arxiv: 2605.05454 · v1 · submitted 2026-05-06 · 🌌 astro-ph.CO · gr-qc

Recognition: unknown

Backreaction and the Role of Spatial Curvature in the Cosmic Neighborhood

Marco Galoppo , Thomas Buchert , Pierre Mourier

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Pith reviewed 2026-05-08 15:47 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords backreactionspatial curvaturelocal universescalar averagingcosmic neighborhoodenergy budgetLambda-CDMkinematical backreaction

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The pith

Average spatial curvature contributes at the 10% level to the local universe's energy budget on scales up to 300 Mpc/h.

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

The paper performs the first direct computation of averaged dynamical quantities in the local universe using a reconstruction of density and velocity fields. It extracts domain averages for density, expansion rate, spatial curvature, and backreaction over regions around our galaxy up to 300 Mpc/h. Significant O(10%) contributions from spatial curvature are found across all scales, while backreaction is smaller at O(1%) maximum. The local structure shows nested features like a large void shell, and averages do not match the global Lambda-CDM model within the probed range.

Core claim

What carries the argument

Covariant scalar averaging formalism applied to the Cosmicflows-4++ reconstructed density and velocity fields to compute domain averages of expansion rate, spatial curvature, and kinematical backreaction.

If this is right

The local cosmic web exhibits scale-dependent variations in averaged dynamics that reflect its nested structure including a large-scale void shell.
Average spatial curvature supplies a substantial O(10%) contribution to the domain-averaged energy budget on all probed scales.
Kinematical backreaction remains at most an O(1%) contribution even on the smallest scales of 30 Mpc/h.
Convergence of the domain averages to the global Lambda-CDM background does not occur within the surveyed range up to 300 Mpc/h.

Where Pith is reading between the lines

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

Local curvature effects may require adjustments when interpreting nearby measurements of cosmological parameters such as the Hubble constant.
Extending the same averaging procedure to larger reconstructions could identify the scale at which the global background is recovered.
The void-shell feature suggests the local universe may systematically bias certain distance-ladder or supernova analyses.

Load-bearing premise

The Cosmicflows-4++ reconstruction provides an accurate enough representation of the three-dimensional density and velocity fields on scales from 30 to 300 Mpc/h for the averaging procedure to yield reliable domain-averaged quantities.

What would settle it

An independent reconstruction or simulation of local density and velocity fields that yields average spatial curvature contributions well below 10% or kinematical backreaction well above 1% on the same scales would falsify the main results.

Figures

Figures reproduced from arXiv: 2605.05454 by Marco Galoppo, Pierre Mourier, Thomas Buchert.

**Figure 1.** Figure 1: Expansion rate (Θ/3, top-left panel), shear scalar amplitude (√ 2σ2, top-right panel), matter density (ρ, bottomleft panel), and spatial scalar curvature (R, bottom-right panel) on the Supergalactic plane as functions of the Supergalactic comoving-distance coordinates SGX, SGY, from the CF4++ reconstruction. In all panels, the white cross marker at the center indicates the observer’s position. pendix A, a… view at source ↗

**Figure 2.** Figure 2: The present-day averaged energy budget as measured by the ΩD,0 i parameters, and computed within the CF4++ reconstruction for concentric spherical averaging domains D of varying comoving radius r, from r = 30 Mpc/h to r = 300 Mpc/h. The green and yellow horizontal dashed lines represent the ΛCDM background values used in the reconstruction for Ω 0 M and Ω0 Λ, respectively. A nonnegligible ΩD,0 R is observe… view at source ↗

**Figure 3.** Figure 3: The present-day averaged energy budget, as measured by the ΩD,0 i parameters, and computed within concentric spherical averaging domains D of varying comoving radius r, from r = 30 Mpc/h to r = 300 Mpc/h, within various robustness tests. The green and yellow horizontal dashed lines in each panel represent the ΛCDM background values used in the reconstruction for Ω0 M and Ω0 Λ, respectively. The zoomed-in … view at source ↗

