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REVIEW 4 major objections 7 minor 86 references

CMB lensing tomography with clustering estimation of lens redshift distributions

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By reconstructing the lensing kernel of NVSS+SUMSS radio galaxies through clustering redshifts, the paper measures $\sigma_8=0.86^{+0.12}_{-0.09}$ from Planck PR4 CMB lensing cross-correlation, consistent with Planck's $\sigma_8=0.812$.

desk verdict Useful clustering-z route to CMB lensing tomography, but the headline sigma_8 rests on a residual-kernel model choice the paper's own chi-squared comparison does not support. read the letter →

arxiv 2502.10168 v2 pith:ANBW34IH submitted 2025-02-14 astro-ph.CO astro-ph.GA

Shun Arai , John A. Peacock , Hironao Miyatake , Atsushi J. Nishizawa This is my paper · ORCID
classification astro-ph.COastro-ph.GA
keywords CMBlensingtomographyclusteringredshiftsgalaxykernelfunctionradiogalaxiessigma8PlanckPR4NVSS-SUMSSlarge-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 show that CMB lensing tomography does not need a catalogue of individual galaxy redshifts: the lens kernel $K_g(z)=b_g(z)(d\bar n_g/dz)/\bar n_g$ can be reconstructed from angular cross-correlations between the lens sample and reference samples with known redshift distributions. The authors demonstrate the pipeline on all-sky NVSS+SUMSS radio galaxies, combining spectroscopic and photometric references spanning $0\lesssim z\lesssim 3$. Correlating the resulting kernel with the Planck PR4 convergence map yields $\sigma_8=0.86^{+0.12}_{-0.09}$ at fixed $\Omega_m=0.315$, consistent with the Planck normalisation $\sigma_8=0.812$. If correct, this makes the large, deep, all-sky radio surveys usable as tomographic probes of structure growth even though the member galaxies lack measured redshifts.

What carries the argument

The central object is the reconstructed kernel $K_u(z)$, the product of the linear bias and the normalised redshift distribution of the lens galaxies. The estimators in Eqs. (18), (22), (34) and (35) all take the form of a cross-correlation divided by the square root of an auto-correlation, with the matter correlation function cancelling, so the kernel is obtained point-wise in angle or multipole without modelling $b_g$ and $d\bar n_g/dz$ separately. A second load-bearing component is the split of the predicted $C_{g\kappa}$ into a reconstructed part $\Sigma_{g\kappa}$ and a high-redshift residual $S_{g\kappa}$; the paper handles the residual with a fitted double-lognormal kernel whose parameters are marginalised over in the $\sigma_8$ fit.

What would settle it

Construct a flux-limited spectroscopic sample of NVSS-SUMSS radio galaxies that overlaps the reference-sample redshift range, measure $b_g\,d\bar n_g/dz$ directly, and compare with the reconstructed kernel; then extend to $z>2.3$ and recompute $\sigma_8$ with the measured tail instead of the double-lognormal model. A mismatch larger than the quoted $1\sigma$ errors would falsify either the linear-bias clustering reconstruction or the residual model.

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Extended reading notes

Core claim

Within a scale-independent linear bias model, the paper derives estimators for the lens kernel $K_u(z)$ from the ratios of cross- and auto-correlation measurements, with both configuration-space and harmonic-space versions, in which the matter correlation function cancels so that no separation of bias from redshift distribution is required. The reconstruction is binned into eleven redshift slices across $0\lesssim z\lesssim3$ using narrow slices (LOWZ-CMASS, eBOSS DR16 LRGs) and broad slices (2MPZ, Gaia-unWISE QSOs), with two poorly behaved bins excluded. Combining the reconstructed kernel with a double-lognormal model for the uncalibrated $z>2.3$ tail, the predicted angular cross-spectrum $C_{g\kappa}$ matches the measured Planck PR4 $\times$ NVSS-SUMSS spectrum with $\chi^2/\mathrm{d.o.f.}\approx 0.97$, and the amplitude ratio yields $\sigma_{8,\mathrm{est}}/\sigma_{8,\mathrm{fid}}=1.07^{+0.16}_{-0.12}$, i.e. $\sigma_8=0.86^{+0.12}_{-0.09}$.

