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REVIEW 3 major objections 4 minor 76 references

A novel test of gravity: Does spacetime geometry track matter density?

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper constructs a null test that vanishes in general relativity if and only if the growth of the Weyl potential matches the growth of matter density, and finds no deviation in current data.

desk verdict A clean new null test for modified gravity, but the 'if and only if' claim is conditional on scale-independence and the current-data precision is method-dependent. read the letter →

arxiv 2506.04387 v1 pith:BLXECEGU submitted 2025-06-04 astro-ph.CO gr-qc

Camille Bonvin , Nastassia Grimm , Isaac Tutusaus This is my paper
classification astro-ph.COgr-qc
keywords nulltestmodifiedgravityWeylpotentialgrowthofstructuregalaxy-galaxylensingredshift-spacedistortionsgeneralrelativitystage-IVsurveys
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 introduces a null test of general relativity that compares how the geometry of spacetime grows with how matter density grows. In GR both are governed by the same growth function, so the test is identically zero; any nonzero value would signal a modified theory of gravity that changes how light propagates. Applied to current lensing and galaxy-clustering data, the test gives values consistent with zero, and future surveys would tighten the bound to a 2–4% mismatch. The test is deliberately blind to dark-energy background changes and to extra forces on dark matter, making it a sharp probe of gravity itself.

What carries the argument

The key object is the null-test combination $N_{\rm grow}(z)$, built from two observationally accessible functions: $\hat f(z)\equiv f(z)\sigma_8(z)$, the growth rate times the clustering amplitude measured from redshift-space distortions, and $\hat J(z)\equiv J(z)\sigma_8(z)/D_1(z)$, a function reconstructing the growth of the Weyl potential from the ratio of galaxy-galaxy lensing to galaxy clustering. In GR one has $\hat J(z)=\Omega_m(z)\sigma_8(z)$, so its redshift derivative is locked to $\hat f(z)/(1+z)$; the identity $N_{\rm grow}=0$ follows from the common growth function $D_1$. The test works by measuring the derivative of $\hat J/\Omega_m$ from discrete measurements of $\hat J$ and comparing it with the independent measurement of $\hat f$.

What would settle it

Measure $N_{\rm grow}(z)$ from the joint DESI and LSST-like data; a detection of $|N_{\rm grow}|$ larger than the forecasted uncertainty of about 0.005 at any redshift in the range $0.25<z<1.85$—equivalently a mismatch between $\tilde D_1$ and $D_1$ exceeding $2\!-\!4\%$—would disprove the paper's central claim. A more direct check is to compare the three-node result with a kde-based interpolation of $\hat J$: the paper already shows that the alternative method shifts $N_{\rm grow}$ by up to $1.5\sigma$, so a measurement with better redshift resolution that makes the two methods disagree by more than the stated uncertainty would signal that the derivative reconstruction is not robust.

Watch

Extended reading notes

Core claim

The central claim is that the combination $N_{\rm grow}(z) \equiv \frac{d}{dz}\left(\frac{\hat J(z)}{\Omega_m(z)}\right) + \frac{\hat f(z)}{1+z}$ is identically zero in GR, where $\hat J$ measures the growth of the Weyl potential (the sum of the two metric potentials) and $\hat f$ measures the growth rate of matter density fluctuations. The paper argues that this quantity deviates from zero if and only if the evolution of the Weyl potential decouples from that of the density, which happens precisely in modified-gravity theories that alter the propagation of light. Using measurements of $\hat J$ from DES lensing and $\hat f$ from redshift-space distortion surveys, the authors find $N_{\rm grow} = \{0.08\pm0.22,\, 0.04\pm0.11,\, -0.01\pm0.26\}$ at redshifts $z=\{0.35, 0.53, 0.71\}$, i.e., no evidence of deviation. Rewriting the result in terms of the growth functions, they constrain the Weyl potential's evolution to track the density evolution to within 33% from $z_*=10$ down to $z\sim0.53$.

Load-bearing premise

The clean interpretation of $N_{\rm grow}$ assumes that the measured $\hat f$ and $\hat J$ are scale-independent, that $\Omega_m(z)$ follows the $\Lambda$CDM background, and that the GR power spectrum at $z_*$ is recovered; if any of these fails, a real mismatch between geometry and matter growth could be masked or a spurious mismatch generated.

