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REVIEW 3 major objections 8 minor 43 references

Cosmography for a General Spacetime Centred at Arbitrary Redshift

T0 review · 3 major / 8 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper derives a model-independent Taylor expansion of the angular-diameter distance about any redshift in any four-dimensional Lorentzian spacetime, and tests its convergence in two LTB models.

desk verdict Genuinely new analytic extension of covariant cosmography to arbitrary redshift, with a solid derivation but a numerical benchmark that quietly drops Weyl focusing. read the letter →

arxiv 2608.07008 v1 pith:SRTNWRCI submitted 2026-08-07 astro-ph.CO

Jonas Broe Bendtsen , Asta Heinesen , Sofie Marie Koksbang This is my paper · ORCID
classification astro-ph.CO
keywords cosmographyangulardiameterdistancearbitraryredshiftLTBmodelsinhomogeneouscosmologyTaylorexpansionluminosityconvergence
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

The paper's central contribution is a model-independent cosmographic expansion: a Taylor series for the angular-diameter distance $d_A(z)$ centred at any chosen redshift $z_*$, valid in an arbitrary four-dimensional Lorentzian spacetime. The first three coefficients are written explicitly in terms of the light beam's isotropic expansion, the line-of-sight Hubble rate $H$ and its derivatives, the Ricci focusing term $k^\mu k^\nu R_{\mu\nu}$, and the Weyl-shear term; the third-order coefficient is new. The authors then test the third-order expansion against exact redshift-distance relations in two Lemaître–Tolman–Bondi (LTB) models, one void and one overdensity. They find that the expansion breaks down quickly when centred near steep density gradients, performs far better in smoother regions, and that stitching together several expansions is not automatically an improvement. The upshot is that coefficients fitted to real data in inhomogeneous spacetimes should be read as effective parameters averaged over the probed redshift range, not as strictly local geometric quantities.

What carries the argument

The central object is the Taylor expansion of the angular-diameter distance along a null geodesic, using redshift as the expansion variable. The machinery is a chain of optical and kinematic identities: $dz/d\lambda=-(1+z)^2 E_o H$ converting affine parameter to redshift, the focusing equation $d\hat\theta/d\lambda=-\tfrac12\hat\theta^2-2|\hat\sigma|^2-k^\mu k^\nu R_{\mu\nu}$ for the beam, the area law $dd_A/d\lambda=\tfrac12\hat\theta d_A$, and the observer-defined line-of-sight Hubble rate $H=\tfrac13\theta-e^\mu a_\mu+e^\mu e^\nu\sigma_{\mu\nu}$. Chaining these identities produces $d'_A$, $d''_A$, and the new $d'''_A$ in terms of local geometry; the FLRW limit of each coefficient is recovered in Appendix A as a check.

What would settle it

Recompute the exact angular-diameter distance for the same observers and light rays in the two LTB models including the full Weyl (shear) contribution to the optical Jacobian, by parallel transporting a screen basis along each ray; if the resulting $d_A$ differs from the Ricci-only $d_A$ by an amount comparable to the reported third-order expansion residuals, the benchmark used to quantify convergence is not reliable.

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Extended reading notes

Core claim

The discovery is that the standard cosmographic expansion, previously tied to $z=0$ and to Friedmann–Lemaître–Robertson–Walker (FLRW) geometry, can be repositioned to any redshift in a general spacetime, with the angular-diameter distance written as $d_A(z)=d_A(z_*)+d'_A(z_*)\Delta z+\tfrac12 d''_A(z_*)\Delta z^2+\tfrac16 d'''_A(z_*)\Delta z^3+\dots$, where $\Delta z=z-z_*$. The authors derive $d'''_A$ explicitly in terms of the beam expansion $\hat\theta$, the line-of-sight Hubble rate $H$ and its derivatives $H'$ and $H''$, the shear $|\hat\sigma|^2$, the Ricci term $k^\mu k^\nu R_{\mu\nu}$, and the Weyl term $k^\alpha k^\beta C_{\mu\alpha\nu\beta}\hat\sigma^{\mu\nu}$, and they show how the series maps to the luminosity distance through the Etherington relation. Applying the series to LTB models, they establish that its convergence is governed chiefly by the local density gradient at the expansion centre: third-order truncations fail near sharp contrasts and improve markedly in smooth regions, and piecewise stitching helps only when each centre sits in a well-behaved region.

