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Reconstructing the shape of the non-linear matter power spectrum using CMB lensing and cosmic shear

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A joint CMB-lensing and cosmic-shear fit finds a roughly 2-sigma small-scale suppression in the matter power spectrum.

desk verdict Joint ACT-DR6 + DES-Y3 P(k) reconstruction is a genuinely new diagnostic, but the headline ~2σ scale-dependence rests on one low-PTE fit and a redshift-independent amplitude assumption the paper flags but does not quantify. read the letter →

arxiv 2502.06687 v1 pith:2KYQAHDJ submitted 2025-02-10 astro-ph.CO

Karen Perez Sarmiento , Alex Laguë , Mathew Madhavacheril , Bhuvnesh Jain , Blake Sherwin This is my paper · ORCID
classification astro-ph.CO
keywords matterpowerspectrumreconstructionCMBlensingcosmicshearS8tensionbaryonicfeedbackultralightaxiondarkACTDR6DESY3
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 is trying to establish that the non-linear matter power spectrum can be reconstructed directly from gravitational lensing data without assuming a particular baryonic feedback or dark matter model. It fits five multiplicative amplitudes, each rescaling the gravity-only Planck prediction in a separate wavenumber bin, to ACT DR6 CMB lensing and DES Y3 cosmic shear jointly. The joint fit is consistent with the early-universe prediction on large scales ($k < 0.2\,\mathrm{Mpc}^{-1}$) but shows a roughly $2\sigma$ deviation from scale-independence when the fit extends to $k = 10\,h/\mathrm{Mpc}$. The paper argues this is the kind of scale-dependent suppression that would resolve the $S_8$ tension, and shows that both baryonic feedback models and an ultralight axion model can reproduce the inferred shape and amplitude.

What carries the argument

The load-bearing object is the binned amplitude parameterization of Eq. (2): $P(k,z) = \sum_i \alpha_i B(k, k_l^i, k_h^i) P_{\mathrm{fid}}(k,z)$, where $B$ is a top-hat that selects wavenumber bins ($k \le 0.10$, $0.10$--$0.32$, $0.32$--$1.00$, $1.00$--$3.16$, $k > 3.16$ in $h\,\mathrm{Mpc}^{-1}$) and $P_{\mathrm{fid}}$ is a gravity-only non-linear power spectrum from the Mead2020 halofit model with Planck 2018 parameters. These $\alpha_i$ enter the Limber projection integral (Eq. 4) for the lensing convergence spectra, so the fit compares the rescaled power spectrum against both the CMB lensing kernel and the cosmic shear kernels of the DES source bins. The paper quantifies scale-independence by taking the best-fit $\alpha_i$ as data points, fitting a constant amplitude, and reporting the probability-to-exceed; a low probability-to-exceed is the evidence for a scale-dependent departure. A secondary mechanism is the two-parameter $A_\mathrm{mod}$ variation (Eq. 12) used to check whether an overall amplitude offset plus a non-linear suppression can describe the $\alpha_i$.

What would settle it

Re-fit the model with the DES source galaxies split into high- and low-redshift bins and test whether the recovered $\alpha_i$ parameters agree between bins; if they differ significantly, the redshift-independent assumption behind Eq. (2) fails and the apparent $k$-dependence is at least partly a projection of redshift-dependent suppression.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the lensing data do not support a single constant rescaling of the gravity-only Planck matter power spectrum across all fitted scales. With DES cosmic shear alone, the reconstruction sits below the Planck-based prediction on all fitted scales, at an overall amplitude of $0.85 \pm 0.05$. Adding ACT DR6 CMB lensing brings the large-scale bins back to $\alpha \sim 0.98$, but the five-bin joint fit still has a probability-to-exceed of only $0.07$ for a constant amplitude, indicating a mild scale-dependent departure; dropping the smallest-scale bin raises the probability-to-exceed to $0.40$. Fitting instead against a model with moderate baryonic feedback removes the preference for scale dependence (probability-to-exceed $0.20$). The paper also claims that the recovered suppression shape can be matched by reasonable variations of baryonic feedback and by a best-fit ultralight axion model, so lensing data alone cannot separate those explanations.

