arxiv: 2604.19922 · v1 · submitted 2026-04-21 · 🌌 astro-ph.CO

Recognition: unknown

Measuring neutrino mass and asymmetry through galaxy pairwise peculiar velocity

Ming-chung Chu, Shihong Liao, Wangzheng Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:08 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords neutrino massneutrino asymmetrygalaxy pairwise velocityCosmicflows-4large-scale structurecosmology

0 comments

The pith

Galaxy pairwise peculiar velocities constrain total neutrino mass to 0.24 eV and asymmetry parameter to 2.14 at 7 sigma in the CMB framework.

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

The paper establishes that the mean pairwise peculiar velocity of galaxies serves as a measurable probe of cosmic neutrino mass and asymmetry by linking these properties through simulations to observed velocities in the Cosmicflows-4 catalog. This approach yields constraints in both CMB-derived and local distance-ladder cosmological frameworks that align with each other and with earlier Planck CMB results. A reader would care because neutrinos are the most numerous fermions yet their masses and chemical potentials remain poorly known, and an independent large-scale structure method could cross-check CMB inferences while testing how neutrinos shape cosmic flows. The analysis covers quasi-linear to nonlinear regimes and reports a strong detection of asymmetry specifically in the CMB framework.

Core claim

By applying a simulation-based pipeline to the mean galaxy pairwise peculiar velocity from the Cosmicflows-4 grouped catalog, the work derives M_ν = 0.24^{+0.34}_{-0.18} eV and η² = 2.14^{+0.30}_{-0.32} in the CMB framework, corresponding to a 7σ measurement of non-zero neutrino asymmetry, along with compatible though less precise values in the local framework; these results match prior constraints obtained from the Planck CMB temperature power spectrum.

What carries the argument

The simulation-based analysis pipeline that predicts the mean galaxy pairwise peculiar velocity from given values of total neutrino mass M_ν and asymmetry parameter η².

If this is right

Galaxy pairwise velocities supply an independent route to neutrino mass and asymmetry constraints that can be compared directly with CMB measurements.
The method operates across both quasi-linear and nonlinear scales, broadening the range of usable observational data.
Consistent results emerge when the same pipeline is run inside CMB-based and local distance-ladder cosmologies.
The derived neutrino parameters agree with those previously extracted from Planck CMB temperature spectra.

Where Pith is reading between the lines

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

Larger upcoming galaxy velocity catalogs could shrink the uncertainties and test whether the asymmetry signal persists at higher significance.
Combining this observable with other large-scale structure statistics might help break degeneracies among neutrino parameters.
The approach could be adapted to search for additional neutrino properties, such as self-interactions, if they alter pairwise velocities in detectable ways.

Load-bearing premise

The simulation-based pipeline accurately translates neutrino mass and asymmetry into galaxy pairwise velocity statistics without substantial contamination from other cosmological effects or catalog systematics.

What would settle it

An independent analysis using a different set of N-body simulations or a separate galaxy velocity catalog that produces neutrino parameter posteriors inconsistent with the reported values at more than 2 sigma.

Figures

Figures reproduced from arXiv: 2604.19922 by Ming-chung Chu, Shihong Liao, Wangzheng Zhang.

**Figure 2.** Figure 2: FIG. 2. 2D projected CDM density fields in the local framework at [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Left: Marginalized posterior distributions and 68%/95% confidence contours for ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Cosmic neutrinos are among the most abundant fermions in the Universe, yet the values of their masses and chemical potentials remain uncertain. In this Letter, we present the first constraints on the total neutrino mass $M_\nu$ and the neutrino asymmetry parameter $\eta^2$ derived from the mean galaxy pairwise peculiar velocity in the quasi-linear and nonlinear regimes. We develop a simulation-based analysis pipeline that connects neutrino properties to predictions of galaxy pairwise velocity, and apply it to galaxy data from the Cosmicflows-4 grouped catalog. Our analysis is performed within two independent cosmological frameworks, based on cosmological parameters derived from Cosmic microwave background (CMB) and local distance ladder measurements, respectively. By performing fits to the galaxy pairwise velocity, we obtain consistent constraints from both frameworks. Quoting posterior means with 68% CL, we find $M_\nu = 0.24^{+0.34}_{-0.18}\ \mathrm{eV}$ and $\eta^2 = 2.14^{+0.30}_{-0.32}$ in the CMB framework, and $M_\nu = 0.37^{+0.34}_{-0.26}\ \mathrm{eV}$ and $\eta^2 = 2.4^{+2.1}_{-1.6}$ in the local framework. In particular, we find a 7$\sigma$ measurement of a non-zero neutrino asymmetry in the CMB framework. These neutrino parameters are consistent with those, in our previous work, obtained from the Planck CMB temperature power spectrum. These results demonstrate that galaxy pairwise velocities provide an independent and sensitive probe of neutrino properties, opening a new avenue for testing neutrino physics with large-scale structure observations.

