arxiv: 2601.02886 · v2 · submitted 2026-01-06 · 🌌 astro-ph.CO · gr-qc

Recognition: 2 theorem links

· Lean Theorem

The velocity coherence scale: a novel probe of cosmic homogeneity and a potential standard ruler

Leonardo Giani , Cullan Howlett , Chris Blake , Ryan J. Turner , Tamara M. Davis

Authors on Pith no claims yet

Pith reviewed 2026-05-16 17:25 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords velocity coherence scalecosmic homogeneitystandard rulerpeculiar velocitiesmatter-radiation equalityLambda-CDMSDSS

0 comments

The pith

The velocity coherence scale is tightly connected to the matter-radiation equality scale and remains constant in comoving coordinates across redshifts in standard cosmologies.

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

The paper introduces the velocity coherence scale R_v as the radius at which the spherical average of galaxy velocity correlations along separation vectors shifts from correlated to anti-correlated motion. It shows this scale links directly to the matter-radiation equality scale and holds a fixed value in comoving units at any redshift under standard models. This link positions R_v as a potential standard ruler and a physically motivated marker for the transition to cosmic homogeneity. The authors derive an estimator using the bulk in spheres B_R and test it on SDSS peculiar velocity data to obtain a value near 132 Mpc/h. Limited precision in current peculiar velocity measurements sets the main practical barrier to tighter results.

Core claim

In standard ΛCDM cosmologies the velocity coherence scale is tightly connected to the matter-radiation equality scale k_eq, and its value in comoving coordinates is redshift-independent. The scale is estimated using the bulk in spheres B_R, a velocity-field analogue of mean scale counts, where for statistically homogeneous matter distributions the logarithmic derivative of B_R shares the same asymptotic behaviour as the correlation dimension D_2 and therefore tracks the transition to statistical homogeneity.

What carries the argument

The bulk in spheres B_R, defined as a velocity-field analogue of mean scale counts, whose logarithmic derivative identifies the velocity coherence scale R_v at which average motions along separation vectors change from correlated to anti-correlated.

If this is right

R_v supplies an independent estimate of the scale at which the velocity field reaches statistical homogeneity.
Its fixed comoving size tied to k_eq allows R_v to function as a standard ruler for cosmic distance measurements.
Peculiar velocity surveys can use R_v to place new constraints on the matter density and expansion history.
Higher-precision PV data from current and future surveys will reduce the uncertainty on the measured value of R_v.

Where Pith is reading between the lines

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

Repeated measurements of R_v in independent peculiar velocity samples could provide a direct test of the statistical homogeneity assumption.
Comparison of the observed R_v with the predicted k_eq scale in large N-body simulations would check the robustness of the link beyond linear theory.
Application of the same B_R estimator to velocity fields reconstructed from galaxy surveys at higher redshift could extend the test of redshift independence.

Load-bearing premise

For a statistically homogeneous matter distribution, the logarithmic derivative of B_R and the correlation dimension D2 share the same asymptotic behaviour.

What would settle it

A precise measurement that finds the comoving value of R_v changing with redshift or deviating substantially from the scale set by k_eq would falsify the claimed connection.

read the original abstract

We introduce the velocity coherence scale $R_v$, the scale at which the spherical volume average of the trace of the velocity correlation tensor transitions from scaling faster than the sphere radius to scaling more slowly. This corresponds to the radius at which the average motion of galaxies along their separation vectors transitions from correlated to anti-correlated. We derive a theoretical estimator for $R_v$ by defining the bulk in spheres $B_R$, a velocity-field analogue of the mean scale counts used in density-field correlation analyses. We show that, for a statistically homogeneous matter distribution, the logarithmic derivative of $B_R$ and the correlation dimension $D_2$ share the same asymptotic behaviour and therefore can be used to estimate the scale of transition to statistical homogeneity. Furthermore, we show that in standard $\Lambda$CDM cosmologies the velocity coherence scale is tightly connected to the matter-radiation equality scale $k_{eq}$, and that its value in comoving coordinates is redshift-independent. These results highlight the potential of $R_v$ both as a standard ruler and as a physically motivated scale characterising the onset of cosmic homogeneity. We present a proof of concept using measurements of the PV correlation functions from SDSS. We show that the main challenge in determining $R_v$ is the limited precision of PV measurements compared to density ones, as they typically rely on smaller samples with larger uncertainties that scale roughly linearly with survey depth. Fitting our theoretical estimators for $R_v$, we obtain $R_v \approx 132^{+29}_{-51}\,\mathrm{Mpc}/h$. Finally, we show that more precise determinations should be achievable with current and upcoming peculiar velocity 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.

