{"id":"dadcd369-5db3-4747-8863-fa060c3c8f5f","arxiv_id":"2411.09571","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A new density-momentum cross-power spectrum estimator applied to SDSSv yields f sigma8 = 0.413 +0.050 -0.058, consistent with GR plus Planck LCDM.","lead":"Astronomers used a new combination of galaxy positions and velocities from the SDSSv survey to measure the cosmic growth rate, finding f sigma8 = 0.413 at redshift 0.073, which matches the standard model of dark matter and dark energy. The cross-power spectrum technique they introduce shrinks the statistical error on this measurement by about 55 percent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Model systematics at kmax=0.3 are acknowledged, but the recovery test fits only the mean of 2048 mocks; a per-mock fit is needed to confirm systematics are subdominant for a single SDSSv realization.","rationale":"The reader's weakest assumption is the accuracy of the perturbation-theory model at kmax=0.3 h/Mpc for SDSSv's volume. I agree that this is the load-bearing point. The paper has strong supporting evidence: public code and data, 2048 realistic mocks, a good real-data chi2/d.o.f of 0.923, a Galilean robustness check, and an explicit kmax-dependence test. The authors also honestly flag in Section 5.2 that the model is not accurate at 2048 times the survey volume. Quantifying their reported chi2 for the mean mock fit shows that the residual model systematic at kmax=0.3 contributes only about 4.3 units of chi2 against roughly 79 from sample variance for a single realization, about a 5% effect. This supports the central claim rather than undermining it. The residual gap is that this estimate comes from fitting the mean of 2048 mocks, not from fitting individual mocks; a per-mock recovery test would directly establish that the systematic is not realization-dependent and that the Box-Cox plus Sellentin-Heavens pipeline gives unbiased fsigma8 with accurate uncertainty coverage. Because the implied bias is small relative to the quoted 0.05 error, and the GR plus Planck prediction of 0.448 lies well within the measurement's 68% interval, this concern does not change the accept verdict. The proposed check is a confirmatory robustness test that would make the validation complete.","tokens_in":22525,"tokens_out":14889,"duration_ms":152570,"concrete_test":"Run the full MCMC pipeline (same Box-Cox transformation, covariance from the other mocks, Sellentin-Heavens likelihood) on at least 100 of the 2048 SDSSv mocks individually, and form the distribution of best-fit fsigma8. Accept the model as validated if the mean of the per-mock best fits is within 0.02 of the fiducial 0.432 and the standard deviation is within about 20% of the average MCMC uncertainty; if the mean is offset by more than about 0.03 or the scatter significantly exceeds the quoted error, the real-data central value or error bars should be revised, or kmax reduced to 0.25.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central claim depends on the perturbation-theory model (Eqs. 31-33 with updated P00/P02/P12) being unbiased at kmax=0.3 h/Mpc for SDSSv's volume. Section 5.2 admits the model is not accurate at 2048 times the volume, and Fig. 5 shows fsigma8 beginning to drift for kmax>=0.35, so kmax=0.3 is a compromise. The only parameter-recovery test (Table 1, Fig. 4) fits the mean of 2048 mocks with single-realization errors. For the combined fit, the reported chi2=4.348 for 79 dof is about 110 times the ~0.039 expected for a perfect model averaged over 2048 realizations, so a small model systematic is present even at kmax=0.3. The implied single-realization systematic is only ~4.3 units of chi2 against ~79 from sample variance, and the recovered fsigma8=0.441 is within 0.009 of the fiducial 0.432. The missing piece is therefore not the existence of systematics, but a direct demonstration that they remain subdominant and do not bias the quoted error bars for one realization, especially after the Box-Cox and Sellentin-Heavens steps.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper, the third in a series on the redshift-space momentum power spectrum, derives an estimator for the density-momentum cross power spectrum multipoles and combines it with the galaxy density and momentum auto-power spectra to measure the growth rate from the SDSS DR14 peculiar velocity catalogue (SDSSv). After modelling the three power spectra with perturbation theory, including the survey window function, and applying a Box-Cox Gaussianization and a Sellentin-Heavens likelihood, the authors validate the pipeline on 2048 mock catalogues and then measure fσ8 = 0.413 +0.050/−0.058 at z_eff = 0.073. This is consistent with the GR+Planck prediction of 0.448 at the 68% confidence level, and including the cross-power spectrum reduces the statistical error on fσ8 by about 55% relative to the two auto-spectra alone.","tokens_in":22800,"tokens_out":10698,"duration_ms":106103,"significance":"If the measurement is unbiased, this is a valuable low-redshift growth-rate constraint obtained from a new 3x2-point combination of density and momentum fields, directly complementing existing peculiar-velocity