Testing local position invariance with odd multipoles of galaxy clustering statistics

Atsushi Taruya; Shohei Saga; Takuya Inoue; Teppei Okumura

REVIEW 3 major objections 3 minor 1 cited by

Testing local position invariance with odd multipoles of galaxy clustering statistics

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Forecast: adding the octupole moment of the galaxy cross-correlation function to the dipole improves expected constraints on the LPI-violating parameter $\alpha$ by 11% for conservative scale cuts ($s_{\rm min}=15\,{\rm Mpc}/h$) and by 6%…

desk verdict Clean, honest Fisher forecast extending the LPI dipole analysis to the octupole; the 11% gain is real but rests on an octupole model the authors themselves have not yet validated with simulations. read the letter →

arxiv 2412.13701 v2 pith:GFWIZTS6 submitted 2024-12-18 astro-ph.CO gr-qc

Takuya Inoue , Teppei Okumura , Shohei Saga , Atsushi Taruya This is my paper · ORCID

classification astro-ph.COgr-qc

keywords localpositioninvariancegravitationalredshiftgalaxycross-correlationfunctionoctupolemomentdipoleFisherforecastEinsteinequivalenceprincipleredshift-spacedistortions

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

The paper argues that a higher-order statistical pattern in how two different galaxy populations are distributed, the octupole moment, carries extra information about a possible violation of local position invariance, the idea that non-gravitational physics is the same everywhere. It forecasts how well upcoming surveys would measure the violation parameter $\alpha$, which scales deviations of the gravitational redshift from general relativity. The central result is that adding the octupole to the dipole tightens the expected measurement by about 11 percent when only larger separations are used, and by 6 percent when small separations are included. Because this improvement requires no additional data, it makes higher-order multipoles worth including in future equivalence-principle tests.

What carries the argument

The central object is the asymmetric part of the redshift-space cross-correlation function $\xi^{XY}_{\rm asym}$, expanded in odd Legendre multipoles; the dipole ($\ell=1$) and octupole ($\ell=3$) are the two terms used. Its model, Eq. (5), combines a plane-parallel piece from gravitational redshift, generated by a nonlinear halo potential, with standard Doppler terms and a leading $(s/d)$ wide-angle correction. The argument is carried by a Fisher matrix over five parameters, namely $\alpha$, the off-centering radius $R_{\rm off}$, and the two galaxy biases, with an analytic Gaussian covariance and survey-overlap volumes. The octupole's key property is that it lacks the $b_X b_Y$ term that dominates the dipole's real-space contribution, giving it a complementary bias dependence and therefore additional constraining power.

What would settle it

Use two galaxy populations in an N-body simulation with realistic halo potentials, measure the octupole moment of their cross-correlation between 5 and 30 Mpc/h, and compare it with Eq. (5); if the measured octupole differs from the model by more than the forecast's 1-sigma band, the predicted 11% improvement would not be realized on those scales.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the octupole moment of the cross-correlation function between two galaxy samples is a useful and independent probe of the LPI-violating parameter $\alpha$. This moment, which arises from the Doppler effect mixing with gravitational redshift from nonlinear halo potentials plus a leading wide-angle correction, has a different dependence on galaxy bias than the dipole does, so it breaks parameter degeneracies rather than merely repeating the dipole. In Fisher forecasts for surveys like DESI, Euclid, PFS, and SKA, adding the octupole to the dipole improves the expected $1\sigma$ constraint on $\alpha$ by an average of 6% for $s_{\rm min}=5\,{\rm Mpc}/h$ and 11% for $s_{\rm min}=15\,{\rm Mpc}/h$; combining all survey pairs yields $\sigma_\alpha \approx 0.028$ and $\approx 0.14$, respectively. The authors present this as the first use of the small-scale octupole in tests of local position invariance.

Load-bearing premise

The octupole's predicted signal, taken from an analytic model that the paper itself says has not yet been verified with a proper simulation setup, is accurate on the 5-30 Mpc/h scales used; if it is not, the forecasted improvement changes.