Load-bearing premise

The exact distances used to judge the expansions are computed without the image-distorting part of spacetime curvature (the Weyl term), on the strength of an earlier result that it is very small in this type of LTB model; if that earlier result does not hold for the particular rays and density gradients studied here, the reported convergence errors would change.

Editorial extensions

If this is right

  • A third-order general cosmographic expansion can be constructed around any redshift in any four-dimensional Lorentzian spacetime, not only at $z=0$.
  • The third-order coefficient $d'''_A$ contains explicit Weyl-shear and Ricci-focusing terms, so distance measurements at successive orders can in principle probe local curvature and shear.
  • In the LTB void and overdensity models, the third-order expansion error stays around ten percent or less in the studied maps when the expansion centre avoids steep density gradients, but grows rapidly when the centre lies in a steep gradient.
  • Piecewise stitching of expansions is not automatically beneficial: the deciding factor is where the expansion centres sit relative to density fluctuations.
  • Coefficients fitted to observations in inhomogeneous spacetimes do not necessarily equal local geometric quantities; they should be interpreted as effective parameters reflecting the finite redshift range probed.

Reading between the lines

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

  • Because the arbitrary-centre expansion loses the finite multipole truncation that makes $z=0$ cosmography tractable, practical use will require explicitly patching data in redshift and angle; the paper gestures at this but does not develop a full data-analysis prescription.
  • The formalism could be turned into a model-independent consistency test for probes naturally centred at $z>0$, such as BAO, by comparing their distance anchors with local $z_*$ expansions; the paper mentions such cross-checks without working out the statistical procedure.
  • A direct test of the $d'''_A$ formula in a shear-dominated or vacuum spacetime, where the Weyl contribution is not negligible, would extend the validation beyond the LTB regime studied here.
  • The term ratio found along the fiducial rays suggests that in smooth regions the second-order coefficient mostly constrains the combination $-\hat\theta H'/H$; a forecast could quantify how well future surveys can isolate that product.
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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

3 major / 8 minor

Summary. This paper presents a general cosmographic framework in which the angular diameter distance is Taylor-expanded about an arbitrary redshift z* in a 4-dimensional Lorentzian spacetime. Section 2 derives the first three redshift derivatives of d_A in terms of local beam expansion, line-of-sight Hubble rate H, Ricci focusing, shear, and Weyl curvature (Eqs. (6), (8), (9)/(10)), and gives the corresponding expansion for the luminosity distance via Etherington reciprocity (Eq. (13)). The framework is then applied to two LTB models, one void and one overdensity, with observers placed in high-density-gradient regions. The authors compare third-order truncations of the d_A expansion with numerically integrated 'exact' distance-redshift relations along many rays, construct sky maps of the expansion coefficients, and study piecewise expansions with varying numbers of expansion centres. Their main numerical conclusions are that low-order cosmography fails near strong density gradients, that piecewise stitching does not automatically improve convergence unless centres are chosen in smooth regions, and that fitted coefficients in inhomogeneous spacetimes behave as effective range-dependent parameters rather than strictly local quantities.