analyses such as Lai et al. (2023). The paper is strong on reproducibility: the analysis code is public, the SDSSv catalogue and mocks are released, and the cross-spectrum estimator, window-function treatment, and Galilean robustness check are clearly presented. The main caveat is that the central mock validation is performed on the mean of 2048 realizations rather than on individual realizations, so the behaviour of the pipeline for a single survey volume—including the nonlinear Box-Cox step—is not directly demonstrated. This is an addressable gap rather than a demonstrated error.","major_comments":[{"comment":"The Box-Cox transformation is applied to both the measured power spectrum and the model power spectrum (T^c_m), but for a nonlinear transformation E[T(P)] is not equal to T(E[P]). The paper does not correct for this bias or quantify its size, and the validation in §5.2 fits only the mean of 2048 mocks. Because the offset between T(E[P]) and E[T(P)] does not average down with the number of mocks, the recovered offset of 0.009 in fσ8 is only a partial check and does not validate the quoted error bars for a single SDSSv-like realization. I request either a per-mock fit to demonstrate that the distribution of best-fit fσ8 is centred on the fiducial value with the expected scatter, or an explicit bias correction/justification (e.g., showing that the estimated λ values are close to unity or that the bias term is negligible).","section":"§5.1, Eqs. (44)–(46)"},{"comment":"The choice kmax = 0.3 h/Mpc is based on the mean-mock fit, and the text itself states that the model is not accurate for a volume 2048 times larger. Since the real-data analysis is a single realization, the current test does not directly show that model systematics remain subdominant at the chosen scale cut for one realization. Fitting a subset of the 2048 mocks individually and reporting the mean, scatter, and coverage of the recovered fσ8 would directly address this concern. This is not a reason to doubt the central value, but it is needed to support the quoted error bars.","section":"§5.2, Fig. 5"}],"minor_comments":[{"comment":"The effective redshift z_eff = 0.073 is quoted but never defined; please state how it is computed from the survey selection function or galaxy redshifts.","section":"§6.1"},{"comment":"The fit for σ_SDv in Eq. (4) is given without uncertainties; reporting the fit covariance would be useful for reproducibility.","section":"§2, Eq. (4)"},{"comment":"The y-axis labels are inconsistent between panels (kP0(k), Pp0(k), kP^p_1(k)) and lack units; please harmonize them and include units such as h Mpc^-1 and km^2 s^-2 as appropriate.","section":"Figs. 4 and 6"},{"comment":"The intrinsic scatter floor of 300 km/s is stated to not affect results, citing Paper II; a one-sentence reminder of that test would aid the self-containedness of this paper.","section":"§5.1, Eq. (29)"},{"comment":"Entries like '< 2.565' for b2σ8 in the momentum-only fit are effectively upper limits; consider presenting them as such in the text or table notes for clarity.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically sound and the central measurement is likely correct, but the single-realization validation gap—especially around the nonlinear Box-Cox step and the kmax = 0.3 scale cut—is the one substantive issue. I would not reject; the requested per-mock validation is well within the scope of the existing 2048 mocks. The self-citation of Paper I and Paper II is appropriate for a series, and the code/data release is a clear strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the density-momentum cross-power spectrum estimator, with its shot-noise treatment and window-function convolution derived in Appendix B. That completes the 3x2-point set for density and momentum fields, and the paper applies it for the first time to SDSSv. The headline result, f sigma8 = 0.413 +0.050/-0.058 at z_eff = 0.073, is consistent with the GR+Planck prediction of 0.448, and including the cross-spectrum shrinks the error by ~55% compared to fitting the two auto-spectra alone.\n\nThe paper does several things well. The derivation is careful, and the validation is serious: the estimator recovers the fiducial f sigma8 = 0.432 on 2048 SDSSv mocks, with a single-realisation error that looks honest. The real-data fit has chi2/dof = 0.923, and the Galilean robustness check in Appendix A returns a bulk velocity consistent with zero. The code and data are public, which is a real plus. The authors also correct a few small errors in Paper I's model expressions and show these corrections don't change previous results.\n\nThe soft spots are real but minor. The stress-test note is right: the mock recovery test fits only the mean of 2048 mocks, not each mock individually. The chi2 = 4.348 for 79 dof on the mean fit is larger than the ~0.039 expected if the model were perfect at 2048 times the survey volume, so a small model systematic is present even at kmax = 0.3. But the implied single-realisation systematic is only ~4.3 chi2 units against ~79 from sample variance, and the recovered f sigma8 is within 0.009 of the fiducial value, so it is likely subdominant. Still, a per-mock fit would have closed the gap, especially given the Box-Cox and Sellentin-Heavens steps in the likelihood. The kmax=0.3 choice is also somewhat empirical: Fig. 5 shows the fitted value starting to drift at kmax >= 0.35, so the chosen cut sits where model systematics begin to grow. That is a reasonable compromise, not a fatal flaw.