Editorial extensions

If this is right

Expected LPI constraints improve by about 11% for $s_{\rm min}=15\,{\rm Mpc}/h$ and 6% for $s_{\rm min}=5\,{\rm Mpc}/h$ when the octupole is added to the dipole, without requiring new observations.
Cross-correlations between low-bias galaxy samples, such as DESI-BGS with SKA1/2, benefit the most from the octupole.
Combining all considered survey pairs gives $\sigma_\alpha \approx 0.14$ for conservative scales and $\sigma_\alpha \approx 0.028$ for aggressive scales.
The octupole alone can constrain $\alpha$ below roughly 0.5 for some survey pairs when small scales are used.
Future equivalence-principle tests based on galaxy clustering should include odd multipoles beyond the dipole.

Reading between the lines

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

If N-body simulations confirm the octupole model, a natural next step is to include the $\ell=5$ triakontadipole, which the paper neglects, in the same Fisher forecasts.
The octupole's sign behavior, which unlike the dipole shows no sign flip, could be used as a consistency check to separate a genuine $\alpha$ signal from systematics.
Because the improvement is largest when small scales are excluded, the real gain depends on nonlinear modeling; better simulations could make aggressive scale cuts usable and yield improvements larger than the reported 6%.
The same Fisher setup can be converted into a likelihood for actual overlapping survey data, making the forecast directly testable with real measurements.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read this one if you care about equivalence-principle tests from galaxy clustering. It is a straightforward Fisher forecast that adds the octupole moment of the cross-correlation function to the dipole-only analysis of Saga et al. 2023. The new number is an 11% average improvement in the forecasted error on the LPI-violating parameter alpha when smin = 15 Mpc/h, and 6% at smin = 5 Mpc/h. That is the honest takeaway: a modest, not transformative, gain.

What the paper does well: the Fisher formalism is laid out cleanly, the covariance matrix is specified, and the nuisance parameters (bias, off-centering) are handled with priors taken from prior work. The dipole-only results reproduce Ref. [42], which is a good sanity check. The authors also present the scale-cut dependence honestly, distinguishing the aggressive smin = 5 Mpc/h from the conservative smin = 15 Mpc/h. They explicitly note that non-Gaussian and nonlinear contributions could affect the smaller-scale results. No overfitting, no magical thinking.

The soft spot is the one the authors themselves flag in the Conclusions: the octupole model, Eq. (5), has not been verified with N-body simulations. The dipole model has been checked; the octupole has not. Since the claimed improvement is only 11%, even a moderate 20-30% error in the octupole's alpha-response could shrink that gain below significance. This is a real limitation, and the stress-test note is right that the central numerical claim rests on an unvalidated ingredient. It is not a fatal flaw for a forecast, but it means the forecast should be read as conditional on the model.

The self-citation pattern is not a problem here: the model and survey setup come from the authors' prior work, but the dipole has been simulation-tested, and the forecast is a legitimate extrapolation. The paper is honest about what is new and what is not.

Who is this for? Specialists in relativistic effects in galaxy clustering who are planning LPI or gravitational-redshift measurements with DESI, PFS, Euclid, or SKA. They will find the numbers useful. General cosmologists can skip it.

Recommendation: send it to peer review. The octupole validation issue should be explicitly raised as a required caveat or a point to address in revision, but it is not grounds for rejection. The paper is a competent, clearly written forecast that knows its own limits.

Best, [You]

Referee Report

3 major / 3 minor

Summary. This Letter extends a dipole-only Fisher forecast of the LPI-violating parameter alpha in galaxy cross-correlations to include the octupole moment. Using the analytic asymmetric-correlation model of Eq. (5), a Gaussian, plane-parallel covariance (Eq. 8), and survey specifications for DESI, Euclid, PFS, and SKA1/2, the authors find that adding the octupole to the dipole improves the marginalized 1-sigma error on alpha by 6% for smin=5 Mpc/h and 11% for smin=15 Mpc/h on average. The combined constraints are sigma_alpha ~ 0.028 and 0.14 for the two scale cuts. The authors present the smin=15 Mpc/h result as the conservative case and explicitly state in the Conclusions that the octupole model has not yet been verified with N-body simulations.