Load-bearing premise

The load-bearing premise is that the five multiplicative amplitudes $\alpha_i$ are independent of redshift, so the same rescaling of the matter power spectrum applies at every epoch along the line of sight; if the suppression actually varies with redshift, the recovered scale dependence could be an artifact of the line-of-sight projection in Eq. (4).

Editorial extensions

If this is right

  • If the joint-fit suppression is real, a single overall amplitude shift cannot account for the lensing data; the matter power spectrum must be suppressed on small scales relative to large scales.
  • A moderate baryonic feedback model removes the evidence for scale dependence, so current data are consistent with baryonic physics alone explaining the small-scale behaviour.
  • The best-fit ultralight axion model from Lyman-alpha and CMB analyses reproduces the inferred suppression, so cosmic shear and CMB lensing do not by themselves rule out non-standard dark matter.
  • The 2.2-sigma preference for an overall amplitude below unity in DES-only fits indicates cosmic shear disagrees with Planck even on linear scales, while adding CMB lensing restores agreement there.
  • Separating baryonic feedback from dark matter physics will require redshift-resolved lensing measurements and complementary probes such as thermal and kinematic Sunyaev-Zel'dovich observations.

Reading between the lines

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

  • The paper qualitatively raises but does not quantify the redshift independence of the $\alpha_i$ parameters: because Eq. (4) integrates over a wide redshift range, a suppression that grows toward low redshift could masquerade as the recovered $k$-dependence.
  • A concrete test would be to split the DES source galaxies into redshift bins and re-fit the $\alpha_i$ per bin; significant differences between bins would show the reported scale dependence is partly a projection of redshift-dependent growth.
  • If the suppression is genuine, Stage-4 surveys should drive the probability-to-exceed for a constant amplitude well below $0.01$ as small-scale bins tighten, while a high probability-to-exceed would point to systematics or redshift-dependent effects.
  • Because baryonic feedback and ultralight axions produce similar suppression shapes at $z \sim 0$, the most promising discriminator is the redshift evolution of the suppression, which differs between the two classes; a future $P(k,z)$ grid could separate them.
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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 / 6 minor

Summary. The paper introduces a phenomenological reconstruction of the non-linear matter power spectrum in five wavenumber bins using multiplicative amplitudes alpha_i applied to a fiducial Planck-2018 LambdaCDM power spectrum, and fits these amplitudes to ACT DR6 CMB lensing and DES Y3 cosmic shear data, separately and jointly. The DES-only fit finds a roughly scale-independent suppression relative to the CMB prediction, while the joint ACT+DES fit is consistent on large scales but shows a ~2 sigma deviation from scale-independence when small-scale bins are included. The authors compare the reconstructed suppression with the Amod parameterization, with SP(k) baryonic feedback models, and with an ultralight axion model, and discuss the need for joint k- and z-dependent mapping with future data.

Significance. If the headline result is robust, the paper provides a useful agnostic route to testing whether the S8 tension is accompanied by a scale-dependent suppression of the matter power spectrum, and it demonstrates the complementary power of CMB lensing and cosmic shear for this purpose. The analysis uses public likelihoods, a transparent binned model, standard MCMC machinery, and explicit comparisons with existing phenomenological models. The main statistical evidence for scale dependence is, however, only at the PTE=0.07 level in one row of Table IV, and the analysis leaves an acknowledged degeneracy between scale dependence and redshift dependence unquantified, so the central claim needs additional support before it can be regarded as established.

