Pith. sign in

REVIEW 2 major objections 4 minor 115 references

Multipolar structure of the local expansion rate from incomplete sky data

T0 review · 2 major / 4 minor · reviewed 2026-07-10 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Local expansion-rate dipole from Cosmicflows-4 exceeds ΛCDM predictions at 3.3σ, while quadrupole and octupole do not.

desk verdict Solid multipole reconstruction of CF4 η that cleanly recovers all 2ℓ+1 degrees of freedom and finds a 3.3σ dipole excess, but the excess sits in the exact redshift bin where unmodeled Malmquist bias is strongest. read the letter →

arxiv 2607.07821 v1 pith:6HBVVDTO submitted 2026-07-08 astro-ph.CO

João G. Vicente , Thiago S. Pereira , Ricardo G. Rodrigues , Sandro D.P. Vitenti , Vitoria M. Gomes This is my paper · ORCID
classification astro-ph.CO
keywords Cosmicflows-4expansion-ratefluctuationbulkflowmultipolevectorsincompleteskyΛCDMpeculiarvelocitiescosmologicalprincipledipoleanisotropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reconstructs the dipole, quadrupole, and octupole of the local expansion-rate fluctuation field from Cosmicflows-4 galaxies and supernovae in the redshift window 0.01–0.05. Incomplete sky coverage is handled with pixel masks; full-sky multipoles are recovered by two independent methods and checked with simulations. The quadrupole and octupole amplitudes sit inside the 95 percent envelope expected from linear and mildly nonlinear peculiar velocities in a standard ΛCDM universe. The dipole amplitude does not: it is 3.3σ high, points toward galactic (l, b) ≈ (290°, −4°), and is driven mainly by the higher-redshift half of the sample. Using the full set of multipole vectors rather than only the peaks of each multipole, the authors find no statistically significant alignments among the three multipoles. The result therefore isolates a bulk-flow-like excess confined to the dipole and to a specific redshift slice, while showing that the rest of the low-multipole structure is consistent with ordinary gravitational flows.

What carries the argument

The expansion-rate fluctuation field η ≡ log(z/d_L) − M(z), expanded both in spherical harmonics and in symmetric trace-free tensors whose multipole vectors give every direction associated with each multipole; full-sky coefficients are recovered from a pixel mask via a multipole-coupling-kernel pseudo-inverse and by maximum-likelihood estimation.

What would settle it

An independent distance catalog covering the same sky and z ∈ [0.03, 0.05], analyzed with the identical mask and reconstruction pipeline, that returns a dipole amplitude consistent with the ΛCDM prediction within 1σ.

Watch

Extended reading notes

Core claim

From masked Cosmicflows-4 data the reconstructed dipole power of the expansion-rate fluctuation field is inconsistent with ΛCDM linear-plus-mild-nonlinear predictions at 3.3σ and points at (l, b) = (290°, −4°) ± 5°; the excess is sourced predominantly by objects in z ∈ [0.03, 0.05]. The quadrupole and octupole amplitudes remain consistent with the same predictions at 95 percent confidence, and the complete multipole-vector structure shows no evidence of alignments.

Load-bearing premise

Nonlinear velocities can be treated as an uncorrelated 300 km/s Gaussian dispersion added to distance errors, and residual selection biases in the higher-redshift bin do not invent the dipole excess.

Editorial extensions

If this is right

  • The bulk-flow signal reported in Cosmicflows-4 is concentrated in the dipole and in the outer half of the 0.01–0.05 window rather than being a broadband multipolar anomaly.
  • Higher multipoles of the local expansion rate can be treated as consistent with standard linear theory once incomplete-sky coupling is inverted.
  • Maxima of successive multipoles can appear aligned even when the full multipole-vector sets show no statistically significant alignment.
  • Tomographic splits of future distance catalogs will be needed to decide whether the excess is residual systematics or a genuine large-scale flow.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the excess survives independent tracers, the preferred direction of the local bulk flow becomes a concrete target for Ellis–Baldwin-style tests against the CMB kinematic dipole.
  • The same mask-plus-kernel pipeline can be applied to other low-redshift distance indicators to test whether the 3.3σ tension is catalog-specific.
  • Absence of multipole-vector alignments weakens claims that the local expansion-rate field shares the same preferred axes sometimes discussed for CMB low multipoles.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper reconstructs the multipoles ℓ=1,2,3 of the expansion-rate fluctuation field η (Eq. 2) from Cosmicflows-4 galaxies in z∈[0.01,0.05] (CMB frame). It employs both spherical-harmonic coefficients and an equivalent symmetric-trace-free tensor basis that yields one amplitude plus ℓ multipole vectors per multipole. Incomplete sky coverage is handled by three pixel masks (Nside=4,8,16) that excise poorly sampled pixels and a galactic strip; the full-sky multipoles are recovered by a Moore–Penrose pseudo-inverse of the coupling kernel (Eq. 11) and by an independent MCMC maximum-likelihood fit. The reconstructed C1 is found to exceed the linear-theory ΛCDM prediction (Eq. 29, Appendix A) at 3.3σ with direction (l,b)=(290°,-4°)±5°, the excess arising mainly from the sub-bin z∈[0.03,0.05]; C2 and C3 remain consistent with the same prediction at 95% CL. Alignment statistics Sℓ,ℓ' and Tℓ,ℓ' built from the multipole vectors show no significant alignments.