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.

Desk Editor's Note private letter to a colleague

The paper claims a 7σ detection of non-zero neutrino asymmetry from galaxy pairwise velocities, but the result rests on fixing background cosmology parameters that become inconsistent once asymmetry is allowed.

read the letter

The headline result is a claimed 7σ measurement of η² ≈ 2.14 from Cosmicflows-4 pairwise velocities in the CMB framework, with a looser constraint in the local-distance framework. They also report M_ν values around 0.2–0.4 eV. This is the first time anyone has tried to extract both parameters from this particular observable using a simulation pipeline on real velocity data, and they run the fit in two separate cosmological setups and get broadly consistent numbers. That part is straightforward and worth noting as a new data point in the neutrino-LSS space. The work also ties back to their earlier Planck analysis and says the numbers line up. Credit for actually applying the method to catalog data instead of just forecasting. The main soft spot is exactly the one the stress test flags. Once you let η² be non-zero, the neutrino energy density rises, N_eff shifts, and the expansion history changes. Yet the CMB-framework run holds Ω_m, H_0 and the rest fixed at the standard Planck values that assumed η = 0. That mismatch can push the fit toward non-zero asymmetry to compensate, which would inflate the significance. The local framework shows much larger errors on η², which is consistent with the idea that the tight CMB result is partly an artifact of the fixed background. The abstract gives no detail on whether the simulation pipeline recomputes the background or marginalizes over the induced changes in radiation density, so it is impossible to tell how large the effect is. Without those steps or a clear covariance treatment, the 7σ number is hard to trust at face value. This paper is aimed at people who already work on neutrino mass bounds from large-scale structure and velocity statistics. A reader who knows Cosmicflows-4 and pairwise-velocity modeling will find the pipeline idea useful to look at, even if they end up disagreeing with the final claim. It is worth sending to a serious referee so the methods section can be checked for exactly how the background is handled and whether the claimed consistency with Planck survives once the parameters are allowed to adjust together. Desk reject would be too quick; the observable itself is worth testing.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to provide the first constraints on the total neutrino mass M_ν and neutrino asymmetry parameter η² from the mean galaxy pairwise peculiar velocity measured in the Cosmicflows-4 grouped catalog. Using a simulation-based pipeline applied in two independent frameworks (one with cosmological parameters fixed from CMB data and one from local distance ladder measurements), the authors report posterior means of M_ν = 0.24^{+0.34}_{-0.18} eV and η² = 2.14^{+0.30}_{-0.32} (claimed 7σ from zero) in the CMB framework, with consistent but less precise values in the local framework, and state that these are consistent with their prior Planck CMB analysis.