Editorial extensions

If this is right

  • If $N_{\rm grow}$ stays zero at the precision of stage-IV surveys (DESI and LSST), it will rule out any modified-gravity model that changes the Weyl-potential evolution by more than $2\!-\!4\%$ between $z\sim0.2$ and $z\sim1.8$.
  • A nonzero $N_{\rm grow}$ would unambiguously signal a deviation in light propagation, because the test is insensitive to changes in the background expansion and to fifth forces acting on dark matter.
  • The test provides a clean separation from the $E_G$ statistic: $E_G$ is sensitive to both matter-motion and light-propagation changes, while $N_{\rm grow}$ isolates the latter, so combining the two can break degeneracies between dark-matter-sector and gravity-sector modifications.
  • Within the $\mu-\Sigma$ parametrization, the test directly constrains $\Sigma(z)\neq1$; current data yield $\Sigma_0=-0.008\pm0.07$ for a standard redshift evolution, and future data would tighten this to $\pm0.011$.
  • The test's insensitivity to the growth-rate change induced by $\mu\neq1$ means it can be used to separate changes in Poisson's equation for the Weyl potential from changes in the force law for matter.

Reading between the lines

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

  • Because $N_{\rm grow}$ depends on a redshift derivative of $\hat J$, the test's constraining power is governed by how finely $\hat J$ can be measured in redshift; a natural extension is to apply the same construction to lensing measurements with many narrow redshift bins, which would reduce the derivative uncertainty more efficiently than adding more total survey area.
  • The scale-independence assumption is the most fragile part of the current implementation; if future data reveal a scale-dependent $\hat f$ or $\hat J$ (e.g., through a growth-index measurement as a function of scale), the clean interpretation of $N_{\rm grow}$ would need to be replaced by a scale-resolved version, and the present null result would need re-evaluation.
  • The test's blindness to dark-matter fifth forces holds only when the continuity equation linking $f$ to $D_1$ is preserved; dark-matter models that break this link could produce a spurious $N_{\rm grow}$ signal, so a detected deviation should be cross-checked against independent measurements of the growth rate and of $\hat J$ before attributing it to light propagation.
  • A direct observational falsifier is to measure $N_{\rm grow}$ using an independent reconstruction of $\Omega_m(z)$ from baryon acoustic oscillations and supernovae instead of the $\Lambda$CDM background; if the result shifts by more than the forecasted uncertainty, the assumption of a $\Lambda$CDM background is not innocuous.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a null test of gravity, N_grow(z) = d/dz(Jhat/Omega_m) + fhat/(1+z), which is constructed to vanish in general relativity under three explicit assumptions: (1) the GR/Planck power spectrum is recovered at high redshift, (2) the background expansion is effectively Lambda-CDM when computing Omega_m(z), and (3) fhat and Jhat are scale-independent. The test is applied to DES measurements of Jhat and a compilation of 22 growth-rate measurements of fhat, giving N_grow = {0.08 +/- 0.22, 0.04 +/- 0.11, -0.01 +/- 0.26} at z = {0.35, 0.53, 0.71}, i.e. no evidence of a deviation. The authors also forecast that LSST and DESI will improve the precision by a factor of 20-50, and they illustrate the response of the test to several redshift-dependent forms of Sigma(z). The central conceptual claim is that N_grow deviates from zero if and only if the growth of the Weyl potential and the growth of matter density are mismatched, and that matter-sector fifth forces and background changes do not generate a signal.

Significance. If the central claim survives scrutiny, this test fills a genuine gap: it is designed as a model-independent discriminator that isolates modifications to the propagation of light (Sigma != 1) from modifications to the motion of matter (mu != 1 or dark-matter fifth forces), making it complementary to the EG statistic and to direct growth-rate tests. The derivation of the null test is transparent, and the data analysis is conducted with care: the authors use published, peer-reviewed measurements, propagate correlations by using the underlying MCMC chains, use AIC to choose reconstruction nodes, and provide an explicit alternative reconstruction method for comparison. The forecast is also carefully constructed, including a Jacobian transformation of the Fisher matrix to the node basis. The current no-deviation result is interesting in itself, and the forecasted improvement is plausible if the quoted precision is robust to the methodological choices discussed below.