Significance. If the 3.3σ dipole excess survives a more complete treatment of selection systematics, the result supplies an independent, low-redshift probe of a bulk-flow anomaly that is already reported in the literature and that may bear on the Hubble tension and tests of the Cosmological Principle. Methodological strengths that strengthen the claim include (i) two fully independent reconstruction pipelines that agree across three mask resolutions, (ii) an explicit linear-theory derivation of Cℓ with no free amplitude fitted to the data, (iii) 10^5 ΛCDM mocks that incorporate both cosmic variance and the catalog’s distance-modulus errors, and (iv) the use of the complete multipole-vector structure rather than only the locations of multipole maxima. These elements make the analysis a useful advance over earlier CF3/CF4 studies that reported only power spectra or single preferred directions.

major comments (2)
  1. §V.B and footnote 6 (also Conclusions): the 3.3σ tension on C1 is driven almost entirely by the higher-redshift sub-bin z∈[0.03,0.05]. The mocks of §IV.B keep observed angular positions fixed, assign each galaxy the mean pixel redshift as its “cosmic” redshift, add a linear velocity drawn from Eq. 29 plus an uncorrelated 300 km s⁻¹ Gaussian, and add only Gaussian noise to the distance modulus. They therefore omit the distance-dependent selection function of CF4 (TF/FP magnitude limits, spectroscopic completeness, Zone-of-Avoidance cuts). Because Malmquist bias couples distance errors to that selection function and grows with redshift, residual bias can preferentially inflate the reconstructed dipole precisely in the bin that sources the claimed excess. The authors correctly flag the effect as unmodeled; without a controlled injection of a realistic selection function into the same pipeli
  2. §II.A Eq. (7) and §IV.B step 3: nonlinear peculiar velocities are modeled solely as an uncorrelated Gaussian dispersion σz=300 km s⁻¹ added in quadrature to the distance-modulus error. While the authors note that raising σz to 500 km s⁻¹ does not change the dipole intensity, the assumption that residual non-linear correlations average to zero inside the large pixels remains untested. A modest coherent non-linear flow on the scale of the Nside=8 pixels could systematically shift the reconstructed C1; at minimum the paper should quantify the residual bias by injecting a non-linear velocity field (e.g., from a constrained N-body realization) into the same mock pipeline.
minor comments (4)
  1. Table I: the MCMC and pseudo-inverse Cℓ values agree well, but the asymmetric MCMC errors are quoted without stating whether they are highest-density or equal-tailed intervals; a brief clarification would help the reader.
  2. Fig. 6 and Table II: the multipole-vector error bars are Fréchet standard deviations that ignore the known non-independence of vectors belonging to the same multipole (and residual multipole–multipole correlations induced by the mask). The text already notes this limitation; adding a short quantitative estimate of the neglected covariance (even from the existing mocks) would strengthen the alignment tests of Table III.
  3. §IV.A: the three mask criteria (minimum objects per pixel, 10° galactic strip, iterative neighbor masking) are reasonable but somewhat ad hoc. A short sensitivity test showing that modest changes in these thresholds leave C1 and the dipole direction stable would be reassuring.
  4. Appendix A: the geometric-mean approximation for the unequal-time power spectrum and the top-hat window are standard, yet a one-sentence check that replacing the top-hat by the actual redshift histogram of Fig. 3 changes Cℓ by ≲ few percent would close a minor loophole.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: C_ℓ theory is independent linear-theory derivation; data comparison and mocks do not reduce to fitted inputs or self-definition.

