{"id":"b2f5d31b-137e-45b3-8498-ed1d73c81d00","arxiv_id":"2502.10168","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":15,"one_line_summary":"Clustering redshifts reconstruct the NVSS+SUMSS radio galaxy kernel, giving sigma_8 = 0.86 (+0.12/-0.09) from Planck PR4 CMB lensing cross-correlation, consistent with Planck.","lead":"This paper reconstructs the distance distribution of radio galaxies using clustering redshifts, then uses that to measure how much matter bends Planck's CMB light. The result, sigma_8 = 0.86 (+0.12/-0.09), agrees with the standard cosmological model, and the method can push CMB lensing tomography to higher redshifts.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quoted sigma8=0.86 is obtained only from the double-lognormal residual kernel, yet the paper's own chi2/d.o.f. values (0.669 for constant residual vs 0.967 for double lognormal) do not support that model choice, so the headline amplitude is model-dependent.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing sensitivity: the double-lognormal model for the unmeasured high-redshift tail of the NVSS-SUMSS kernel sets the residual term S_{u kappa} and therefore the final sigma8. My read agrees and adds a precise internal check: the paper's own chi2/d.o.f. values contradict its stated preference for the double lognormal model. With the constant-residual model the central sigma8 is about 0.67, and with the double lognormal it is about 0.87; the difference is larger than the quoted uncertainty. The estimator derivation itself is internally consistent, and the paper includes useful scale-cut and selection-function checks in the appendices, so the concern is not about the method in general but about the specific model choice used to produce the headline number. Because both models remain consistent with Planck within their quoted errors, and because the method is clearly documented, the appropriate verdict remains CONDITIONAL: the authors should justify the residual-kernel choice with a proper model-selection statistic and demonstrate that sigma8 is stable under an empirical high-z kernel such as CENSORS.","tokens_in":29971,"tokens_out":5796,"duration_ms":65993,"concrete_test":"Build one joint likelihood over C_gk and the reconstructed kernel points, and compare three nested models: constant K_res, the double lognormal of Eq. (56), and a double lognormal with a free constant tail above zmax. Apply AIC/BIC or a likelihood-ratio test to determine which residual model is actually preferred. If the constant or mixed model is preferred, report sigma8 marginalized over models and check whether the central value shifts by more than the quoted 0.09-0.12 error. Separately, replace the z > 2.3 tail with the empirical CENSORS redshift distribution (Brookes et al. 2008), converted to a kernel under the same linear-bias treatment, and refit sigma8. If the resulting shift in sigma8 exceeds the quoted statistical error, the headline result is dominated by the assumed residual kernel shape.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's sigma8=0.86 comes only from the double-lognormal residual kernel model (Eq. 56, Sec. 5.4.2), whose extrapolation to z > zmax sets S_{u kappa} in Eq. (37) and therefore controls sigma8 through Eq. (39). Replacing it with the constant-residual model (Sec. 5.4.1) changes sigma8/sigma8,fid from 1.07^{+0.16}_{-0.12} (central sigma8 ~ 0.87) to 0.82^{+0.33}_{-0.18} (central sigma8 ~ 0.67). The paper argues that the double lognormal model is preferred, but the reported values are chi2/d.o.f. = 12.05/18 = 0.669 for the constant model and 15.48/16 = 0.967 for the double lognormal; the double lognormal is worse by Delta chi2 = 3.43 while adding three parameters. Thus the stated model preference is not supported by the reported statistics. The high-z tail of the double lognormal is not independently calibrated: the CENSORS comparison in Fig. 8 is qualitative, and two reference bins are removed post hoc. The clustering-redshift estimator itself is not the weak point; the weak point is the selection and extrapolation of the residual kernel, on which the headline sigma8 directly depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30317,"tokens_out":13501,"duration_ms":132790,"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":[{"comment":"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.","section":"Sec. 5.4, Tables 4-5, Eq. (52), Eq. (56)"},{"comment":"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.","section":"Sec. 5.4.2, Tables 4-5"},{"comment":"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.","section":"Sec. 5.4.2, Eq. (56), Eq. (37), Fig. 8, Table 2"},{"comment":"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.","section":"Sec. 5.4.2, Eq. (58), Sec. 5.5, Eq. (59)-(60)"}],"minor_comments":[{"comment":"The subsection title 'Double lognormal model for the NVSS-NVSS kernel' should read 'NVSS-SUMSS kernel'.","section":"Sec. 5.4.2, title"},{"comment":"The heading 'eBOSS DR16 LBGs' is inconsistent with the text, which discusses LRGs; the heading should be corrected to 'eBOSS DR16 LRGs'.","section":"Sec. 4.2.2, heading"},{"comment":"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.","section":"Sec. 5.4.1, Eq. (52)"},{"comment":"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.","section":"Sec. 5.4.1"},{"comment":"The reference list entry 'Ferreira; Ishak et al. 2019; 2019' appears garbled and should be cleaned up.","section":"Introduction, references"},{"comment":"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.","section":"Sec. 4.4"},{"comment":"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.","section":"Sec. 5.5, Eq. (59)"}],"recommendation":"major_revision","confidential_remarks":"The methodological core of the paper is sound and interesting, but the headline sigma_8 needs substantial revision: the model-selection statistics as reported do not support the chosen residual-kernel model, and the two residual models give central values differing by about 0.2 in sigma_8. I would recommend major revision with a focus on presenting sigma_8 as conditional on the residual kernel, adding robustness tests against alternative high-z kernel shapes, and correcting the model-preference argument. The paper is publishable in principle if these load-bearing points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the demonstration that you can do CMB lensing tomography with clustering-redshift kernel reconstruction on an all-sky radio sample, combining NVSS+SUMSS with 2MPZ, LOWZ-CMASS, eBOSS LRGs, and Gaia-unWISE QSOs. The harmonic-space cross-noise fitting (Eqs 50-51) is a nice practical addition, and the reconstructed kernel from the reference samples is a plausible route to z~2-3.\n\nThat said, the headline sigma_8 = 0.86^{+0.12}_{-0.09} depends on the double lognormal residual kernel model (Eq 56). The alternative constant residual model yields sigma_8/sigma8fid = 0.82^{+0.33}_{-0.18}, i.e. central sigma8 ~ 0.67. The paper argues the double lognormal is preferred based on chi2/d.o.f., but the reported numbers are 12.05/18 = 0.669 for the constant model against 15.48/16 = 0.967 for the double lognormal. The double lognormal adds three parameters for a Delta chi2 of 3.43. That is not a preference; if anything, it is a penalty. The statement in Sec 5.4.2 that the constant model is 'insufficient' misreads which chi2 is better. The paper's own Sec 5.4.1 acknowledges the constant model gives a smaller sigma8 and attributes it to systematics of the assumed kernel—which is exactly the problem: the residual kernel model determines the answer.\n\nAlso, two reference bins are removed post hoc (Sec 5.1), and the double lognormal parameters are fitted to the same C_gk measurement they then predict, so the 'prediction' is not independent. The CENSORS comparison in Fig 8 is qualitative.\n\nWhere does this leave the paper? The clustering-z estimator itself is not the weak point; the formalism is clearly derived, and the data combination is new. The concern is the model selection and extrapolation of the high-z tail. Both models are consistent with Planck within errors, so this doesn't overturn anything, but the quoted 12-15% sigma_8 precision is not robust to the residual-model choice.