Cosmicflows-4 and the Cosmological Consistency of Large-Scale Motions

Abstract: We study coherent flows in Cosmicflows-4 (CF4) and its constituent catalogs. We fit monopole and dipole moments in nonoverlapping radial bins using a log-distance radial-velocity estimator. For each sample, the Planck 2018 $Λ$CDM prediction accounts for the actual positions, statistical weights, velocity covariance, and measurement covariance. Across six bins extending to $300,h<sup>{-1},\mathrm{Mpc}$, the All Individual sample is broadly consistent with $Λ$CDM, with dipole and joint deviations of $2.29σ$ and $2.36σ$. The strongest localized feature occurs at $120$--$160,h<sup>{-1},\mathrm{Mpc}$, where $|V_{\rm dip}|=628\pm82,\mathrm{km,s<sup>{-1}}$ and the dipole deviation reaches $3.40σ$. The excess is concentrated toward negative supergalactic X and depends strongly on the survey window. TFR and 6dFGS favor larger dipoles, while SDSS and SN favor smaller ones. Removing 6dFGS reduces the upper-bin dipole deviation to $1.56σ$, while removing SN raises it to $3.77σ$. Nevertheless, the SN and 6dFGS dipoles remain mutually consistent when their covariance is included. SDSS is consistent with $Λ$CDM through $160,h<sup>{-1},\mathrm{Mpc}$ but shows a separate rise at larger radii whose significance depends on the treatment of the monopole and shared distance-scale covariance. Overall, CF4 does not show uniformly anomalous motion across its full radial range. Instead, it contains localized and window-dependent features that should be tested with selection-matched mock catalogs and correlated calibration uncertainties.

• The paper finds Cosmicflows-4 broadly consistent with Planck 2018 ΛCDM, with joint deviations below 2.4σ across six radial bins extending to 300 h⁻¹ Mpc.
• The analysis identifies a localized coherent-motion excess at 100–160 h⁻¹ Mpc, including a 628 ± 82 km/s dipole and a 3.87σ joint deviation concentrated toward negative supergalactic SGX.
• The paper shows that catalog composition and estimator choices strongly affect the signal—especially 6dFGS, monopole treatment, and distance-scale systematics—so the result motivates selection-matched mocks rather than a challenge to ΛCDM.

Overview

Nusser and Tully analyze coherent large-scale motions in Cosmicflows-4 (CF4), testing whether the catalog's peculiar velocities are consistent with the velocity field expected in a Planck 2018 $\Lambda$CDM cosmology. Rather than estimating a bulk flow within an idealized spherical window, the authors fit monopole and dipole moments in nonoverlapping radial shells using a log-distance radial-velocity estimator, evaluating the $\Lambda$CDM covariance at the actual object positions, inverse-variance weights, and angular coverage of each catalog. The central finding is that CF4 is broadly consistent with $\Lambda$CDM when all radial bins are combined jointly—dipole and joint deviations of $2.29\sigma$ and $2.36\sigma$ over six bins through $300\,h^{-1}$Mpc—but contains a localized excess of coherent motion at $R_{\rm eff}\simeq100$–$150\,h^{-1}$Mpc, concentrated in the negative supergalactic SGX direction, whose significance is strongly dependent on the constituent catalog and the treatment of the monopole.

Method

The analysis uses the CF4 All Individual compilation (55,749 objects), decomposed into its main distance-indicator components: the SDSS and 6dFGS Fundamental Plane samples, the Tully–Fisher (TFR) sample, supernova (SN Ia and SN II) distances, and smaller FP and TRGB samples. Pairwise overlap between component catalogs is small in every bin, though all are subsets of All Individual. Objects are placed at their CMB-frame redshift positions, $$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$, rather than at their measured distances; this redshift-space binning deliberately reduces sensitivity to distance- and velocity-Malmquist effects, a known pathology of distance-based analyses.

Radial peculiar velocities are estimated from the distance-modulus difference $\Delta_i = {\rm DM}_i - {\rm DM}_{z,i}$ via a first-order log-distance conversion $\Lambda$0, where the redshift-dependent factor $\Lambda$1 tends to $\Lambda$2 at low redshift and the conversion preserves the approximately Gaussian character of distance-modulus errors. A $\Lambda$3 measurement-error floor prevents very small reported distance errors from dominating the inverse-variance weights. Within each shell the authors fit three nested models: monopole only, dipole only ($\Lambda$4), and a joint monopole-plus-dipole fit, via weighted least squares.

The $\Lambda$5CDM comparison uses the linear-theory radial velocity covariance $\Lambda$6 projected from the Cartesian velocity covariance tensor computed from a Planck-normalized $\Lambda$7 linear power spectrum (Eisenstein–Hu transfer function). Propagating $\Lambda$8 through each catalog's fitting operator yields the cosmic-variance covariance $\Lambda$9, and adding the diagonal measurement covariance gives $\Lambda$0. Predictive $\Lambda$1-values are computed for the monopole ($\Lambda$2, 1 dof), dipole ($\Lambda$3, 3 dof), and joint fit ($\Lambda$4, 4 dof), with cross-bin and cross-catalog covariances constructed from the same linear machinery. The authors emphasize that these are predictive tail probabilities under $\Lambda$5CDM, not probabilities that $\Lambda$6CDM is false, and that the two-sided $\Lambda$7 conversion does not reduce multidimensional tests to one-dimensional amplitudes.

