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Off-center observers in a spherical LTB overdensity get accurate cosmographic distances up to δc≈2.5 near the structure, while linear perturbation theory is better beyond ~3Rs; a gauge dictionary links the two.

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2026-08-04 11:39 UTC pith:KQZIZC4C

load-bearing objection Solid analytic machinery and a useful gauge dictionary, but the headline accuracy thresholds are over-generalized from a single density profile.

arxiv 2510.03517 v2 pith:KQZIZC4C submitted 2025-10-03 astro-ph.CO gr-qc

Covariant cosmography in the presence of local structures: comparing exact solutions and perturbation theory

Maharshi Sarma , Christian Marinoni , Basheer Kalbouneh , Chris Clarkson , Roy Maartens This is my paper
classification astro-ph.CO gr-qc
keywords anisotropiescovariantlocalanalysisapproachcomparecosmographicdistance
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a toy universe called an LTB model: spherically symmetric, with one dense clump in an otherwise smooth expanding background. An observer sits off to one side of this clump. The authors compute the true distance to galaxies by tracing light rays through the curved spacetime using the Sachs equation. They then compare this exact answer with two low-cost approximations: a Taylor-series 'covariant cosmography' expansion of distance in redshift, and linear perturbation theory around a smooth universe.

They find that near the clump, cosmography with terms through jerk and curvature stays within about 10 percent error up to a central overdensity of about 2.5 times the background density, while perturbation theory already fails at overdensity about 1. Far from the clump, perturbation theory wins: it stays accurate for overdensities up to about 3, while cosmography errors climb above 10 percent beyond central contrasts of about 1.5. They also derive a dictionary connecting the LTB language to the perturbed-FLRW language, resolving a mismatch in the monopole of the Hubble parameter as a gauge artifact, and correcting a typo in an earlier paper's quadrupole formula.

The main caveat: all thresholds come from one analytic density profile and one fixed size parameter; the generalized statements in the abstract assume this profile is representative.

Core claim

For a spherical overdensity, linear perturbation theory already exceeds 10% error in the luminosity distance toward the center for δc ≳ 1 when the observer lies within the typical size of the structure. In contrast, cosmographic reconstruction maintains better than 10% accuracy up to δc ≲ 2.5. At larger radii the situation reverses: LPT remains accurate up to δc ≲ 3, while CC already exceeds 10% error for δc ≳ 1.5. (Abstract and Conclusions.)

Load-bearing premise

The quantitative 10%-accuracy thresholds are computed for a single analytic density profile, Eq. (4.2), with the shape parameter fixed and Rs held at 37.4 Mpc throughout the parameter scans in Figure 6. The abstract and conclusions generalize these thresholds to 'a spherical overdensity' without testing profile-shape dependence, multiple structures, or nonzero Λ. If other realistic profiles shift the boundaries, the load-bearing regime statements would not hold as stated.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claims rest on general relativity, standard Sachs optics, and a specific ad hoc density profile. No new particles, forces, or geometric entities are introduced; the only adjustable inputs are the model parameters δc, Rs, χo, and H∞.

free parameters (4)
  • δc = 2.5 (model M1); -0.22 illustrative H0 example
    Central density contrast of the profile (4.2), chosen by hand; drives the nonlinearity and the reported thresholds.
  • Rs = 37.4 Mpc
    Characteristic radius of the density profile, taken from model M1 of [43]; kept fixed in parameter scans.
  • χo = 200 Mpc (≈5.35 Rs) in fiducial runs, varied in scans
    Observer distance from the center of the inhomogeneity; controls the reported regimes.
  • H∞ = 70 km/s/Mpc
    Asymptotic EdS expansion rate used to set t0 and normalize the background; a chosen input rather than a fitted constant.
axioms (5)
  • domain assumption Gravity is described by general relativity with pressureless dust and Λ=0 in the LTB model.
    The entire LTB construction and the comparison with EdS background rely on this; stated in §2.
  • standard math The angular diameter distance is given by the Sachs equation with the optical tidal tensor of Eq. (A.23).
    The exact distance used as ground truth in §4 is computed from this standard geodesic-deviation framework, derived in Appendix A.
  • ad hoc to paper The matter profile is the specific analytic form δ(χ)=δc(1+(χ/Rs)^2)^(-3/2) asymptoting to EdS.
    Eq. (4.2). All quantitative reliability claims depend on this shape; no other profile shapes are tested.
  • domain assumption The LTB metric can be linearized around an EdS background with |δa/a|≪1 and |k|≪1.
    Eqs. (5.21)–(5.22). This is the basis of the linearized-LTB dictionary and the LPT comparison.
  • domain assumption In the Newtonian gauge the perturbations have no anisotropic stress (Ψ=Φ), a time-independent potential, and a growing-mode velocity field with the integration constant C2=0.
    §5.2, Eqs. (5.32)–(5.41). These assumptions are needed to derive the dictionary and the Hubble-monopole gauge correction.

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read the original abstract

Recent observational evidence of axially symmetric anisotropies in the local cosmic expansion rate motivates an investigation of whether they can be accounted for within the Lema\^itre-Tolman-Bondi (LTB) framework with an off-center observer. Within this setting, we compute the exact relativistic luminosity distance via the Sachs equation and compare it with the approximate expression obtained from the covariant cosmographic approach (including Hubble, deceleration, jerk and curvature parameters). This comparison allows us to identify the regimes in which the covariant cosmographic method remains reliable. In addition, we compare the LTB relativistic distance for small inhomogeneities with the corresponding result derived from linear perturbation theory (LPT) in the standard cosmological model. This analysis establishes a precise correspondence between the LTB and LPT approaches, offering a consistent dictionary for the interpretation of the observed anisotropies of the large-scale gravitational field. This analysis will be instrumental in interpreting expansion-rate anisotropies, facilitating investigations of the local Universe beyond the FLRW framework with a fully non-perturbative metric approach.

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