major comments (3)
  1. [The null test, Eq. (5), and assumption (3)] The cancellation that makes N_grow vanish in GR, and the mapping from a nonzero N_grow to a modification of light propagation, both rely on the assumption that fhat and Jhat are scale-independent, i.e. that a single D1(z) describes density growth and a single J(z) describes the Weyl potential. This is load-bearing for the abstract's 'if and only if' claim and for the claimed insensitivity to dark-matter fifth forces. In screened f(R), chameleon, and finite-Compton-wavelength dark-matter fifth-force models, growth is k-dependent; RSD and galaxy-galaxy lensing then weight the k-dependent growth and velocity kernels with different survey-window functions, so fhat and Jhat are not simple functions of z alone. A scale-dependent mu with Sigma = 1 could in principle produce a nonzero N_grow even when the local Weyl-potential/density relation is the GR one, and a genuine Sigma signal could be diluted by window-function averaging. The authors state this as an assumption, but the headline claims are not restricted to the scale-independent class. I request either a clear restriction of the central claim to scale-independent theories, or a demonstration with a scale-dependent mock (e.g. mu(k,z) with Sigma = 1) that the reconstruction of fhat and Jhat does not bias N_grow away from zero.
  2. [Supplemental Materials, Figs. 4-6 and Table II] The headline current-data precision and the quoted 33% constraint between z* = 10 and z = 0.53 come from the three-node reconstruction, for which sigma_Ngrow(0.53) = 0.11. The alternative interpolation/four-node method gives sigma_Ngrow(0.53) about 0.32 and a result that is only compatible with zero at the 1.5-sigma level, and the authors note that the AIC is similar for three and four nodes. Because the node-count choice is not decisively selected by the stated criterion, the current-data error bars and the derived 'within 33%' bound are not robust to this methodological choice. Please report the more conservative four-node/interpolation uncertainties as the primary current-data constraint, or validate the three-node choice with reconstruction tests on mocks, and adjust the abstract and the dev values accordingly.
  3. [The null test and Deviations in the null test] The statement that changes in the background expansion 'induce no deviations' in N_grow is only correct if the Omega_m(z) entering Eq. (5) is the true matter-density parameter. In the current implementation Omega_m(z) is computed assuming a Lambda-CDM background (assumption 2), and if the true background differs, N_grow acquires a term proportional to sigma8(z) d/dz ln[Omega_m_true(z)/Omega_m_LCDM(z)], which need not vanish. The manuscript notes that Omega_m(z) can be reconstructed from BAO and supernova data, but the abstract's unqualified claim that alternative dark-energy models do not affect the test is stronger than what the current fixed-background analysis supports. Please qualify the claim and, ideally, quantify the bias for a representative non-Lambda-CDM background.
minor comments (4)
  1. [Discussion and conclusion] The discussion states that 'we measured N_grow at four redshifts,' but Table II lists three nodes at z = 0.35, 0.53, and 0.71; please correct this inconsistency.
  2. [Formalism, Eq. (4)] The statement that in GR Jhat reduces to Omega_m sigma8 implicitly assumes a normalization convention (e.g. Omega_m(z*) about 1). Please state the normalization explicitly so that Eq. (4) and Eq. (5) are unambiguous.
  3. [Forecasts] The forecasted precision inherits assumption (3) that fhat and Jhat are scale-independent; this should be stated explicitly in the forecast section so that readers do not mistake the forecast for a test that is already robust to scale-dependent modified-gravity theories.
  4. [Throughout] The notation 'N grow' is typeset inconsistently; please use N_grow uniformly, and fix the spacing in '1 sigma' in figure captions and tables.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the null test is derived from GR field equations and confronted with independent published measurements; the main limitations are explicit physical assumptions, not recycled inputs.

full rationale

The derivation of N_grow is not circular. Equation (5) is constructed from the GR relation (2), in which the density contrast and the Weyl potential share one growth function D1, and the identity N_grow = 0 follows from the continuity-equation relation d sigma8/dz = -f sigma8/(1+z) noted in footnote [44], rather than from any fitted parameter. The measurements of J-hat and f-hat are external inputs, namely DES lensing/clustering data from [37] and the RSD compilation from [36,47-58]; these are published, peer-reviewed data sets, not parameters fitted in the present paper, so the self-citations are not load-bearing arguments. The 'dev' constraint in Eqs. (6)-(7) is a translation of the measured uncertainty into a statement about relative growth, not a prediction recycled from the fit. The explicit assumptions, namely recovery of the Planck/GR power spectrum at z*, a Lambda-CDM background for Omega_m(z), and scale independence of f-hat and J-hat, limit the scope of the 'if and only if' claim, and the footnote about massive neutrinos quantifies a small known, residual GR violation of the exact null relation; these are stated limitations of generality and precision, not circular definitions. No step in the derivation reduces to its own input.

