    work page doi:10.1093/mnras/stab1622 2011

  38. [38]

    and Schaap, W

    van de Weygaert, R. and Schaap, W. , title =. 2010 , doi =

    work page 2010

  39. [39]

    Cosmological Constraints from Multiple Probes in the Dark Energy Survey , author =. Phys. Rev. Lett. , volume =. 2019 , month =. doi:10.1103/PhysRevLett.122.171301 , url =

    work page doi:10.1103/physrevlett.122.171301 2019

  40. [40]

    Mellier, et al

    Euclid - I. Overview of the Euclid mission★ , DOI= "10.1051/0004-6361/202450810", url= "https://doi.org/10.1051/0004-6361/202450810", journal =

    work page doi:10.1051/0004-6361/202450810

  41. [41]

    Ivezi ´c, S

    LSST: From Science Drivers to Reference Design and Anticipated Data Products. , keywords =. doi:10.3847/1538-4357/ab042c , archivePrefix =. 0805.2366 , primaryClass =

    work page doi:10.3847/1538-4357/ab042c

  42. [42]

    2025 , eprint=

    Data Release 1 of the Dark Energy Spectroscopic Instrument , author=. 2025 , eprint=

    work page 2025

  43. [43]

    Describing statistics of the non-linear cosmic velocity field

    Evolution mapping - II. Describing statistics of the non-linear cosmic velocity field. , keywords =. doi:10.1093/mnras/stae2351 , archivePrefix =. 2406.08539 , primaryClass =

    work page doi:10.1093/mnras/stae2351

  44. [44]

    arXiv e-prints , keywords =

    Evolution mapping III: A new recipe for the halo mass function. arXiv e-prints , keywords =. doi:10.48550/arXiv.2511.16730 , archivePrefix =. 2511.16730 , primaryClass =

    work page doi:10.48550/arxiv.2511.16730

  45. [45]

    arXiv e-prints , keywords =

    Aletheia: Emulating the non-linear matter power spectrum in the context of evolution mapping. arXiv e-prints , keywords =. doi:10.48550/arXiv.2511.13826 , archivePrefix =. 2511.13826 , primaryClass =

    work page doi:10.48550/arxiv.2511.13826

  46. [46]

    and Crocce, M

    Sefusatti, E. and Crocce, M. and Scoccimarro, R. and Couchman, H. M. P. , title =. , volume =. 2016 , month =. doi:10.1093/mnras/stw1229 , url =

    work page doi:10.1093/mnras/stw1229 2016

  47. [47]

    , keywords =

    Illustrating the consequences of a misuse of 8 in cosmology. , keywords =. doi:10.1103/rpf5-ldks , archivePrefix =. 2501.11571 , primaryClass =

    work page doi:10.1103/rpf5-ldks

  48. [48]

    , keywords =

    Clustering in real space and in redshift space. , keywords =. doi:10.1093/mnras/227.1.1 , adsurl =

    work page doi:10.1093/mnras/227.1.1

  49. [49]

    A test of the nature of cosmic acceleration using galaxy redshift distortions

    A test of the nature of cosmic acceleration using galaxy redshift distortions. , keywords =. doi:10.1038/nature06555 , archivePrefix =. 0802.1944 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature06555 1944

  50. [50]

    arXiv e-prints , keywords =

    The DTFE public software: The Delaunay Tessellation Field Estimator code. arXiv e-prints , keywords =. doi:10.48550/arXiv.1105.0370 , archivePrefix =. 1105.0370 , primaryClass =

    work page doi:10.48550/arxiv.1105.0370

  51. [51]

    Fourier imaging of non-linear structure formation , volume=

    Brandbyge, Jacob and Hannestad, Steen , year=. Fourier imaging of non-linear structure formation , volume=. /Journal of Cosmology and Astroparticle Physics , publisher=. doi:10.1088/1475-7516/2017/04/032 , number=

    work page doi:10.1088/1475-7516/2017/04/032 2017

  52. [52]

    and Eastwood, James W

    Hockney, Roger W. and Eastwood, James W. , title =. 1988 , publisher =. doi:10.1201/9780367806934 , isbn =

    work page doi:10.1201/9780367806934 1988

  53. [53]

    Extending evolution mapping to massive neutrinos with COMET , author =. Phys. Rev. D , volume =. 2025 , month =. doi:10.1103/vy3h-p92n , url =

    work page doi:10.1103/vy3h-p92n 2025

  54. [54]

    The MillenniumTNG Project: high-precision predictions for matter clustering and halo statistics , volume=

    Hernández-Aguayo, César and Springel, Volker and Pakmor, Rüdiger and Barrera, Monica and Ferlito, Fulvio and White, Simon D M and Hernquist, Lars and Hadzhiyska, Boryana and Delgado, Ana Maria and Kannan, Rahul and Bose, Sownak and Frenk, Carlos , year=. The MillenniumTNG Project: high-precision predictions for matter clustering and halo statistics , volu...

    work page doi:10.1093/mnras/stad1657

  55. [55]

    , volume =

    Maksimova, Nina A and Garrison, Lehman H and Eisenstein, Daniel J and Hadzhiyska, Boryana and Bose, Sownak and Satterthwaite, Thomas P , title =. , volume =. 2021 , month =. doi:10.1093/mnras/stab2484 , url =

    work page doi:10.1093/mnras/stab2484 2021

  56. [56]

    The Flagship galaxy mock catalogue: A comprehensive simulation for the Euclid mission

    Euclid: V. The Flagship galaxy mock catalogue: A comprehensive simulation for the Euclid mission. , keywords =. doi:10.1051/0004-6361/202450853 , archivePrefix =. 2405.13495 , primaryClass =

    work page doi:10.1051/0004-6361/202450853

  57. [57]

    , title =

    Hamann, Jan and Hannestad, Steen and Lesgourgues, Julien and Rampf, Cornelius and Wong, Yvonne Y.Y. , title =. 2010 , month =. doi:10.1088/1475-7516/2010/07/022 , url =

    work page doi:10.1088/1475-7516/2010/07/022 2010

  58. [58]

    , volume =

    Lange, Johannes U , title = ". , volume =. 2023 , month =. doi:10.1093/mnras/stad2441 , url =

    work page doi:10.1093/mnras/stad2441 2023

  59. [59]

    Fingers of God

    A critique of Rees's theory of primordial gravitational radiation. , keywords =. doi:10.1093/mnras/156.1.1P , archivePrefix =. 0810.3908 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/156.1.1p

  60. [60]

    , keywords =

    A statistical method for determining the cosmological density parameter from the redshifts of a complete sample of galaxies. , keywords =. doi:10.1086/182362 , adsurl =

    work page doi:10.1086/182362