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A low-redshift test of structure growth finds S8 consistent with Planck but a 2.8-sigma low matter density.

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2026-08-04 10:26 UTC pith:6DDA2D5R

load-bearing objection Solid low-z growth measurement with a careful robustness analysis, but the 2.8σ low-Ωm headline softens to ~1σ under the paper's own alternative bias model. the 2 major comments →

arxiv 2510.09563 v2 pith:6DDA2D5R submitted 2025-10-10 astro-ph.CO

Low-redshift constraints on structure growth from CMB lensing tomography

Andrea Rubiola , Matteo Zennaro , Carlos Garc\'ia-Garc\'ia , David Alonso , Raul Angulo This is my paper
classification astro-ph.CO
keywords S8structure growthCMB lensing tomographygalaxy clusteringhybrid effective field theorylow redshiftmatter densityBAO prior
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper targets the least-explored epoch for structure-growth measurements: redshifts below about 0.3, where dark energy dominates the expansion and where late-time deviations in the growth of matter fluctuations would be most visible. Combining the projected clustering of three low-redshift galaxy samples with their cross-correlation against CMB lensing, and modelling galaxy bias with a hybrid effective field theory valid on mildly non-linear scales, the authors try to measure the growth amplitude S8 without relying on higher-redshift probes. With a BAO prior on the matter density they find S8=0.79±0.06, consistent with Planck and with other lensing tomography analyses. Without that prior, the data prefer Ωm=0.245±0.024, 2.8σ below Planck, a preference driven by the broadband shape of the highest-redshift galaxy auto-correlation and flagged as potentially sensitive to theoretical uncertainties in the bias templates. The low-redshift growth history they reconstruct is compatible with Planck, with a weak hint of steeper growth at the lowest redshifts.

Core claim

The central claim is that CMB lensing tomography at z≲0.3 can constrain the amplitude of matter fluctuations well enough to test Planck's growth prediction, provided the galaxy bias is modelled with a hybrid effective field theory. Using three low-redshift galaxy bins and CMB lensing from Planck, the paper reports S8=0.79±0.06 (with a BAO prior on Ωm) and σ8=0.80±0.06, both in agreement with Planck; when Ωm is left free, the data prefer Ωm=0.245±0.024, a 2.8σ disagreement. It further reconstructs σ8(z) at z≈0.07, 0.18 and 0.30, finding values compatible with Planck, and shows that the inferred HEFT bias parameters sit within 1σ of coevolution relations calibrated on galaxy formation simulati

What carries the argument

The load-bearing machinery is the hybrid effective field theory (HEFT) bias expansion, which writes the galaxy overdensity as an advected combination of the linear density, its square, the tidal field, and a derivative operator, with free bias coefficients; the operator cross-spectra come from an N-body emulator valid up to k≲0.6 h/Mpc. To go outside the emulator's training range in Ωm, the paper extrapolates the ratio of each operator spectrum to the matter spectrum using a first-order Taylor expansion, which introduces up to about 4% error near the BAO scale. Around this model sits a pseudo-Cℓ likelihood with analytic Gaussian, connected non-Gaussian, shot-noise, and redshift-distribution

Load-bearing premise

The result rests on the HEFT power-spectrum templates remaining accurate when extrapolated in matter density down to Ωm≈0.245, a regime where the emulator was not trained and where the extrapolation error reaches roughly 4% near the BAO scale.

What would settle it

Retrain the HEFT operator templates on N-body simulations run at Ωm≈0.245 and re-fit the same data vector: if the preferred Ωm moves back toward 0.30, the 2.8σ low-density hint is a template artifact; if it stays near 0.245, the preference is robust.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, low-redshift structure growth is consistent with Planck's ΛCDM prediction, leaving little room for new physics in growth at z<0.3.
  • The low Ωm preference, if real, would add a new low-redshift large-scale-structure anomaly driven by the shape of the galaxy power spectrum rather than by the BAO standard ruler.
  • Full HEFT modelling out to k≈0.42 Mpc⁻¹ does not yield smaller σ8 errors than a linear bias model cut at k≈0.11 Mpc⁻¹, since the extra small-scale modes mostly pay for the freedom in bias parameters.
  • The per-bin σ8(z) values provide a growth-history anchor at z≈0.1–0.3 that can be combined with higher-redshift lensing tomography to map the full growth curve.
  • The inferred bias parameters support the coevolution relations between linear and higher-order bias, potentially justifying simulation-based priors in future low-redshift analyses.