Significance. If the forecast is robust, this is a useful, low-cost forecast: it shows that a higher-order odd multipole can tighten an equivalence-principle test without new data. The Fisher formalism is transparent, the survey setup is clearly described, and the authors are commendably explicit about the main limitation. The central idea is worth publishing, but the numerical size of the claimed improvement rests on an unvalidated octupole model, and the combined constraints assume independence between overlapping survey pairs. These issues are addressable but are load-bearing rather than cosmetic.

major comments (3)

[Eq. (5) and Conclusions] The central improvement claim rests on the octupole model in Eq. (5), which the authors state in the Conclusions has not been verified with N-body simulations ('a verification of octupole prediction requires a more proper simulation setup'). Because the octupole is a subdominant signal and its alpha-response is not tested against simulations, a moderate error in its amplitude or scale dependence could erase or even reverse the 11% improvement. I ask the authors to either provide a simulation-level validation of the octupole model on the scales used (5-30 Mpc/h), or to demonstrate quantitatively that the forecast is robust to plausible mismodeling (e.g., by perturbing the octupole alpha-response within a range allowed by off-centering, velocity dispersion, and nonlinear power-spectrum uncertainties). Without one of these, the quoted 6-11% improvements are conditional on an unvalidated ingredient.
[Eq. (8) and Results] The covariance matrix in Eq. (8) is derived under the Gaussian and plane-parallel approximations, while the signal model in Eq. (5) includes wide-angle corrections. For the smin=5 Mpc/h case, the included separations lie in a regime where non-Gaussian and nonlinear contributions to the covariance are expected to be important; the authors caution that these 'could affect the constraints,' but they still quote sigma_alpha ~ 0.028 as a headline result. For smin=15 Mpc/h the scales are more benign, but the model still depends on nonlinear halo-potential and velocity-dispersion terms. Please quantify the impact of non-Gaussian covariance, or clearly demote the smin=5 Mpc/h result and present smin=15 Mpc/h as the headline.
[Results (Fig. 4)] The combined constraints sigma_alpha ~ 0.028 and 0.14 are obtained by summing inverse variances from different cross-correlation pairs. These pairs are not independent: they share the same survey volume and, in several cases, one of the two galaxy samples (e.g., SKA2 appears in DESI-BGS x SKA2, DESI-LRG x SKA2, DESI-ELG x SKA2, and SKA2 x Euclid). The cross-covariance between the different pair measurements is nonzero, so simple inverse-variance addition can overstate the combined precision. Please either construct a joint Fisher matrix that includes cross-pair covariances, or present the pair-by-pair results as separate forecasts without an 'All combined' number.

minor comments (3)

[Model and Forecast formalism] The parameter alpha does not appear explicitly in Eq. (2) or Eq. (5); please state how the LPI-violating parameter enters the model (presumably through the gravitational-redshift term in epsilon_NL) so that the Fisher derivatives d xi / d alpha are unambiguous.
[Results] The 'average improvement of 11%' is not defined; please specify whether it is the unweighted mean, inverse-variance weighted mean, or median of the ratios in the bottom panels of Fig. 4, and state over which survey pairs the average is taken.
[Setup] There are minor formatting and notation issues: 'smax = 30, Mpc/h' and 'smin = 5Mpc /h' should be typeset consistently, and the notation Roff,X/Y in the parameter vector theta is confusing (presumably Roff,X and Roff,Y).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the 11% improvement is a model-derived Fisher forecast, not a fit; self-citations and the unvalidated octupole model are robustness concerns, not tautologies.

full rationale

I find no step in which a predicted quantity is equivalent to an input by construction. The central claim, that adding the octupole to the dipole improves constraints on alpha by about 11% for smin = 15 Mpc/h, is obtained from a Fisher matrix (Eq. 6) in which alpha is a free parameter with fiducial value zero. The ratio sigma_dipole+octupole/sigma_dipole is a derived function of model derivatives and the covariance, not a fit to alpha or a renamed input. The octupole model in Eq. (5) is imported from Ref. [33], a self-citation with overlapping authors (Saga and Taruya), and the survey setup follows Ref. [42] by the same authors; this is load-bearing in the sense that the forecast would change if the model changed. However, the model is presented with explicit physical ingredients (NFW halo potentials from the Poisson equation, virial and halo velocity dispersions, and linear peak theory), and the dipole version has been checked against N-body simulation measurements as the paper notes: 'While the theoretical model for the dipole has been well confirmed with simulation-based measurements [33], a verification of octupole prediction requires a more proper simulation setup' (Conclusions). That explicit statement is a model-accuracy limitation, not a circular reduction: the octupole prediction is not assumed equal to the improvement ratio, and no fitted parameter is renamed as a prediction. There is no uniqueness theorem invoked from the authors' prior work, no ansatz smuggled in solely by citation, and no known empirical result merely renamed. The self-citations are normal reliance on prior work, and the unvalidated octupole signal is a correctness risk that the authors themselves flag, but it does not make the derivation circular. Score 2 reflects the minor, non-tautological self-citation load-bearing in the forecast ingredients rather than any definitional or constructional circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The forecast uses existing analytic models with survey-specific fiducial biases and an off-centering parameter; no new particles, forces, or physical entities are introduced.