Significance. If the simulation pipeline accurately maps neutrino mass and asymmetry to pairwise velocities across the relevant regimes without unaccounted systematics or modeling biases, this would constitute a novel large-scale structure probe of neutrino properties independent of the CMB. The dual-framework consistency and use of quasi-linear to nonlinear velocity data are positive aspects. The claimed 7σ detection of non-zero asymmetry would be a notable result if the underlying cosmology is handled self-consistently, but the current presentation leaves open questions about robustness.

major comments (1)

[Abstract and CMB framework] Abstract and CMB framework analysis: the background parameters (Ω_m, H_0, etc.) are fixed to Planck-derived values obtained under the assumption of standard neutrinos with η=0, while the fit allows η² ≈ 2.14. Non-zero neutrino asymmetry increases the effective relativistic degrees of freedom and neutrino energy density, shifting the radiation-matter transition, sound horizon, and expansion history. This fixed-background approach creates an internal inconsistency that can bias the recovered η² away from zero to compensate for the mismatch between the assumed cosmology and the velocity predictions. The manuscript should either marginalize over adjusted background parameters or demonstrate quantitatively that the effect on pairwise velocities is negligible at the claimed precision.

minor comments (1)

[Abstract] The abstract states a '7σ measurement' but does not indicate whether this is derived from the posterior width, a likelihood ratio test, or another method; this detail should be clarified for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting an important consistency issue regarding the fixed background cosmology in the CMB framework. We provide a point-by-point response below and will revise the manuscript to include a quantitative assessment addressing the concern.

read point-by-point responses

Referee: Abstract and CMB framework analysis: the background parameters (Ω_m, H_0, etc.) are fixed to Planck-derived values obtained under the assumption of standard neutrinos with η=0, while the fit allows η² ≈ 2.14. Non-zero neutrino asymmetry increases the effective relativistic degrees of freedom and neutrino energy density, shifting the radiation-matter transition, sound horizon, and expansion history. This fixed-background approach creates an internal inconsistency that can bias the recovered η² away from zero to compensate for the mismatch between the assumed cosmology and the velocity predictions. The manuscript should either marginalize over adjusted background parameters or demonstrate quantitatively that the effect on pairwise velocities is negligible at the claimed precision.

Authors: We agree this is a substantive point that requires clarification. The CMB framework fixes background parameters to Planck 2018 values derived under standard neutrino assumptions (η=0). Non-zero η² does alter N_eff and the early expansion history. However, our simulation-based pipeline models pairwise velocities primarily through late-time gravitational evolution in the quasi-linear and nonlinear regimes, where the sensitivity to early-universe shifts is reduced. To address the referee's recommendation, we will add a quantitative test in the revised manuscript: we will recompute the mean pairwise velocity predictions using background cosmologies adjusted for η² ≈ 2.14 (incorporating the corresponding change in effective relativistic degrees of freedom) and compare them directly to the fixed-Planck case. We will demonstrate that the difference lies well within the statistical uncertainties of the Cosmicflows-4 measurement. This check will confirm that any bias is negligible at the reported precision. We will also update the abstract to note this robustness test if the results warrant it. Full marginalization over background parameters is beyond the scope of this Letter but can be pursued in follow-up work. revision: yes

Circularity Check

0 steps flagged

No significant circularity: independent velocity data and simulation pipeline yield direct constraints

full rationale

The paper derives its neutrino mass and asymmetry constraints from a simulation-based pipeline applied to independent Cosmicflows-4 galaxy pairwise velocity data within two separate cosmological frameworks (CMB and local distance ladder). The central results—M_ν and η² posteriors—are obtained by fitting the observed velocities, with no reduction of the fitted quantities to prior inputs by construction. The single reference to consistency with the authors' previous Planck analysis is a post-hoc statement and does not serve as a load-bearing premise or input for the current derivation; the velocity-based measurement stands on its own data and modeling.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the simulation pipeline accurately captures neutrino effects on pairwise velocities and that the two cosmological frameworks are appropriate; no new entities are introduced.

free parameters (2)

M_ν
Total neutrino mass fitted to the velocity data in both frameworks.
η²
Neutrino asymmetry parameter fitted to the velocity data.

axioms (1)

domain assumption Neutrino properties affect the growth of cosmic structure and thus galaxy pairwise velocities in a manner captured by the simulations
Invoked when connecting neutrino parameters to the observable in the analysis pipeline.

pith-pipeline@v0.9.0 · 5605 in / 1268 out tokens · 47315 ms · 2026-05-10T01:08:38.316788+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 51 canonical work pages · 8 internal anchors

[1]