Significance. If the Section 2 derivation is correct, the paper provides a genuinely more general tool than the standard z=0 cosmography: it works at arbitrary redshift and in arbitrary Lorentzian spacetimes, and it maps d_A coefficients to d_L. The FLRW limits in Appendix A reproduce the standard distance expansion, which is a reassuring consistency check. The numerical experiments are also valuable as concrete stress tests of low-order cosmography in inhomogeneous models, and the piecewise-expansion comparison is a practical question for future surveys. However, the strength of the numerical conclusions is currently limited by the unverified omission of Weyl focusing in the benchmark LTB light propagation (Section 3.1), and by the lack of code/data release; these issues are load-bearing for the claims about convergence and coefficient interpretation, even though they do not affect the analytic core.

major comments (3)
  1. [3.1] The exact d_A used as benchmark is computed with the Weyl term in the optical tidal matrix omitted, on the basis of an unreferenced assertion that Weyl distortion is 'highly sub-dominant' in these models. The third-order expansion in Eq. (9) however contains the explicit Weyl term k^α k^β C_{μανβ} σ̂^{μν}. The paper never specifies whether the numerical expansion coefficients are evaluated from Eq. (9) (in which case the benchmark omits a term that the expansion includes, so the reported discrepancies are not pure Taylor truncation error) or by differentiating the Ricci-only exact d_A (in which case the Weyl part of the formalism is never exercised). Either way, the quantitative claims in Section 4—that third-order cosmography fails near strong gradients and that piecewise coefficients are range-dependent effective parameters—are not established as stated. A demonstration of Weyl sub-dominance for the specific rays and density contrasts studied, or a computation including the Weyl term, is required.
  2. [4] The computational pipeline is underspecified. The reader is not told how the coefficients d_A(z*), d'_A(z*), d''_A(z*) and d'''_A(z*) entering Figures 3–9 are obtained: from the analytic expressions (6), (8), (10) using quantities integrated along each ray, or by numerical differentiation of the exact d_A. This matters because the first option requires the screen-space basis and shear, whose computation Section 3.1 explicitly avoids when omitting the Weyl term. Please state the procedure precisely and, if the geometric expressions are used, list which terms (Ricci, shear, Weyl, derivatives of H) are included in each coefficient.
  3. [6] The concluding claim that cosmographic coefficients 'provide meaningful descriptions of the local expansion and geometry on appropriately smoothed scales' goes beyond what is demonstrated: the numerical evidence is limited to two specific LTB models with a Weyl-truncated exact solution and no quantified comparison of the omitted term. Please either soften this claim or support it with the full optical computation.
minor comments (8)
  1. [Figure 3 caption] The caption reads 'z∗ =0.0015' where it should read 'z∗ =0.015'.
  2. [Figure 4 caption] The caption reads 'z∗ =0.0006' where it should read 'z∗ =0.006'.
  3. [4.1] The word 'dispalyed' in the text should be 'displayed'.
  4. [4.2] The phrase 'This in in agreement' should be 'This is in agreement'.
  5. [6] The sentence 'The expansion centred at arbitrary redshift lose the finite multipole structure' should read 'loses'.
  6. [3.1] The sentence 'we do not need to set initial conditions for and parallel transport the screen space basis vectors' is ungrammatical; suggest 'we do not need to set initial conditions for, or to parallel-transport, the screen-space basis vectors'.
  7. [References] Reference [35] is cited as a 2026 work that appears unpublished; please provide an arXiv identifier or publication status.
  8. [General] The absence of a code/data release is not an error, but for a numerical paper of this type a reproducibility statement would help; at minimum, the parameter values and the exact procedure for evaluating coefficients should be listed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Section 2 derivation and d_L mapping are self-contained; the LTB benchmark is an independent convergence test. A Weyl-truncation caveat affects the numerical examples but does not make the central claim reduce to its inputs.