Load-bearing premise

The load-bearing premise is that the unobserved high-redshift tail of the NVSS-SUMSS kernel ($z\gtrsim2.3$) follows the double-lognormal model used for the residual; replacing it with a constant residual changes the best-fit $\sigma_8$ by about 0.2.

Editorial extensions

If this is right

  • Large-area radio catalogues such as NVSS+SUMSS can act as lens samples for CMB lensing tomography without spectroscopic redshifts for member galaxies.
  • The reconstructed kernel feeds a prediction of $C_{g\kappa}$ whose shape is not assumed, moving systematic uncertainty from redshift calibration to the linear-bias and nonlinear-matter-spectrum model.
  • The derived $\sigma_8=0.86^{+0.12}_{-0.09}$ agrees with Planck's $\sigma_8=0.812$, showing no significant tension in this radio-galaxy lensing probe.
  • Tomographic information is separable up to $z_{\rm sep}\approx2$ at the current noise level, while the kernel cumulant extends to $z\approx4$, so deeper CMB lensing maps and radio surveys can push tomographic slices to higher redshift.

Reading between the lines

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

  • If the method's scale-cancellation holds for other tracers, the same clustering-kernel estimator could attach CMB lensing tomography to photometric galaxy samples and cosmic-shear source catalogues that currently rely on uncertain photo-$z$ priors.
  • The spread between the constant-residual and double-lognormal residual models indicates that the high-redshift tail of the radio kernel, not the statistical power of $C_{g\kappa}$, currently sets the systematic floor; a direct spectroscopic census of the $z>2.3$ radio population would test whether this floor is real.
  • Combining the reconstructed kernel with the auto-power-spectrum estimator sketched in Appendix A would give an independent $\sigma_8$ constraint from the same radio sample, potentially sharpening the comparison with Planck.
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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

4 major / 7 minor

Summary. The paper develops a clustering-redshift estimator for the lensing kernel K_g(z)=b_g(z)(1/\bar n_g)d\bar n_g/dz of a galaxy sample without per-galaxy redshifts, using cross-correlations with spectroscopic and photometric reference samples. Estimators are derived in configuration and harmonic space for both narrow and broad redshift slices, and the reconstructed kernel is then used to model the CMB lensing cross-spectrum C_{g\kappa}. As a proof of concept, the authors reconstruct the NVSS+SUMSS radio-galaxy kernel from 2MPZ, LOWZ-CMASS, eBOSS DR16 LRG, and Gaia-unWISE cross-correlations over 0<z<~2.3, measure C_{g\kappa} with the Planck PR4 convergence map, and constrain sigma_8. The headline result is sigma_8=0.86^{+0.12}_{-0.09}, obtained with a double-lognormal parametric model for the radio kernel, reported as consistent with Planck 2018.

Significance. If the method holds up, it offers a valuable route to CMB lensing tomography for all-sky radio samples that lack spectroscopic redshifts, and the estimator construction is clean: the reconstructed kernel points are grounded in external cross-correlations and the paper includes useful validation appendices on linear bias, cross-noise, and sample cuts. The proof-of-concept measurement is plausible, but the headline sigma_8 is not robust to the treatment of the high-redshift residual kernel, and the reported model-comparison statistics do not support the adopted double-lognormal model over the constant-residual model. With a careful revision that honestly presents the model dependence and strengthens the robustness tests, the paper would be a solid methodological contribution.