**Figure 4.** Figure 4: The present-day averaged energy budget inferred using either a second-order (solid orange lines) or fourth-order (purple dashed lines) finite-difference schemes for computing the PV gradients. Each panel shows a specific ΩD,0 i parameter: Ω D,0 Q , ΩD,0 R , ΩD,0 M , and ΩD,0 Λ , respectively, from left to right and top to bottom view at source ↗

read the original abstract

We present the first direct computation of spatially averaged dynamical quantities in the local Universe, employing the Cosmicflows-4++ reconstruction and a covariant scalar averaging formalism. We extract the domain-averaged density, expansion rate, spatial curvature, and kinematical backreaction over cosmologically relevant domains around our Galaxy, extending up to a comoving radius of $300~\mathrm{Mpc}/h$. The resulting domain-averaged present-day energy budget features nontrivial variations with scale that reflect a nested structure within the cosmic neighborhood, including a large-scale void shell encompassing the local cosmic web. Remarkably, we find significant contributions to this energy budget from the average spatial curvature at the $\mathcal{O}(10\%)$ level on all probed scales. By contrast, the kinematical backreaction remains much smaller throughout the surveyed volume, reaching at most a $\mathcal{O}(1\%)$ contribution on the smallest scales considered, i.e., $30~\mathrm{Mpc}/h$. Convergence to the global $\Lambda$CDM background is not observed within this range of scales.

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.

Desk Editor's Note private letter to a colleague

This paper averages Cosmicflows-4++ fields with the covariant formalism and reports O(10%) curvature with small backreaction, but the fiducial LCDM reconstruction likely biases those numbers.

read the letter

This paper applies the covariant scalar averaging formalism to the Cosmicflows-4++ reconstruction and extracts domain-averaged density, expansion rate, spatial curvature, and kinematical backreaction over scales from 30 to 300 Mpc/h. The main result is that average spatial curvature contributes at the O(10%) level to the local energy budget on all those scales, while backreaction stays below O(1%) even on the smallest domains, and the averages do not converge to the global Lambda-CDM background. They also note a nested structure with a large-scale void shell around the local web. The work is the first to perform this direct numerical averaging on that particular observational catalog, which moves the discussion from simulations or analytic models to reconstructed real data. The scale-dependent plots give a clear picture of how the terms vary with domain size. The formalism itself is applied in a straightforward way, and the energy-budget decomposition follows the standard Buchert equations without obvious algebraic errors. The soft spot is the input reconstruction. Cosmicflows-4++ uses linear theory and a fiducial flat Lambda-CDM background to convert redshifts to distances and model velocities. If genuine curvature or backreaction effects exist at these scales, that assumption can fold them into the fields in a way that either suppresses backreaction or inflates the curvature term. The paper reports no tests against mocks that include known non-FLRW behavior or comparisons to other reconstruction pipelines, so the quoted percentages and the lack of convergence rest on an unverified assumption. Error analysis and sensitivity to reconstruction choices are not detailed in the available text. This work is aimed at cosmologists who track the backreaction literature and want to see what the numbers look like when real local data are plugged in. A reader already comfortable with the covariant formalism will get the most from the scale-dependent results. It deserves peer review because the data application is new and the topic matters for interpreting local observables, even though referees will need to press on the reconstruction biases and ask for validation steps.

Referee Report

3 major / 2 minor

Summary. The manuscript applies a covariant scalar averaging formalism to the Cosmicflows-4++ reconstructed 3D density and velocity fields to compute domain-averaged present-day density, expansion rate, spatial curvature, and kinematical backreaction over spherical domains centered on the Milky Way with comoving radii from 30 to 300 Mpc/h. It reports that average spatial curvature contributes O(10%) to the local energy budget on all probed scales while kinematical backreaction remains O(1%) or smaller, with no convergence to the global flat Lambda-CDM background observed within this range.

Significance. If the quoted percentages are robust, the work would provide the first direct observational quantification of backreaction and curvature effects in the local universe, demonstrating scale-dependent deviations from FLRW and a nested void-shell structure in the cosmic neighborhood. The use of an independent reconstruction catalog rather than a fitted model is a methodological strength that avoids circularity in the central numbers.

major comments (3)

[Methods] Methods section (reconstruction pipeline): Cosmicflows-4++ employs linear theory and a fiducial flat Lambda-CDM background for redshift-distance conversion and velocity modeling. This assumption risks suppressing genuine non-FLRW curvature or backreaction signals in the reconstructed fields, which could artificially produce the reported O(10%) curvature dominance and small backreaction; no validation against mocks containing known non-zero backreaction or curvature is described.
[Results] Results (energy-budget figures): The O(10%) curvature and O(1%) backreaction contributions are presented without propagated uncertainties from the reconstruction or sensitivity tests to the fiducial cosmology, rendering it impossible to confirm that these percentages are stable against the reconstruction uncertainties acknowledged in the abstract.
[Discussion] Discussion (convergence claim): The statement that convergence to the global Lambda-CDM background is not observed up to 300 Mpc/h is central to the interpretation, yet the manuscript provides no comparison to larger-volume simulations or alternative reconstruction pipelines that would test whether the lack of convergence is physical or an artifact of the limited domain and reconstruction assumptions.

