Axial Bianchi IX meets Gaia data

Thomas Sch\"ucker

REVIEW 3 major objections 3 minor 20 references

Axial Bianchi IX universes produce the same comoving-source drift as axial Bianchi I when anisotropy and curvature are small, so drift data alone cannot distinguish them.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 08:30 UTC pith:DS6KM3B2

load-bearing objection New linear-order result: axial Bianchi IX drift equals Bianchi I in the double limit; interesting but the linearization may not hold over the Gaia redshift range. the 3 major comments →

arxiv 2608.02371 v1 pith:DS6KM3B2 submitted 2026-08-03 astro-ph.CO

Axial Bianchi IX meets Gaia data

Thomas Sch\"ucker This is my paper

classification astro-ph.CO

keywords cosmological parametersBianchi universescosmic driftcomoving sourcesanisotropyspatial curvatureLemaître–Hubble diagramcosmological principle

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.

This paper asks whether adding a small amount of positive spatial curvature to an already mildly anisotropic (axial Bianchi I) universe changes the predicted drift of distant sources, the quantity measured by quasar astrometry. Working in axial Bianchi IX universes—the curved counterparts of axial Bianchi I—with a positive cosmological constant and comoving dust, the author first computes the drift without assuming Einstein's equations, then linearizes in both small anisotropy and small curvature. The result is that, to first order, the curvature parameter disappears from the drift formula entirely; the drift is identical to the axial Bianchi I drift. That means cosmic-drift observations of the kind used to fit Bianchi I models cannot tell a curved, anisotropic universe from a flat one, even though the Lemaître–Hubble (luminosity–redshift) relation does distinguish them. The paper also shows that the limits of small anisotropy and small curvature commute, and flags that the anisotropy grows with redshift, so the linear statement may need care for high-redshift samples.

Core claim

On the paper's own terms, the central discovery is an identity: in axial Bianchi IX universes with Λ > 0 and comoving dust, the drift of a comoving source, linearized jointly in the anisotropy η(t) = (c−a)/(a+c/2) and in the dimensionless curvature ν = −4Ωκ/(3ΩΛ), reduces to δcosθ = (3/2) sin²θ cosθ η'_0 [ (4/(3H_F0)) sqrt(ΩΛ)/(1−ΩΛ)(coth(½√(3Λ)t_e)−1/sqrt(ΩΛ)) − a_F0 χ_e0f ] T_D/(a_F0 χ_e0f) + O(ν², η², νη), with no standalone ν term. The author therefore concludes that in the linear regime the drift in Bianchi IX coincides with the drift in Bianchi I, that the two limits commute, and that drifts cannot distinguish the two families, whereas Lemaître–Hubble diagrams can.

What carries the argument

The derivation is carried by the four Killing vectors of the axial Bianchi IX metric, which has two scale factors a(t) and c(t); these produce four Noether-conserved quantities along every geodesic. The conserved quantities reduce the photon geodesic equations to quadratures, and comparing two infinitesimally close null geodesics gives the redshift and, from the angle between arrival directions, the drift. The anisotropy η enters the geodesic equations through the combination W(t) and V(x), while dynamics enters through the linearized Einstein equation η'' + 3H_F η' + (8/a_F²)η = 0. Its solution ties η_e − η_0 to today's Hubble stretch η'_0/H_F0, and because the Friedman-background drift van

Load-bearing premise

The load-bearing assumption is that both the anisotropy and the curvature stay small enough over the observed redshift range that first-order linearization is valid; the paper itself notes that a Gaia-like fit produces η(t) = −23% at z = 3, just outside the perturbative regime, so the degeneracy may break down at the highest observed redshifts.