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

The null test itself has no free parameters; its construction is a derivative of the equations of GR. However, the measurement relies on the authors' earlier reconstruction of J-hat, which depends on Planck priors and a Lambda-CDM background, and the reconstruction of f-hat uses node choices that affect the error budget. The illustrative Sigma0 constraints are auxiliary.

free parameters (3)
  • Redshift nodes for J-hat reconstruction (three nodes at z=0.35, 0.53, 0.71) = z = 0.35, 0.53, 0.71
    Chosen by AIC minimization; the number of nodes affects the reconstructed derivative and hence the headline Ngrow uncertainties. The alternative four-node/interpolation method yields larger error bars.
  • Redshift nodes for f-hat reconstruction (four nodes) = z in [0.001, 1.944]
    Chosen by AIC; used to interpolate f-hat at the nodes for constructing Ngrow.
  • Sigma0 for illustrative modified-gravity evolutions = e.g., -0.008 +/- 0.07 for standard evolution
    Constrained from the measured Ngrow for three ad-hoc redshift evolutions; illustrative only and not part of the null test itself.
assumptions (5)
  • domain assumption GR holds at high redshift z* so that the initial matter power spectrum is the GR/Planck one
    Assumption (1) in the Measurements section; excludes early dark energy models from the test.
  • domain assumption Background expansion follows Lambda-CDM, used to compute Omega_m(z) in the null test
    Assumption (2) in the Measurements section; the test is claimed to be background-insensitive only if the true Omega_m(z) is known.
  • domain assumption f-hat and J-hat are scale-independent
    Assumption (3); needed to relate measurements at different scales to the single growth functions D1 and J.
  • domain assumption Quasi-static approximation and scale-independent mu and eta in the mu-Sigma parametrization
    Used in the illustrative deviation section; not needed for the null test itself.
  • standard math D1(z)/sigma8(z) is constant for z less than about 10, neglecting radiation and massive neutrinos
    Footnote 44; residual effects produce a 0.001 deviation in Ngrow, smaller than forecast errors.

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Pith. "Pith review of A novel test of gravity: Does spacetime geometry track matter density?." pith.science (2026). https://pith.science/paper/BLXECEGU

@misc{pith2026250604387,
  author       = {Pith},
  title        = {Pith review of: A novel test of gravity: Does spacetime geometry track matter density?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLXECEGU}},
  note         = {Machine review of arXiv:2506.04387}
}
abstract

We propose a novel test of gravity that combines galaxy clustering with gravitational lensing. In general relativity, the evolution of matter density fluctuations and of the Weyl potential -- the sum of spatial and temporal distortions of the geometry -- are governed by the same growth function. In contrast, alternative theories of gravity that modify the relation between geometry and matter content can lead to differences in these two growths. Exploiting a recent method to directly measure the Weyl potential, we construct a null test that deviates from zero if and only if there is a mismatch between the growth rate of density and that of geometry distortions. We show that changes in the background expansion due to alternative dark energy models and additional forces in the dark matter sector induce no deviations in this test, making it a robust probe for detecting departures from general relativity. Applying the test to current data, we find no evidence of deviation. From an initial $z_*=10$ to $z\sim 0.5$, we constrain the evolution of the Weyl potential to track that of the density to within 33\%. Combining stage-IV surveys will improve the precision across a broad redshift range, limiting differences between the two evolutions to below $2-4\%$.

Figures

Figures reproduced from arXiv: 2506.04387 by the authors.

Figure 1
Figure 1. Measurements of N grow, from current data sets, together with the 1σ uncertainties. We show the results at the three nodes (blue points) and the reconstructed N grow over the whole redshift range. The black crosses indicate the position of the DES redshifts where Jˆ is measured. Importantly, both ˆf and Jˆ are measured in a model￾independent way, without assuming any specific theory of gravity or dark matter model, … view at source ↗
Figure 5
Figure 5. The results are however also compatible with GR [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 2
Figure 2. Forecasted mean and 1σ uncertainties for N grow obtained by combining LSST and DESI. the precision on N grow achievable by combining LSST￾and DESI-like data. For Jˆ, we adopt the LSST forecast from [41], which provides Jˆ and its covariance matrix in ten redshift bins between z = 0.25 and z = 2.1. For ˆf, we assume measurements at 16 redshifts between z = 0.25 and z = 1.85. The 1σ uncertainties correspond to fore￾ca… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: , we understand that having measurements of Jˆ in a larger number of redshift bins, from stage-IV surveys, will significantly reduce the uncertainty on the derivative of Jˆ and thus on the null test. 0.3 0.4 0.5 0.6 0.7 z 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 J CDM D…
Figure 5
Figure 5. Figure 5: Reconstruction of Jˆ (left panel) and of d(J/ˆ Ωm)/dz (right panel) from interpolation. The red points are the measurements from DES with their 1σ uncertainties. The blue region is the reconstruction obtained from interpolation. The black line and gray region show the …
Figure 6
Figure 6. Figure 6: Measurements of N grow from current data sets, together with the 1σ uncertainties, obtained from interpolation of Jˆ. We show the reconstructed N grow over the whole redshift range. The black crosses indicate the position of the DES redshifts where Jˆ is measured. Addi…

