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arxiv: 2605.00450 · v1 · submitted 2026-05-01 · 🌌 astro-ph.CO

Measuring fσ₈ and BAO scale in the Local Universe: a joint real and redshift space analysis from CosmicFlows-4++

C. Franco , H. M. Courtois , A. Bernui This is my paper

Pith reviewed 2026-05-09 19:12 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords BAO scaleredshift-space distortionsgrowth ratef sigma 8local universecorrelation functionCosmicFlowsKaiser formalism
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The pith

Joint real and redshift space analysis of local galaxy distances measures the BAO scale at 132 Mpc and constrains the growth rate fσ8 to 0.344.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper uses the CosmicFlows-4++ catalog of 65,331 galaxy distances, which supplies both real-space positions and redshift-space coordinates, to study large-scale clustering out to z=0.1. An empirical fit to the two-point correlation function locates the baryon acoustic oscillation feature in real space at 132±8 h^{-1} Mpc and in redshift space at 139±7 h^{-1} Mpc, both at mean redshift 0.07. Modeling the excess clustering amplitude with the Kaiser linear distortion formula then yields a constraint on the growth-rate combination fσ8 = 0.344±0.105. A reader would care because these quantities anchor the expansion history and structure growth at the lowest redshifts accessible to galaxy surveys, providing a nearby reference point for consistency checks against higher-redshift data.

Core claim

Fitting an empirical model to the measured correlation function in the CosmicFlows-4++ sample recovers the BAO scale r_BAO^real = 132±8 h^{-1} Mpc in real space and r_BAO^z = 139±7 h^{-1} Mpc in redshift space at z=0.07; the same data, interpreted through the Kaiser formalism, give fσ8 = 0.344±0.105. The simultaneous extraction of geometric and dynamical information from one catalog supplies a self-consistent description of the local large-scale structure.

What carries the argument

The two-point correlation function measured separately in real-space and redshift-space coordinates, with an empirical template for the BAO peak and the Kaiser linear redshift-space distortion model for the amplitude enhancement.

If this is right

  • The same dataset simultaneously constrains both the cosmic expansion scale and the growth of structure at low redshift.
  • Real-space and redshift-space BAO measurements can be compared directly within one catalog to test for systematic offsets.
  • The derived fσ8 value supplies a local-universe benchmark for consistency tests with upcoming surveys such as DESI and 4MOST.
  • Joint detection of BAO and RSD features demonstrates that real and redshift-space information can be combined without separate surveys.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If future surveys repeat the analysis at slightly higher redshift, any evolution in the measured r_BAO or fσ8 could be compared directly against this z=0.07 anchor.
  • A mismatch between the real-space and redshift-space BAO scales larger than the quoted errors would point to unmodeled velocity-field effects or catalog systematics.
  • The reported precision suggests that adding more distance indicators could tighten the fσ8 error bar enough to begin discriminating among modified-gravity models at low redshift.

Load-bearing premise

The empirical template correctly isolates the BAO peak position while the linear Kaiser formula fully accounts for redshift-space distortions at z=0.07 without higher-order corrections.

What would settle it

An independent analysis of a larger local galaxy sample that recovers a BAO peak position differing by more than ~15 h^{-1} Mpc or an fσ8 value outside the 0.24–0.45 range would contradict the reported measurements.

Figures

Figures reproduced from arXiv: 2605.00450 by A. Bernui, C. Franco, H. M. Courtois.

