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The Redshift-Space Momentum Power Spectrum III: measuring the growth rate from the SDSSv survey using auto- and cross- power spectrum of the galaxy density and momentum fields

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.09571 v3 pith:FOYHE3DQ submitted 2024-11-14 astro-ph.CO

Fei Qin , Cullan Howlett , David Parkinson This is my paper · ORCID
classification astro-ph.CO
keywords cosmologylarge-scalestructurepeculiarvelocitiesredshift-spacedistortionsgrowthratemomentumpowerspectrumSDSSvelocitysurveyperturbationtheory
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 5 minor

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.

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 (2)
  1. [§5.1, Eqs. (44)–(46)] 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).
  2. [§5.2, Fig. 5] 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.
minor comments (5)
  1. [§6.1] 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.
  2. [§2, Eq. (4)] The fit for σ_SDv in Eq. (4) is given without uncertainties; reporting the fit covariance would be useful for reproducibility.
  3. [Figs. 4 and 6] 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.
  4. [§5.1, Eq. (29)] 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.
  5. [Table 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f sigma_8 measurement is validated against mock-truth and an external GR+Planck prediction, and the model inputs are from perturbation theory rather than constructed from the data.

full rationale

The central claim, f sigma_8 = 0.413^{+0.050}_{-0.058}, is obtained by fitting perturbation-theory models (Eqs. 31-36) to the SDSSv density and momentum power spectra. The model components P_mn originate from McDonald & Roy (2009), Vlah et al. (2012, 2013), and Okumura et al. (2014), with expressions carried over from Paper I but checked and updated in this work; these are independent external derivations, not fits to the SDSSv measurement. The method is tested on 2048 mocks with known fiducial parameters, and the recovered f sigma_8 = 0.441^{+0.041}_{-0.038} is compared to the simulation truth 0.432, giving an independent validation that does not reduce to the data being analysed. The final result is compared to the external GR+Planck prediction of 0.448. The use of data-derived velocity scatter in constructing random-catalogue weights is a nuisance input for the window function, not a parameter whose fitted value is subsequently reported as a prediction of f sigma_8. Self-citations to Paper I and Paper II are present for model coefficients and fitting conventions, but they are not load-bearing in a circular way: the cited prior work was itself tested on independent surveys and simulations, and the current paper's mock recovery is an external check. No equation or fitted parameter in the paper is equivalent by construction to the claimed growth-rate measurement.

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The central measurement rests on a perturbation-theory model inherited from Paper I, the fidelity of the SDSSv mocks, and several hand-chosen scale and weighting parameters (kmax, FKP pivots, the 300 km/s velocity floor). No new physical entities are introduced.

free parameters (9)
  • f sigma8 = 0.413 +0.050 -0.058
    Target growth-rate parameter fitted to the combined SDSSv power spectra.
  • b1 sigma8 = 1.078 +0.086 -0.087
    Linear galaxy bias times sigma8, fitted simultaneously with f sigma8.
  • b2 sigma8 = -0.810 +0.141 -0.150
    Second-order galaxy bias parameter fitted as a nuisance in the perturbation theory model.
  • b3nl sigma8 = 0.711 +0.261 -0.260
    Third-order nonlocal bias parameter fitted as a nuisance.
  • sigma_v^2 = 82.0 +24.0 -14.1 km^2 s^-2
    Nonlinear velocity dispersion fitted as a nuisance with a flat prior up to 250^2 km^2 s^-2.
  • sigma_SDv relation slope and intercept = slope 0.206, intercept 117.182 km/s
    Linear relation fitted to the measured redshift-dependent velocity scatter (Eq. 4), used to assign velocities to the random catalogue.
  • FKP power spectrum pivots = P_delta_FKP = 1600 h^-3 Mpc^3, P_p_FKP = 5e9 h^-3 Mpc^3 km^2 s^-2
    Hand-chosen pivots following Paper II and Turner et al. (2023) to set the optimal FKP weights.
  • Box-Cox lambda per k-bin = not tabulated; fitted from 2048 mocks per k-bin
    Gaussianization exponent estimated by maximum likelihood from mock power spectra; the shift Delta is manually chosen and stated not to change the covariance.
  • Intrinsic velocity scatter floor in <v^2> = 300 km/s
    Hand-chosen constant added to the velocity error model (Eq. 29), following Paper II; the authors state results are insensitive to it.
assumptions (8)
  • domain assumption The density and momentum fields share the same galaxy bias and velocity dispersion parameters (no velocity bias).
    Section 4.1 states that no distinction is made for bias and velocity dispersion between the density and momentum fields because both are estimated from the same galaxy catalogue.
  • domain assumption The perturbation theory model of the power spectrum (Eqs 31-33 with P_mn from Paper I and the corrected Eqs 34-36) accurately describes SDSSv power spectra up to kmax = 0.3 h/Mpc.
    The model is inherited from Paper I and the McDonald-Roy and Vlah et al. frameworks; the authors test it on mocks and note in Section 5.2 that it would be inaccurate for a survey volume 2048 times larger.
  • domain assumption The survey window function convolution matrix for the cross-power spectrum is correctly symmetrized via the replacement in Eqs 42-43.
    The window treatment follows Blake et al. (2018) with a symmetrized combination of the density and momentum window functions for the cross spectrum.
  • domain assumption The mock catalogues faithfully reproduce the SDSSv selection function, Fundamental Plane relation, and galaxy clustering.
    The covariance matrix and the validation of the estimators rest on the 2048 mocks from Howlett et al. (2022); any mismatch between mocks and the real survey would bias both the error bars and the validation.
  • domain assumption The Box-Cox transformation with mock-fitted lambda Gaussianizes the power spectrum likelihood in each k-bin.
    The likelihood in Eq. 46 assumes the transformed data vector is Gaussian and the covariance is estimated from the transformed mocks; this is inherited from Paper I.
  • domain assumption The line-of-sight peculiar velocities estimated from Fundamental Plane log-distance ratios via Eqs 2-3 are unbiased.
    The velocity estimator is standard in the literature, but any systematic in the Fundamental Plane or the low-redshift approximation propagates directly into the momentum field.
  • domain assumption The local plane-parallel approximation holds for the SDSSv geometry on the scales used.
    The cross-power spectrum estimator in Eq. 14 is derived under this approximation, and wide-angle effects are neglected throughout.
  • domain assumption A flat LCDM fiducial cosmology with Planck-like parameters is used to generate the linear power spectrum and distances.
    Section 1 lists the fiducial parameters (ns = 0.9653, Om = 0.3121, Ob = 0.0491, sigma8 = 0.8150, h = 0.6751); the fitted f sigma8 depends on this choice through the model power spectrum and distance conversions.