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Works this paper leans on

76 extracted references · 15 canonical work pages

  1. [1]

    A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201. 9

    arXiv 1998

  2. [2]

    Perlmutter, G

    S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson,...

    arXiv 1999

  3. [3]

    Bertone and D

    G. Bertone and D. Hooper, Rev. Mod. Phys. 90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO]

    arXiv 2018

  4. [4]

    G. R. Blumenthal, S. M. Faber, J. R. Primack, and M. J. Rees, Nature 311, 517 (1984)

    1984

  5. [5]

    Davis, G

    M. Davis, G. Efstathiou, C. S. Frenk, and S. D. M. White, Astrophys. J. 292, 371 (1985)

    work page 1985

  6. [6]

    Verde, T

    L. Verde, T. Treu, and A. G. Riess, Nature Astron. 3, 891 (2019), arXiv:1907.10625 [astro-ph.CO]

    arXiv 2019

  7. [7]

    Di Valentino, O

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO]

    arXiv 2021

  8. [8]

    Di Valentino, L

    E. Di Valentino, L. A. Anchordoqui, ¨O. Akarsu, Y. Ali-Haimoud, L. Amendola, N. Arendse, M. Asgari, M. Ballardini, S. Basilakos, E. Battistelli, M. Benetti, S. Birrer, F. R. Bouchet, M. Bruni, E. Calabrese, D. Camarena, S. Capozziello, A. Chen, J. Chluba, A. Chudaykin, E. ´O. Colg´ ain, F.-Y. Cyr-Racine, P. de Bernardis, J. de Cruz P´ erez, J. Delabrouill...

    arXiv 2021

Show all 76 references
  1. [9]

    R. C. Nunes and S. Vagnozzi, Mon. Not. Roy. Astron. Soc. 505, 5427 (2021), arXiv:2106.01208 [astro-ph.CO]

    2021 arXiv

  2. [10]

    N. J. Secrest, S. von Hausegger, M. Rameez, R. Mohayaee, and S. Sarkar, Astrophys. J. Lett. 937, L31 (2022), arXiv:2206.05624 [astro-ph.CO]

    2022 arXiv

  3. [11]

    Dalang and C

    C. Dalang and C. Bonvin, Mon. Not. Roy. Astron. Soc. 512, 3895 (2022), arXiv:2111.03616 [astro-ph.CO]

    2022 arXiv

  4. [12]

    A. G. Adame et al. (DESI), JCAP 02, 021 (2025), arXiv:2404.03002 [astro-ph.CO]

    2025 arXiv

  5. [13]

    Abdul Karim et al

    M. Abdul Karim et al. (DESI), (2025), arXiv:2503.14738 [astro-ph.CO]

    2025 arXiv

  6. [14]

    Berti, E

    M. Berti, E. Bellini, C. Bonvin, M. Kunz, M. Viel, and M. Zumalacarregui, (2025), arXiv:2503.13198 [astro-ph.CO]

    2025 arXiv

  7. [15]

    Castello, N

    S. Castello, N. Grimm, and C. Bonvin, Phys. Rev. D 106, 083511 (2022), arXiv:2204.11507 [astro-ph.CO]

    2022 arXiv

  8. [16]

    Bonvin and L

    C. Bonvin and L. Pogosian, Nature Astron. 7, 1127 (2023), arXiv:2209.03614 [astro-ph.CO]

    2023 arXiv

  9. [17]

    Castello, Z

    S. Castello, Z. Wang, L. Dam, C. Bonvin, and L. Pogosian, (2024), arXiv:2404.09379 [astro-ph.CO]

    2024 arXiv

  10. [18]

    B. A. Bassett and M. Kunz, Phys. Rev. D 69, 101305 (2004), arXiv:astro-ph/0312443

    2004 arXiv

  11. [19]