full rationale

The load-bearing claim (dipole C1 inconsistent with ΛCDM at 3.3σ, sourced mainly by z∈[0.03,0.05]) rests on (i) the pixelized estimator of η from CF4 distance moduli (Eqs. 5–8), (ii) mask-deconvolution via pseudo-inverse or MCMC of the coupling kernel (Eqs. 11–12, 24–28), and (iii) comparison of the resulting C_ℓ to the independent theoretical spectrum derived in Appendix A from the continuity equation, linear Pm(k) via CLASS defaults, and a top-hat redshift window (Eq. 29). No free amplitude is fitted to the CF4 multipoles and then re-used as a “prediction.” The STF/multipole-vector reparametrization (Eqs. 13–18) and PolyMV code are a known, frame-independent rewriting of the same 2ℓ+1 degrees of freedom already present in the η_ℓm; they are used only for directional diagnostics and alignment statistics that return null results, not for the amplitude tension. Prior η-field papers cited for the observable definition are by non-overlapping authors. Self-citations of the authors’ earlier multipole-vector work supply a convenience tool, not a uniqueness theorem or ansatz that forces the 3.3σ result. The mock pipeline that sets the significance injects the theoretical C_ℓ plus uncorrelated 300 km s⁻¹ noise; it does not circularly re-inject the measured multipoles. The derivation chain is therefore self-contained against an external benchmark.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central 3.3σ claim rests on standard linear-theory velocity–density relations, a public catalog, and a small set of analysis choices (mask thresholds, nonlinear dispersion). No new physical entities are postulated; free parameters are conventional analysis knobs rather than fitted cosmological amplitudes.

free parameters (3)
  • nonlinear velocity dispersion σ_z = 300 km s⁻¹
    Fixed at 300 km s⁻¹ (tested up to 500 km s⁻¹) and added in quadrature to distance errors; directly enters the mock catalogs that set the significance of C_1.
  • minimum objects per pixel = 14/5/3
    Hand-chosen thresholds (14, 5, 3 for N_side = 4, 8, 16) that define which pixels are masked; alters sky fraction and therefore the reconstructed multipoles.
  • galactic-plane strip width = 10°
    Fixed 10° equatorial strip removed to avoid the Zone of Avoidance; choice affects mask geometry and coupling kernel.
assumptions (3)
  • domain assumption Line-of-sight peculiar velocity is related to the density contrast by the linear continuity equation on sub-horizon scales.
    Used to derive the theoretical C_ℓ in Appendix A; standard but not exact once mild nonlinearities are present.
  • domain assumption Distance-modulus errors are Gaussian and uncorrelated with the nonlinear velocity component.
    Enters the pixel estimator (Eq. 6–7) and the Monte-Carlo error model; stated without independent verification for the CF4 subsample.
  • ad hoc to paper The multipole-coupling kernel of a binary mask can be inverted by a Moore–Penrose pseudo-inverse or by a Gaussian likelihood up to ℓ_max = 3.
    Core reconstruction step (§IV.A); validated only by the authors’ own simulations, not by an external theorem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multipolar structure of the local expansion rate from incomplete sky data." pith.science (2026). https://pith.science/paper/6HBVVDTO

@misc{pith2026260707821,
  author       = {Pith},
  title        = {Pith review of: Multipolar structure of the local expansion rate from incomplete sky data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HBVVDTO}},
  note         = {Machine review of arXiv:2607.07821}
}
abstract

Using the Cosmicflows-4 data, we reconstruct the first multipolar moments of a general function describing the local expansion rate. In addition to the conventional harmonic approach, we employ a basis of symmetric and trace-free tensors to characterize the anisotropies of the expansion rate, allowing us to identify all directions associated with each of its multipoles. Focusing on objects in $z\in[0.01,0.05]$ in the CMB rest frame, we derive all $2\ell+1$ degrees of freedom in the multipoles $\ell=1,2$ and 3, which are split into one amplitude and $\ell$ unit vectors per multipole. To mitigate anisotropies induced by incomplete sky coverage, we introduce a pixel-based mask that removes poorly sampled pixels. The full-sky expansion rate is reconstructed using two independent approaches: a pseudo-inverse of the multipole-coupling kernel induced by the mask, and a maximum-likelihood estimate of the underlying full-sky field. These approaches are validated through simulations that explore different mask resolutions, cosmic variance and statistical noise. We find that the quadrupole and octupole amplitudes are consistent (at $95\%$ C.L.) with the expectations of a $\Lambda$CDM universe with linear and mild nonlinear perturbations, where the anisotropies of the expansion rate result from small peculiar velocities. The dipole amplitude, however, is inconsistent with $\Lambda$CDM predictions at 3.3$\sigma$, with a direction $(l, b) = (290^\circ, -4^\circ) \pm 5^\circ$ consistent with a bulk flow. This signal comes predominantly from sources in $z\in[0.03,0.05]$. Finally, we conduct alignment tests between the dipole, quadrupole, and octupole vectors. We confirm recent findings showing that the maxima of these multipoles are approximately located at $(290^\circ,-4^\circ)$. However, detailed tests using the complete vector structure of these multipoles reveal no evidence of alignments.