Results

The per-bin tests show that the clearest local deviations occur at intermediate radii. In the $\Lambda$8–$\Lambda$9Mpc bin, the All Individual joint fit yields $2.29\sigma$0 and $2.29\sigma$1, with dipole and joint deviations of $2.29\sigma$2 and $2.29\sigma$3. In the $2.29\sigma$4–$2.29\sigma$5Mpc bin, TFR alone reaches a $2.29\sigma$6 dipole deviation, the largest single value in the analysis, and All Individual gives $2.29\sigma$7. Component decomposition shows that SGX is the least consistent direction: All Individual's cross-bin SGX test gives $2.29\sigma$8, with marginal per-bin deviations of $2.29\sigma$9 and $2.36\sigma$0 in the two discrepant bins, while SGY, SGZ, and the monopole remain consistent.

Crucially, these local deviations do not accumulate into a strong global result. The cross-bin All Individual tests give at most $2.36\sigma$1 over the four common bins and $2.36\sigma$2 over all six bins, indicating that the anomaly is localized rather than a uniformly anomalous flow across the full radial range. The authors argue this is a materially different claim from a general failure of the predicted velocity field, and the cross-bin covariance is fully included, so the conclusion does not rely on treating shells as independent.

The component-catalog analysis reveals that no single anomalous mode is shared among samples. TFR and 6dFGS favor larger dipoles toward negative SGX; SDSS's distinctive contribution is positive SGY; SN is more unusual in its monopole ($2.36\sigma$3 in the $2.36\sigma$4–$2.36\sigma$5 bin) than in its dipole. Leave-one-out tests show that removing 6dFGS from All Individual reduces the upper-bin dipole deviation from $2.36\sigma$6 to $2.36\sigma$7 (and the joint deviation to $2.36\sigma$8), while removing SN raises it to $2.36\sigma$9 ($300\,h^{-1}$0 joint). Yet a direct covariance-aware comparison of the standalone SN and 6dFGS fits finds them mutually consistent ($300\,h^{-1}$1-values of 0.64–0.95 in the discrepant bins). The authors stress that these removals measure catalog influence, not an additive decomposition of the flow, since each removal simultaneously changes the window, weights, and predicted cosmic variance.

The SDSS outer bins illustrate how strongly estimator choice matters. SDSS alone is fully consistent with $300\,h^{-1}$2CDM through $300\,h^{-1}$3Mpc (dipole deviations of $300\,h^{-1}$4–$300\,h^{-1}$5 across four bins), but in the $300\,h^{-1}$6–$300\,h^{-1}$7Mpc range its nearly degenerate monopole and $300\,h^{-1}$8 (measurement correlation $300\,h^{-1}$9) produce results that shift from $R_{\rm eff}\simeq100$0 with $R_{\rm eff}\simeq100$1 fixed to $R_{\rm eff}\simeq100$2–$R_{\rm eff}\simeq100$3 with $R_{\rm eff}\simeq100$4 free. An illustrative shared 2% distance-scale uncertainty lowers the joint combined deviation to $R_{\rm eff}\simeq100$5. A quadrupole extension does not significantly improve the outer-bin fits, so no angular structure beyond a dipole is required, though the authors caution that this does not establish that the fitted outer dipole is a physical bulk motion.

Relation to prior work and limitations

The localized excess at $R_{\rm eff}\simeq100$6–$R_{\rm eff}\simeq100$7Mpc is qualitatively consistent with earlier CF4 bulk-flow analyses reporting $R_{\rm eff}\simeq100$8 (3.8$R_{\rm eff}\simeq100$9) at $150\,h^{-1}$0Mpc and $150\,h^{-1}$1 (3.3$150\,h^{-1}$2) at $150\,h^{-1}$3Mpc, and with the Wiener-filter result that 6dFGS dominates the flow profile beyond $150\,h^{-1}$4Mpc toward the Shapley Concentration direction. The present work differs in the measured quantity—window-specific shell moments rather than cumulative or minimum-variance bulk flows—and adds the identification of which components and directions drive the signal. The negative-SGX concentration is suggestive of Shapley, but the authors note that confirming an association requires a velocity–density comparison, not a velocity-only consistency test.

The paper is explicit about its limitations. The covariance model omits shared distance-calibration errors, cross-catalog measurement covariance, repeated use of the same distance measurement, and correlated nonlinear velocities; the 100 km s$150\,h^{-1}$5 floor does not model correlated small-scale dispersion. The $150\,h^{-1}$6CDM covariance is unconditional on the actual local density field, evaluated at $150\,h^{-1}$7, and neglects redshift evolution. Catalog selection and Malmquist corrections are inherited from CF4 without refitting. The linearized log-distance conversion should be checked with an exact distance likelihood, particularly for the TFR result. Most importantly, no multiple-comparisons correction is applied across the many correlated catalogs, bins, and components examined, and the global rows of different catalogs must not be combined as independent measurements.

Conclusion

Using window-aware monopole–dipole fits with catalog-specific $150\,h^{-1}$8CDM covariances, the authors find that Cosmicflows-4 as a whole is broadly consistent with the Planck 2018 $150\,h^{-1}$9CDM velocity field, with full-range joint deviations below $$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$0. The notable exception is a localized excess of coherent motion at $$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$1–$$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$2Mpc—$$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$3 in the $$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$4–$$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$5Mpc bin—concentrated in negative SGX and most sensitive to the 6dFGS sample, though the SN and 6dFGS dipoles are themselves mutually consistent. Because the quoted significances are unadjusted for multiple testing and the covariance model excludes calibration and cross-catalog correlations, the authors conclude that the results motivate a selection-matched mock-catalog analysis with predefined statistics rather than a claim of a robust challenge to $$\boldsymbol{x}_i = hD_C(z_{\rm CMB})\hat{\boldsymbol{r}}_i$$6CDM. The open question left by the paper is whether the negative-SGX feature survives a unified treatment of observational and cosmological uncertainties.