What would settle it

Numerically integrate the full unlinearized geodesic equations for an axial Bianchi IX model with ν = 9.5% and η'_0/H_F0 = 5% and compare the drift with equation (80) at z = 3; a difference exceeding the stated ~1% linearization error would show the degeneracy is an artifact of truncation. Equivalently, an all-sky quasar drift survey with redshift bins reaching z ≈ 3 that detects a curvature-dependent pattern in the quadrupole of δcosθ would contradict the linear prediction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Quasar drift datasets that fit axial Bianchi I models will fit axial Bianchi IX models equally well at linear order; adding spatial curvature does not alter the predicted drift signal.
The Lemaître–Hubble diagram does change with curvature at first order, so supernova distance data remain a viable way to look for positive curvature even if drift data cannot.
Predictions from flat anisotropic and curved anisotropic models can be compared directly after the same linearization; first-order luminosity distances are 'additive' in anisotropy and curvature, but the drift is not.
The paper argues that the current mismatch between CMB-based and drift-based anisotropy estimates is not explained by positive curvature at linear order, and suggests that new drift and supernova observations are the route forward.

Where Pith is reading between the lines

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

A natural next step is to compute the O(η², ν², ην) drift; if a second-order curvature term appears, the degeneracy is an artifact of the truncation and might become visible at z near 3, where the paper notes η reaches −23%.
The absence of a curvature term means any observed drift pattern can be interpreted purely as anisotropy; joint fits combining drift with distance moduli or CMB geometric priors would be needed to break the degeneracy.
The cancellation may or may not survive for other matter content, such as radiation or a scalar field; testing the linearization there would show whether the result is tied to the dust-plus-Λ fluid or is a purely kinematic property of Bianchi IX geodesics.
Because the privileged z-direction persists in Bianchi IX, drift maps could in principle localize the symmetry axis of a slightly curved universe even while failing to determine its curvature.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

New linear-order result: axial Bianchi IX drift equals Bianchi I in the double limit; interesting but the linearization may not hold over the Gaia redshift range.

read the letter

This paper computes the drift of comoving sources in axial Bianchi IX universes with dust and a positive cosmological constant, then linearizes in both small anisotropy and small curvature. The new result is that, to first order in both, the drift equals the drift in axial Bianchi I, while the Lemaître-Hubble diagram does not share that degeneracy. That contrast is a genuine and nontrivial addition to Schücker's own line of work, and the observational implication—that Gaia drift fits cannot distinguish the two in linear approximation—is clearly stated.

The derivation is careful in structure: Noether constants, two-photon setup, then a double limit. The paper is also commendably honest. It states outright in §3.2 that for the Gaia fit with a 5% Hubble stretch, η(t) runs from 0 to -23% as z approaches 3, 'slightly outside the perturbative domain.' That is the load-bearing weak spot, and the author names it himself. The stress-test note is right: at z≈3, η² is about 5% and νη about 2%, both comparable to the leading Hubble-stretch term. So the equality of drifts to linear order does not by itself cover the redshift range where Gaia actually has data. The conclusion that drifts cannot distinguish Bianchi I from IX is a linear-order statement; the paper does not compute second-order terms and does not establish that they stay negligible over 0

There are also two places where the algebra is asserted rather than shown: the four coefficients in eq. (38) are called 'straight-forward, but long,' and the cancellation that removes curvature dependence from the drift in eq. (80) is not demonstrated. Given the paper's reliance on a specific cancellation, a referee should ask for those steps, or at least a notebook.

None of this kills the central claim. The computation is new, the linear-order degeneracy is plausible and internally consistent, and the author has not overclaimed—he explicitly says he has not computed second-order terms. The paper fits into his established program, and the self-citations are to published work that this paper builds on.

I would send this to a serious referee. The referee should ask for the omitted algebra and a more careful discussion of domain of validity, including some estimate of the neglected η² and νη terms over the Gaia window. It is a conditional accept, not a reject. I'd also cite it if I worked on drift in anisotropic cosmologies, though I would quote the linear-order caveat.