    Avgoustidis, C

    A. Avgoustidis, C. Burrage, J. Redondo, L. Verde, and R. Jimenez, JCAP 10, 024 (2010), arXiv:1004.2053 [astro-ph.CO]

    2010 arXiv

  12. [20]

    F. Keil, S. Nesseris, I. Tutusaus, and A. Blanchard, (2025), arXiv:2504.01750 [astro-ph.CO]

    2025

  13. [21]

    F. Yang, X. Fu, B. Xu, K. Zhang, Y. Huang, and Y. Yang, Eur. Phys. J. C 85, 186 (2025), arXiv:2502.05417 [astro-ph.CO]

    2025 arXiv

  14. [22]

    N. J. Secrest, S. von Hausegger, M. Rameez, R. Mohayaee, S. Sarkar, and J. Colin, Astrophys. J. Lett. 908, L51 (2021), arXiv:2009.14826 [astro-ph.CO]

    2021 arXiv

  15. [23]

    Nadolny, R

    T. Nadolny, R. Durrer, M. Kunz, and H. Padmanabhan, JCAP 11, 009 (2021), arXiv:2106.05284 [astro-ph.CO]

    2021 arXiv

  16. [24]

    Bonvin and P

    C. Bonvin and P. Fleury, JCAP 05, 061 (2018), arXiv:1803.02771 [astro-ph.CO]

    2018 arXiv

  17. [25]

    Castello, Z

    S. Castello, Z. Zheng, C. Bonvin, and L. Amendola, (2024), arXiv:2412.08627 [astro-ph.CO]

    2024 arXiv

  18. [26]

    F. O. Franco, C. Bonvin, and C. Clarkson, Mon. Not. Roy. Astron. Soc. 492, L34 (2020), arXiv:1906.02217 [astro-ph.CO]

    2020 arXiv

  19. [27]

    Arjona, S

    R. Arjona, S. Nesseris, I. Tutusaus, D. S. Blanco, and C. Bonvin, (2024), arXiv:2409.15170 [astro-ph.CO]

    2024 arXiv

  20. [28]

    Zhang, M

    P. Zhang, M. Liguori, R. Bean, and S. Dodelson, Phys. Rev. Lett. 99, 141302 (2007), arXiv:0704.1932 [astro-ph]

    2007 arXiv

  21. [29]

    S. Alam, H. Miyatake, S. More, S. Ho, and R. Mandelbaum, Mon. Not. Roy. Astron. Soc. 465, 4853 (2017), arXiv:1610.09410 [astro-ph.CO]

    2017 arXiv

  22. [30]

    Amon et al

    A. Amon et al. , Mon. Not. Roy. Astron. Soc. 479, 3422 (2018), arXiv:1711.10999 [astro-ph.CO]

    2018 arXiv

  23. [31]

    Reyes, R

    R. Reyes, R. Mandelbaum, U. Seljak, T. Baldauf, J. E. Gunn, L. Lombriser, and R. E. Smith, Nature 464, 256 (2010), arXiv:1003.2185 [astro-ph.CO]

    2010 arXiv

  24. [32]

    Blake et al

    C. Blake et al. , Mon. Not. Roy. Astron. Soc. 462, 4240 (2016), arXiv:1608.02668 [astro-ph.CO]

    2016 arXiv

  25. [33]

    A. R. Pullen, S. Alam, S. He, and S. Ho, Mon. Not. Roy. Astron. Soc. 460, 4098 (2016), arXiv:1511.04457 [astro-ph.CO]

    2016 arXiv

  26. [34]

    Wenzl, R

    L. Wenzl, R. Bean, S.-F. Chen, G. S. Farren, M. S. Madhavacheril, G. A. Marques, F. J. Qu, N. Sehgal, B. D. Sherwin, and A. van Engelen, Phys. Rev. D 109, 083540 (2024), arXiv:2401.12971 [astro-ph.CO]

    2024 arXiv

  27. [35]

    M. M. Abidi, C. Bonvin, M. Jalilvand, and M. Kunz, Phys. Rev. D 107 (2023), 10.1103/physrevd.107.063514, 2208.10419

    2023 arXiv

  28. [36]

    Grimm, C

    N. Grimm, C. Bonvin, and I. Tutusaus, Phys. Rev. Lett. 133, 211004 (2024), arXiv:2403.13709 [astro-ph.CO]

    2024 arXiv

  29. [37]

    Tutusaus, C

    I. Tutusaus, C. Bonvin, and N. Grimm, Nature Commun. 15, 9295 (2024), arXiv:2312.06434 [astro-ph.CO]

    2024 arXiv

  30. [38]