Each realization includes four variants: standard, paired, fixed, and paired-and- fixed, following the method of Angulo and Pontzen [45] and Klypinet al.[46]

in both frameworks. Each realization includes four variants: standard, paired, fixed, and paired-and- fixed, following the method of Angulo and Pontzen [45] and Klypinet al.[46]. B. Pairwise velocity From these simulations, we compute the halo–halo pairwise velocity by vhh(r) =⟨[v 1(r1)−v 2(r2)]·ˆr⟩,(3) wherer 1,2 (v1,2) are the positions (peculiar veloci...
[2]

Esteban, M

I. Esteban, M. C. Gonzalez-Garcia, M. Maltoniet al., The fate of hints: updated global analysis of three-flavor neutrino oscillations, Journal of High Energy Physics 7 Mν [eV]η 2 H0 [km s−1 Mpc−1] Ω m Ωbh2 ns As [10−9] 0.00 0.0 68.03 0.3070 0.02241 0.9661 2.100 0.06 0.0 67.73 0.3105 0.02242 0.9667 2.102 0.06 0.4 68.77 0.3075 0.02255 0.9728 2.120 0.06 0.8 ...

work page arXiv 2020
[3]

Katrin Collaboration, M. Aker, A. Beglarianet al., Direct neutrino-mass measurement with sub-electronvolt sensi- tivity, Nature Physics18, 160 (2022), arXiv:2105.08533 [hep-ex]

work page arXiv 2022
[4]

Katrin Collaboration, M. Aker, D. Batzleret al., Direct neutrino-mass measurement based on 259 days of KA- TRIN data, Science388, 180 (2025), arXiv:2406.13516 [nucl-ex]

work page arXiv 2025
[5]

Ashtari Esfahani, S

Project 8 Collaboration, A. Ashtari Esfahani, S. B¨ oser et al., The Project 8 Neutrino Mass Experiment, arXiv e- prints , arXiv:2203.07349 (2022), arXiv:2203.07349 [nucl- ex]

work page arXiv 2022
[6]

Ashtari Esfahani, S

Project 8 Collaboration, A. Ashtari Esfahani, S. B¨ oser et al., Tritium Beta Spectrum Measurement and Neu- trino Mass Limit from Cyclotron Radiation Emission Spectroscopy, Phys. Rev. Lett.131, 102502 (2023), arXiv:2212.05048 [nucl-ex]

work page arXiv 2023
[7]

Planck 2018 results. VI. Cosmological parameters

Planck Collaboration, N. Aghanim, Y. Akramiet al., Planck 2018 results. VI. Cosmological parameters, A&A 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO]

work page internal anchor Pith review arXiv 2018
[8]

C., Jense, H

E. Calabrese, J. C. Hill, H. T. Jenseet al., The Atacama Cosmology Telescope: DR6 constraints on extended cos- mological models, J. Cosmology Astropart. Phys.2025, 063 (2025), arXiv:2503.14454 [astro-ph.CO]

work page arXiv 2025
[9]

SPT-3G D1: CMB temperature and polarization power spectra and cosmology from 2019 and 2020 observations of the SPT-3G Main field

E. Camphuis, W. Quan, L. Balkenholet al., SPT-3G D1: CMB temperature and polarization power spectra and cosmology from 2019 and 2020 observations of the SPT- 3G Main field, arXiv e-prints , arXiv:2506.20707 (2025), arXiv:2506.20707 [astro-ph.CO]

work page internal anchor Pith review arXiv 2019
[10]

Elbers, A

W. Elbers, A. Aviles, H. E. Noriegaet al., Con- straints on neutrino physics from DESI DR2 BAO and DR1 full shape, Phys. Rev. D112, 083513 (2025), arXiv:2503.14744 [astro-ph.CO]

work page arXiv 2025
[11]

KiDS-Legacy: Constraining dark energy, neutrino mass, and curvature

R. Reischke, B. St¨ olzner, B. Joachimiet al., KiDS- Legacy: Constraining dark energy, neutrino mass, and curvature, arXiv e-prints , arXiv:2512.11041 (2025), arXiv:2512.11041 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[12]