\n\nWho is this for? CMB lensing tomographers and radio survey cosmologists. It deserves a serious referee—the method will be useful for CMB-S4/SKA-era analyses—but it needs major revision: a principled model-selection argument, a systematic error budget for the residual kernel, and ideally released analysis code. I'd engage with it, and I'd send it to review, but I would not take the headline sigma_8 at face value until the residual-model dependence is resolved.","headline":"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.","tokens_in":30995,"tokens_out":4261,"would_cite":true,"duration_ms":38175,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $\\sigma_8=0.812$.","keywords":["CMB lensing tomography","clustering redshifts","galaxy kernel function","radio galaxies","sigma8","Planck PR4","NVSS-SUMSS","large-scale structure"],"falsifier":"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.","tokens_in":29656,"feed_emoji":"📡","tokens_out":10018,"duration_ms":92410,"temperature":0.7,"pith_summary":"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.","feed_headline":"Radio galaxies without per-galaxy redshifts yield sigma8 = 0.86","feed_subtitle":"Clustering cross-correlations reconstruct the lensing kernel, matching Planck's structure-growth amplitude.","key_machinery":"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.","core_discovery":"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. $\\sigma_8=0.86^{+0.12}_{-0.09}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the clustering-redshift principle that cross-correlation with known-redshift samples measures the unknown redshift distribution; the paper's kernel estimators build on this.","marker":"Newman 2008"},{"why":"Refines clustering redshifts to reconstruct the lensing kernel and supplies the broad-slice validity condition used by the harmonic-space estimators.","marker":"Rahman et al. 2015"},{"why":"Supplies the CMB lensing convergence kernel and the Limber projection formalism used to predict $C_{g\\kappa}$.","marker":"Lewis & Challinor 2006"},{"why":"Earlier all-sky CMB lensing tomography with the 2MPZ sample; sets the harmonic-space scale cuts and pseudo-$C(\\ell)$ treatment adopted for 2MPZ.","marker":"Peacock & Bilicki 2018"},{"why":"Provides the Planck PR4 convergence map whose cross-correlation with NVSS-SUMSS is the measurement being fit.","marker":"Carron et al. 2022"},{"why":"Source of the NVSS catalogue, the main all-sky lens sample.","marker":"Condon et al. 1998"},{"why":"Source of the SUMSS southern-sky catalogue combined with NVSS to form the lens sample.","marker":"Mauch et al. 2003"},{"why":"Provides the Gaia-unWISE QSO reference catalogue and selection function used for the $z\\simeq0.8$-$2.3$ kernel bins.","marker":"Storey-Fisher et al. 2024"},{"why":"CENSORS spectroscopic subsample used as an external check of the reconstructed radio-galaxy kernel at high redshift.","marker":"Brookes et al. 2008"},{"why":"Nonlinear matter power spectrum model used to validate the scale independence of the linear bias assumption.","marker":"Smith et al. 2003; Takahashi et al. 2012"}],"fun_headline_variants":["No redshifts needed: radio galaxies pin down sigma8 = 0.86","Clustering redshifts reconstruct lensing kernel, sigma8 = 0.86","CMB lensing tomography from clustering redshifts, sigma8=0.86","Cross-correlations without photo-z: radio galaxies yield sigma8=0.86","Radio galaxies alone: no photo-z needed, sigma8 = 0.86"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No redshifts needed: radio galaxies pin down sigma8 = 0.86","Clustering redshifts reconstruct lensing kernel, sigma8 = 0.86","CMB lensing tomography from clustering redshifts, sigma8=0.86","Cross-correlations without photo-z: radio galaxies yield sigma8=0.86","Radio galaxies alone: no photo-z needed, sigma8 = 0.86"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000875,"raw_usage":{"total_tokens":3825,"prompt_tokens":1026,"completion_tokens":2799,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":2696}},"tokens_in":642,"tokens_out":2799,"duration_ms":17078,"temperature":1.0,"reasoning_tokens":2696,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:09:39.788069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A., Bilicki M., 2018, @doi [Mon","cited_arxiv_id":null,"evidence_quote":"Earlier all-sky CMB lensing tomography with the 2MPZ sample; sets the harmonic-space scale cuts and pseudo-$C(\\ell)$ treatment adopted for 2MPZ."}],"review_version":1}