Where Pith is reading between the lines

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

  • If the Ωm preference is physical rather than a template artifact, it would join other low-Ωm hints from cosmic shear and intermediate-redshift lensing tomography, pointing either to a subtle shape-calibration issue in projected clustering or to a real tension between the BAO ruler and the broadband power-spectrum shape.
  • The ~4% template extrapolation error near the BAO scale is of the same order as the shift that separates Ωm=0.245 from 0.30, so a direct N-body validation at low Ωm would settle whether the 2.8σ result is evidence or systematics.
  • The weak constraining power of the two lowest-redshift bins suggests that pushing lensing tomography deep into the dark-energy epoch will require magnification information, smaller physical scales, or higher-density samples rather than simply more sky area.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper presents low-redshift (z ≲ 0.3) measurements of galaxy auto-power spectra and galaxy–CMB lensing cross-power spectra using 2MPZ and WISE×SuperCOSMOS galaxies in three tomographic bins, combined with Planck CMB lensing. Galaxy bias is modeled with a hybrid effective field theory (HEFT) approach whose template spectra are taken from the BACCO N-body emulator. Within flat ΛCDM, the authors report S8 = 0.79 ± 0.06 when a DESI BAO prior is imposed on Ωm, in good agreement with Planck; when Ωm is left free they find Ωm = 0.245 ± 0.024, a 2.8σ departure from Planck, and S8 = 0.82 ± 0.07. They also reconstruct σ8(z) at three redshifts, compare bias parameters with coevolution relations, and test robustness to scale cuts, a linear bias model, p(z) uncertainties, and volume effects via analytic marginalization. Appendix A repeats the Ωm measurement with an empirical bias model to isolate shape information.

Significance. If the S8 result is correct, it is a useful independent confirmation of Planck-compatible growth at very low redshifts using a sophisticated bias model and a complementary data combination. The paper has real strengths: the S8 value is stable across several analysis choices, volume effects are treated explicitly through the AAM procedure, and the Appendix A empirical-bias reanalysis is a valuable internal cross-check that explicitly probes the origin of the Ωm preference. The main weakness is that the headline 2.8σ low-Ωm anomaly is not stable under a change of bias model: Appendix A itself reduces the significance to roughly 1.5σ. The paper is honest about the HEFT extrapolation caveat, but the abstract and conclusions still foreground the 2.8σ number, which overstates the robustness of that anomaly.

major comments (2)
  1. [§4.1, Table 2; Appendix A, Table 4] The abstract's headline claim that the data favour Ωm = 0.245 ± 0.024, 2.8σ below Planck, is not robust to the bias model. The empirical bias model of Eq. (A.1), applied to the same data, gives Ωm = 0.264 ± 0.035 for all bins (PTE 62%) and Ωm = 0.250 ± 0.036 for Bin 3, i.e. only ~1.5σ and ~1.8σ below Planck. Since Appendix A shows the Ωm constraint is driven by the broadband shape of the Bin 3 galaxy auto-spectrum, the smaller significance from a model that does not depend on HEFT operator templates should be reported alongside the fiducial HEFT number in the abstract and conclusions. The 2.8σ significance overstates the current robustness of the low-Ωm preference.
  2. [§3.4.2, Eqs. (3.13)–(3.15), Fig. 2] The free-Ωm posterior extends to Ωm ≈ 0.245, outside the baccoemu training range, and the first-order Taylor extrapolation of Rij shows up to ~4% errors at k ~ 0.1 Mpc^{-1}, precisely the BAO-scale shape that drives the Ωm preference. The paper acknowledges this in the abstract, but it does not propagate the extrapolation uncertainty into the Ωm constraint. I request either (i) explicit validation of the emulator at low Ωm, (ii) a systematic error term in the Ωm measurement, or (iii) removal of the 2.8σ significance claim from the abstract. Without one of these, the free-Ωm result should be described only as preliminary.
minor comments (4)
  1. [§5] There is a duplicated word in the sentence “various types of galaxies galaxies”; please fix.
  2. [Fig. 5] The caption states that black points and error bars mark the mean and 68% intervals. For the skewed posteriors shown, reporting the median or mode would be less sensitive to tails; please clarify or change the reported statistic.
  3. [Eqs. (3.7), (3.10)] The symbol N^{gg}_ℓ is used for the analytic shot-noise spectrum in Eq. (3.10) while ~N^{ab}_ℓ denotes the noise pseudo-spectrum in Eq. (3.7). Please distinguish these notations explicitly to avoid confusion.
  4. [References] Reference [68] appears as “ThBeyond-2pt Collaboration”; this is likely a typo for “The Beyond-2pt Collaboration”.