Figure 1
Figure 1. Figure 1: Footprint of the distribution of the CF4++ galaxies in equatorial coordinates. The colorbar represents the redshift range of each galaxy in the catalogue. For the purposes of the present analysis, we employ the catalogue in two representations. In the first, we adopt the re￾construct comoving distances, which, by construction, are free from the line-of-sight displacements induced by peculiar veloc￾ities an… view at source ↗
Figure 2
Figure 2. Figure 2: Two-dimensional slice of the large-scale structures in supergalactic coordinates, together with the overdensity field derived from CF4++. Positions from each catalogue are overplotted as the blue, green, and orange points, respectively. 0 50 100 150 200 250 300 Distance [h −1 Mpc] 0.000 0.001 0.002 0.003 0.004 PDF CF4++ MDPL2 Random 0 500 1000 1500 2000 2500 Peculiar velocity [km/s] 0.0000 0.0002 0.0004 0.… view at source ↗
Figure 3
Figure 3. Figure 3: Left: Radial distances of the catalogues. Middle: Peculiar velocity magnitude inferred from the velocity field at the galaxy positions. Right: Overdensity field δ evaluated at the positions of galaxies in each catalogue. In linear theory, redshift space distortions arise from coher￾ent peculiar velocities generated by gravitational infall, which introduce fluctuations in the correlation function that depen… view at source ↗
Figure 4
Figure 4. Figure 4: CF4++ two-point correlation functions in real (light blue cir￾cles) and redshift (dark blue squares) space, together with the corre￾sponding best-fit model (solid lines). Dashed curves show the mean of 1000 MDPL2 mocks processed with the same pipeline, also in real (light green) and redshift (dark green) space. The correlation function is shown in the commonly r 2 ξ(r) representation, which enhances the vi… view at source ↗
Figure 5
Figure 5. Figure 5: Posterior distributions of the fσ8 and bσ8 parameters obtained through the CF4++ data. As discussed in Section 3.4, there is a degeneracy between the parameters f and b; and this is evident in the elongated shape of the joint posterior contours of the view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between measurements of σ8 as a function of z, compared to the prediction of the ΛCDM model (solid black line). Our measurement using CF4++ is shown as the red star. Measurements within the same redshift have been slightly shifted for better visualiza￾tion. Points in dark (light) purple are from redshift (real) space datasets. where D(z) is the linear growth factor normalized to unity at z = 0. … view at source ↗
read the original abstract

The large-scale clustering of galaxies encodes both geometric and dynamical information about the Universe. The Baryon Acoustic Oscillations (BAO) phenomenon provides a standard ruler that constrains the cosmic expansion history, while Redshift Space Distortions (RSD) probe the growth of structure through the peculiar velocity field. In this work, we present a joint analysis of BAO and growth rate parameter, $f\sigma_{8}$, in the Local Universe out to $z = 0.1$, using the $65,331$ galaxy distances of CosmicFlows-4++ database. A distinctive property of this catalogue is the availability of real space galaxy positions in addition to the redshift space coordinates. Fitting an empirical model to the measurements we obtain $r_{\rm{BAO}}^{\rm{real}} = 132\pm 8\,h^{-1}\,{\rm Mpc}$ in real space, and $r_{\rm{BAO}}^{z} = 139 \pm 7\,h^{-1}\,{\rm Mpc}$ in redshift space, at redshift $z = 0.07$. Modeling the enhancement of the correlation function within the Kaiser formalism, we derive a constraint on the growth rate parameter $f\sigma_8 = 0.344 \pm 0.105$. This analysis demonstrates how the combination of real and redshift space clustering measurements enables a simultaneous probe of important observables of the large-scale structure. Their joint detection in the same dataset, therefore, provides a self consistent view of the structure and evolution of the Local Universe. This study may be used for consistency analyses of upcoming surveys, as DESI and 4MOST, that will also provide data in both real and redshift space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analyzes the CosmicFlows-4++ catalog (65,331 galaxies) to measure the BAO scale in both real and redshift space out to z=0.1, reporting r_BAO^real = 132±8 h^{-1} Mpc and r_BAO^z = 139±7 h^{-1} Mpc at z=0.07 via fits of an empirical model to the correlation functions. It then uses the amplitude enhancement between the two spaces, modeled with the linear Kaiser formula, to constrain fσ8 = 0.344±0.105. The work emphasizes the value of having both real- and redshift-space positions in the same dataset for a self-consistent local-universe probe of geometry and growth.

Significance. If the modeling assumptions hold, the result supplies a low-redshift anchor for fσ8 and the BAO scale that can serve as a consistency test for upcoming surveys (DESI, 4MOST) that will also deliver real-space data. The distinctive use of a catalog with direct real-space distances is a clear strength, allowing a direct comparison that is rarely available at higher redshift.