    Kaiser, Mon

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

    1987

  31. [39]

    A. J. S. Hamilton and M. Culhane, Mon. Not. Roy. Astron. Soc. 278, 73 (1996), arXiv:astro-ph/9507021

    1996 arXiv

  32. [40]

    Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003)

    S. Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003)

    2003

  33. [41]

    Tutusaus, D

    I. Tutusaus, D. Sobral-Blanco, and C. Bonvin, Phys. Rev. D 107 (2023), 10.1103/physrevd.107.083526, arXiv:2209.08987 [astro-ph.CO]

    2023 arXiv

  34. [42]

    W. J. Percival and M. White, Mon. Not. Roy. Astron. Soc. 393, 297 (2009), arXiv:0808.0003 [astro-ph]

    2009 arXiv

  35. [43]

    Beutler, C

    F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, G. B. Poole, L. Campbell, Q. Parker, W. Saunders, and 10 F. Watson, Mon. Not. Roy. Astron. Soc. 423, 3430 (2012), arXiv:1204.4725 [astro-ph.CO]

    2012 arXiv

  36. [44]

    In practice, both the residual radiation present at z∗ = 10 and the presence of massive neutrinos induce a small time evolution of this ratio

    Here we have used that at late time, once radiation is negligible, D1(z)/σ8(z) is constant. In practice, both the residual radiation present at z∗ = 10 and the presence of massive neutrinos induce a small time evolution of this ratio. This results in a deviation of about 0.001...

  37. [45]

    Amendola, M

    L. Amendola, M. Pietroni, and M. Quartin, JCAP 11, 023 (2022), arXiv:2205.00569 [astro-ph.CO]

    2022

  38. [46]

    A. P. Schirra, M. Quartin, and L. Amendola, (2024), arXiv:2406.15347 [astro-ph.CO]

    2024

  39. [47]

    Howlett, L

    C. Howlett, L. Staveley-Smith, P. J. Elahi, T. Hong, T. H. Jarrett, D. H. Jones, B. S. Koribalski, L. M. Macri, K. L. Masters, and C. M. Springob, Mon. Not. Roy. Astron. Soc. 471, 3135 (2017), arXiv:1706.05130 [astro-ph.CO]

    2017 arXiv

  40. [48]

    Huterer, D

    D. Huterer, D. Shafer, D. Scolnic, and F. Schmidt, JCAP 05, 015 (2017), arXiv:1611.09862 [astro-ph.CO]

    2017 arXiv

  41. [49]

    M. J. Hudson and S. J. Turnbull, Astrophys. J. Lett. 751, L30 (2013), arXiv:1203.4814 [astro-ph.CO]

    2013 arXiv

  42. [50]

    S. J. Turnbull, M. J. Hudson, H. A. Feldman, M. Hicken, R. P. Kirshner, and R. Watkins, Mon. Not. Roy. Astron. Soc. 420, 447 (2012), arXiv:1111.0631 [astro-ph.CO]

    2012 arXiv

  43. [51]

    Davis, A

    M. Davis, A. Nusser, K. Masters, C. Springob, J. P. Huchra, and G. Lemson, Mon. Not. Roy. Astron. Soc. 413, 2906 (2011), arXiv:1011.3114 [astro-ph.CO]

    2011 arXiv

  44. [52]

    Song and W

    Y.-S. Song and W. J. Percival, JCAP 10, 004 (2009), arXiv:0807.0810 [astro-ph]

    2009 arXiv

  45. [53]

    Blake et al

    C. Blake et al. , Mon. Not. Roy. Astron. Soc. 436, 3089 (2013), arXiv:1309.5556 [astro-ph.CO]

    2013 arXiv

  46. [54]

    S. Alam, M. Aubert, S. Avila, C. Balland, J. E. Bautista, M. A. Bershady, D. Bizyaev, M. R. Blanton, A. S. Bolton, J. Bovy, J. Brinkmann, J. R. Brownstein, E. Burtin, S. Chabanier, M. J. Chapman, P. D. Choi, C.-H. Chuang, J. Comparat, M.- C. Cousinou, A. Cuceu, K. S. Dawson, S...