Desk Editor's Note private letter to a colleague

The paper defines a new velocity coherence scale R_v tied to k_eq with a clean theoretical estimator, but the SDSS measurement is too loose to test it and the velocity-tensor version of the D2 equivalence needs explicit checking.

read the letter

The main thing here is a new scale R_v extracted from the point where the spherical average of the trace of the velocity correlation tensor switches from growing faster than the radius to slower. They call this the bulk B_R and show that in standard LCDM it sits right at the matter-radiation equality scale and stays fixed in comoving coordinates. That link is the cleanest part of the work and is not just a restatement of existing homogeneity scales. They also give an explicit estimator and run it on SDSS peculiar-velocity correlations to get roughly 132 Mpc/h, with the expected large error bar. The derivation looks internally consistent on paper and they are upfront that peculiar-velocity samples are smaller and noisier than density ones, which is fair. The redshift independence in comoving units is a nice feature if it survives closer scrutiny. The soft spot is the step that equates the logarithmic derivative of B_R to the usual correlation dimension D2. The stress-test note is right to flag that the velocity tensor has a sign change from correlated to anti-correlated motions and the trace can cancel in ways that scalar density counts do not. The abstract claims the asymptotic behaviour is the same for any statistically homogeneous field, but without seeing the full algebra it is not obvious that the tensor structure and the zero-crossing do not shift the transition scale. The SDSS fit itself is too loose to check this; the errors are bigger than the central value difference one would care about. So the idea is worth following, but the current data point does not yet constrain the model or confirm the claimed link. This is the sort of paper that belongs in a cosmology journal after a referee has checked the velocity-tensor asymptotics and perhaps asked for a forecast with next-generation surveys. I would bring it to a reading group to see whether the B_R definition really inherits the D2 properties without extra assumptions.

Referee Report

2 major / 2 minor

Summary. The paper introduces the velocity coherence scale R_v as the transition radius where the spherical average of the trace of the velocity correlation tensor shifts from correlated to anti-correlated galaxy motions. It defines the bulk velocity estimator B_R, argues that its logarithmic derivative shares the asymptotic behaviour of the correlation dimension D2 for statistically homogeneous distributions, and shows that in standard ΛCDM cosmologies R_v is tightly linked to the matter-radiation equality scale k_eq with a redshift-independent comoving value. A proof-of-concept fit to SDSS peculiar-velocity correlation functions yields R_v ≈ 132^{+29}_{-51} Mpc/h, positioning R_v as a potential standard ruler and homogeneity probe.

Significance. If the derivation and equivalence hold, the result supplies a physically motivated, observationally accessible scale tied directly to k_eq that is constant in comoving coordinates, offering a new route to test homogeneity and a potential standard ruler independent of density-field methods. The SDSS measurement demonstrates practical feasibility despite large uncertainties, and the suggestion that future peculiar-velocity surveys can tighten constraints is a concrete strength. The work is novel in extending correlation-dimension ideas to the velocity tensor.

major comments (2)

[Theoretical estimator] Theoretical estimator section: The central claim that the logarithmic derivative of B_R and the correlation dimension D2 share identical asymptotic behaviour rests on an unproven extension from scalar density counts to the trace of the velocity correlation tensor. Because the trace incorporates a sign change between correlated and anti-correlated velocities, cancellations could modify the derivative even in the homogeneous limit; an explicit derivation or counter-example ruling out this possibility is required before the transition scale can be identified with D2.
[SDSS measurement] SDSS proof-of-concept and fit: The reported R_v = 132^{+29}_{-51} Mpc/h carries uncertainties comparable to the central value itself, and the manuscript provides neither the full covariance matrix nor a quantitative assessment of how sample-size and depth-dependent errors propagate into the estimator; this weakens the claim that R_v can already serve as a competitive probe.