Pitrou, A

C. Pitrou, A. Coc, J.-P. Uzanet al., Precision big bang nucleosynthesis with improved Helium-4 predic- 8 tions, Phys. Rep.754, 1 (2018), arXiv:1801.08023 [astro- ph.CO]

work page arXiv 2018
[13]

Burns, T.M.P

A.-K. Burns, T. M. P. Tait, and M. Valli, Indications for a Nonzero Lepton Asymmetry from Extremely Metal- Poor Galaxies, Phys. Rev. Lett.130, 131001 (2023), arXiv:2206.00693 [hep-ph]

work page arXiv 2023
[14]

Escudero, A

M. Escudero, A. Ibarra, and V. Maura, Primordial lep- ton asymmetries in the precision cosmology era: Current status and future sensitivities from BBN and the CMB, Phys. Rev. D107, 035024 (2023), arXiv:2208.03201 [hep- ph]

work page arXiv 2023
[15]

Barenboim, W

G. Barenboim, W. H. Kinney, and W.-I. Park, Flavor versus mass eigenstates in neutrino asymmetries: impli- cations for cosmology, European Physical Journal C77, 590 (2017), arXiv:1609.03200 [astro-ph.CO]

work page arXiv 2017
[16]

Yeung, K

S. Yeung, K. Lau, and M. C. Chu, Relic neutrino de- generacies and their impact on cosmological parame- ters, J. Cosmology Astropart. Phys.2021, 024 (2021), arXiv:2010.01696 [astro-ph.CO]

work page arXiv 2021
[17]

Yeung, W

S. Yeung, W. Zhang, and M.-c. Chu, Refitting Cosmo- logical Data with Neutrino Mass and Degeneracy, ApJ 990, L45 (2025), arXiv:2403.11499 [hep-ph]

work page arXiv 2025
[18]

Zhao, D.-M

Z.-C. Zhao, D.-M. Xia, and S. Wang, Constraints on the lepton asymmetry from DESI DR2 BAO data, Phys. Rev. D112, 123527 (2025), arXiv:2504.09624 [astro-ph.CO]

work page arXiv 2025
[19]

Z. Zeng, S. Yeung, and M.-c. Chu, Effects of neutrino mass and asymmetry on cosmological structure forma- tion, J. Cosmology Astropart. Phys.2019, 015 (2019), arXiv:1808.00357 [astro-ph.CO]

work page arXiv 2019
[20]

Y. Liu, W. Zhang, S. Liaoet al., Halo abundance and clustering in cosmologies with massive and asymmet- ric neutrinos, arXiv e-prints , arXiv:2603.12801 (2026), arXiv:2603.12801 [astro-ph.CO]

work page arXiv 2026
[21]

H. W. Wong and M.-c. Chu, Effects of neutrino masses and asymmetries on dark matter halo assem- bly, J. Cosmology Astropart. Phys.2022, 066 (2022), arXiv:2109.00303 [astro-ph.CO]

work page arXiv 2022
[22]

Juszkiewicz, P

R. Juszkiewicz, P. G. Ferreira, H. A. Feldmanet al., Ev- idence for a Low-Density Universe from the Relative Ve- locities of Galaxies, Science287, 109 (2000), arXiv:astro- ph/0001041 [astro-ph]

work page arXiv 2000
[23]

Feldman, R

H. Feldman, R. Juszkiewicz, P. Ferreiraet al., An Esti- mate of Ωm without Conventional Priors, ApJ596, L131 (2003), arXiv:astro-ph/0305078 [astro-ph]

work page arXiv 2003
[24]

Y.-Z. Ma, M. Li, and P. He, Constraining cosmology with pairwise velocity estimator, A&A583, A52 (2015), arXiv:1509.06413 [astro-ph.CO]

work page arXiv 2015
[25]

J. E. Garc´ ıa-Farieta and H. J. Hort´ ua, Parameter sensitivity of cosmic pairwise velocities in the non- linear regime of structure formation, arXiv e-prints , arXiv:2509.15223 (2025), arXiv:2509.15223 [astro- ph.CO]

work page arXiv 2025
[26]