free parameters (3)

alpha (LPI-violating parameter) = 0 (fiducial), target of forecast
The parameter whose future 1-sigma error is forecast; it is varied in the Fisher matrix, not fitted to data.
Roff (off-centering parameter) = 0.2 rvir (fiducial), Gaussian prior 0.01 rvir
Model parameter for galaxy offset from halo center; affects halo potential and velocity dispersion. Chosen as fiducial with a prior, not fitted.
bX, bY (linear galaxy biases) = Survey-dependent fiducial values, e.g., DESI-BGS and SKA2; priors from even-multipole Fisher analysis
The bias difference drives the odd multipole amplitude; values are adopted from survey specifications and Ref. [42], not derived in this paper.

assumptions (5)

domain assumption Flat LCDM with WMAP7 fiducial cosmological parameters
Used for the matter power spectrum and growth rate; stated in the introduction.
domain assumption Linear theory for the density field and Gaussian covariance
The signal uses the linear matter power spectrum and the covariance assumes a Gaussian density field; the authors note nonlinear corrections are neglected.
domain assumption Analytic model of Ref. [33] for odd multipoles, including halo potential and velocity dispersion
The octupole model is adopted from Ref. [33] and is not newly derived; its small-scale validity is not simulation-verified.
domain assumption Plane-parallel approximation for the covariance matrix
The covariance is derived under the plane-parallel approximation, stated to be valid below 190 Mpc/h per Ref. [69].
domain assumption Sheth-Tormen mass function and the bias-mass relation
Used to relate bias to halo mass and to compute Mmin and halo properties; standard but a modeling choice.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Testing local position invariance with odd multipoles of galaxy clustering statistics." pith.science (2026). https://pith.science/paper/GFWIZTS6

@misc{pith2026241213701,
  author       = {Pith},
  title        = {Pith review of: Testing local position invariance with odd multipoles of galaxy clustering statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFWIZTS6}},
  note         = {Machine review of arXiv:2412.13701}
}

abstract

We investigate cosmological constraints on local position invariance (LPI), a key aspect of the Einstein equivalence principle (EEP), through asymmetric galaxy clustering. The LPI asserts that the outcomes of the non-gravitational experiments are identical regardless of location in spacetime and has been tested through measurements of the gravitational redshift effect. Therefore, measuring the gravitational redshift effect encoded in galaxy clustering provides a powerful and novel cosmological probe of the LPI. Recent work by Saga et al. proposed its validation using the cross-correlation function between distinct galaxy samples, but their analysis focused solely on the dipole moment. In this paper, we extend their work by further analyzing a higher-order odd multipole moment, the octupole moment, in the constraints on the LPI-violating parameter, $\alpha$, expected from galaxy surveys such as Dark Energy Spectroscopic Instrument, Euclid space telescope, Subaru Prime Focus Spectrograph, and Square Kilometre Array. We demonstrate that combining the octupole and dipole moments significantly improves the constraints, particularly when the analysis is restricted to larger scales, characterized by a large minimum separation $s_{\rm min}$. For a conservative setup with $s_{\rm min}=15 {\rm Mpc}/h$, we find an average improvement of 11$\%$ compared to using the dipole moment alone. Our results highlight the importance of higher-order multipoles in constraining $\alpha$, providing a more robust approach to testing the EEP on cosmological scales.

Figures

Figures reproduced from arXiv: 2412.13701 by the authors.

**Figure 2.** FIG. 2: Octupole moments along with their 1 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: 1 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gravitational redshift from large-scale structure: nonlinearities, antisymmetries, and the dipole
astro-ph.CO 2025-06 conditional novelty 6.0 of 10

A wide-angle streaming model including gravitational redshift, lightcone and kinematic effects explains the dipole turnover at ~20 h^-1 Mpc as an advection-like shift driven by the density-weighted pairwise potential ...