minor comments (2)

[Abstract] The abstract states that R_v is redshift-independent in comoving coordinates but does not reference the specific equation or figure that demonstrates this independence.
[Notation] Notation for the velocity correlation tensor and its trace should be introduced once with a clear definition before being used in the estimator B_R.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us identify areas where the manuscript can be clarified and strengthened. We address each major comment point by point below.

read point-by-point responses

Referee: Theoretical estimator section: The central claim that the logarithmic derivative of B_R and the correlation dimension D2 share identical asymptotic behaviour rests on an unproven extension from scalar density counts to the trace of the velocity correlation tensor. Because the trace incorporates a sign change between correlated and anti-correlated velocities, cancellations could modify the derivative even in the homogeneous limit; an explicit derivation or counter-example ruling out this possibility is required before the transition scale can be identified with D2.

Authors: We thank the referee for highlighting the need for greater explicitness in the derivation. The manuscript defines B_R explicitly as the spherical average of the trace of the velocity correlation tensor (Eq. 7), which incorporates the sign change by construction: positive contributions arise from correlated motions and negative from anti-correlated. In the homogeneous limit, the velocity field has vanishing mean, so the trace average scales linearly with volume in the same manner as the scalar count used for D2; the logarithmic derivative therefore approaches the same asymptotic value of 3. We will add an expanded appendix with the intermediate algebraic steps showing that the sign change does not alter the leading-order scaling once the homogeneous limit is taken, thereby making the equivalence fully explicit. revision: yes
Referee: SDSS proof-of-concept and fit: The reported R_v = 132^{+29}_{-51} Mpc/h carries uncertainties comparable to the central value itself, and the manuscript provides neither the full covariance matrix nor a quantitative assessment of how sample-size and depth-dependent errors propagate into the estimator; this weakens the claim that R_v can already serve as a competitive probe.

Authors: We agree that the present SDSS constraints are limited by the precision of existing peculiar-velocity data, as already noted in the manuscript. The reported uncertainties reflect the linear scaling of PV errors with depth and the modest sample size. To address the referee's request, we will include the covariance matrix employed in the fit (derived from the PV correlation function measurements) and add a short quantitative discussion of error propagation in the revised text, while retaining the proof-of-concept framing that future surveys will yield tighter constraints. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained and independent of inputs

full rationale

The paper defines R_v directly from the spherical average B_R of the trace of the velocity correlation tensor and derives its asymptotic link to the correlation dimension D2 for homogeneous fields as a mathematical property of the two-point statistics. The connection to k_eq and redshift independence is obtained by substituting the standard ΛCDM velocity power spectrum into the estimator, without fitting parameters to the target scale itself. The SDSS analysis is an observational fit to measured peculiar-velocity correlations and is not presented as a model prediction forced by the same data. No load-bearing step reduces by construction to a self-citation, a fitted input renamed as output, or an ansatz smuggled via prior work; the central claims remain externally falsifiable against independent density-field homogeneity scales.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claims rest on the assumption of statistical homogeneity to equate the B_R derivative with the correlation dimension, plus the standard Lambda-CDM background for the k_eq link; no new free parameters or invented entities are introduced beyond the definition of R_v itself.

axioms (2)

domain assumption Statistical homogeneity of the matter distribution
Invoked to show that the logarithmic derivative of B_R shares asymptotic behaviour with the correlation dimension D2
domain assumption Standard Lambda-CDM cosmology
Used to establish the tight connection between R_v and the matter-radiation equality scale k_eq

pith-pipeline@v0.9.0 · 5615 in / 1314 out tokens · 42983 ms · 2026-05-16T17:25:05.658912+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We identify R_H as the scale at which the slope of the function B_R changes... the zero crossing of Ψ∥ signals that no external potential gradient is sourcing coherent flows
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

for a statistically homogeneous matter distribution, the logarithmic derivative of B_R and the correlation dimension D2 share the same asymptotic behaviour

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

51 extracted references · 51 canonical work pages · 24 internal anchors

[1]

Martinez and P

V.J. Martinez and P. Coles,Correlations and Scaling in the QDOT Redshift Survey,Astrophys. J.437(1994) 550

work page 1994
[2]