major comments (3)
  1. [Eq. (2), Eq. (4), Section VI] The headline scale-dependence claim rests on the assumption in Eq. (2) that the five alpha_i amplitudes are redshift-independent. Because the observables in Eq. (4) are line-of-sight integrals with kernels that peak at very different redshifts for DES cosmic shear and CMB lensing, a redshift-dependent suppression can masquerade as apparent scale dependence in the recovered alpha_i. Section VI acknowledges this qualitatively ('a redshift-dependent suppression effect could manifest as scale-dependence in our analysis') but does not quantify it. Please add a quantitative test, for example by injecting a redshift-dependent suppression into the likelihood and showing the recovered alpha_i, or by reporting the effective redshift sensitivity of each k-bin. Without such a demonstration, the abstract's ~2 sigma scale-dependence claim is not separable from z-dependence.
  2. [Table IV, Section V] The evidence for the central claim is weaker than the abstract suggests. The joint ACT+DES alpha_1...alpha_5 fit has PTE=0.07, and the DES-only alpha_1...alpha_5 fit has PTE=0.12; neither alone is strong evidence, and the table does not report the Delta-chi^2 or the number of degrees of freedom for the constant-amplitude fit, nor any correction for the multiple rows of tests. Please report the full test statistic (Delta-chi^2, dof) and either provide a properly defined significance for the scale-dependence claim or soften the abstract and conclusions accordingly.
  3. [Section VI] The DES-only result that P(k) departs from the early-universe CMB prediction on all scales may be partly an artifact of fixing the fiducial cosmology to Planck, since the paper itself notes that DES cosmic shear prefers Omega_m = 0.290^{+0.039}_{-0.063}, lower than the adopted Omega_m = 0.315. Because the alpha_i are fit with Planck Omega_m held fixed, the apparent all-scale suppression could be an Omega_m mismatch rather than a genuine departure in P(k). Please quantify the effect by marginalizing over Omega_m or by showing the alpha_i posteriors when Omega_m is allowed to vary; this is load-bearing for the DES-only claim.
minor comments (6)
  1. [Eq. (2)] The notation 'NαX' appears to be a typo for a summation symbol; please write the sum explicitly as sum_{i=1}^{N_alpha}.
  2. [Eq. (11)] The binning conditions use inconsistent units: 'k [h/Mpc]' should be written 'k/(h/Mpc)' (or equivalently in Mpc^-1 if the h-dependence is absorbed), to avoid implying that k is measured in h/Mpc.
  3. [Table II] The column header 'Halofit model' is inaccurate for the Mead2020 model, which is an HMcode-type prescription; please rename the column, for example to 'Non-linear model'.
  4. [Fig. 2 caption] The sentence 'we also we also overplot SP(k)' contains a duplicated phrase; please correct it.
  5. [Eq. (12), Table V] The two-parameter Amod variant is described as adding 'an additional Amod,1' but the equation contains both Amod,1 and Amod,2; please define both parameters explicitly and state their priors, and mark the rows of Table V where Amod,1 is fixed to unity more clearly.
  6. [Abstract and Section V] The abstract uses 'k < 0.2 Mpc^{-1}' while the text and bins use h/Mpc; please harmonize the units throughout, including the 'k = 10 h/Mpc' statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the α_i amplitudes are free parameters fit directly to external ACT DR6 and DES-Y3 lensing data; the Section VI redshift degeneracy is a limitation, not a definitional reduction.

full rationale

The derivation chain is self-contained against external data. Equation (2) defines P(k,z) as the fiducial spectrum multiplied by free step-function amplitudes α_i, and these α_i are fit directly to the ACT DR6 and DES-Y3 lensing likelihoods through Equations (4)–(7). The reported α_i values and the constant-amplitude PTE in Table IV are therefore a summary of the fit, not a quantity derived from the claim being tested; the headline scale-dependence is a goodness-of-fit statistic on fitted parameters, with no fitted input renamed as a prediction. The Amod mapping in Equation (10) is an explicit algebraic equivalence between parameterizations and is used only for comparison, not to generate the α_i. The Section VI statement that "a redshift-dependent suppression effect could manifest as scale-dependence in our analysis" flags a genuine redshift–k degeneracy, but this is a modeling limitation rather than circularity: the α_i are not defined in terms of the conclusion they support, and the degeneracy does not make the fit self-referential. Self-citations to the ACT DR6 likelihood papers [9,10] and axion constraints [44,45] involve present authors, but these are public data products and external constraints, not unverified theoretical premises that carry the argument. No circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The reconstruction is mostly a fit: five α_i amplitudes and the DES nuisance parameters are free. The fixed Planck cosmology, the gravity-only Mead2020 nonlinear model, the Limber approximation, and the redshift independence of α_i are assumptions carried into the fit. No new physical entities are introduced.