\n\nI don't think any of this undermines the central measurement. The paper is a legitimate methodological step forward and the result is credible. It deserves a serious referee, and I would expect acceptance after minor revisions. The audience is people working on peculiar-velocity surveys, RSD, and growth-rate measurements.","headline":"Solid methods paper: a genuinely new density-momentum cross-power spectrum estimator, carefully tested on 2048 mocks, yields a low-redshift growth-rate constraint consistent with GR+Planck; the main gap is the lack of per-mock recovery tests, but the central claim holds up.","tokens_in":23402,"tokens_out":2530,"would_cite":true,"duration_ms":23938,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Using the density–momentum cross-power spectrum of 34,059 SDSS galaxies, this paper measures the cosmic growth rate $f\\sigma_8 = 0.413^{+0.050}_{-0.058}$ at $z=0.073$, consistent with general relativity and the Planck $\\Lambda$CDM…","keywords":["cosmology","large-scale structure","peculiar velocities","redshift-space distortions","growth rate","momentum power spectrum","SDSS peculiar velocity survey","perturbation theory"],"falsifier":"A direct test: apply the full pipeline to each of the 2048 SDSSv mocks individually, not to the mean, and check the distribution of recovered $f\\sigma_8$ values. If the median deviates from the mock fiducial 0.432 by more than the typical 68% error, or if the scatter exceeds the quoted 0.04–0.05, the model–likelihood combination is biased at the survey volume and the central claim would not survive.","tokens_in":22264,"feed_emoji":"🌌","tokens_out":11134,"duration_ms":94808,"temperature":0.7,"pith_summary":"To measure how fast cosmic structure grows at low redshift, this paper adds a missing piece to the standard pair of power spectra: the cross-power spectrum between the galaxy density field and the galaxy momentum field. The authors derive an estimator for its multipoles, build a perturbation-theory model complete with survey-window convolution, and validate the whole pipeline on 2048 mocks of the SDSS DR14 peculiar-velocity catalogue before applying it to the real data. From the combined density monopole, momentum monopole, and cross dipole, they obtain $f\\sigma_8 = 0.413^{+0.050}_{-0.058}$ at $z_{\\mathrm{eff}}=0.073$. That value is consistent with the prediction of general relativity in a Planck-based $\\Lambda$CDM cosmology (0.448), and including the cross spectrum tightens the constraint by roughly 55% relative to the two auto-spectra alone.","feed_headline":"Galaxy motions put cosmic growth rate at 0.413","feed_subtitle":"Adding the density-momentum cross-power spectrum tightens the low-redshift test of general relativity","key_machinery":"The central object is the density–momentum cross-power spectrum multipole, in particular the dipole $P^{\\delta p}_1(k)$. Because flipping both $\\delta$ and $v$ leaves the physics unchanged, the cross spectrum must be purely imaginary, which selects odd multipoles and makes the dipole the first non-trivial signal. The estimator symmetrises the two fields, subtracts a shot-noise term proportional to the mean velocity, and is computed with local plane-parallel approximations and FFT-based multipole estimators. The model uses standard perturbation-theory integrals $P_{mn}$ with three corrected terms, convolved with the survey window function via a matrix built from random catalogues. Fitting is done with Box-Cox Gaussianised data vectors and a t-distribution likelihood that accounts for the finite number of mocks used to estimate the covariance.","core_discovery":"The central claim is that the density–momentum cross-power spectrum can be measured, modelled, and used to extract cosmological parameters, completing the full set of 3×2-point statistics available from the density and momentum fields. In redshift space the cross spectrum is purely imaginary and has only odd multipoles, so its dipole carries information complementary to the monopoles of the auto-spectra. The paper presents the estimator, the window-function treatment, and corrected perturbation-theory expressions for $P_{00}$, $P_{02}$, and $P_{12}$, and shows on mocks that the recovered growth rate matches the fiducial value up to $k_{\\max}=0.3\\,h\\,\\mathrm{Mpc}^{-1}$. Applied to SDSSv, the pipeline returns $f\\sigma_8 = 0.413^{+0.050}_{-0.058}$ at $z_{\\mathrm{eff}}=0.073$, consistent with the GR plus Planck $\\Lambda$CDM expectation of 0.448 and with the independent maximum-likelihood-field measurement on the same catalogue.","pith_inferences":["If the modest deficit of low-redshift growth measurements relative to the Planck plus GR prediction persists as more volume is added, the cross-spectrum pipeline offers a clean way to test whether that deficit is real; this paper does not claim such a deficit, but its own