Searching for the scale of homogeneity

V.J. Martinez, M.-J. Pons-Borderia, R.A. Moyeed and M.J. Graham,Searching for the scale of homogeneity,Mon. Not. Roy. Astron. Soc.298(1998) 1212 [astro-ph/9804073]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[3]

The scale of homogeneity in the Las Campanas Redshift Survey

L. Amendola and E. Palladino,The scale of homogeneity in the las campanas redshift survey, Astrophys. J. Lett.514(1999) L1 [astro-ph/9901420]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[4]

Large-scale Cosmic Homogeneity from a Multifractal Analysis of the PSCz Catalogue

J. Pan and P. Coles,Large scale cosmic homogeneity from a multifractal analysis of the pscz catalogue,Mon. Not. Roy. Astron. Soc.318(2000) L51 [astro-ph/0008240]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[5]

Testing homogeneity on large scales in the Sloan Digital Sky Survey Data Release One

J. Yadav, S. Bharadwaj, B. Pandey and T.R. Seshadri,Testing homogeneity on large scales in the Sloan Digital Sky Survey Data Release One,Mon. Not. Roy. Astron. Soc.364(2005) 601 [astro-ph/0504315]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[6]

The scale of homogeneity of the galaxy distribution in SDSS DR6

P. Sarkar, J. Yadav, B. Pandey and S. Bharadwaj,The scale of homogeneity of the galaxy distribution in SDSS DR6,mnras399(2009) L128 [0906.3431]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[7]

Large-scale fluctuations in the distribution of galaxies from the Two Degree Field Galaxy Redshift Survey

F. Sylos Labini, N.L. Vasilyev and Y.V. Baryshev,Large-scale fluctuations in the distribution of galaxies from the two-degree galaxy redshift survey,Astron. Astrophys.496(2009) 7 [0902.0229]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[8]

Absence of anti-correlations and of baryon acoustic oscillations in the galaxy correlation function from the Sloan Digital Sky Survey DR7

F. Sylos Labini, N.L. Vasilyev, Y.V. Baryshev and M. L´ opez-Corredoira,Absence of anti-correlations and of baryon acoustic oscillations in the galaxy correlation function from the Sloan Digital Sky Survey data release 7,Astron. Astrophys.505(2009) 981 [0903.0950]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[9]

Testing the Copernican and Cosmological Principles in the local universe with galaxy surveys

F.S. Labini and Y.V. Baryshev,Testing the Copernican and Cosmological Principles in the local universe with galaxy surveys,JCAP06(2010) 021 [1006.0801]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[10]

Very large scale correlations in the galaxy distribution

F.S. Labini,Very large scale correlations in the galaxy distribution,EPL96(2011) 59001 [1110.4041]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[11]

Exploring cosmic homogeneity with the BOSS DR12 galaxy sample

P. Ntelis et al.,Exploring cosmic homogeneity with the BOSS DR12 galaxy sample,JCAP06 (2017) 019 [1702.02159]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[12]

Scaling in the Universe

S. Borgani,Scaling in the universe,Phys. Rept.251(1995) 1 [astro-ph/9404054]. – 12 –

work page internal anchor Pith review Pith/arXiv arXiv 1995
[13]

Clustering statistics in cosmology

V.J. Martinez and E. Saar,Clustering statistics in cosmology,Proc. SPIE Int. Soc. Opt. Eng. 4847(2002) 86 [astro-ph/0209208]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[14]

X. Shao, R. Gon¸ calves, C.A.P. Bengaly, G.C. Carvalho and J. Alcaniz,Cosmic homogeneity: the effect of redshift-space distortions and bias and cosmological constraints,2507.18720

work page arXiv
[15]

Accessing the homogeneity scale with 21 cm intensity mapping surveys

B.B. Bizarria, C.P. Novaes, F. Avila, R. Mokeddem, H.H. da Costa, C.A. Wuensche et al., Accessing the homogeneity scale with 21 cm intensity mapping surveys,2511.13931

work page internal anchor Pith review Pith/arXiv arXiv
[16]

The scale of cosmic homogeneity as a standard ruler

P. Ntelis, A. Ealet, S. Escoffier, J.-C. Hamilton, A.J. Hawken, J.-M. Le Goff et al.,The scale of cosmic homogeneity as a standard ruler,JCAP12(2018) 014 [1810.09362]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[17]

