Testing General Relativity using Large Scale Structures Photometric Redshift Surveys and Cosmic Microwave Background Lensing Effect

Jun-Qing Xia; Shang Li

REVIEW 3 major objections 5 minor 68 references

Testing General Relativity using Large Scale Structures Photometric Redshift Surveys and Cosmic Microwave Background Lensing Effect

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read General relativity passes a new DES-Planck lensing test

desk verdict New E_G measurements from DES MagLim plus Planck lensing are broadly consistent with GR, but the fourth bin sits ~3σ low and the paper's consistency claim rests on an untested bias assumption. read the letter →

arxiv 2501.02852 v1 pith:JKIN5ZZA submitted 2025-01-06 astro-ph.CO

Shang Li , Jun-Qing Xia This is my paper · ORCID

classification astro-ph.CO

keywords E_GstatisticgeneralrelativitymodifiedgravityweakgravitationallensingCMBgalaxyclusteringphotometricredshiftsurveyslarge-scalestructure

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 sets out to test whether general relativity holds on cosmological scales by measuring the $E_G$ statistic, which compares the gravitational-lensing signal of matter with the growth of cosmic structure, using photometric galaxies from the Dark Energy Survey and the Planck 2018 CMB lensing map. Because photometric redshifts are imprecise, the authors reconstruct the growth rate $f\sigma_8$ with an artificial neural network and take the galaxy bias $b\sigma_8$ from DES chains, forming the redshift-space-distortion parameter $\beta = f\sigma_8/b\sigma_8$. They report $E_G = 0.354 \pm 0.146$, $0.452 \pm 0.092$, $0.414 \pm 0.069$, and $0.296 \pm 0.069$ at $z = 0.30$, $0.47$, $0.63$, and $0.80$, all consistent with $\Lambda$CDM. A forecast with the China Space Station Telescope and CMB-S4 indicates $E_G$ could be measured to roughly 1% precision, which would separate general relativity from several modified-gravity models.

What carries the argument

The central object is the $E_G$ statistic introduced by Zhang et al. (2007), estimated here as $E_G(\ell,\bar{z}) = \Gamma(\bar{z})\, C_\ell^{g\kappa} / [\beta(\bar{z})\, C_\ell^{gg}]$, where $C_\ell^{g\kappa}$ is the galaxy-CMB lensing cross-power spectrum, $C_\ell^{gg}$ the galaxy auto-power spectrum, and $\beta$ the redshift-space-distortion parameter. The paper's operational move is $\beta = f\sigma_8/b\sigma_8$: the numerator comes from an ANN fit (ReFANN) to 66 spectroscopic growth-rate measurements, and the denominator from model-independent DES Y3 chains. The calibration factor $\Gamma(z)$ corrects for broad redshift distributions, the lensing kernel, and scale-dependent bias, while jackknife resampling supplies the covariance with Hartlap and Percival corrections for the inverse covariance.

What would settle it

Measure $b\sigma_8$ at $z \approx 0.8$ from an independent probe, for example the MagLim auto-correlation alone or ACT DR4 lensing with cosmology held fixed; if it comes out significantly above $0.865 \pm 0.034$, the fourth-bin $E_G$ rises and may land above the $\Lambda$CDM prediction, directly testing the paper's bias-underestimation explanation.

Watch

Extended reading notes

Core claim

The central claim is that a photometric-redshift survey can deliver meaningful $E_G$ constraints once the redshift-space-distortion parameter is rewritten as $\beta = f\sigma_8/b\sigma_8$, with $f\sigma_8$ reconstructed from the existing spectroscopic growth-rate compilation and $b\sigma_8$ taken from DES Y3 chains. Applying this to four MagLim tomographic bins yields $E_G = 0.354 \pm 0.146$, $0.452 \pm 0.092$, $0.414 \pm 0.069$, and $0.296 \pm 0.069$ at $z = 0.30$, $0.47$, $0.63$, $0.80$, consistent with $\Lambda$CDM and with earlier estimates. The paper also claims that the same pipeline applied to future CSST and CMB-S4 data will reach roughly 1% precision, enough to distinguish general relativity from chameleon and $f(R)$ models.

Load-bearing premise

The analysis assumes that the $b\sigma_8$ values taken from the DES Y3 3x2pt chains are unbiased in every tomographic bin, especially the highest one, where the chains give an unusually low value.