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Pith. "Pith review of The Redshift-Space Momentum Power Spectrum III: measuring the growth rate from the SDSSv survey using auto- and cross- power spectrum of the galaxy density and momentum fields." pith.science (2026). https://pith.science/paper/FOYHE3DQ

@misc{pith2026241109571,
  author       = {Pith},
  title        = {Pith review of: The Redshift-Space Momentum Power Spectrum III: measuring the growth rate from the SDSSv survey using auto- and cross- power spectrum of the galaxy density and momentum fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOYHE3DQ}},
  note         = {Machine review of arXiv:2411.09571}
}
abstract

The large-scale structure of the Universe and its evolution over time contains an abundance of cosmological information. One way to unlock this is by measuring the density and momentum power spectrum from the positions and peculiar velocities of galaxies, and fitting the cosmological parameters from these power spectrum. In this paper, we will explore the cross power spectrum between the density and momentum fields of galaxies. We derive the estimator of the density-momentum cross power spectrum multipoles. The growth rate of the large-scale-structure, $f\sigma_8$ is measured from fitting the combined density monopole, momentum monopole and cross dipole power spectrum. The estimators and models of power spectrum as well as our fitting method have been tested using mock catalogues, and we find that they perform well in recovering the fiducial values of the cosmological parameters of the simulations, and we also find that the errors of the parameters can be largely reduced by including the cross-power spectrum in the fit. We measure the auto-density, auto-momentum and cross power spectrum using the Sloan Digital Sky Survey Data Release 14 peculiar velocity catalogue. The fit result of the growth rate $f\sigma_8$ is $f\sigma_8=0.413^{+0.050}_{-0.058}$ at effective redshift $z_{\mathrm{eff}}=0.073$, and our measurement is consistent with the prediction of the $\Lambda$ Cold Dark Matter cosmological model assuming General Relativity.

Figures

Figures reproduced from arXiv: 2411.09571 by the authors.

Figure 1
Figure 1. The sky coverage of the SDSSv galaxies. The color of the dot indicates the redshift of the galaxy according to the color bar. 0 5000 10000 15000 20000 25000 cz [ km s 1 ] 0 200 400 600 800 1000 1200 1400 N [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The redshift distribution of the SDSSv galaxies. where 𝑧mod is given by (Davis & Scrimgeour 2014; Watkins & Feldman 2015) 𝑧mod = 𝑧  1 + 1 2 (1 − 𝑞0)𝑧 − 1 6 (1 − 𝑞0 − 3𝑞 2 0 + 1)𝑧 2  , (3) and where the acceleration parameter is 𝑞0 = 0.5(Ω𝑚 − 2ΩΛ) = −0.532. This relation uses the low-redshift approx￾imation of the log-distance ratio and assumes that the true peculiar velocities of the galaxies are much less than th… view at source ↗
Figure 4
Figure 4. The power spectrum and parameter fit results of the SDSSv mocks. In the left panels, the yellow filled circles in the top, middle and bottom panels are the measured density monopole, momentum monopole and cross dipole power spectrum, respectively. These are the average of the measurements of the 2048 mocks, with error bars representative of the error on a single realisation. The blue curves are the model power spect… view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: The blue filled-squares show the estimated growth rate 𝑓 𝜎8 as a function of cut-off wave number 𝑘max. The pink dashed￾line is the fiducial value 𝑓 𝜎8=0.432. In all cases, the minimum fitting scale is set to 𝑘 = 0.025ℎ Mpc−1 . redshift. This has a slightly larger error…
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: The growth rate 𝑓 𝜎8 as a function of redshift 𝑧. The blue curve is computed using the equations 47 to 51 assuming GR and a Planck Collaboration et al. (2020)-based ΛCDM cosmology. The red dot is the measurement of this paper. The purple filled square (L23) is Lai et a…
Figure 8
Figure 8. Figure 8: The green-colored histograms and 2D-contours show the fit results of the parameters by adding 𝜖. The blue-colored histograms and 2D-contours show the fit results without 𝜖 (same as Fig.6). Therefore, under the Galilean transformation, the momentum power spectrum of Eq.…

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