Circularity Check

0 steps flagged

No significant circularity: HEFT templates come from N-body simulations independent of the target data, and the low-Ωm preference is not forced by construction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The theoretical model uses Hybrid Effective Field Theory with Pij templates measured in the BACCO N-body simulations and accelerated by an emulator [48,51,52]. These templates are calibrated on simulations, not on the 2MPZ/WISE×SuperCOSMOS or Planck lensing data used in the likelihood, so the cosmological constraints are not equivalent to a fitted input relabeled as a prediction. The bias parameters are free nuisance parameters fitted to the data, but the quantities reported as constraints—S8, Ωm, σ8—are not definitions of the fitted parameters. The free-Ωm preference, Ωm=0.245±0.024, is driven by the broadband shape of the galaxy auto-spectrum; the paper explicitly tests this interpretation with a different, empirical bias model in Appendix A, which reproduces a low-Ωm value (0.264±0.035), showing that the preference is not an artifact of the HEFT parametrization. The self-citations to the BACCO/HEFT program, the AAM marginalization procedure, and the empirical bias model are to externally validated tools: N-body simulations, statistical methods, and a different bias model, none of which contains the 2MPZ/WIxSC data or the Planck lensing measurements as inputs. The abstract's caveat that the low-Ωm result 'may be affected by theoretical uncertainties in the HEFT power spectrum templates' is a modeling-robustness concern, not evidence that the derivation is circular. The BAO-prior S8 result uses an external DESI prior, and the growth reconstruction is a derived quantity from standard ΛCDM, not a redefinition of the input. I find no step where the claimed output reduces by construction to the input.

Axiom & Free-Parameter Ledger

4 free parameters · 8 axioms · 0 invented entities

The ledger separates the scientific targets (σ8, Ωm, S8) from the nuisance parameters. The only genuinely ad hoc ingredient is the first-order Taylor extrapolation of the HEFT emulator outside its training range (Section 3.4.2); all other inputs are standard operational choices, validated approximations, or external data products. No new physical entities are introduced.

free parameters (4)
  • b1,i, b2,i, bs,i, b∇2,i (i=1,2,3) = Posterior means not explicitly quoted; broad Gaussian priors N(0,5) for b1,b2,bs and N(0,10 Mpc²) for b∇2
    Twelve HEFT bias nuisance parameters entering Eq. 3.4; marginalized analytically via AAM for cosmology and explicitly sampled for bias inference.
  • ASN,i (residual shot-noise amplitude per bin) = Not quoted; marginalized with σ_SN=0.1 Gaussian prior
    Accounts for non-Poisson stochastic contributions to galaxy auto-spectra (Section 3.3, Eq. 3.10).
  • p(z) bin amplitudes = Not quoted; bootstrap covariance from DIR calibration
    Marginalized via Eq. 3.11; assumes Gaussian errors and no inter-bin covariance.
  • Mmin,i and M1,i (HOD parameters for cNG covariance) = Not quoted; fitted to each bin's auto-spectrum
    Used to estimate the connected non-Gaussian covariance; subdominant on scales used; α=1, M0=Mmin, σlnM=0.4 fixed.
axioms (8)
  • domain assumption Flat ΛCDM background metric used for distances, kernels, and growth factor (Section 3.1, Eq. 3.2).
    All inferences assume a cosmological constant; no w0-wa or modified-gravity growth model is tested, so the growth reconstruction is model-dependent.
  • standard math Limber approximation (Eq. 3.3) is valid for these projected spectra.
    Authors test this and find <2% effect on S8, so it is not load-bearing.
  • domain assumption The Lagrangian bias expansion truncated at second order (Eq. 3.4) with operators δ, δ², s², ∇²δ is sufficient for k<0.42 Mpc⁻¹.
    HEFT validation in refs [48-50,54,67,68] supports this; failure would bias b1 and S8.
  • ad hoc to paper First-order Taylor extrapolation of HEFT templates outside the baccoemu training range is accurate (Eqs. 3.13-3.15).
    Free-Ωm posteriors reach Ωm≈0.245 outside the emulator range; Fig. 2 shows up to ~4% deviations, and the paper treats the low-Ωm result as possibly affected by this.
  • domain assumption The analytical covariance (Eq. 3.9: iNKA Gaussian + halo-model cNG + analytic shot-noise and p(z) marginalization) describes the true error distribution.
    Mis-estimated covariance would rescale quoted errors and the significance of tensions.
  • domain assumption DIR-calibrated redshift distributions and bootstrap uncertainties are unbiased and have uncorrelated bin-to-bin errors (Section 2.1).
    N(z) errors propagate via Eq. 3.11; the authors assume no correlation between bins, which may be optimistic.
  • domain assumption Empirical bias model of Appendix A (Eq. A.1) is valid to k=0.3 Mpc⁻¹.
    Used to show the Ωm preference is not an artifact of HEFT; relies on ref. [94].
  • domain assumption Fixed auxiliary parameters (Ωb h², n_s, Ωm h³=0.09633) are correct.
    Values taken from Planck/DESI; varying neutrino mass within bounds changes results by <1%, so impact is small.