Referee Report

3 major / 3 minor

Summary. The paper computes the cosmological drift of comoving extragalactic sources in axial Bianchi IX universes, first kinematically from the geodesic equations with the Noether constants, then dynamically by adding Einstein's equations with a positive cosmological constant and comoving dust. It linearizes both the drift and the Lemaitre-Hubble diagram in a small anisotropy parameter and a small curvature parameter. The central result is Eq. (80): to first order in both small parameters, the drift in axial Bianchi IX equals the previously known drift in axial Bianchi I, while the Lemaitre-Hubble diagram retains curvature corrections. The paper relates this to the Gaia quasar drift data through a Hubble-stretch fit from earlier work, and concludes that drift measurements cannot distinguish axial Bianchi I from IX in the linear approximation.

Significance. If the derivation is correct, the result is a useful no-go statement: the drift of comoving sources, a promising cosmological observable, is degenerate between axial Bianchi I and axial Bianchi IX to first order in small anisotropy and small curvature. The paper is careful to frame this as a linear-order result and explicitly acknowledges the growth of the anisotropy with redshift. The calculation is analytic, builds on exact Friedman solutions and earlier work on axial Bianchi universes, and is presented with transparent notation. However, two load-bearing pieces of algebra are not shown, and the applicability of the conclusion to the Gaia redshift range is not quantitatively established. These issues are fixable but currently leave the central claim only partially verified.

major comments (3)

[§2.4, Eq. (38)] The four coefficients ex, τ, S2, τz in the arrival-direction formula are called 'straight-forward, but long' and are not shown. Since Eq. (38) is the foundation for the subsequent drift formula and ultimately for Eq. (80), the reader cannot check the derivation without redoing it. Please include the computation in an appendix or in a supplementary file, even in compressed form. The same applies to the combination that leads to Eq. (35), where the text says 'subtracts 2 times equation (34)' but the displayed algebra uses the factor s².
[§3.2, Eq. (80)] The cancellation of all curvature-dependent terms in the drift is asserted rather than demonstrated. The step from Eq. (54) plus the linearized Einstein solution (66)-(67) to the curvature-independent Eq. (80) is the main physical conclusion, so the O(ν) terms should be tracked explicitly. Please show how χe0, tanχe0, and the solution for η combine so that every ν term drops out to O(ν², νη, η²). This is central and cannot be left as an implicit calculation.
[Introduction and §3.2] The conclusion that 'drifts cannot distinguish axial Bianchi I from IX' is stated in the Introduction as 'We will show that they cannot', but the proof is only first-order in the anisotropy and curvature. The paper itself notes that for the Gaia fit with a 5.0% Hubble stretch, η(t) reaches about −23% as z approaches 3, 'slightly outside the perturbative domain'. At z≈3, η² ≈ 5% and νη ≈ 2% are of the same order as the leading Hubble-stretch term, so second-order corrections could easily break the degeneracy over the actual Gaia window. Please either estimate or bound the neglected O(η²) and O(νη) terms for z<3, or explicitly restrict the no-distinguish claim to the range where these terms are numerically negligible.

minor comments (3)

[§3] The typo note 'the 2/3 in front of (Hc − H) in the last equation here is correct' is confusing in context; clarify whether it refers to a typo in [4] only and make the equation self-contained.
[Appendix A.3] The notation ν is introduced as a dimensionless curvature parameter, but earlier κ is used for spatial curvature; please define the relation between κ and ν explicitly at first use to avoid confusion with the elliptic modulus k².
[References] Reference [14] is cited as arXiv:2605.30579, which is after the submission date of this paper; please confirm the reference details and, if available, include the journal version's full bibliographic information.