major comments (4)
  1. [Sec. 5.4, Tables 4-5, Eq. (52), Eq. (56)] The headline sigma_8=0.86 is obtained exclusively from the double-lognormal residual-kernel model (Table 5: sigma_8/sigma_8,fid=1.07^{+0.16}_{-0.12}). The alternative constant-residual model in Sec. 5.4.1 gives sigma_8/sigma_8,fid=0.82^{+0.33}_{-0.18}, i.e. a central sigma_8 of about 0.67. This is a shift of roughly 0.2 in sigma_8, larger than the quoted lower error bar of 0.09. Because the two models are not strongly disfavoured relative to each other by the reported statistics, the paper should present the sigma_8 constraint as conditional on the assumed residual-kernel model, and the abstract and conclusions should not present the double-lognormal result as the unique outcome.
  2. [Sec. 5.4.2, Tables 4-5] The text states that the double-lognormal model is preferred and that the constant-residual model 'seems insufficient', but the reported chi-squared values imply the opposite: chi^2/d.o.f. = 12.05/18 = 0.669 for the constant model versus 15.48/16 = 0.967 for the double-lognormal model. The double-lognormal model has a larger chi^2 and a worse chi^2 per degree of freedom; Delta chi^2 = 3.43 for two fewer degrees of freedom is not evidence in its favour. Both values being below unity also suggest that the covariance may be overestimated. As reported, the model selection argument is not supported and is load-bearing because the model choice changes the central sigma_8 by about 0.2.
  3. [Sec. 5.4.2, Eq. (56), Eq. (37), Fig. 8, Table 2] The residual term S_{u\kappa} in Eq. (37) is set by extrapolating the four-parameter double-lognormal form of Eq. (56) from z_max~2.3 to the last scattering surface. This extrapolation is not independently calibrated: the highest reconstructed bin (Gaia-unWISE bin3, z_mean=2.348) has b_r K_g = 1.27 +/- 1.25 (Table 3), so it is essentially uninformative; the two removed bins in Table 2 (LOWZ-CMASS bin5 and eBOSS DR16 LRG bin2) are excluded post hoc and appear as low open circles in Fig. 8; and the CENSORS comparison is qualitative. The paper should test alternative residual shapes, such as a truncated kernel, a power-law tail, or a free K_res with a broad prior, and report how sigma_8 changes. As it stands, the quoted sigma_8 depends on an unvalidated extrapolation of a chosen functional form.
  4. [Sec. 5.4.2, Eq. (58), Sec. 5.5, Eq. (59)-(60)] The double-lognormal parameters are fitted simultaneously to the measured C_{g\kappa}(l) and to the reconstructed kernel points in the likelihood of Eq. (58), so the resulting C_{g\kappa} is a joint fit rather than an independent prediction. The claim in Sec. 5.5 that residual high-redshift signal does not affect sigma_8 is also model-dependent, because the SNR calculation uses the same double-lognormal K_model to construct C_{g\kappa}(l,z_sep). To validate the method as predictive, the authors should either calibrate K_model on the clustering-redshift data alone and then compare with the measured C_{g\kappa}, or explicitly quantify how many degrees of freedom the joint fit consumes and show that the conclusions are unchanged under alternative high-z kernel shapes.
minor comments (7)
  1. [Sec. 5.4.2, title] The subsection title 'Double lognormal model for the NVSS-NVSS kernel' should read 'NVSS-SUMSS kernel'.
  2. [Sec. 4.2.2, heading] The heading 'eBOSS DR16 LBGs' is inconsistent with the text, which discusses LRGs; the heading should be corrected to 'eBOSS DR16 LRGs'.
  3. [Sec. 5.4.1, Eq. (52)] The symbol Sigma is used both for the summation in Eq. (37) and as a fitted amplitude parameter in Eq. (52) and Table 4; a different symbol, such as A or K_Sigma, would avoid confusion.
  4. [Sec. 5.4.1] The sentence 'The smaller sigma_8, however, seems not to be cosmological but to be systematics of the NVSS-SUMSS kernel assumed here' is unclear, since the constant model is one of the two models being tested; please rephrase to specify which aspect of the model causes the downward shift.
  5. [Introduction, references] The reference list entry 'Ferreira; Ishak et al. 2019; 2019' appears garbled and should be cleaned up.
  6. [Sec. 4.4] The PR4 lensing map is attributed to Akrami et al. (2020), but the relevant PR4 analysis is Carron et al. (2022), which is already cited elsewhere; please correct the citation.
  7. [Sec. 5.5, Eq. (59)] The quantity defined in Eq. (59) is a chi-squared-like sum rather than a conventional signal-to-noise ratio; please define the meaning of 'SNR' and the threshold SNR=1 more explicitly.

Circularity Check

1 steps flagged · score 4.0 of 10

The clustering-z estimator is self-contained, but the 'predicted' C_{gκ} is computed with a double-lognormal model fitted to the same C_{gκ}, and the fitted high-z tail controls the quoted σ8.