free parameters (4)
  • Scale-bin amplitudes α1..α5 = Joint best-fit constant-amplitude 0.98±0.02; DES-only 0.85±0.05; individual bin values shown in Fig. 1
    Five multiplicative factors in Eq. (2) reshape the fiducial P(k,z) and are fit to the lensing data; they are the reconstructed quantities.
  • DES photo-z shifts Δz1..Δz4 = Marginalized with Gaussian priors from Table III (means 0, widths 0.011-0.018)
    Nuisance parameters sampled when DES cosmic shear is used; they affect the source redshift distributions in the lensing kernel.
  • DES shear calibration m1..m4 = Marginalized with Gaussian priors from Table III (means -0.006 to -0.037, widths 0.008-0.009)
    Multiplicative shear calibration uncertainties in DES Y3, marginalized in all DES fits.
  • Intrinsic alignment TATT parameters a1,a2,η1,η2,bTA = Marginalized with uniform priors (a,η: U(-5,5); bTA: U(0,2))
    Intrinsic alignment model for cosmic shear; standard DES-Y3 nuisance set.
assumptions (5)
  • domain assumption Limber approximation is accurate for the lensing multipoles used (L>40)
    Lensing power spectra in Eq. (4) use the Limber approximation to relate C(k) to P(k,z); the approximation is standard and accurate here, but it is still an approximate projection.
  • domain assumption Planck 2018 TT,TE,EE+lowE ΛCDM parameters are the correct fiducial cosmology
    Cosmological parameters are fixed to Table I; any mismatch in Ωm or σ8 is absorbed into the α_i amplitudes, which the paper acknowledges in Section VI for the DES-only result.
  • domain assumption Mead2020 with no baryonic feedback gives the correct gravity-only non-linear P(k)
    The baseline non-linear matter power spectrum ignores baryonic feedback; inaccuracies in the halo model would masquerade as scale-dependent α_i.
  • ad hoc to paper The five α_i amplitudes are redshift-independent
    Eq. (2) applies the same multiplicative bin amplitudes at all redshifts, while lensing kernels integrate over a wide redshift range; a z-dependent effect would be misinterpreted as k-dependence.
  • domain assumption TATT intrinsic alignment model and DES-Y3 nuisance priors correctly describe source systematics
    Used for cosmic shear; errors in the IA model or prior choices could bias the small-scale α_i constraints.

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Cite this review

Pith. "Pith review of Reconstructing the shape of the non-linear matter power spectrum using CMB lensing and cosmic shear." pith.science (2026). https://pith.science/paper/2KYQAHDJ

@misc{pith2026250206687,
  author       = {Pith},
  title        = {Pith review of: Reconstructing the shape of the non-linear matter power spectrum using CMB lensing and cosmic shear},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2KYQAHDJ}},
  note         = {Machine review of arXiv:2502.06687}
}
abstract

We reconstruct the non-linear matter power spectrum $P(k)$ using a joint analysis of gravitational lensing of the cosmic microwave background (CMB) and lensing of galaxies. This reconstruction is motivated by the $S_8$ tension between early-universe CMB predictions and late-time observables. We use CMB lensing data from the Atacama Cosmology Telescope DR6 and cosmic shear data from the Dark Energy Survey (DES) Y3 release to perform a gravity-only (i.e. no baryonic feedback) fit to $P(k)$ in bins of wave-number, within $\rm{\Lambda CDM}$. We find that with DES cosmic shear data alone, $P(k)$ departs from the early-universe CMB prediction on all scales. The joint fit with CMB lensing is consistent on large scales $k<0.2 \;{\rm Mpc}^{-1}$ but shows a $\sim 2 \sigma$ deviation from scale-independence when extending to $k = 10 \;h/\mathrm{Mpc}$. We compare our agnostic $P(k)$ reconstruction to baryonic feedback models and non-standard dark matter models: reasonable variations of both scenarios can recover the shape and amplitude of the suppression. We discuss the advances needed to disentangle these physical effects with a full mapping of $P(k,z)$.

Figures

Figures reproduced from arXiv: 2502.06687 by the authors.

Figure 1
Figure 1. FIG. 1. The non-linear matter power spectrum (at redshift [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Inferred deviation of the non-linear matter power spectrum from our gravity-only baseline shown against various [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Marginalized 2d posteriors of the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Correlation matrices for the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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