Circularity Check

0 steps flagged

No circularity: the BIX/BI drift equality is a genuine double-limit computation, not a fit or self-citation reduction.

full rationale

The central claim—that, to first order in both anisotropy η and curvature ν, the drift in axial Bianchi IX equals that in axial Bianchi I (Eq. 80)—is derived by an independent kinematic calculation in §2 and a linearized Einstein equation in §3. No Gaia data are fitted in this paper; the numerical values quoted (5% Hubble stretch, η reaching −23% at z≈3) come from prior fits [14, 15] and are used only to delimit the perturbative domain, not to define the formula. The self-citations to [4] and [9] are parameter-free derivations with stated assumptions (Bianchi I/IX metrics, dust + Λ) that do not include the target equality. The equality itself is not a renaming or an ansatz: the paper computes the BIX drift directly and shows that the ν term vanishes because isotropic Friedmann universes have zero drift, while the η term is governed by the same linearized shear evolution as in Bianchi I. The appendix independently linearizes the Friedman solutions using Edwards [17]. The acknowledged limitation in §3.2—η varies from 0 to −23% over z∈[0,3], slightly outside the perturbative domain—is a domain-of-validity caveat, not a circular step. Therefore no specific reduction of Eq. (80) to its own inputs by construction is present.

Axiom & Free-Parameter Ledger

3 free parameters · 8 axioms · 0 invented entities

The central claim rests on earlier results by the same group ([4] for metric, geodesics, Einstein equations; [9] for the BI drift; [14] for the Gaia fit values). These are parameter-free derivations rather than fitted results, so the dependence is not circular in the pernicious sense, but the present paper does not reproduce all of that upstream machinery. No new physical entities are introduced.

free parameters (3)

Hubble stretch η'_0/HF0 (anisotropy expansion parameter) = ≈5.0% ± 0.7% from Gaia fit [14]; not fitted in this paper
Defines the small-anisotropy expansion; the value is used only to assess the domain of validity, not to derive the equality.
curvature parameter ν = -4 Ωκ/(3 ΩΛ) = e.g. 9.5% for ΩΛ=0.7, -Ωκ=0.05; not fitted in this paper
Small-curvature expansion parameter; assumed small (≤10%) by the authors.
observer peculiar velocity V
Added as a small vector in eqs. (81)-(82) to model the kinematic dipole; not fitted here, but assumed to be at most order 10%.

axioms (8)

domain assumption The axial Bianchi IX metric with two scale factors a(t), c(t) in eq. (1) is the correct minimal symmetry-breaking ansatz.
Taken from the author's earlier paper [4]; not re-derived here.
domain assumption The Noether conserved quantities and reduced geodesic equations (2)-(10) from [4] are correct and complete.
The drift derivation starts from these equations; an error there would propagate into the central claim.
domain assumption Einstein's equations with comoving dust and positive cosmological constant take the form (61)-(62), with ρ0 ≈ ρF0 to zeroth order.
Used to derive the linearized anisotropy evolution equation (63); the form is inherited from [4].
standard math Farnsworth's theorem: axial Bianchi V universes are incompatible with Einstein's equations and comoving dust unless maximally symmetric.
Used in the introduction to justify considering only Bianchi I and IX; cited as [3].
standard math Wald's theorem controls the backward growth of anisotropy in the presence of a positive cosmological constant.
Used in §3.2 to estimate η(t) over the redshift range and to flag the breakdown near z=3; cited as [16].
standard math Edwards' exact Jacobi-elliptic solutions of Friedman's equation with dust and positive curvature, and their small-ν linearization, are valid; γ²−ν³ > 0.
Used in the appendix and §3.1 to linearize the Friedman scale factor and the Lemaître-Hubble diagram.
ad hoc to paper All functions encountered are analytic and the double limit of small anisotropy and small curvature is locally continuous; convergence must be checked post-fit.
Stated in §3.2; the authors defer domain-of-convergence control, so the linear-order equality is not established outside the presumed analytic domain.
domain assumption A comoving observer at the equator and an emitter at the North pole capture the generic drift by homogeneity; the longer calculation for arbitrary observer positions is skipped.
The paper relies on homogeneity of Bianchi IX and skips the general-position computation (§2.1-§2.2).

pith-pipeline@v1.3.0-daily-deepseek · 12469 in / 15812 out tokens · 127516 ms · 2026-08-04T08:30:33.609574+00:00 · methodology