Figures

Figures reproduced from arXiv: 2607.07821 by the authors.

Figure 1
Figure 1. FIG. 1. Absolute value of the multipole-coupling kernel [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Spatial distribution of all 55,877 objects in the CF4 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Redshift distributions of sources in the range [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Masks adopted in this work, for different sky resolutions. For [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Fluctuation field [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Reconstructed dipole (dots), quadrupole (diamonds) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between the theoretical (green line) and measured (red dots) angular power spectrum using the pseudo [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

115 extracted references · 115 canonical work pages

  1. [1]

    Multipolar structure of the local expansion rate from incomplete sky data

    for a recent review). Indications of large-angle statis- tical anomalies in the CMB date back to COBE [23], and a plethora of explanations based on deviations from the CP have been proposed—see Refs. [11–13, 24] for com- prehensive reviews. Although they have been reported with lower statistical significances in comparison to the Hubble tension, they are ...

    work page Pith review arXiv 2026

  2. [2]

    and CF4 [69, 70] data. However, a reconstruction of all the angular degrees of freedom of each multipole, allowing, in particular, for a thorough assessment of pos- sible alignments between them, has not been carried out so far. We introduce two important methodological improve- ments over existing analyses. The first of them relies on the choice of a com...

    work page

  3. [3]

    provides the ideal catalog for this task. It consists of the largest compiled catalog of galaxy distances and peculiar velocities to date, and has been used to map the dynamics of the local universe [85], as well as to study its large scale structure [86]. CF4 contains 55,877 individ- ual galaxy distances, measured from multiple and inde- pendent methods,...

    work page

  4. [4]

    This method is very efficient since, for a given mask, the pseudo-inverse matrix needs to be computed only once

    Pseudo-inverse A possible solution to the problem above is to con- struct a pseudo-inverse matrix for the kernelK ℓmℓ′m′. This method is very efficient since, for a given mask, the pseudo-inverse matrix needs to be computed only once. LetK + ℓmℓ′m′ be the pseudo-inverse ofK ℓ′m′ℓ′′m′′, such that the absolute value of the difference X ℓ′,m′ K+ ℓmℓ′m′Kℓ′m′ℓ...

    work page

  5. [5]

    MCMC To check the robustness of our results, we perform an alternative inference to the pseudo-inverse method. Since pseudo-inverse matrices offer a least-squares solution to an overdetermined system of equations, we extend this approach to include the statistical errors in the inversion procedure. That is, we introduce theχ 2 statistic: χ2 = X ℓm X ℓ′m′ ...

    work page

  6. [6]

    cos- mological

    For each galaxyiin a pixelp, we define its “cos- mological” redshift to be the mean redshift of all galaxies in pixelp

    work page

  7. [7]

    (29), we usehealpyto generate a new realization of the mapη(ˆn)

    From Eq. (29), we usehealpyto generate a new realization of the mapη(ˆn). For this step, it is enough to work withℓ max = 6, since the theoret- ical power spectrum falls steeply with increasing ℓ, as can be easily checked. From this map we compute the peculiar velocity through Eq. (4), us- ing the CF4 angular positions and the cosmological redshifts compu...

    work page

  8. [8]

    As discussed above Eq. (7), to this linear velocity we add a non-linear velocity component drawn from a normal distributionN(0, σ 2 z), whereσ z is a 1D ve- locity dispersion usually taken to be 300 km s −1/c. To check the impact of this step, we have increased σz up to 500 km s −1/c, finding that it does not al- ter the dipole intensity but slightly incr...

    work page

Show all 115 references
  1. [9]

    To each computed distance, we add Gaussian noise using the reported uncertainties onµfrom the CF4 catalog

    Using CF4 redshifts, we compute the distance mod- ulus predicted by the ΛCDM model for each galaxy. To each computed distance, we add Gaussian noise using the reported uncertainties onµfrom the CF4 catalog. These steps effectively produce a new ‘catalog’. In to- tal, we have g...

  2. [10]

    area vectors

    The dashed gray circle defines the equator having the dipole vectorv (1,1) as thez-axis. The points and dashed circles in the associated southern hemisphere correspond to the antipodes of each direction. with the theoretical predictions. In order to isolate the source of this ...