Reference graph

Works this paper leans on

69 extracted references · 64 canonical work pages · cited by 1 Pith paper

[42]

S. Saga, A. Taruya, Y. Rasera, and M.-A. Breton, Mon. Not. Roy. Astron. Soc. 524, 4472 (2023)

work page 2023
[1]

W. L. W. Sargent and E. L. Turner, Astrophys. J. Lett. 212, L3 (1977)

work page 1977
[2]

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

work page 1980
[3]

Kaiser, Mon

N. Kaiser, Mon. Not. Roy. Astron. Soc. 227, 1 (1987)

1987
[4]

E. V. Linder, Astroparticle Physics 29, 336 (2008)

work page 2008
[5]

Guzzo, M

L. Guzzo, M. Pierleoni, B. Meneux, E. Branchini, O. Le F` evre, C. Marinoni, B. Garilli, J. Blaizot, G. De Lucia, A. Pollo, et al., Nature (London) 451, 541 (2008)

work page 2008
[6]

S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H. Chuang, et al., Mon. Not. Roy. Astron. Soc. 470, 2617 (2017)

work page 2017
[7]

Sasaki, Mon

M. Sasaki, Mon. Not. Roy. Astron. Soc. 228, 653 (1987)

work page 1987

Show all 69 references

[8]

Matsubara, Astrophys

T. Matsubara, Astrophys. J. Lett. 537, L77 (2000)

2000
[9]

J. Yoo, A. L. Fitzpatrick, and M. Zaldarriaga, Phys. Rev. D 80, 083514 (2009)

2009
[10]

Bonvin and R

C. Bonvin and R. Durrer, Phys. Rev. D 84, 063505 (2011)

2011
[11]

Challinor and A

A. Challinor and A. Lewis, Phys. Rev. D 84, 043516 (2011)

2011
[12]

J. Yoo, N. Hamaus, U. c. v. Seljak, and M. Zaldarriaga, Phys. Rev. D 86, 063514 (2012)

2012
[13]

Jeong, F

D. Jeong, F. Schmidt, and C. M. Hirata, Phys. Rev. D 85, 023504 (2012)

2012
[14]

Yoo, Classical and Quantum Gravity 31, 234001 (2014)

J. Yoo, Classical and Quantum Gravity 31, 234001 (2014)

2014
[15]

Raccanelli, D

A. Raccanelli, D. Bertacca, D. Jeong, M. C. Neyrinck, and A. S. Szalay, Physics of the Dark Universe 19, 109 (2018)

2018
[16]

McDonald, J

P. McDonald, J. Cosmol. Astropart. Phys. 11, 026 (2009)

2009
[17]

R. A. C. Croft, Mon. Not. Roy. Astron. Soc. 434, 3008 (2013)

2013
[18]

Bonvin, Classical and Quantum Gravity 31, 234002 (2014)

C. Bonvin, Classical and Quantum Gravity 31, 234002 (2014)

2014
[19]

Bonvin, L

C. Bonvin, L. Hui, and E. Gazta˜ naga, Phys. Rev. D 89, 083535 (2014)

2014
[20]

Kodwani and H

D. Kodwani and H. Desmond, Phys. Rev. D 100, 064030 (2019)

2019
[21]

Bonvin and P

C. Bonvin and P. Fleury, J. Cosmol. Astropart. Phys. 5, 061 (2018)

2018
[22]

Bonvin, F

C. Bonvin, F. Oliveira Franco, and P. Fleury, J. Cosmol. Astropart. Phys. 2020, 004 (2020)

2020
[23]

O. Umeh, K. Koyama, and R. Crittenden, J. Cosmol. Astropart. Phys. 2021, 049 (2021)

2021
[24]

Wojtak, S

R. Wojtak, S. H. Hansen, and J. Hjorth, Nature (London) 477, 567 (2011)

2011
[25]

Sadeh, L

I. Sadeh, L. L. Feng, and O. Lahav, Phys. Rev. Lett. 114, 071103 (2015)

2015
[26]