0 comments

read the original abstract

The aim of the present work is two-fold: {\it(i)} Compute the drift of comoving extragalactic sources in axial Bianchi IX universes, in particular in those satisfying Einstein's equations with positive cosmological constant and comoving dust. {\it(ii)} Linearize this drift (and for comparison the previously calculated Lema{\^i}tre-Hubble diagram) simultaneously in small anisotropy and in small positive curvature. We find that the drift in this linear approximation coincides with the drift in axial Bianchi I universes. This is not true for Lema{\^i}tre-Hubble diagrams.

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 1 canonical work pages

[1]

Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movi- menti,

Luigi Bianchi, “Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movi- menti,” Memorie di Matematica e di Fisica della Societa Italiana delle Scienze11(1898) 267352; English translation by R.T. Jantzen,“On the three-dimensional spaces which admit a continuous group of motions,” Gen. Rel. Grav.33(2001) 2171, http://www34.homepage.villanova....

2001
[2]

Sugli spazii che ammettono un gruppo continuo de movimenti,

Guido Fubini, “Sugli spazii che ammettono un gruppo continuo de movimenti,” Annali di Mat., ser. 3,8(1903) 54. 14

1903
[3]

Some New General Relativistic Dust Metrics Possessing Isometries,

D. L. Farnsworth, “Some New General Relativistic Dust Metrics Possessing Isometries,” J. Math. Phys.8(1967) 2315

1967
[4]

Axial Bianchi IX and its Lema ˆ ıtre-Hubble diagram,

G. Valent, A. Tilquin and T. Sch¨ ucker, “Axial Bianchi IX and its Lema ˆ ıtre-Hubble diagram,” [arXiv:2211.04844 [gr-qc]]; and a short version, “The Lema ˆ ıtre–Hubble diagram in axial Bianchi IX universes with comoving dust,” Class. Quant. Grav.41(2024) no.1, 015034 doi:10.1088/1361-6382/ad13c1

Pith/arXiv arXiv 2024
[5]

L’univers en expansion,

George Lema ˆ ıtre, “L’univers en expansion,” Annales Soc. Sci. Bruxelles A53(1933), 51-85 doi:10.1023/A:1018855621348; English translation by M. A. H. MacCallum, “The expanding universe,” Gen. Rel. Grav.29(1997) 641

doi:10.1023/a:1018855621348 1933
[6]

Bianchi I meets the Hubble diagram,

T. Sch¨ ucker, A. Tilquin and G. Valent, “Bianchi I meets the Hubble diagram,” Mon. Not. Roy. Astron. Soc.444(2014) 2820 doi:10.1093/mnras/stu1656 [arXiv:1405.6523 [astro-ph.CO]]

Pith/arXiv arXiv 2014
[7]

Light propagation in a homogeneous and anisotropic universe,

P. Fleury, C. Pitrou and J. P. Uzan, “Light propagation in a homogeneous and anisotropic universe,” Phys. Rev. D91(2015) 043511 doi:10.1103/PhysRevD.91.043511 [arXiv:1410.8473 [gr-qc]]

Pith/arXiv arXiv 2015
[8]

Direction and redshift drifts for general observers and their applications in cosmology,

O. H. Marcori, C. Pitrou, J. P. Uzan and T. S. Pereira, “Direction and redshift drifts for general observers and their applications in cosmology,” Phys. Rev. D98(2018) no.2, 023517 doi:10.1103/PhysRevD.98.023517 [arXiv:1805.12121 [astro-ph.CO]]

Pith/arXiv arXiv 2018
[9]

Axial Bianchi I meets drifting extragalactic sources,

T. Sch¨ ucker, “Axial Bianchi I meets drifting extragalactic sources,” Class. Quant. Grav. 42(2025) no.7, 075002 doi:10.1088/1361-6382/adbda5 [arXiv:2407.02889 [astro-ph.CO]]