    2024

  3. [11]

    Hinshawet al., The Astrophysical Journal Supple- ment Series208, 19 (2013)

    G. Hinshawet al., The Astrophysical Journal Supple- ment Series208, 19 (2013)

    2013

  4. [12]

    Komatsuet al., The Astrophysical Journal Supple- ment Series192, 18 (2011)

    E. Komatsuet al., The Astrophysical Journal Supple- ment Series192, 18 (2011)

    2011

  5. [13]

    Aghanimet al., Astron

    N. Aghanimet al., Astron. Astrophys641, A6 (2020)

    2020

  6. [14]

    Akramiet al., arXiv preprint arXiv:1807.06205 (2018)

    Y. Akramiet al., arXiv preprint arXiv:1807.06205 (2018)

    2018 arXiv

  7. [15]

    D. G. Yorket al., The Astronomical Journal120, 1579 (2000)

    2000

  8. [16]

    D. H. Joneset al., Monthly Notices of the Royal Astro- nomical Society399, 683 (2009)

    2009

  9. [17]

    Giovanelliet al., The astronomical journal130, 2598 (2005)

    R. Giovanelliet al., The astronomical journal130, 2598 (2005)

    2005

  10. [18]

    Zhanget al., Science China Physics, Mechanics & Astronomy67, 219511 (2024)

    C.-P. Zhanget al., Science China Physics, Mechanics & Astronomy67, 219511 (2024)

    2024

  11. [19]

    Saidet al., Monthly Notices of the Royal Astronom- ical Society539, 3627 (2025)

    K. Saidet al., Monthly Notices of the Royal Astronom- ical Society539, 3627 (2025)

    2025

  12. [20]

    P. J. E. Peebles, Annals of Physics447, 169159 (2022)

    2022

  13. [21]

    Abdallaet al., JHEAp34, 49 (2022), [2203.06142]

    E. Abdallaet al., JHEAp34, 49 (2022), [2203.06142]

    2022 arXiv

  14. [22]

    P. K. Aluriet al., Class. Quant. Grav.40, 094001 (2023), [2207.05765]

    2023 arXiv

  15. [23]

    Di Valentinoet al., Phys

    CosmoVerse Network, E. Di Valentinoet al., Phys. Dark Univ.49, 101965 (2025), [2504.01669]

    2025 arXiv

  16. [24]

    A. G. Riesset al., The Astrophysical journal letters 934, L7 (2022)

    2022

  17. [25]

    Breuvalet al., The Astrophysical Journal973, 30 (2024)

    L. Breuvalet al., The Astrophysical Journal973, 30 (2024)

    2024

  18. [26]

    Krishnan, R

    C. Krishnan, R. Mohayaee, E. ´O. Colg´ ain, M. M. Sheikh-Jabbari and L. Yin, Class. Quant. Grav.38, 184001 (2021), [2105.09790]

    2021 arXiv

  19. [27]

    Luongo, M

    O. Luongo, M. Muccino, E. ´O. Colg´ ain, M. M. Sheikh- Jabbari and L. Yin, Phys. Rev. D105, 103510 (2022), [2108.13228]

    2022 arXiv

  20. [28]

    Krishnan, R

    C. Krishnan, R. Mohayaee, E. ´O. Colg´ ain, M. M. Sheikh-Jabbari and L. Yin, Phys. Rev. D105, 063514 (2022), [2106.02532]

    2022 arXiv

  21. [29]

    Camarena, V

    D. Camarena, V. Marra, Z. Sakr and C. Clarkson, Class. Quant. Grav.39, 184001 (2022), [2205.05422]

    2022 arXiv

  22. [30]

    Mc Conville and E

    R. Mc Conville and E. ´O. Colg´ ain, Phys. Rev. D108, 123533 (2023), [2304.02718]

    2023 arXiv

  23. [31]

    Jia, J.-L

    J.-Y. Jia, J.-L. Niu, D.-C. Qiang and H. Wei, Phys. Rev. D112, 043507 (2025), [2504.13380]

    2025 arXiv

  24. [32]

    Marra, Tensions in Cosmology: Interpreting Them Through Inhomogeneous Models, 2026, [2601.22278]

    V. Marra, Tensions in Cosmology: Interpreting Them Through Inhomogeneous Models, 2026, [2601.22278]

    2026

  25. [33]

    Hinshawet al., Astrophys

    G. Hinshawet al., Astrophys. J. Lett.464, L25 (1996), [astro-ph/9601061]

    1996 arXiv

  26. [34]

    Bullet al., Phys

    P. Bullet al., Phys. Dark Univ.12, 56 (2016), [1512.05356]

    2016 arXiv

  27. [35]

    Pitrou, A

    C. Pitrou, A. Coc, J.-P. Uzan and E. Vangioni, Phys. Rept.754, 1 (2018), [1801.08023]

    2018 arXiv

  28. [36]

    Barrow, Monthly Notices of the Royal Astronomical Society175, 359 (1976)

    J. Barrow, Monthly Notices of the Royal Astronomical Society175, 359 (1976)

    1976

  29. [37]