Editorial extensions

If this is right

General relativity remains consistent with the combined DES and Planck lensing data at redshifts 0.30 to 0.80, with no significant scale dependence in $E_G$.
Photometric surveys can test gravity using the $\beta = f\sigma_8/b\sigma_8$ split instead of direct redshift-space-distortion measurements.
The low $E_G$ in the highest redshift bin likely reflects an underestimated $b\sigma_8$ from the DES chains rather than new physics; a corrected bias would move it toward the $\Lambda$CDM prediction.
Future CSST and CMB-S4 data could reach roughly 1% precision on $E_G$, separating general relativity from chameleon gravity at the 5$\sigma$ level and from $f(R)$ gravity at the 13$\sigma$ level for $B_0 > 10^{-7}$.
Magnification bias shifts the forecast $E_G$ by up to roughly 6%, so future high-precision analyses must include it explicitly.

Reading between the lines

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

If the fourth-bin bias is indeed underestimated, an independent $b\sigma_8$ measurement from the MagLim auto-correlation alone or from ACT lensing would shift $E_G(z\approx0.8)$ upward, providing a direct test of the paper's explanation.
The ANN reconstruction inherits the selection of the 66-point growth-rate compilation; swapping in a stricter or updated RSD sample would quantify how much of the result depends on that choice.
The same $\beta$-split pipeline transfers naturally to other photometric surveys such as LSST and Euclid, where magnification-bias corrections will matter at the few-percent level.
Combining future CMB-S4 lensing with CSST clustering may also break the current degeneracy between galaxy bias and growth in the highest tomographic bin.

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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

You should know this paper before you delegate it: it is a legitimate application of the existing E_G estimator to DES MagLim plus Planck lensing, producing four new E_G measurements that are consistent with GR within the uncertainties — except possibly the fourth bin, where the paper's own consistency claim is on thin ice.

What is actually new: the numbers in Eq. (31), i.e., E_G = 0.354 ± 0.146, 0.452 ± 0.092, 0.414 ± 0.069, 0.296 ± 0.069 at z = 0.30, 0.47, 0.63, 0.80 for the DES MagLim sample combined with Planck 2018 lensing. The methodology is taken from Pullen et al. (2015) and Wenzl et al. (2024), including the β = fσ8/bσ8 formulation — which is an algebraic identity with f/b, not a new idea, and the paper's wording "new definition" is overstatement. The mass of the paper is a careful application: NaMaster pseudo-Cℓ, jackknife covariances with Hartlap and Percival corrections, conservative scale cuts, and a reasonable treatment of magnification bias as a systematic. That part is solid.

Now the soft spots. The main issue is the fourth bin. The stress-test is right: with the paper's own fiducial cosmology, the ΛCDM prediction at z = 0.80 is about 0.51, and the measured 0.296 ± 0.069 is ~3σ low. The paper notices and suggests bσ8(z4) from the DES chains is underestimated, citing Marques et al. (2024). But it never tests that hypothesis by recomputing E_G with a different bias. If bσ8(z4) is unbiased, the fourth bin is a real tension and the abstract's blanket consistency statement fails. If bσ8(z4) is biased as suspected, then the quoted E_G(z4) is itself the wrong number. Either way, the headline result is not robust until this is resolved. The authors should be required to either drop the fourth bin, rerun with an alternative bias, or explicitly present the result as conditional. Second, the ANN-derived fσ8 errors in Table 2 are implausibly small: ~0.01 on fσ8 from input measurements with ~0.05–0.1 errors. The ANN output error should account for scatter in the training data; as reported, it looks like an artifact of the regression. This doesn't change the E_G values much, but it understates the error budget. Third, the forecast of ~1% precision sits on top of the paper's own 6% magnification-bias estimate, so the claim "achievable constraint of approximately 1%" needs a caveat about unmodeled systematics.

Overall: this is a useful data point, not a breakthrough. The measurement methodology is sound; the presentation is honest about the fourth-bin issue, but the paper's own claim goes beyond what it demonstrates. A serious referee should require the bσ8(z4) resolution and a softening of the abstract before publication. I would send it to review with a request for major revision.