  1. fitted input called prediction [Sec. 5.4.2, Eqs. (56)-(58); cf. Abstract and Fig. 9.]
    "Then C_radioκ is derived as C_{gκ}(ℓ)≡ (σ8,est./σ8,fid.) ∫_0^{z*} dz [K_model_g(z) K_κ(z)/(χ^2 c/H) P_m(...)]. ... We adopt the maximum likelihood estimation of the best-fit parameters with the following log-likelihood, ln(L)≡−Σ_ℓ ((Ĉ_gκ(ℓ)−C_model_gκ(ℓ))^2/(2σ̂^2_gκ(ℓ))) − Σ_i ((K_g(z_i)−K_model_g(z_i))^2/(2σ^2_K(z_i))) ."

    The model used for the 'theoretical prediction' of C_{gκ} is not fixed a priori: Eq. (56) defines K_model with free shape parameters (α1, α2, β1, β2), and Eq. (58) fits these parameters to the measured Ĉ_{gκ} in the first term of the log-likelihood. Therefore the C_{gκ} curve shown as the 'prediction' in Fig. 9 is a best fit to the very data it is said to predict; its agreement carries no independent confirmation of the reconstructed kernel. Moreover, the residual term S_{uκ} in Eq. (37), which enters the σ8 estimate through Eq. (39), is obtained by extrapolating this same fitted K_model beyond z_max≈2.3, where no reconstructed kernel points exist.

full rationale

The clustering-redshift estimator (Eqs. 18/22/34/35) is self-contained: K_rec is formed from measured w_rr/w_ur or C_rr/C_ur with external reference samples, and no C_{gκ} data enter the kernel reconstruction. The linear-bias framework and HALOFIT modeling are standard external inputs. There is no load-bearing self-citation: the Peacock & Bilicki (2018), Alonso et al. (2023), and Rahman et al. (2015) citations supply scale cuts and estimator conventions, not the result itself. The central circular concern is localized to Sec. 5.4.2: the double-lognormal K_model is fit to the measured C_{gκ} (Eq. 58), so the 'theoretical prediction' in Eq. 57 and Fig. 9 is a fitted curve, and the residual S_{uκ} — the part of the model that most affects σ8 — is an extrapolation of that fitted shape beyond the reconstructed range. The paper's own χ²/d.o.f. comparison (15.48/16 for double lognormal vs 12.05/18 for constant residual) does not support its claim that the double lognormal is preferred, which reinforces that the headline σ8 is model-dependent. This is partial circularity in presentation, not a derivation that is equivalent to its inputs: the low-redshift kernel is still externally anchored by clustering redshifts, and σ8 is a fitted parameter from an independent C_{gκ} measurement. Score 4.

Assumptions & free parameters 15 free parameters · 9 assumptions · 0 invented entities

The central result rests on a large set of modeling choices: the flat LCDM background with Planck parameters, linear scale-independent bias, the Limber approximation, HALOFIT matter power spectrum, and the two clustering-redshift approximations (narrow slices and slow-variation broad slices). The double lognormal functional form is ad hoc to this paper and directly controls the high-z residual that shifts sigma_8 by about 0.2. Fifteen fitted amplitudes (sigma_8, four double-lognormal parameters, two alternative-model normalizations, four reference biases, and four cross-noise amplitudes) absorb the data; only sigma_8 is interpreted cosmologically, while the rest are nuisance parameters. No new physical entities are introduced.