Howlett, L

C. Howlett, L. Staveley-Smith, P. J. Elahiet al., 2MTF - VI. Measuring the velocity power spectrum, MNRAS 471, 3135 (2017), arXiv:1706.05130 [astro-ph.CO]

work page arXiv 2017
[27]

Dupuy, H

A. Dupuy, H. M. Courtois, and B. Kubik, An estimation of the local growth rate from Cosmicflows peculiar veloc- ities, MNRAS486, 440 (2019), arXiv:1901.03530 [astro- ph.CO]

work page arXiv 2019
[28]

Bhattacharya and A

S. Bhattacharya and A. Kosowsky, Dark energy con- straints from galaxy cluster peculiar velocities, Phys. Rev. D77, 083004 (2008), arXiv:0712.0034 [astro-ph]

work page arXiv 2008
[29]

Bhattacharya, A

S. Bhattacharya, A. Kosowsky, J. A. Newmanet al., Galaxy peculiar velocities from large-scale supernova sur- veys as a dark energy probe, Phys. Rev. D83, 043004 (2011), arXiv:1008.2560 [astro-ph.CO]

work page arXiv 2011
[30]

Mueller, F

E.-M. Mueller, F. de Bernardis, R. Beanet al., Con- straints on Gravity and Dark Energy from the Pairwise Kinematic Sunyaev-Zel’dovich Effect, Astrophys. J.808, 47 (2015), arXiv:1408.6248 [astro-ph.CO]

work page arXiv 2015
[31]

Running FLRW

A. Bibiano and D. J. Croton, Pairwise velocities in the “Running FLRW” cosmological model, MNRAS467, 1386 (2017), arXiv:1701.04453 [astro-ph.CO]

work page arXiv 2017
[32]

Jaber, W

M. Jaber, W. A. Hellwing, J. E. Garc´ ıa-Farietaet al., Dy- namics of pairwise motions in the fully nonlinear regime in lcdm and modified gravity cosmologies, Phys. Rev. D 109, 123528 (2024)

2024
[33]

K. H. Luo, M.-c. Chu, and W. Zhang, Probing the Type 3 interacting dark-energy model using matter pair- wise velocity, arXiv e-prints , arXiv:2508.04312 (2025), arXiv:2508.04312 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[34]

Hu, M.-c

R. Hu, M.-c. Chu, S. Yeunget al., Impact of light sterile neutrinos on cosmological large scale struc- ture, J. Cosmology Astropart. Phys.2025, 014 (2025), arXiv:2501.16908 [astro-ph.CO]

work page arXiv 2025
[35]

Zhang, M.-c

W. Zhang, M.-c. Chu, R. Huet al., Measuring neu- trino mass and asymmetry with matter pairwise veloc- ities, MNRAS529, 360 (2024), arXiv:2312.04278 [astro- ph.CO]

work page arXiv 2024
[36]

Zhang, M.-c

W. Zhang, M.-c. Chu, S. Liaoet al., Measuring the Hubble Constant through the Galaxy Pairwise Peculiar Velocity, ApJ978, L6 (2025), arXiv:2412.04660 [astro- ph.CO]

work page arXiv 2025
[37]

Springel, MNRAS364, 1105 (2005), arXiv:astro- ph/0505010 [astro-ph]

V. Springel, The cosmological simulation code GADGET-2, MNRAS364, 1105 (2005), arXiv:astro- ph/0505010 [astro-ph]

work page arXiv 2005
[38]

Ali-Ha¨ ımoud and S

Y. Ali-Ha¨ ımoud and S. Bird, An efficient implementation of massive neutrinos in non-linear structure formation simulations, MNRAS428, 3375 (2013), arXiv:1209.0461 [astro-ph.CO]

work page arXiv 2013
[39]

Crocce, S

M. Crocce, S. Pueblas, and R. Scoccimarro, Transients from initial conditions in cosmological simulations, MN- RAS373, 369 (2006), arXiv:astro-ph/0606505 [astro-ph]

work page arXiv 2006
[40]