Referee Report

3 major / 5 minor

Summary. This paper measures the E_G statistic, a ratio of gravitational lensing to galaxy clustering, as a test of general relativity on cosmological scales. The authors use the DES Y3 MagLim photometric galaxy sample and the Planck 2018 CMB lensing map, estimating the angular power spectra C_ℓ^gg and C_ℓ^gκ with the NaMaster pseudo-Cℓ estimator. To handle the photometric-redshift limitation, they introduce a new RSD parameter β = fσ8/bσ8, where fσ8 is reconstructed from 66 spectroscopic growth-rate measurements using an artificial neural network (ReFANN) and bσ8 is taken from the DES Y3 3×2pt chains. They obtain E_G = 0.354 ± 0.146, 0.452 ± 0.092, 0.414 ± 0.069, and 0.296 ± 0.069 in four redshift bins centered at z = 0.30, 0.47, 0.63, and 0.80, and claim consistency with ΛCDM. They also forecast E_G constraints for the CSST photometric survey combined with CMB-S4 lensing, projecting ~1% precision.

Significance. If the measurements are robust, the paper provides new E_G constraints from a photometric survey combined with CMB lensing, extending the redshift range of E_G tests and demonstrating a novel ANN-based treatment of the RSD parameter. The use of public codes, conservative scale cuts, and Hartlap/Percival covariance corrections are strengths, as is the explicit acknowledgment of the fourth-bin bias issue in Section 4.4. However, the headline consistency claim is sensitive to the fourth redshift bin, where the measured E_G is about 3σ below the ΛCDM prediction under the paper's own fiducial cosmology. The paper's suggested explanation (underestimated bσ8 at z≈0.80) is plausible but untested, and the ANN-derived fσ8 errors appear implausibly small given the input data errors. These issues, if resolved, would make the paper a useful contribution; as it stands, the central claim is not yet fully supported.

major comments (3)

[§4.4 and Eq. (31)] The fourth-bin measurement E_G(z4) = 0.296 ± 0.069 is approximately 3σ below the ΛCDM prediction computed from the paper's own fiducial cosmology (Ω_m0 = 0.336, f(z) from GR), as the text itself notes in §4.4. The only resolution offered is the suggestion that bσ8(z4) = 0.865 ± 0.034 from the DES Y3 chains is underestimated, which would inflate β and suppress E_G. This explanation is not tested: the paper never recomputes E_G with an alternative bias value or adds a systematic error to absorb the discrepancy. The abstract's blanket statement that all four measurements 'are consistent with the predictions of the standard ΛCDM model' is therefore not supported by the presented analysis. The authors should either provide a quantitative test of the bias hypothesis or qualify the consistency claim to exclude or downgrade bin 4.
[Table 2 and §4.1.1] The ANN-reconstructed fσ8 values have quoted uncertainties of ~0.010–0.015, which are much smaller than the typical errors of the 66 input measurements (many are 0.03–0.18, see Table 1). The paper gives no description of how these uncertainties are computed (e.g., bootstrap, network variance, or covariance with the input data) and no validation that the ANN error estimate is unbiased. Since β = fσ8/bσ8 and the E_G estimate depends linearly on β, an underestimated fσ8 error would propagate into underestimated β and E_G errors, potentially affecting the significance of the consistency claims. Please provide details of the ReFANN error estimation and a comparison with a more standard reconstruction (e.g., a Gaussian process with explicit prior sensitivity checks).
[§2.3, Eqs. (11)–(15)] The calibration factor Γ(ℓ,z) is computed using theoretical C^{mg}_ℓ and Q^{mg}_ℓ that depend on the fiducial ΛCDM parameters (Ω_m0, σ8, h, etc.) and on the assumed linear matter power spectrum. The paper states that E_G is independent of galaxy bias and σ8, but this only holds if the calibration factor is sufficiently precise and if the bias and growth inputs are mutually consistent. The bσ8 values are taken from DES Y3 3×2pt chains that assume a particular background cosmology and lensing kernel, so the claimed 'model-independent' status of the bσ8 input is overstated. The paper should state whether the calibration factor's theoretical uncertainty is propagated into the E_G error budget, and if not, justify why it is negligible.

minor comments (5)