Pith/arXiv arXiv 2025
[10]

WMAP data and the curvature of space,

J. P. Uzan, U. Kirchner and G. F. R. Ellis, “WMAP data and the curvature of space,” Mon. Not. Roy. Astron. Soc.344(2003) L65 doi:10.1046/j.1365-8711.2003.07043.x [arXiv:astro-ph/0302597 [astro-ph]]

arXiv 2003
[11]

Curvature tension: evidence for a closed universe,

W. Handley, “Curvature tension: evidence for a closed universe,” Phys. Rev. D 103(2021) no.4, L041301 doi:10.1103/PhysRevD.103.L041301 [arXiv:1908.09139 [astro- ph.CO]]

Pith/arXiv arXiv 2021
[12]

Planck evidence for a closed Universe and a possible crisis for cosmology,

E. Di Valentino, A. Melchiorri and J. Silk, “Planck evidence for a closed Universe and a possible crisis for cosmology,” Nature Astron.4(2019) no.2, 196 doi:10.1038/s41550- 019-0906-9 [arXiv:1911.02087 [astro-ph.CO]]

Pith/arXiv arXiv 2019
[13]

Gaia Data Release 3 - The extragalactic content,

C. A. L. Bailer-Joneset al.[Gaia], “Gaia Data Release 3 - The extragalactic content,” Astron. Astrophys.674(2023) A41 doi:10.1051/0004-6361/202243232 [arXiv:2206.05681 [astro-ph.GA]]

Pith/arXiv arXiv 2023
[14]

Mapping the Universe as a Bianchi I cosmology with Gaia data,

Valeri V. Makarov, A. Heinesen and T. Sch¨ ucker, “Mapping the Universe as a Bianchi I cosmology with Gaia data,” Class. Quantum Grav.43(2026) 125012 doi: 10.1088/1361- 6382/ae76b3, [arXiv:2605.30579 [astro-ph.CO]]. 15

Pith/arXiv arXiv 2026
[15]

The Ellipsoidal Universe in the Planck Satellite Era,

P. Cea, “The Ellipsoidal Universe in the Planck Satellite Era,” Mon. Not. Roy. Astron. Soc.441(2014) no.2 1646 doi:10.1093/mnras/stu687 [arXiv:1401.5627 [astro-ph.CO]]

Pith/arXiv arXiv 2014
[16]

Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant,

R. M. Wald, “ Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant,” Phys. Rev. D28(1983) 2118

1983
[17]

Exact expressions for the properties of the zero-pressure Friedmann mod- els,

D. Edwards, “Exact expressions for the properties of the zero-pressure Friedmann mod- els,” Mon. R. Astron. Soc.159(1972) 51

1972
[18]

Massive and massless Yang-Mills and gravitational fields,

H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys. B22(1970) 397. doi:10.1016/0550-3213(70)90416-5

work page doi:10.1016/0550-3213(70)90416-5 1970
[19]

Linearized gravitation theory and the graviton mass,

V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett.12 (1970) 312

1970
[20]

A posteriori correction of Gaia CRF proper motions using condi- tional probability and zero-parallax prior

Valeri V. Makarov, “A posteriori correction of Gaia CRF proper motions using condi- tional probability and zero-parallax prior”, to appear. A Exact solutions of Friedman’s equation with comoving dust in the limit of small curvature We use notations and results of reference [4], Ωm := 8π G ρF0 3H 2 F0 ,Ω Λ := Λ 3H 2 F0 >0,Ω κ := −κ a2 F0 H 2 F0 = 1−Ω m −Ω ...

[1] [1]

Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movi- menti,

Luigi Bianchi, “Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movi- menti,” Memorie di Matematica e di Fisica della Societa Italiana delle Scienze11(1898) 267352; English translation by R.T. Jantzen,“On the three-dimensional spaces which admit a continuous group of motions,” Gen. Rel. Grav.33(2001) 2171, http://www34.homepage.villanova....