Jimeno, T

P. Jimeno, T. Broadhurst, J. Coupon, K. Umetsu, and R. Lazkoz, Mon. Not. Roy. Astron. Soc. 448, 1999 (2015)

2015
[27]

S. Alam, H. Zhu, R. A. C. Croft, S. Ho, E. Giusarma, and D. P. Schneider, Mon. Not. Roy. Astron. Soc. 470, 2822 (2017)

2017
[28]

Gaztanaga, C

E. Gaztanaga, C. Bonvin, and L. Hui, J. Cosmol. As- tropart. Phys. 1, 032 (2017)

2017
[29]

H. Zhu, S. Alam, R. A. C. Croft, S. Ho, and E. Giusarma, Mon. Not. Roy. Astron. Soc. 471, 2345 (2017)

2017
[30]

Breton, Y

M.-A. Breton, Y. Rasera, A. Taruya, O. Lacombe, and S. Saga, Mon. Not. Roy. Astron. Soc. 483, 2671 (2019)

2019
[31]

Beutler and E

F. Beutler and E. Di Dio, J. Cosmol. Astropart. Phys. 7, 048 (2020)

2020
[32]

S. Saga, A. Taruya, M.-A. Breton, and Y. Rasera, Mon. Not. Roy. Astron. Soc. 498, 981 (2020)

2020
[33]

S. Saga, A. Taruya, Y. Rasera, and M.-A. Breton, Mon. Not. Roy. Astron. Soc. 511, 2732 (2022)

2022
[34]

Giusarma, S

E. Giusarma, S. Alam, H. Zhu, R. A. C. Croft, and S. Ho, arXiv:1709.07854 (2017)

2017 arXiv
[35]

Di Dio and U

E. Di Dio and U. Seljak, J. Cosmol. Astropart. Phys. 4, 050 (2019)

2019
[36]

Takada, R

M. Takada, R. S. Ellis, M. Chiba, J. E. Greene, H. Ai- hara, N. Arimoto, K. Bundy, J. Cohen, O. Dor´ e, G. Graves, et al., Publ. Astron. Soc. Japan66, R1 (2014)

2014
[37]

Aghamousa, J

DESI Collaboration, A. Aghamousa, J. Aguilar, S. Ahlen, S. Alam, L. E. Allen, C. Allende Prieto, J. An- nis, S. Bailey, C. Balland, et al., arXiv:1611.00036 (2016)

2016 arXiv
[38]

Mellier, Abdurro’uf, J

Euclid Collaboration, Y. Mellier, Abdurro’uf, J. A. Acevedo Barroso, A. Ach´ ucarro, J. Adamek, R. Adam, G. E. Addison, N. Aghanim, M. Aguena, et al., arXiv:2405.13491 (2024)

2024
[39]

Square Kilometre Array Cosmology Science Working Group, D. J. Bacon, R. A. Battye, P. Bull, S. Camera, P. G. Ferreira, I. Harrison, D. Parkinson, A. Pourtsidou, M. G. Santos, et al., Publications of the Astronomical 8 Society of Australia 37, e007 (2020)

2020
[40]

Bonvin, F

C. Bonvin, F. Lepori, S. Schulz, I. Tutusaus, J. Adamek, and P. Fosalba, Mon. Not. Roy. Astron. Soc. 525, 4611 (2023)

2023
[41]

Lepori, S

F. Lepori, S. Schulz, I. Tutusaus, M. A. Breton, S. Saga, C. Viglione, J. Adamek, C. Bonvin, L. Dam, P. Fosalba, et al., arXiv:2410.06268 (2024)

2024
[43]

R. V. Pound and G. A. Rebka, Phys. Rev. Lett. 3, 439 (1959)

1959
[44]

R. V. Pound and J. L. Snider, Physical Review140, B788 (1965)

1965
[45]

C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 2018), 2nd ed

2018
[46]

J. D. Bekenstein, Phys. Rev. D 25, 1527 (1982)

1982
[47]

H. B. Sandvik, J. D. Barrow, and J. Magueijo, Phys. Rev. Lett. 88, 031302 (2002)

2002
[48]

J. D. Barrow and J. Magueijo, Physics Letters B 443, 104 (1998)

1998
[49]

J. D. Barrow, Phys. Rev. D 59, 043515 (1999)

1999
[50]