Avila, A

F. Avila, A. Bernui, R.C. Nunes, E. de Carvalho and C.P. Novaes,The homogeneity scale and the growth rate of cosmic structures,Mon. Not. Roy. Astron. Soc.509(2021) 2994 [2111.08541]

work page arXiv 2021
[18]

The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity

M. Scrimgeour et al.,The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity,Mon. Not. Roy. Astron. Soc.425(2012) 116 [1205.6812]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[19]

Aluri et al.,Is the observable Universe consistent with the cosmological principle?,Class

P.K. Aluri et al.,Is the observable Universe consistent with the cosmological principle?,Class. Quant. Grav.40(2023) 094001 [2207.05765]

work page arXiv 2023
[20]

Watkins et al.,Analysing the large-scale bulk flow using cosmicflows4: increasing tension with the standard cosmological model,Mon

R. Watkins et al.,Analysing the large-scale bulk flow using cosmicflows4: increasing tension with the standard cosmological model,Mon. Not. R. Astron. Soc.524(2023) 1885–1892

work page 2023
[21]

Whitford, C

A.M. Whitford, C. Howlett and T.M. Davis,Evaluating bulk flow estimators for CosmicFlows-4 measurements,2306.11269

work page arXiv
[22]

Watkins and H.A

R. Watkins and H.A. Feldman,The Origins of the Bulk flow,2512.03168

work page arXiv
[23]

Clifton and N

T. Clifton and N. Hyatt,A Radical Solution to the Hubble Tension Problem,preprint(2024) [2404.08586]

work page arXiv 2024
[24]

Giani, C

L. Giani, C. Howlett, K. Said, T. Davis and S. Vagnozzi,An effective description of Laniakea: impact on cosmology and the local determination of the Hubble constant,JCAP01(2024) 071 [2311.00215]

work page arXiv 2024
[25]

Giani, R

L. Giani, R. Von Marttens and R. Camilleri,Novel Approach to Cosmological Nonlinearities as an Effective Fluid,Phys. Rev. Lett.135(2025) 071004 [2410.15295]

work page arXiv 2025
[26]

Giani, R

L. Giani, R. Von Marttens and O.F. Piattella,The matter with(in) CPL,2505.08467

work page arXiv
[27]

Z.G. Lane, A. Seifert, R. Ridden-Harper and D.L. Wiltshire,Cosmological foundations revisited with Pantheon+,Mon. Not. Roy. Astron. Soc.536(2025) 1752 [2311.01438]

work page arXiv 2025
[28]

Camarena, K

D. Camarena, K. Greene, J. Houghteling and F.-Y. Cyr-Racine,Designing concordant distances in the age of precision cosmology: The impact of density fluctuations,Phys. Rev. D 112(2025) 083526 [2507.17969]

work page arXiv 2025
[29]

An effective $\boldsymbol{\Lambda}$-Szekeres modelling of the local Universe with Cosmicflows-4

M. Galoppo, L. Giani, M. Hills and A. Valade,An effectiveΛ-Szekeres modelling of the local Universe with Cosmicflows-4,2512.16591

work page internal anchor Pith review Pith/arXiv arXiv
[30]

Lyall, C

S. Lyall, C. Blake and R.J. Turner,Constraining modified gravity scenarios with the 6dFGS and SDSS galaxy peculiar velocity data sets,Mon. Not. Roy. Astron. Soc.532(2024) 3972 [2407.18684]

work page arXiv 2024
[31]

Howlett, K

C. Howlett, K. Said, J.R. Lucey, M. Colless, F. Qin, Y. Lai et al.,The sloan digital sky survey peculiar velocity catalogue,Mon. Not. Roy. Astron. Soc.515(2022) 953 [2201.03112]

work page arXiv 2022
[32]

Gorski,On the Pattern of Perturbations of the Hubble Flow,apjl332(1988) L7

K. Gorski,On the Pattern of Perturbations of the Hubble Flow,apjl332(1988) L7

work page 1988
[33]

Groth, R

E.J. Groth, R. Juszkiewicz and J.P. Ostriker,An Estimate of the Velocity Correlation Tensor: Cosmological Implications,Astrophys. J.346(1989) 558

work page 1989
[34]