free parameters (15)
  • sigma_8 = 0.86^{+0.12}_{-0.09}
    Amplitude of matter fluctuations; the main fitted parameter. It is marginalized over kernel-shape nuisance parameters (Sec 5.4.2).
  • alpha_1 = 1.22^{+0.41}_{-0.37}
    Amplitude of the primary (AGN) lognormal term in Eq (56); fitted jointly to the reconstructed kernel points and C_gk.
  • alpha_2 = 0.14^{+0.04}_{-0.03}
    Amplitude of the secondary (star-forming) lognormal term; fitted jointly.
  • beta_1 = 0.45^{+0.04}_{-0.03}
    Width of the primary lognormal term; fitted jointly.
  • beta_2 = 1.27^{+0.23}_{-0.17}
    Width parameter of the secondary term; fitted jointly.
  • K_res (constant residual model) = 0.68^{+0.82}_{-0.53}
    Constant amplitude of the high-z residual kernel in the alternative model (Table 4); its value drives the lower sigma_8 solution.
  • Sigma (constant model) = 1.14 +/- 0.07
    Amplitude of the reconstructed kernel sum in the constant-residual model (Table 4); degenerate with sigma_8 and K.
  • b_r (2MPZ) = 0.99 +/- 0.007
    Reference sample linear bias fitted in the harmonic cross-noise model (Table 3).
  • b_r (Gaia-unWISE bin1) = 1.42 +/- 0.22
    Reference sample bias for the first Gaia-unWISE bin (Table 3).
  • b_r (Gaia-unWISE bin2) = 2.30 +/- 0.56
    Reference sample bias for the second Gaia-unWISE bin (Table 3).
  • b_r (Gaia-unWISE bin3) = 3.34 +/- 1.73
    Reference sample bias for the third Gaia-unWISE bin (Table 3).
  • a_gr (2MPZ) = 0.46^{+0.28}_{-0.26}
    Cross-noise amplitude between radio sample and 2MPZ, fitted (Table 3).
  • a_gr (Gaia-unWISE bin1) = 0.37 +/- 0.12
    Cross-noise amplitude, first Gaia bin (Table 3).
  • a_gr (Gaia-unWISE bin2) = 0.19 +/- 0.11
    Cross-noise amplitude, second Gaia bin (Table 3).
  • a_gr (Gaia-unWISE bin3) = 0.38 +/- 0.08
    Cross-noise amplitude, third Gaia bin (Table 3).
assumptions (9)
  • domain assumption Flat LCDM background with Planck 2018 parameters and Omega_m = 0.315 fixed.
    Used throughout for H(z), chi(z), and P_m; secs 1-2 and 5.4. If the background cosmology is wrong, the kernel reconstruction and sigma_8 normalization shift.
  • domain assumption Linear, scale-independent galaxy bias (Eq 2).
    Assumed for radio and reference samples; validated only approximately in App B with 10-50% precision.
  • standard math Limber approximation for projected correlation functions and power spectra (Eqs 8, 11).
    Standard small-angle approximation; valid for the scales used, ell >= 20.
  • domain assumption HALOFIT model for the nonlinear matter power spectrum.
    Used to compute theoretical w(z,theta) and C(z,ell); the reconstructed kernel inherits any HALOFIT inaccuracies at the scales used.
  • domain assumption Reference redshift distributions are approximately constant within each narrow bin (Eq 13).
    Required for Eqs 16-18 and 19-22; the authors exclude two bins that violate this, after seeing the data.
  • domain assumption Slow variation condition Eq (29) holds for broad slices (2MPZ and Gaia-unWISE).
    Needed for Eqs 30-35; not directly verified for these samples.
  • ad hoc to paper Double lognormal functional form for the radio kernel, Eq (56).
    A fitting formula with four free parameters; the extrapolated high-z tail directly controls S_uk and hence sigma_8. No external validation is provided.
  • domain assumption NVSS and SUMSS redshift distributions are identical despite different selection frequencies.
    Stated in Sec 4.1.2; if false, the combined kernel is biased.
  • domain assumption Cross-shot-noise between radio and reference samples is described by a single amplitude a_gr (Eq 49-51).
    Assumes the only correlated noise is from overlapping objects; a_gr is fitted, not measured by cross-matching.

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Pith. "Pith review of CMB lensing tomography with clustering estimation of lens redshift distributions." pith.science (2026). https://pith.science/paper/ANBW34IH

@misc{pith2026250210168,
  author       = {Pith},
  title        = {Pith review of: CMB lensing tomography with clustering estimation of lens redshift distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ANBW34IH}},
  note         = {Machine review of arXiv:2502.10168}
}
abstract