full rationale

The Section 2 derivation is self-contained: Eqs. (6)-(10) are obtained by differentiating the standard optical scalar relations (2)-(4) with respect to redshift, with no use of the final Taylor expansion as an input. The d_L mapping in Section 2.1 is an algebraic identity from Etherington reciprocity and the polynomial (1+z)^2. The agreement of d''_A with [35] is a check, not a load-bearing citation, and the third-order expression is derived here. The LTB numerical test is an external convergence test rather than a fit: the exact d_A is obtained by direct integration of the geodesic and optical transport equations, and the Taylor approximations are evaluated from local quantities along the same solutions; the reported discrepancies are therefore genuine truncation/convergence effects. The one substantive caveat is in Section 3.1, where the benchmark omits Weyl focusing on the authority of an unreported earlier result by one of the authors; this could bias the quantitative convergence errors and the piecewise-error conclusions, but it does not make the expansion's predictions equal to its inputs. No equation in the paper is defined in terms of the quantity it is used to predict, and no fitted parameter is renamed as a prediction. The self-citations present (e.g., [35]) do not carry the derivation.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central expansion itself uses only standard Lorentzian geometry, the null-geodesic assumption, and the Sachs optical equations. The numerical convergence conclusions additionally rely on the chosen LTB models, their hand-picked parameters, and an unshown Weyl-sub-dominance assumption.

free parameters (7)
  • LTB1 curvature amplitude k_max = 5.4e-8
    Hand-chosen to construct a compensated void model; affects the density gradients that drive the convergence test but not the analytic expansion.
  • LTB1 boundary radius r_b = 40 Mpc
    Hand-chosen size of the inhomogeneous LTB region in LTB1.
  • LTB1 shape exponents (m, n) = m = 4, n = 4
    Hand-chosen profile shape parameters for the void in LTB1.
  • LTB2 overdensity amplitude delta_0 = 1e-3
    Hand-chosen amplitude of the central Gaussian overdensity in LTB2.
  • LTB2 width r_sigma = 7 Mpc
    Hand-chosen Gaussian width in the LTB2 curvature function.
  • Background FLRW parameters = Omega_m,0 = 0.30, H0 = 70 km/s/Mpc
    Chosen flat Lambda CDM background used for initial conditions and as the FLRW reference in skymap comparisons.
  • Observer positions in the LTB models = not stated numerically; shown only by vertical lines in Figure 1
    Observers are deliberately placed in high density-gradient regions, and the exact coordinates are not given, which limits exact reproduction of the skymaps.
assumptions (6)
  • domain assumption Spacetime is a 4-dimensional Lorentzian manifold.
    Stated in the Introduction as the minimal assumption; the Taylor expansion and optical equations rely on Lorentzian geometry.
  • domain assumption Light follows null geodesics and redshift as a function of affine parameter is invertible.
    Stated in the Introduction as assumptions (i) and (ii); used to express dz/dlambda and to define z as an expansion variable.
  • standard math Sachs optical equations for the beam expansion, shear, and angular diameter distance.
    Equations (2)-(4) are quoted from Ellis, Maartens and MacCallum [34] and are the input relations from which d'_A, d''_A, and d'''_A are derived.
  • domain assumption Etherington reciprocity relation holds.
    Used in Section 2.1 to convert the d_A expansion into a d_L expansion; standard for spacetime optics.
  • ad hoc to paper Weyl focusing is highly sub-dominant to Ricci focusing in the LTB models used as benchmarks.
    Section 3.1 omits Weyl distortion from the deformation matrix, citing prior work by one author instead of demonstrating it for the rays studied here; the exact d_A benchmark depends on this premise.
  • domain assumption LTB dust plus Lambda model with the specified mass function M(r) from Eq. (20).
    Section 3 defines the two LTB models and initial conditions; the exact d_A used for comparison is computed within this model family.

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Pith. "Pith review of Cosmography for a General Spacetime Centred at Arbitrary Redshift." pith.science (2026). https://pith.science/paper/SRTNWRCI

@misc{pith2026260807008,
  author       = {Pith},
  title        = {Pith review of: Cosmography for a General Spacetime Centred at Arbitrary Redshift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRTNWRCI}},
  note         = {Machine review of arXiv:2608.07008}
}
read the original abstract