Y. Wang, C. Rooney, H.A. Feldman and R. Watkins,The peculiar velocity correlation function, Mon. Not. R. Astron. Soc.480(2018) 5332 [1808.07543]. – 13 –

work page internal anchor Pith review Pith/arXiv arXiv 2018
[35]

Y. Wang, S. Peery, H.A. Feldman and R. Watkins,Improved Methods for Estimating Peculiar Velocity Correlation Functions Using Volume Weighting,Astrophys. J.918(2021) 49 [2108.08036]

work page arXiv 2021
[36]

Turner, C

R.J. Turner, C. Blake and R. Ruggeri,A local measurement of the growth rate from peculiar velocities and galaxy clustering correlations in the 6dF Galaxy Survey,Mon. Not. Roy. Astron. Soc.518(2022) 2436 [2207.03707]

work page arXiv 2022
[37]

Blake and R.J

C. Blake and R.J. Turner,On the correlations of galaxy peculiar velocities and their covariance, Mon. Not. Roy. Astron. Soc.527(2023) 501 [2308.15735]

work page arXiv 2023
[38]

Turner,Cosmology with Peculiar Velocity Surveys,2411.19484

R.J. Turner,Cosmology with Peculiar Velocity Surveys,2411.19484

work page arXiv
[39]

F. Qin, C. Howlett and L. Staveley-Smith,The redshift-space momentum power spectrum – II. Measuring the growth rate from the combined 2MTF and 6dFGSv surveys,Mon. Not. Roy. Astron. Soc.487(2019) 5235 [1906.02874]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[40]

Turner, C

R.J. Turner, C. Blake and R. Ruggeri,Improving estimates of the growth rate using galaxy–velocity correlations: a simulation study,Mon. Not. Roy. Astron. Soc.502(2021) 2087 [2101.09026]

work page arXiv 2021
[41]

Peebles,The large-scale structure of the universe(1980)

P.J.E. Peebles,The large-scale structure of the universe(1980)

work page 1980
[42]

M. Li, J. Pan, L. Gao, Y. Jing, X. Yang, X. Chi et al.,Bulk Flow of Halos inΛCDM Simulation,Astrophys. J.761(2012) 151 [1207.5338]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[43]

Cosmology with Peculiar Velocities: Observational Effects

P. Andersen, T.M. Davis and C. Howlett,Cosmology with Peculiar Velocities: Observational Effects,Mon. Not. Roy. Astron. Soc.463(2016) 4083 [1609.04022]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[44]

The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview

J. Lesgourgues,The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview,arXiv e-prints(2011) arXiv:1104.2932 [1104.2932]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[45]

Cosmology with photometric redshift surveys

C. Blake and S. Bridle,Cosmology with photometric redshift surveys,Mon. Not. Roy. Astron. Soc.363(2005) 1329 [astro-ph/0411713]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[46]

emcee: The MCMC Hammer

D. Foreman-Mackey, D.W. Hogg, D. Lang and J. Goodman,emcee: The MCMC Hammer, pasp125(2013) 306 [1202.3665]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[47]

Hinton,Chainconsumer,Journal of Open Source Software1(2016) 45

S. Hinton,Chainconsumer,Journal of Open Source Software1(2016) 45

work page 2016
[48]

Saulder et al.,Target Selection for the DESI Peculiar Velocity Survey,2302.13760

C. Saulder et al.,Target Selection for the DESI Peculiar Velocity Survey,2302.13760

work page arXiv
[49]

Said et al.,DESI Peculiar Velocity Survey – Fundamental Plane,Mon

K. Said et al.,DESI Peculiar Velocity Survey – Fundamental Plane,Mon. Not. Roy. Astron. Soc.539(2025) 3627 [2408.13842]

work page arXiv 2025
[50]

4MOST: Project overview and information for the First Call for Proposals

e.a. de Jong,4MOST: Project overview and information for the First Call for Proposals,The Messenger175(2019) 3 [1903.02464]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[51]

Turner, C

R.J. Turner et al.,The DESI DR1 Peculiar Velocity Survey: growth rate measurements from galaxy and momentum correlation functions,2512.03230. – 14 –

work page arXiv