    J. D. Barrow and R. A. Matzner, Monthly Notices of the Royal Astronomical Society181, 719 (1977)

    1977

  30. [38]

    K. S. Thorne, Astrophys. J.148, 51 (1967)

    1967

  31. [39]

    G. F. R. Ellis and M. A. H. MacCallum, Commun. Math. Phys.12, 108 (1969)

    1969

  32. [40]

    Saunders, Monthly Notices of the Royal Astronomi- cal Society142, 213 (1969)

    P. Saunders, Monthly Notices of the Royal Astronomi- cal Society142, 213 (1969)

    1969

  33. [41]

    Fleury, C

    P. Fleury, C. Pitrou and J.-P. Uzan, Phys. Rev. D91, 043511 (2015), [1410.8473]

    2015 arXiv

  34. [42]

    T. S. Pereira, C. Pitrou and J.-P. Uzan, JCAP09, 006 (2007), [0707.0736]

    2007 arXiv

  35. [43]

    A. E. Gumrukcuoglu, C. R. Contaldi and M. Peloso, JCAP11, 005 (2007), [0707.4179]

    2007 arXiv

  36. [44]

    T. S. Pereira, G. A. M. Marug´ an and S. Carneiro, JCAP 14 07, 029 (2015), [1505.00794]

    2015 arXiv

  37. [45]

    F. O. Franco and T. S. Pereira, JCAP11, 022 (2017), [1709.00007]

    2017 arXiv

  38. [46]

    Saadeh, S

    D. Saadeh, S. M. Feeney, A. Pontzen, H. V. Peiris and J. D. McEwen, Physical review letters117, 131302 (2016)

    2016

  39. [47]

    R. R. Caldwell and A. Stebbins, Phys. Rev. Lett.100, 191302 (2008), [0711.3459]

    2008 arXiv

  40. [48]

    Zhang and A

    P. Zhang and A. Stebbins, Phys. Rev. Lett.107, 041301 (2011), [1009.3967]

    2011 arXiv

  41. [49]

    Kalus, D

    B. Kalus, D. J. Schwarz, M. Seikel and A. Wiegand, Astronomy & Astrophysics553, A56 (2013)

    2013

  42. [50]

    C. A. P. Bengaly, A. Bernui and J. S. Alcaniz, Astro- phys. J.808, 39 (2015), [1503.01413]

    2015 arXiv

  43. [51]

    Andrade, C

    U. Andrade, C. A. P. Bengaly, B. Santos and J. S. Al- caniz, Astrophys. J.865, 119 (2018), [1806.06990]

    2018 arXiv

  44. [52]

    Colin, R

    J. Colin, R. Mohayaee, M. Rameez and S. Sarkar, As- tronomy & Astrophysics631, L13 (2019)

    2019

  45. [53]

    Soltis, A

    J. Soltis, A. Farahi, D. Huterer and C. M. Liberato, Physical Review Letters122, 091301 (2019)

    2019

  46. [54]

    Rahman, R

    W. Rahman, R. Trotta, S. Boruah, M. Hudson and D. Van Dyk, Monthly Notices of the Royal Astronomi- cal Society514, 139 (2022)

    2022

  47. [55]

    Sorrenti, R

    F. Sorrenti, R. Durrer and M. Kunz, Journal of Cos- mology and Astroparticle Physics2023, 054 (2023)

    2023

  48. [56]

    Sorrenti, R

    F. Sorrenti, R. Durrer and M. Kunz, Journal of Cos- mology and Astroparticle Physics2025, 013 (2025)

    2025

  49. [57]

    A. Sah, M. Rameez, S. Sarkar and C. G. Tsagas, The European Physical Journal C85, 596 (2025)

    2025

  50. [58]

    C. A. P. Bengaly, A. Bernui, J. S. Alcaniz and I. S. Ferreira, Mon. Not. Roy. Astron. Soc.466, 2799 (2017), [1511.09414]

    2017 arXiv

  51. [59]

    C. A. P. Bengaly, A. Bernui, J. S. Alcaniz, H. S. Xavier and C. P. Novaes, Mon. Not. Roy. Astron. Soc.464, 768 (2017), [1606.06751]

    2017 arXiv

  52. [60]

    C. A. P. Bengalyet al., Mon. Not. Roy. Astron. Soc. 475, L106 (2018), [1707.08091]

    2018 arXiv

  53. [61]

    Rameez, R

    M. Rameez, R. Mohayaee, S. Sarkar and J. Colin, Mon. Not. Roy. Astron. Soc.477, 1772 (2018), [1712.03444]

    2018 arXiv

  54. [62]

    N. J. Secrestet al., The Astrophysical Journal Letters 908, L51 (2021)

    2021

  55. [63]