Efficient Computation of CMB anisotropies in closed FRW models

A. Lewis, A. Challinor, and A. Lasenby, Efficient Compu- tation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models, Astrophys. J.538, 473 (2000), arXiv:astro-ph/9911177 [astro-ph]

work page Pith review arXiv 2000
[41]

The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics

E. Di Valentino, J. L. Said, A. Riesset al., The Cos- moVerse White Paper: Addressing observational ten- sions in cosmology with systematics and fundamental physics, Physics of the Dark Universe49, 101965 (2025), arXiv:2504.01669 [astro-ph.CO]

work page internal anchor Pith review arXiv 2025
[42]

The Pantheon+ Analysis: Cosmological Constraints

D. Brout, D. Scolnic, B. Popovicet al., The Pantheon+ Analysis: Cosmological Constraints, Astrophys. J.938, 110 (2022), arXiv:2202.04077 [astro-ph.CO]

work page internal anchor Pith review arXiv 2022
[43]

B. D. Fields, K. A. Olive, T.-H. Yehet al., Big-Bang Nucleosynthesis after Planck, J. Cosmology Astropart. Phys.2020, 010 (2020), arXiv:1912.01132 [astro-ph.CO]

work page arXiv 2020
[44]

P. S. Behroozi, R. H. Wechsler, and H.-Y. Wu, The ROCKSTAR Phase-space Temporal Halo Finder and the Velocity Offsets of Cluster Cores, Astrophys. J.762, 109 (2013), arXiv:1110.4372 [astro-ph.CO]

work page arXiv 2013
[45]

G. L. Bryan and M. L. Norman, Statistical Properties of X-Ray Clusters: Analytic and Numerical Comparisons, Astrophys. J.495, 80 (1998), arXiv:astro-ph/9710107 [astro-ph]

work page arXiv 1998
[46]

R. E. Angulo and A. Pontzen, Cosmological N-body simulations with suppressed variance, MNRAS462, L1 9 (2016), arXiv:1603.05253 [astro-ph.CO]

work page arXiv 2016
[47]

Klypin, F

A. Klypin, F. Prada, and J. Byun, Suppressing cos- mic variance with paired-and-fixed cosmological sim- ulations: average properties and covariances of dark matter clustering statistics, MNRAS496, 3862 (2020), arXiv:1903.08518 [astro-ph.CO]

work page arXiv 2020
[48]

R. B. Tully, E. Kourkchi, H. M. Courtoiset al., Cosmicflows-4, Astrophys. J.944, 94 (2023), arXiv:2209.11238 [astro-ph.CO]

work page arXiv 2023
[49]

P. G. Ferreira, R. Juszkiewicz, H. A. Feldmanet al., Streaming Velocities as a Dynamical Estimator of Ω, ApJ 515, L1 (1999), arXiv:astro-ph/9812456 [astro-ph]

work page arXiv 1999
[50]

emcee: The MCMC Hammer

D. Foreman-Mackey, D. W. Hogg, D. Langet al., em- cee: The MCMC Hammer, PASP125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

work page internal anchor Pith review arXiv 2013
[51]

J. D. Hunter, Matplotlib: A 2D Graphics Environment, Computing in Science and Engineering9, 90 (2007)

2007
[52]

GetDist: a Python package for analysing Monte Carlo samples

A. Lewis, GetDist: a Python package for analysing Monte Carlo samples, J. Cosmology Astropart. Phys.2025, 025 (2025), arXiv:1910.13970 [astro-ph.IM]

work page internal anchor Pith review arXiv 2025
[53]

Virtanenet al., SciPy 1.0–Fundamental Algorithms for Scientific Computing in Python, Nat

P. Virtanen, R. Gommers, T. E. Oliphantet al., SciPy 1.0: fundamental algorithms for scientific com- puting in Python, Nature Methods17, 261 (2020), arXiv:1907.10121 [cs.MS]

work page arXiv 2020
[54]

C. R. Harris, K. J. Millman, S. J. van der Waltet al., Array programming with NumPy, Nature (London)585, 357 (2020), arXiv:2006.10256 [cs.MS]

work page arXiv 2020