We develop a clustering-based redshift estimation approach for CMB lensing tomography, focusing on the kernel function of the lensing galaxies. Within a linear galaxy bias framework, we derive estimators for this kernel from two-point cross-correlations between lens mass and reference samples. The reconstructed kernel then enables a theoretical prediction for the angular cross-power spectrum \(C_{g\kappa}\) between CMB lensing convergence and lens galaxies. As a proof of concept, we measure \(C_{g\kappa}\) by correlating the \emph{Planck} PR4 convergence map with NVSS+SUMSS radio galaxies (\(0\lesssim z\lesssim 3\)). We estimate the radio-galaxy kernel by collectively cross-correlating their distribution with spectroscopic and photometric surveys (2MPZ, LOWZ-CMASS, eBOSS DR16 LRGs, and Gaia-unWISE QSOs). From the measured \(C_{g\kappa}\), we obtain \(\sigma_8 = 0.86^{+0.12}_{-0.09}\) when the density parameter is set to the {\it Planck} value of $\Omega_m = 0.315$; this is in good agreement with the \emph{Planck} normalisation of $\sigma_8 = 0.812$.

Figures

Figures reproduced from arXiv: 2502.10168 by the authors.

Figure 1
Figure 1. The probability density of the reference samples in redshifts, showing the binning of redshifts of tracers for the reconstruction of the distribution of lensing galaxies. We apply the narrow redshift slices for the LOWZ-CMASS samples and eBOSS DR16 LRGs, whereas the broad redshift slices for 2MPZ and Gaia-unWISE QSOs. need to infer information about the entire sky from the data within limited masks. In configuration… view at source ↗
Figure 2
Figure 2. Illustrations of sky maps with a Mollweide projection in galactic coordinates. The top figure shows the NVSS-SUMSS map smoothed at 0.5 ◦FWHM spanning between 0 and 3 the mean surface number density 𝑛mean = 24.4 deg−2 , with the masking of radio emis￾sion from the galactic plane at |𝑏| ≤ 10◦ . Note that there are some spots in which fewer galaxies are observed in the Southern celestial hemisphere, whereas the mean su… view at source ↗
Figure 3
Figure 3. The histogram of 𝜒 2 /d.o.f.. Here we collect 132,600 sam￾ples of (𝜎8, 𝛼1, 𝛼2, 𝛽1, 𝛽2 ). The median (red), and the quantile range: 0.16 and 0.84 are shown by the blue dotted and the blue dashed lines, respectively [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Cumulant of reconstructed𝐶𝑔𝜅 (ℓ = 225). The black line is the model computed with the median. The other coloured lines are 10 random samples drawn from the entire 132,600 MCMC realisations [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Separable redshifts as a function of signal to noise ratio. The blue and orange points represent the different 10 subsamples of the entire 132,600 MCMC realisations: the blue points were sampled between 0.16-0.84 quartiles of 𝜒 2 distribution, whereas the orange ones w…
Figure 6
Figure 6. Figure 6: The posterior distributions and the contour maps for the parameters of the constant 𝐾res. 𝑔 model. The blue contours represent confidence regions, corresponding to 1-𝜎 (68%), 2-𝜎 (95%), or 3-𝜎 (99.7%), and 4-𝜎 (99.9973%) credible intervals, respectively. The red-shaded…
Figure 7
Figure 7. Figure 7: The posterior distributions and the contour maps for the parameters of the double lognormal model are shown. The blue contours represent confidence regions, corresponding to 1-𝜎 (68%), 2-𝜎 (95%), or 3-𝜎 (99.7%), and 4-𝜎 (99.9973%) credible intervals, respectively. The …
Figure 8
Figure 8. Figure 8: Comparison of the NVSS-SUMSS kernel function (left) and the kernel of 𝐶𝑔𝜅 (right) with the best-fit lognormal model. The data points with filled circles show the results of reconstruction: 2MPZ (green), LOWZ-CMASS (blue), eBOSS DR16 LRGs (orange), and Gaia-unWISE (red)…
Figure 9
Figure 9. Figure 9: Comparison between the measurement of 𝐶𝑔𝜅 from NVSS-SUMSS × Planck PR4 and two different model of 𝐶𝑔𝜅 . The points are the measured values using a bandwidth of ℓbin = 30. The red and blue solid lines denote the predictions of 𝐶𝑔𝜅 from the constant 𝐾res. 𝑔 model and the…

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Pith tools

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