Fig. 7 shows the pattern in previous data.","Because the model accuracy is validated only for a survey of this volume, future surveys with much larger volumes will need either higher-order perturbation theory or a reduced $k_{\\max}$; the authors' 2048-volume mock test already shows the current model would mis-fit such data.","The cross-spectrum dipole is sensitive to bulk flows through the Galilean-transformation terms; the authors show the fit is insensitive to a constant bulk velocity, but the same formalism could be turned into a dedicated bulk-flow or frame-dependence test."],"forward_implications":["Adding the density–momentum cross-power spectrum to the fit reduces the statistical error on $f\\sigma_8$ by about 55% compared with using the two auto-spectra alone, as measured on the 2048 mocks.","The mock validation recovers $f\\sigma_8 = 0.441^{+0.041}_{-0.038}$ against the fiducial value 0.432, so the estimator–model–fitting chain is unbiased at the SDSSv volume up to $k_{\\max}=0.3\\,h\\,\\mathrm{Mpc}^{-1}$.","The SDSSv result $f\\sigma_8 = 0.413^{+0.050}_{-0.058}$ at $z_{\\mathrm{eff}}=0.073$ is consistent with the GR plus Planck $\\Lambda$CDM prediction 0.448, adding a low-redshift data point to the growth-rate-versus-redshift plane.","At the same $k_{\\max}=0.15\\,h\\,\\mathrm{Mpc}^{-1}$ as an earlier maximum-likelihood-field analysis of the same catalogue, the pipeline returns a comparable central value, suggesting the gain comes from the cross spectrum rather than from a different modelling choice.","The public code for the estimators, models, and window convolution makes the method directly applicable to other peculiar-velocity surveys."],"supporting_citations":[{"why":"Paper I of the series; defines the momentum field estimator and the perturbation-theory model that this paper extends to the density-momentum cross spectrum.","marker":"Howlett 2019"},{"why":"Paper II; first application of the combined density and momentum power spectra to peculiar-velocity surveys, the baseline this work improves on.","marker":"Qin et al. 2019a"},{"why":"Builds the SDSSv catalogue and the 2048 mocks used for validation, covariance estimation, and the final measurement.","marker":"Howlett et al. 2022"},{"why":"Establishes the weighted density field and optimal FKP weights used to form both the density and momentum fields.","marker":"Feldman et al. 1994"},{"why":"Provides the multipole power spectrum estimator that the auto and cross estimators adapt to the two fields.","marker":"Yamamoto et al. 2006"},{"why":"Derives the FFT-based even-multipole estimators used in the numerical implementation.","marker":"Bianchi et al. 2015"},{"why":"Provides the window function convolution matrix used to convert model multipoles to observed ones.","marker":"Blake et al. 2018"},{"why":"Supplies the perturbation-theory expansion from which the $P_{mn}$ model terms are taken.","marker":"McDonald & Roy 2009"},{"why":"Gives the t-distribution likelihood that corrects for the finite number of mocks used in the covariance.","marker":"Sellentin & Heavens 2016"},{"why":"Sets the fiducial $\\Lambda$CDM cosmology and the GR prediction $f\\sigma_8=0.448$ to which the measurement is compared.","marker":"Planck Collaboration et al. 2020"}],"fun_headline_variants":["Cross-power spectrum pins cosmic growth at 0.413","Galaxy cross-power tightens cosmic growth rate to 0.413","New probe of galaxy flows confirms GR at low redshift","Density-momentum cross-spectrum measures fσ8 = 0.413","Cosmic growth rate from cross-power spectrum: 0.413"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the perturbation-theory model for the cross-power spectrum, with the corrected $P_{00}$, $P_{02}$, and $P_{12}$ terms, is accurate up to $k_{\\max}=0.3\\,h\\,\\mathrm{Mpc}^{-1}$ for a survey with this volume; if that accuracy fails, the quoted $f\\sigma_8$ would be biased, and the paper's own mock test at 2048 times the volume and the drift in Fig. 5 beyond $k_{\\max}=0.3$ show the model already knows its limit.","fun_headline_variants_meta":{"raw":{"variants":["Cross-power spectrum pins cosmic growth at 0.413","Galaxy cross-power tightens cosmic growth rate to 0.413","New probe of galaxy flows confirms GR at low redshift","Density-momentum cross-spectrum measures fσ8 = 0.413","Cosmic growth rate from cross-power spectrum: 0.413"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000392,"raw_usage":{"total_tokens":2108,"prompt_tokens":1038,"completion_tokens":1070,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":979}},"tokens_in":654,"tokens_out":1070,"duration_ms":9529,"temperature":1.0,"reasoning_tokens":979,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:30:48.141590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test: apply the full pipeline to each of the 2048 SDSSv mocks individually, not to the mean, and check the distribution of recovered $f\\sigma_8$ values. If the median deviates from the mock fiducial 0.432 by more than the typical 68% error, or if the scatter exceeds the quoted 0.04–0.05, the model–likelihood combination is biased at the survey volume and the central claim would not survive.","supporting_citations":[],"review_version":1}