[Abstract and §6] The phrase 'which are consistent with the predictions of the standard ΛCDM model' should be made conditional on the fourth-bin caveat, or the conclusion should be rephrased to 'consistent within the current large uncertainties, with the possible exception of the highest-redshift bin.'
[§4.1.1, Table 2] The effective redshifts are quoted to two decimal places, but the bin edges are given as 0.20–0.40, 0.40–0.55, etc. It would be clearer to state whether the effective redshift is computed from the weighted n(z) and to provide the redshift distribution of each bin in a table or figure with the effective redshifts marked.
[Figure 3] The right panels show R_ℓ = C^{gκ}_ℓ / C^{gg}_ℓ, but the ordinate ranges for the four bins differ by orders of magnitude. Please use the same scale or explicitly note the different y-axis limits to avoid visual overinterpretation of the fluctuations.
[§5.3, Eq. (38)] The forecast covariance formula uses the standard Gaussian expression, but the shot-noise and beam terms for the CMB-S4 lensing map should be specified more precisely (particularly the N^{κκ}_ℓ term) so that the forecast is reproducible.
[Throughout] There are several typographical errors, including 'preform' for 'perform' in §4.1.1 and inconsistent use of 'EG' vs 'E_G' in the text and figures. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the E_G measurement combines independent external inputs and observed spectra; the flagged fourth-bin bσ8 issue is a robustness limitation, not a circular reduction.

full rationale

The derivation chain is not circular. E_G is measured from Eq. (5) as Γ(ℓ,z) C_gκ/(β C_gg), using observed DES MagLim and Planck lensing angular power spectra. The RSD parameter is built from β = fσ8/bσ8, where fσ8 is an ANN (ReFANN) reconstruction of 66 external spectroscopic RSD measurements and bσ8 comes from DES Y3 chains; neither is fitted to the E_G values. The theoretical comparison E_G = Ωm0/f is an independent ΛCDM prediction; the calibration factor Γ is a redshift-distribution/nonlinearity correction from Pullen et al. (2016)/Yang & Pullen (2018) and does not absorb the measured C_gκ/C_gg ratio. The self-citation to ReFANN (Wang et al. 2020) is a code/method citation, not a circular justification, and the cited DES chains are external data products. The paper itself (Section 4.4 and Conclusions) notes that the fourth-bin E_G is lower than ΛCDM and suggests an underestimated bσ8(z4) from the DES chains; this is an admitted robustness/correctness limitation and an unresolved tension, but it does not make the quoted measurement equivalent to its inputs by construction. The forecast section is a simulation under an assumed fiducial model, so its internal consistency is not a circular 'prediction.' Overall, no prediction reduces to its input; the concern about bσ8(z4) belongs to data robustness, not circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central E_G estimate depends on external inputs: 66 literature fσ8 measurements, DES Y3 3x2pt bσ8 chains, and a fiducial ΛCDM cosmology for the calibration factor. The ANN fit and the DES chains are both data-driven but carry model assumptions that the paper does not fully propagate.

free parameters (3)

ANN fitted fσ8 values = 0.432, 0.438, 0.438, 0.434
Outputs of the ReFANN regression on 66 literature growth-rate measurements; the paper treats these as measured values with ~0.01 errors, but the fit is a model-dependent interpolation with correlated inputs.
Scale cuts ℓmax per bin = 188, 390, 400, 400
Chosen to exclude non-linear scales; affects the amount of data used and therefore the reported uncertainties.
Fiducial cosmology for forecast = Ωm=0.32, ns=0.9665, σ8=0.83, h0=0.72
Used as fiducial values in the CSST/CMB-S4 MCMC forecast; the final E_G uncertainty projection depends on these choices.

assumptions (6)

domain assumption Flat ΛCDM background metric
Section 2.1 Eq. (1) assumes a flat Friedmann-Robertson-Walker metric and scalar perturbations.
standard math Linear perturbation theory and Limber approximation
Section 2.2 Eq. (6) uses the Limber approximation to compute angular power spectra from 3D power spectra.
domain assumption Scale-independent RSD parameter β = f/b
Equation (5) assumes β is constant across the multipole range used in the E_G estimator.
ad hoc to paper Unbiased bσ8 constraints from DES Y3 3x2pt chains
The paper uses bσ8 from Abbott et al. (2023) and itself questions the fourth-bin value in Section 4.4.
ad hoc to paper ANN reconstruction of fσ8 is unbiased
The ReFANN fit on 66 literature measurements is assumed to give a faithful estimate at the DES effective redshifts.
domain assumption Fiducial cosmological parameters from Abbott et al. (2023)
Section 1 states Ωm=0.336, Ωb=0.045, h=0.670, ns=0.959, σ8=0.746 are assumed for self-consistency; used in calibration factor and theoretical predictions.