    N. J. Secrest, S. von Hausegger, M. Rameez, R. Mo- hayaee and S. Sarkar, The Astrophysical journal letters 937, L31 (2022)

    2022

  56. [64]

    L. Dam, G. F. Lewis and B. J. Brewer, Monthly Notices of the Royal Astronomical Society525, 231 (2023)

    2023

  57. [65]

    Migkaset al., Astron

    K. Migkaset al., Astron. Astrophys.636, A15 (2020), [2004.03305]

    2020 arXiv

  58. [66]

    T. S. Pereira, C. Pitrou and J.-P. Uzan, Astron. Astro- phys.585, L3 (2016), [1503.01127]

    2016 arXiv

  59. [67]

    J. Adam, R. Maartens, J. Larena and C. Clarkson, JCAP02, 016 (2025), [2411.08560]

    2025 arXiv

  60. [68]

    J. Adam, R. Maartens, J. Larena, C. Clarkson and R. Durrer, JCAP01, 023 (2026), [2509.07259]

    2026

  61. [69]

    C. G. Tsagas, L. Perivolaropoulos and K. Asvesta, 2510.05340

  62. [70]

    Watkins, H

    R. Watkins, H. A. Feldman and M. J. Hudson, Mon. Not. Roy. Astron. Soc.392, 743 (2009), [0809.4041]

    2009 arXiv

  63. [71]

    Nusser and M

    A. Nusser and M. Davis, Astrophys. J.736, 93 (2011), [1101.1650]

    2011 arXiv

  64. [72]

    Planck, P. A. R. Adeet al., Astron. Astrophys.561, A97 (2014), [1303.5090]

    2014 arXiv

  65. [73]

    Carrick, S

    J. Carrick, S. J. Turnbull, G. Lavaux and M. J. Hud- son, Mon. Not. Roy. Astron. Soc.450, 317 (2015), [1504.04627]

    2015 arXiv

  66. [74]

    G. F. R. Ellis and J. E. Baldwin, Mon. Not. Roy. Astron. Soc.206, 377 (1984)

    1984

  67. [75]

    Secrest, S

    N. Secrest, S. von Hausegger, M. Rameez, R. Mo- hayaee and S. Sarkar, Nature Rev. Phys.7, 68 (2025), [2501.06450]

    2025 arXiv

  68. [76]

    R. B. Tullyet al., Astrophys. J.944, 94 (2023), [2209.11238]

    2023 arXiv

  69. [77]

    R. C. Kraan-Korteweg and O. Lahav, Astron. Astro- phys. Rev.10, 211 (2000), [astro-ph/0005501]

    2000 arXiv

  70. [78]

    Kalbouneh, C

    B. Kalbouneh, C. Marinoni and J. Bel, Phys. Rev. D 107, 023507 (2023), [2210.11333]

    2023 arXiv

  71. [79]

    Kalbouneh, C

    B. Kalbouneh, C. Marinoni and R. Maartens, JCAP 09, 069 (2024), [2401.12291]

    2024 arXiv

  72. [80]

    Salzano, J

    V. Salzano, J. Beltr´ an Jim´ enez, D. Bettoni, P. Brax and A. Valade, 2512.02526

    arXiv

  73. [81]

    K. S. Thorne, Rev. Mod. Phys.52, 299 (1980)

    1980

  74. [82]

    C. J. Copi, D. Huterer and G. D. Starkman, Phys. Rev. D70, 043515 (2004), [astro-ph/0310511]

    2004 arXiv

  75. [83]

    R. A. Oliveira, T. S. Pereira and M. Quartin, Phys. Dark Univ.30, 100608 (2020), [1812.02654]

    2020 arXiv

  76. [84]

    Sarkar, H

    D. Sarkar, H. A. Feldman and R. Watkins, Monthly No- tices of the Royal Astronomical Society375, 691 (2007)

    2007

  77. [85]

    M. I. Scrimgeouret al., Monthly Notices of the Royal Astronomical Society455, 386 (2016)

    2016

  78. [86]

    Zoncaet al., Journal of Open Source Software4, 1298 (2019)

    A. Zoncaet al., Journal of Open Source Software4, 1298 (2019)

    2019

  79. [87]

    Hivonet al., The Astrophysical Journal567, 2 (2002)

    E. Hivonet al., The Astrophysical Journal567, 2 (2002)

    2002

  80. [88]

    A. R. Edmonds,Angular Momentum in Quantum Me- chanics(Princeton University Press, Princeton, 1996)

    1996

  81. [89]

    R. G. Rodrigues, T. S. Pereira and M. Quartin, JCAP 06, 039 (2025), [2411.08087]

    2025 arXiv

  82. [90]