With upcoming surveys providing large volumes of highly precise observational data across a wide range of redshifts, it is increasingly important to have tools that can translate observational data into geometric and dynamical information without imposing a predetermined cosmological model. General cosmographic expansions centred at arbitrary redshift provide exactly such a tool. We here present the formalism for general cosmographic expansions centred at an arbitrary redshift, valid for 4-dimensional Lorentzian spacetimes. We then apply the expansion formalism to test its ability to reproduce the redshift-distance relation in two examples of large-scale cosmic structures (an underdensity and an overdensity) modelled by the Lema\^itre-Tolman-Bondi metric, where we examine the effect of choosing different redshift intervals for the cosmographic series expansions. This quantifies the extent to which cosmographic coefficients inferred from redshift-distance observations retain their interpretation as local geometric and dynamical quantities, as is expected in standard FLRW cosmology. Similarly to earlier results, we here find that in more general spacetimes the coefficients instead become effective parameters reflecting the finite observational range probed. Lastly, we discuss possible strategies for using the expansions to constrain dynamical and geometric quantities.

Figures

Figures reproduced from arXiv: 2608.07008 by the authors.

Figure 2
Figure 2. The density of the two LTB models along a fiducial light ray. 0 20 40 60 80 d A dA exact vs expansion (fiducial ray, LTB1) Exact FLRW z∗ = 0.000 z∗ = 0.005 z∗ = 0.010 z∗ = 0.015 0.000 0.005 0.010 0.015 0.020 −0.5 0.0 0.5 1.0 z (dA,exp − d A)/d A [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Cosmographic expansions of dA in LTB1 centred at z∗ = 0.000, z∗ = 0.005, z∗ = 0.010, and z∗ = 0.0015 compared to the exact solution of a fiducial light ray. 4. GENERAL COSMOGRAPHY IN LTB MODELS: NUMERICAL EXAMPLES We consider one observer in each LTB model. Each observer is placed in a region with large gradient of the density field in order to maximize the effects of the structures. The positions of the observers a… view at source ↗
Figure 4
Figure 4. Cosmographic expansions of dA in LTB2 centred at z∗ = 0.000, z∗ = 0.002, z∗ = 0.004, and z∗ = 0.0006 compared to the exact solution of a fiducial light ray. LTB1. 4.1. Skymaps of expansion coefficients We now move on to consider multiple light rays for the two observers. Figures 5 and 6 show skymaps of the expansion co￾efficients, dA(z∗), d ′ A (z∗), d ′′ A (z∗) and d ′′′ A (z∗) for each observer. The maps were made… view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: Skymaps of the LTB1 model showing the relative deviation of the angular diameter distance compared to the FLRW limit. From left to right, the plots show dA, d ′ A , d ′′ A , and d ′′′ A . The skymaps are evaluated around different redshifts, from top to bottom: z∗ = 0.…
Figure 6
Figure 6. Figure 6: Skymaps of the LTB2 model showing the relative deviation of the angular diameter distance compared to the FLRW limit. From left to right, the plots show dA, d ′ A , d ′′ A , and d ′′′ A . The skymaps are evaluated around different redshifts, from top to bottom: z∗ = 0.…
Figure 7
Figure 7. Figure 7: Relative error of the cosmographic expansions for LTB1 (left) and LTB2 (right) for 6, 11 and 21 numbers of expansion points, respectively, from top to bottom. The solid line indicates mean while the dashed lines show the standard deviation ±σ of the mean relative error…
Figure 8
Figure 8. Figure 8: Skymaps of LTB1 evaluated at z = 0.010. The plots show the relative error of the cosmographic expansion compared to the exact dA to the zeroth, first, second, and third orders (from left to right). The cosmographic expansion is performed around the following redshifts …
Figure 9
Figure 9. Figure 9: Skymaps of LTB2 evaluated at z = 0.002. The plots show the relative error of the cosmographic expansion compared to the exact dA to the zeroth, first, second, and third orders (from left to right). The cosmographic expansion is performed around the following redshifts …
Figure 10
Figure 10. Figure 10: The non-vanishing terms in the second order derivative of dA shown for the two LTB models. In the bottom row, the dAθˆ (1+z) 3EoH term (blue solid line) is multiplied with 10−2 , while − dAθˆH′ 2(1+z) 2EoH2 is multiplied with 10−5 (red dashed line) to show all terms c…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.