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Cite this review

Pith. "Pith review of Testing General Relativity using Large Scale Structures Photometric Redshift Surveys and Cosmic Microwave Background Lensing Effect." pith.science (2026). https://pith.science/paper/JKIN5ZZA

@misc{pith2026250102852,
  author       = {Pith},
  title        = {Pith review of: Testing General Relativity using Large Scale Structures Photometric Redshift Surveys and Cosmic Microwave Background Lensing Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKIN5ZZA}},
  note         = {Machine review of arXiv:2501.02852}
}

abstract

The $E_G$ statistic provides a valuable tool for evaluating predictions of General Relativity (GR) by probing the relationship between gravitational potential and galaxy clustering on cosmological scales within the observable universe. In this study, we constrain the $E_G$ statistic using photometric redshift data from the Dark Energy Survey (DES) MagLim sample in combination with the Planck 2018 Cosmic Microwave Background (CMB) lensing map. Unlike spectroscopic redshift surveys, photometric redshift measurements are subject to significant redshift uncertainties, making it challenging to constrain the redshift distortion parameter $\beta$ with high precision. We adopt a new definition for this parameter, $\beta(z) = {f\sigma_8(z)}/{b\sigma_8(z)}$. In this formulation, we reconstruct the growth rate of structure, $f\sigma_8(z)$, using Artificial Neural Networks (ANN) method, while simultaneously utilizing model-independent constraints on the parameter $b\sigma_8(z)$, directly obtained from the DES collaboration. After obtaining the angular power spectra $C_\ell^{gg}$ (galaxy-galaxy) and $C_\ell^{g\kappa}$ (galaxy-CMB lensing) from the combination of DES photometric data and Planck lensing, we derive new measurements of the $E_G$ statistic: $E_G = 0.354 \pm 0.146$, $0.452 \pm 0.092$, $0.414 \pm 0.069$, and $0.296 \pm 0.069$ (68$\%$ C.L.) across four redshift bins: $z = 0.30, 0.47, 0.63$, and $0.80$, respectively, which are consistent with the predictions of the standard $\Lambda$CDM model. Finally, we forecast the $E_G$ statistic using future photometric redshift data from the China Space Station Telescope, combined with lensing measurements from the CMB-S4 project, indicating an achievable constraint on $E_G$ of approximately 1$\%$, improving the precision of tests for GR on cosmological scales.

Figures

Figures reproduced from arXiv: 2501.02852 by the authors.

**Figure 1.** Redshift distributions of the DES Y3 MagLim sample. The kernel function of the CMB lensing is also shown in the figure. For clarity, all results have been normalized. where ¯n represents the mean number of sources in the unmasked pixels and is calculated as: n¯ = P p Np P p fp . (17) Here, fp denotes the fractional coverage of each pixel, which accounts for the DES mask. The DES mask is provided at a higher resolut… view at source ↗

**Figure 2.** The reconstruction of the function ˆf(z) from the ˆf measurements, shown as the black solid line, along with the 68% confidence level indicated by the light green region. The blue points with error bars represent the ˆf values obtained from the literature, while the red stars denote the inferred ˆf values for the DES MagLim sample. For comparison, the theoretical prediction of ˆf(z) based on the ΛCDM model is also d… view at source ↗

**Figure 3.** The observed power spectra C gg ℓ (left panels), C gκ ℓ (middle panels) and their ratio Rℓ (right panels) for the four redshift bins of the DES MagLim sample are shown. The solid orange lines depict the theoretical predictions evaluated using the best-fit parameters from Abbott et al. (2023). The gray-shaded regions highlight the range of multipoles that were excluded from the analysis due to the non-linearity. powe… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The estimations of the EG statistic for the DES MagLim samples, combined with the Planck lensing measurements at four redshift bins. The blue lines represent the theoretical prediction within the ΛCDM model, which is scale independent. consistency between the observed …

**Figure 5.** Figure 5: The scale-independent measurements of the EG statistic for the four redshift bins of the DES MagLim sample. W.M. and Wo.M. represent measurements with and without the magnification bias effect, respectively. For comparison, previous EG measurements from other studies…

**Figure 6.** Figure 6: The estimations of the EG statistic at redshift bins for the CSST photometric redshift survey, combined with the CMB-S4 lensing measurements. The blue lines represent the theoretical prediction within the ΛCDM model [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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