    Maxwell,A Treatise on Electricity and Magnetism, A Treatise on Electricity and Magnetism No

    J. Maxwell,A Treatise on Electricity and Magnetism, A Treatise on Electricity and Magnetism No. v. 1 (Claren- don Press, 1873)

  83. [91]

    L. R. Abramo, A. Bernui and T. S. Pereira, JCAP12, 013 (2009), [0909.5395]

    2009 arXiv

  84. [92]

    Pinkwart and D

    M. Pinkwart and D. J. Schwarz, Phys. Rev. D98, 083536 (2018), [1803.07473]

    2018 arXiv

  85. [93]

    R. A. Oliveira, T. S. Pereira and M. Quartin, Physics of the Dark Universe30, 100608 (2020)

    2020

  86. [94]

    Fr´ echet, Annales de l’Institut Henri Poincar´ e10, 215 (1948)

    M. Fr´ echet, Annales de l’Institut Henri Poincar´ e10, 215 (1948)

    1948

  87. [95]

    Valadeet al., Nature Astronomy8, 1610 (2024)

    A. Valadeet al., Nature Astronomy8, 1610 (2024)

    2024

  88. [96]

    Courtoiset al., Astronomy & Astrophysics670, L15 (2023)

    H. Courtoiset al., Astronomy & Astrophysics670, L15 (2023)

    2023

  89. [97]

    Howlettet al., Monthly Notices of the Royal Astro- nomical Society515, 953 (2022)

    C. Howlettet al., Monthly Notices of the Royal Astro- nomical Society515, 953 (2022)

    2022

  90. [98]

    Kourkchiet al., The Astrophysical Journal902, 145 (2020)

    E. Kourkchiet al., The Astrophysical Journal902, 145 (2020)

    2020

  91. [99]

    C. M. Springobet al., Monthly Notices of the Royal Astronomical Society445, 2677 (2014)

    2014

  92. [100]

    Hartlap, P

    J. Hartlap, P. Simon and P. Schneider, Astron. Astro- phys.464, 399 (2007), [astro-ph/0608064]

    2007 arXiv

  93. [101]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Good- man, Publications of the Astronomical Society of the Pacific125, 306–312 (2013)

    2013

  94. [102]

    D. Blas, J. Lesgourgues and T. Tram, Journal of Cosmology and Astroparticle Physics2011, 034–034 (2011)

    2011

  95. [103]

    Watkinset al., Mon

    R. Watkinset al., Mon. Not. Roy. Astron. Soc.524, 1885 (2023), [2302.02028]

    2023 arXiv

  96. [104]

    Hoffmanet al., Monthly Notices of the Royal Astro- 15 nomical Society527, 3788 (2024)

    Y. Hoffmanet al., Monthly Notices of the Royal Astro- 15 nomical Society527, 3788 (2024)

    2024

  97. [105]

    Kalbounehet al., arXiv preprint arXiv:2510.02510 (2025)

    B. Kalbounehet al., arXiv preprint arXiv:2510.02510 (2025)

    2025

  98. [106]

    Pedregosaet al., Journal of Machine Learning Re- search12, 2825 (2011)

    F. Pedregosaet al., Journal of Machine Learning Re- search12, 2825 (2011)

    2011

  99. [107]

    M. R. Dennis, J. Phys. A38, 1653 (2005), [math- ph/0410004]

    2005

  100. [108]

    M. R. Dennis and K. Land, Mon. Not. Roy. Astron. Soc.383, 424 (2008), [0704.3657]

    2008 arXiv

  101. [109]

    A. M. Whitford, C. Howlett and T. M. Davis, Monthly Notices of the Royal Astronomical Society526, 3051 (2023)

    2023

  102. [110]

    O. A. Martinet al., Journal of Open Source Software 11, 9889 (2026)

    2026

  103. [111]

    L. R. Abramo, A. Bernui, I. S. Ferreira, T. Villela and C. A. Wuensche, Physical Review D—Particles, Fields, Gravitation, and Cosmology74, 063506 (2006)

    2006

  104. [112]

    C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Stark- man, Monthly Notices of the Royal Astronomical Soci- ety367, 79 (2006)

    2006

  105. [113]

    C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Stark- man, Monthly Notices of the Royal Astronomical Soci- ety449, 3458 (2015)

    2015

  106. [114]

    de Oliveira-Costa, M

    A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga and A. Hamilton, Phys. Rev. D69, 063516 (2004), [astro- ph/0307282]

    2004

  107. [115]

    T. D. Kitching and A. F. Heavens, Phys. Rev. D95, 063522 (2017), [1612.00770]

    2017 arXiv

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.