D. S. Blanco, C. Bonvin, C. Clarkson, and R. Maartens, J. Cosmol. Astropart. Phys. 12, 029 (2024)

2024
[51]

Komatsu, K

E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, et al., Astrophys. J. Suppl. 192, 18 (2011)

2011
[52]

H. Zhao, J. A. Peacock, and B. Li, Phys. Rev. D 88, 043013 (2013)

2013
[53]

Kaiser, Mon

N. Kaiser, Mon. Not. Roy. Astron. Soc. 435, 1278 (2013)

2013
[54]

Y.-C. Cai, N. Kaiser, S. Cole, and C. Frenk, Mon. Not. Roy. Astron. Soc. 468, 1981 (2017)

2017
[55]

J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro- phys. J. 462, 563 (1996)

1996
[56]

E. L. Lokas and G. A. Mamon, Mon. Not. Roy. Astron. Soc. 321, 155 (2001)

2001
[57]

R. K. Sheth and A. Diaferio, Mon. Not. Roy. Astron. Soc. 322, 901 (2001)

2001
[58]

Hikage, R

C. Hikage, R. Mandelbaum, M. Takada, and D. N. Spergel, Mon. Not. Roy. Astron. Soc. 435, 2345 (2013)

2013
[59]

Masaki, C

S. Masaki, C. Hikage, M. Takada, D. N. Spergel, and N. Sugiyama, Mon. Not. Roy. Astron. Soc. 433, 3506 (2013)

2013
[60]

Z. Yan, N. Raza, L. Van Waerbeke, A. J. Mead, I. G. McCarthy, T. Tr¨ oster, and G. Hinshaw, Mon. Not. Roy. Astron. Soc. 493, 1120 (2020)

2020
[61]

R. K. Sheth and G. Tormen, Mon. Not. Roy. Astron. Soc. 308, 119 (1999)

1999
[62]

Taruya, S

A. Taruya, S. Saga, M.-A. Breton, Y. Rasera, and T. Fu- jita, Mon. Not. Roy. Astron. Soc. 491, 4162 (2020)

2020
[63]

Reimberg, F

P. Reimberg, F. Bernardeau, and C. Pitrou, J. Cosmol. Astropart. Phys. 1, 048 (2016)

2016
[64]

Castorina and M

E. Castorina and M. White, Mon. Not. Roy. Astron. Soc. 476, 4403 (2018)

2018
[65]

Castorina and M

E. Castorina and M. White, Mon. Not. Roy. Astron. Soc. 479, 741 (2018)

2018
[66]

Tegmark, Phys

M. Tegmark, Phys. Rev. Lett. 79, 3806 (1997)

1997
[67]

Tegmark, A

M. Tegmark, A. N. Taylor, and A. F. Heavens, Astro- phys. J. 480, 22 (1997)

1997
[68]

Hall and C

A. Hall and C. Bonvin, Phys. Rev. D 95, 043530 (2017)

2017
[69]

Lepori, E

F. Lepori, E. Di Dio, E. Villa, and M. Viel, J. Cosmol. Astropart. Phys. 5, 043 (2018)

2018

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [42]

S. Saga, A. Taruya, Y. Rasera, and M.-A. Breton, Mon. Not. Roy. Astron. Soc. 524, 4472 (2023)

work page 2023

[2] [1]

W. L. W. Sargent and E. L. Turner, Astrophys. J. Lett. 212, L3 (1977)

work page 1977

[3] [2]

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

work page 1980

[4] [3]

Kaiser, Mon

N. Kaiser, Mon. Not. Roy. Astron. Soc. 227, 1 (1987)

1987

[5] [4]

E. V. Linder, Astroparticle Physics 29, 336 (2008)

work page 2008

[6] [5]

Guzzo, M

L. Guzzo, M. Pierleoni, B. Meneux, E. Branchini, O. Le F` evre, C. Marinoni, B. Garilli, J. Blaizot, G. De Lucia, A. Pollo, et al., Nature (London) 451, 541 (2008)

work page 2008

[7] [6]

S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H. Chuang, et al., Mon. Not. Roy. Astron. Soc. 470, 2617 (2017)

work page 2017

[8] [7]

Sasaki, Mon

M. Sasaki, Mon. Not. Roy. Astron. Soc. 228, 653 (1987)

work page 1987