    2021 arXiv

  47. [55]

    Blake et al

    C. Blake et al. , Mon. Not. Roy. Astron. Soc. 425, 405 (2012), arXiv:1204.3674 [astro-ph.CO]

    2012 arXiv

  48. [56]

    Pezzotta, S

    A. Pezzotta, S. de la Torre, J. Bel, B. R. Granett, L. Guzzo, J. A. Peacock, B. Garilli, M. Scodeggio, M. Bolzonella, U. Abbas, C. Adami, D. Bottini, A. Cappi, O. Cucciati, I. Davidzon, P. Franzetti, A. Fritz, A. Iovino, J. Krywult, V. L. Brun, O. L. F` e vre, D. Maccagni, K. ...

    2017 arXiv

  49. [57]

    Okumura et al

    T. Okumura et al. , Publ. Astron. Soc. Jap. 68, 38 (2016), arXiv:1511.08083 [astro-ph.CO]

    2016 arXiv

  50. [58]

    Zhao et al

    G.-B. Zhao et al. (eBOSS), Mon. Not. Roy. Astron. Soc. 482, 3497 (2019), arXiv:1801.03043 [astro-ph.CO]

    2019 arXiv

  51. [59]

    As we will see the uncertainties on N grow are dominated by uncertainties on the Weyl potential, hence adding DESI measurements is not expected to modify the results

    Note that we do not include the recent measurements of ˆf from DESI [75], since they may be correlated with the other measurements, and we do not have access to the covariance. As we will see the uncertainties on N grow are dominated by uncertainties on the Weyl potential, hen...

  52. [60]

    A. R. Liddle, Mon. Not. Roy. Astron. Soc. 377, L74 (2007), arXiv:astro-ph/0701113

    2007 arXiv

  53. [61]

    https://www.desi.lbl.gov

  54. [62]

    https://rubinobservatory.org

  55. [63]

    Aghamousa et al

    A. Aghamousa et al. (DESI), (2016), arXiv:1611.00036 [astro-ph.IM]

    2016 arXiv

  56. [64]

    Amendola, M

    L. Amendola, M. Kunz, and D. Sapone, JCAP 0804, 013 (2008), arXiv:0704.2421 [astro-ph]

    2008 arXiv

  57. [65]

    Bertschinger and P

    E. Bertschinger and P. Zukin, Phys. Rev. D 78, 024015 (2008), arXiv:0801.2431 [astro-ph]

    2008 arXiv

  58. [66]

    Pogosian, A

    L. Pogosian, A. Silvestri, K. Koyama, and G.-B. Zhao, Phys. Rev. D 81, 104023 (2010), arXiv:1002.2382 [astro-ph.CO]

    2010 arXiv

  59. [67]

    P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, E. Battaner, R. Battye, K. Benabed, A. Beno ˆ ı t, A. Benoit-L´ evy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bon...

    2016 arXiv

  60. [68]

    T. M. C. Abbott et al. (DES), Phys. Rev. D 107, 083504 (2023), arXiv:2207.05766 [astro-ph.CO]

    2023 arXiv

  61. [69]

    A. G. Adame et al. (DESI), (2024), arXiv:2411.12022 [astro-ph.CO]

    2024 arXiv

  62. [70]

    changing the link between f and D1, would break N grow

    Note however that dark matter models violating the continuity equation, i.e. changing the link between f and D1, would break N grow. In models where dark matter interacts via a scalar field, such violations are typically negligible [76], but this possibility should be consider...

  63. [71]

    Grimm, C

    N. Grimm, C. Bonvin, and I. Tutusaus, (2025), arXiv:2502.12843 [astro-ph.CO]

    2025 arXiv

  64. [72]

    Gleyzes, D

    J. Gleyzes, D. Langlois, M. Mancarella, and F. Vernizzi, JCAP 08, 054 (2015), arXiv:1504.05481 [astro-ph.CO]

    2015 arXiv

  65. [73]

    Gleyzes, D

    J. Gleyzes, D. Langlois, M. Mancarella, and F. Vernizzi, JCAP 02, 056 (2016), arXiv:1509.02191 [astro-ph.CO]

    2016 arXiv

  66. [74]

    Raveri, L

    M. Raveri, L. Pogosian, K. Koyama, M. Martinelli, A. Silvestri, G.-B. Zhao, J. Li, S. Peirone, and A. Zucca, JCAP 02, 061 (2023), arXiv:2107.12990 [astro-ph.CO]

    2023 arXiv

  67. [75]

    A. G. Adame et al. (DESI), (2024), arXiv:2411.12021 [astro-ph.CO]

    2024 arXiv

  68. [76]

    Bonvin and L

    C. Bonvin and L. E. Pogosian, Nature Astron. 7, 1023 (2023)

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.