minor comments (2)

[Abstract] The abstract and introduction should explicitly define the averaging domain (e.g., whether it is a top-hat sphere or a more complex volume) and the precise normalization of the energy-budget percentages.
[Formalism] Notation for the averaged scalars (e.g., the distinction between <R> and the kinematical backreaction Q_D) is introduced without a dedicated equation block; a single summary equation would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report, including recognition of the methodological strengths. We address each major comment below and indicate revisions to be incorporated in the next version of the manuscript.

read point-by-point responses

Referee: [Methods] Methods section (reconstruction pipeline): Cosmicflows-4++ employs linear theory and a fiducial flat Lambda-CDM background for redshift-distance conversion and velocity modeling. This assumption risks suppressing genuine non-FLRW curvature or backreaction signals in the reconstructed fields, which could artificially produce the reported O(10%) curvature dominance and small backreaction; no validation against mocks containing known non-zero backreaction or curvature is described.

Authors: We acknowledge that Cosmicflows-4++ relies on linear theory and a fiducial flat ΛCDM cosmology, as is standard for velocity-field reconstructions. The covariant averaging formalism is then applied directly to the reconstructed fields. While this does not introduce circularity in the averaging step itself, we agree that the absence of explicit validation against mocks with injected non-FLRW signals is a limitation. In the revised manuscript we will expand the methods section to discuss this assumption explicitly, reference prior mock validations of the Cosmicflows pipeline, and note that any suppression would likely be conservative for the reported deviations. We will also outline plans for future mock-based tests. revision: partial
Referee: [Results] Results (energy-budget figures): The O(10%) curvature and O(1%) backreaction contributions are presented without propagated uncertainties from the reconstruction or sensitivity tests to the fiducial cosmology, rendering it impossible to confirm that these percentages are stable against the reconstruction uncertainties acknowledged in the abstract.

Authors: We agree that explicit uncertainty propagation and sensitivity tests are needed to substantiate the quoted percentages. In the revised manuscript we will add error bands derived from the published Cosmicflows-4++ uncertainty maps to the energy-budget figures and include a new subsection performing sensitivity tests by varying the fiducial cosmological parameters within their observational ranges. These additions will allow readers to assess stability directly. revision: yes
Referee: [Discussion] Discussion (convergence claim): The statement that convergence to the global Lambda-CDM background is not observed up to 300 Mpc/h is central to the interpretation, yet the manuscript provides no comparison to larger-volume simulations or alternative reconstruction pipelines that would test whether the lack of convergence is physical or an artifact of the limited domain and reconstruction assumptions.

Authors: The lack of convergence is reported strictly within the 300 Mpc/h domain accessible to Cosmicflows-4++. We will revise the discussion to include direct comparisons with expectations from large-volume ΛCDM simulations (e.g., Millennium and IllustrisTNG) at comparable scales, and we will cite independent reconstruction pipelines (e.g., 2M++ and SDSS-based) that show qualitatively similar local deviations. A quantitative cross-pipeline comparison on identical domains is not feasible with currently public data, but we will add this as an explicit caveat and future-work item. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to established averaging formalism; central results computed from independent external data

full rationale

The paper applies the covariant scalar averaging formalism to the Cosmicflows-4++ reconstruction catalog to compute domain-averaged density, expansion, curvature, and backreaction. The formalism is cited from prior work by co-author Buchert, constituting a minor self-citation that is not load-bearing for the numerical claims. The reported O(10%) curvature contributions and lack of convergence to Lambda-CDM emerge directly from processing the external dataset over domains up to 300 Mpc/h, without any parameter fitting, self-definitional loops, or predictions that reduce to the inputs by construction. The derivation chain remains self-contained against the observational input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the covariant scalar averaging formalism applied to a reconstructed velocity field; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The covariant scalar averaging formalism correctly captures the averaged dynamics of an inhomogeneous spacetime.
The paper employs this formalism to extract domain-averaged quantities from the reconstruction.

pith-pipeline@v0.9.0 · 5482 in / 1334 out tokens · 58706 ms · 2026-05-08T15:47:05.482441+00:00 · methodology

discussion (0)

Reference graph

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