pith-pipeline@v1.3.0-alltime-deepseek · 24936 in / 18866 out tokens · 174127 ms · 2026-08-04T10:26:34.450706+00:00 · methodology

0 comments
read the original abstract

We present constraints on the amplitude of matter fluctuations from the clustering of galaxies and their cross-correlation with the gravitational lensing convergence of the cosmic microwave background (CMB), focusing on low redshifts ($z\lesssim0.3$), where potential deviations from a perfect cosmological constant dominating the growth of structure could be more prominent. Specifically, we make use of data from the 2MASS photometric survey (\tmpz) and the \wisc galaxy survey, in combination with CMB lensing data from \planck. Using a hybrid effective field theory (HEFT) approach to model galaxy bias we obtain constraints on the combination $S_8=\sigma_8\sqrt{\Omega_m/0.3}$, where $\sigma_8$ is the amplitude of matter fluctuations, and $\Omega_m$ is the non-relativistic matter fraction. Using a prior on $\Omega_m$ based on the baryon acoustic oscillation measurements of DESI, we find $S_8=0.79\pm0.06$, in reasonable agreement with CMB constraints. We also find that, in the absence of this prior, the data favours a value of $\Omega_m=0.245\pm0.024$, that is 2.8$\sigma$ lower than \planck. This result is driven by the broadband shape of the galaxy auto-correlation, and may be affected by theoretical uncertainties in the HEFT power spectrum templates. We further reconstruct the low-redshift growth history, finding it to be compatible with the \planck predictions, as well as existing constraints from lensing tomography. Finally, we study our constraints on the HEFT bias parameters of the galaxy samples studied, finding them to be in reasonable agreement with coevolution predictions.

discussion (0)

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Forward citations

Cited by 3 Pith papers

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

  1. Synergy between the gravitational potential decay rate and other structure growth probes in testing gravity

    astro-ph.CO 2026-06 conditional novelty 5.0

    Combining gravitational-potential decay-rate (DR) data with Σ8 and fσ8 growth measurements tightens modified-gravity parameter constraints by ~1.5–2×, with all results consistent with general relativity.

  2. Biased tracers, Hybrid Effective Field Theory and Modified Gravity

    astro-ph.CO 2026-06 unverdicted novelty 5.0

    Extends HEFT to f(R) gravity for loop-corrected biased power spectra and outlines emulator extensions from LambdaCDM codes.

  3. Synergy between the gravitational potential decay rate and other structure growth probes in testing gravity

    astro-ph.CO 2026-06 unverdicted novelty 4.0

    Tomographic DR data added to Σ8 + fσ8 tightens phenomenological MG parameters (μ0, Σ0, η0) and EFT α coefficients by factors of 1.5–2.

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