major comments (2)
  1. [RSD modeling and fσ8 derivation (abstract and associated sections)] The extraction of fσ8 relies on the linear Kaiser monopole boost factor (1 + 2β/3 + β²/5) applied to the ratio of redshift-space to real-space correlation-function amplitudes. At z=0.07 and on BAO scales (~130 h^{-1} Mpc), nonlinear gravitational evolution, Fingers-of-God damping, and wide-angle effects are expected to modify both the amplitude and scale dependence of the redshift-space correlation function. The reported 7 h^{-1} Mpc offset between the two best-fit r_BAO values lies within the quoted uncertainties but is large enough to suggest residual model mismatch; any unaccounted scale-dependent distortion is absorbed directly into the fσ8 constraint without explicit validation against higher-order RSD models or mocks.
  2. [BAO scale fitting procedure] The empirical BAO template is fitted independently in real and redshift space. Because the template parameters are not required to enforce a common r_BAO value across the two spaces, any mismatch arising from incomplete RSD modeling or from the empirical form itself propagates into the growth-rate parameter. A joint fit that enforces a single BAO scale while allowing only the amplitude to vary under the Kaiser ansatz would provide a stronger internal consistency check.
minor comments (2)
  1. [Abstract and data section] The abstract quotes the full catalog size (65,331 galaxies) but does not state the precise selection cuts, redshift range, or number of objects actually used for the correlation-function measurements; this information should be added for reproducibility.
  2. [Analysis methodology] It would be useful to report the fitting range in separation, the covariance estimation method, and any priors placed on the empirical model parameters so that the robustness of the quoted uncertainties can be assessed.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their insightful comments on our manuscript. We have carefully considered each point and provide our responses below, along with the revisions we intend to incorporate in the updated version.

read point-by-point responses

  1. Referee: [RSD modeling and fσ8 derivation (abstract and associated sections)] The extraction of fσ8 relies on the linear Kaiser monopole boost factor (1 + 2β/3 + β²/5) applied to the ratio of redshift-space to real-space correlation-function amplitudes. At z=0.07 and on BAO scales (~130 h^{-1} Mpc), nonlinear gravitational evolution, Fingers-of-God damping, and wide-angle effects are expected to modify both the amplitude and scale dependence of the redshift-space correlation function. The reported 7 h^{-1} Mpc offset between the two best-fit r_BAO values lies within the quoted uncertainties but is large enough to suggest residual model mismatch; any unaccounted scale-dependent distortion is absorbed directly into the fσ8 constraint without explicit validation against higher-order RSD models or mocks.

    Authors: We agree that a more detailed assessment of potential RSD modeling systematics would strengthen the paper. On BAO scales at low redshift, the linear Kaiser formula is widely used and the nonlinear corrections are expected to be small (less than a few percent in amplitude). The observed offset in r_BAO is statistically insignificant. In the revised manuscript, we have expanded the discussion in Section 4 to include an assessment of nonlinear RSD effects, citing relevant literature, and added a test of the amplitude ratio on non-BAO scales to confirm robustness. Explicit mock validation is not feasible given the unique characteristics of the CosmicFlows-4++ dataset, but the internal consistency between real and redshift space supports the reliability of our fσ8 measurement. revision: partial

  2. Referee: [BAO scale fitting procedure] The empirical BAO template is fitted independently in real and redshift space. Because the template parameters are not required to enforce a common r_BAO value across the two spaces, any mismatch arising from incomplete RSD modeling or from the empirical form itself propagates into the growth-rate parameter. A joint fit that enforces a single BAO scale while allowing only the amplitude to vary under the Kaiser ansatz would provide a stronger internal consistency check.

    Authors: We appreciate this constructive suggestion for improving the robustness of the analysis. We have now performed a joint fit to the real- and redshift-space correlation functions, enforcing a single r_BAO parameter while modeling the amplitude difference with the Kaiser factor to derive fσ8. The results are consistent with our original findings, and this joint analysis is presented as the primary result in the revised manuscript, with the independent fits shown for comparison. revision: yes

standing simulated objections not resolved
  • Full validation of the RSD modeling using mocks specifically constructed for the CosmicFlows-4++ catalog

Circularity Check

0 steps flagged

No circularity: direct empirical fits to observed correlation functions using standard Kaiser RSD model

full rationale

The derivation measures the galaxy correlation function separately in real space and redshift space from the CosmicFlows-4++ catalog, fits an empirical BAO template to each to extract r_BAO values, and then applies the standard linear Kaiser formula to the amplitude enhancement between the two measured functions to constrain fσ8. No step reduces an output quantity to a definition or fit of its own input parameters by construction, no load-bearing self-citations are invoked for uniqueness or ansatzes, and the central results remain independent of the fitted values themselves. This is a conventional observational analysis whose content is not tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of an empirical correlation-function model and the applicability of the Kaiser approximation at low redshift; no new entities are introduced.

free parameters (1)
  • BAO scale in empirical model
    The reported r_BAO values are obtained by fitting the model to the measured correlation function.
axioms (1)
  • domain assumption Kaiser formalism accurately models the linear enhancement of the redshift-space correlation function at z=0.07
    Invoked to convert the observed clustering boost into a constraint on fσ8.

pith-pipeline@v0.9.0 · 5634 in / 1414 out tokens · 44318 ms · 2026-05-09T19:12:02.924698+00:00 · methodology

discussion (0)

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