2001

[2] [2]

Sugli spazii che ammettono un gruppo continuo de movimenti,

Guido Fubini, “Sugli spazii che ammettono un gruppo continuo de movimenti,” Annali di Mat., ser. 3,8(1903) 54. 14

1903

[3] [3]

Some New General Relativistic Dust Metrics Possessing Isometries,

D. L. Farnsworth, “Some New General Relativistic Dust Metrics Possessing Isometries,” J. Math. Phys.8(1967) 2315

1967

[4] [4]

Axial Bianchi IX and its Lema ˆ ıtre-Hubble diagram,

G. Valent, A. Tilquin and T. Sch¨ ucker, “Axial Bianchi IX and its Lema ˆ ıtre-Hubble diagram,” [arXiv:2211.04844 [gr-qc]]; and a short version, “The Lema ˆ ıtre–Hubble diagram in axial Bianchi IX universes with comoving dust,” Class. Quant. Grav.41(2024) no.1, 015034 doi:10.1088/1361-6382/ad13c1

Pith/arXiv arXiv 2024

[5] [5]

L’univers en expansion,

George Lema ˆ ıtre, “L’univers en expansion,” Annales Soc. Sci. Bruxelles A53(1933), 51-85 doi:10.1023/A:1018855621348; English translation by M. A. H. MacCallum, “The expanding universe,” Gen. Rel. Grav.29(1997) 641

doi:10.1023/a:1018855621348 1933

[6] [6]

Bianchi I meets the Hubble diagram,

T. Sch¨ ucker, A. Tilquin and G. Valent, “Bianchi I meets the Hubble diagram,” Mon. Not. Roy. Astron. Soc.444(2014) 2820 doi:10.1093/mnras/stu1656 [arXiv:1405.6523 [astro-ph.CO]]

Pith/arXiv arXiv 2014

[7] [7]

Light propagation in a homogeneous and anisotropic universe,

P. Fleury, C. Pitrou and J. P. Uzan, “Light propagation in a homogeneous and anisotropic universe,” Phys. Rev. D91(2015) 043511 doi:10.1103/PhysRevD.91.043511 [arXiv:1410.8473 [gr-qc]]

Pith/arXiv arXiv 2015

[8] [8]

Direction and redshift drifts for general observers and their applications in cosmology,

O. H. Marcori, C. Pitrou, J. P. Uzan and T. S. Pereira, “Direction and redshift drifts for general observers and their applications in cosmology,” Phys. Rev. D98(2018) no.2, 023517 doi:10.1103/PhysRevD.98.023517 [arXiv:1805.12121 [astro-ph.CO]]

Pith/arXiv arXiv 2018

[9] [9]

Axial Bianchi I meets drifting extragalactic sources,

T. Sch¨ ucker, “Axial Bianchi I meets drifting extragalactic sources,” Class. Quant. Grav. 42(2025) no.7, 075002 doi:10.1088/1361-6382/adbda5 [arXiv:2407.02889 [astro-ph.CO]]

Pith/arXiv arXiv 2025

[10] [10]

WMAP data and the curvature of space,

J. P. Uzan, U. Kirchner and G. F. R. Ellis, “WMAP data and the curvature of space,” Mon. Not. Roy. Astron. Soc.344(2003) L65 doi:10.1046/j.1365-8711.2003.07043.x [arXiv:astro-ph/0302597 [astro-ph]]

arXiv 2003

[11] [11]

Curvature tension: evidence for a closed universe,

W. Handley, “Curvature tension: evidence for a closed universe,” Phys. Rev. D 103(2021) no.4, L041301 doi:10.1103/PhysRevD.103.L041301 [arXiv:1908.09139 [astro- ph.CO]]

Pith/arXiv arXiv 2021

[12] [12]

Planck evidence for a closed Universe and a possible crisis for cosmology,

E. Di Valentino, A. Melchiorri and J. Silk, “Planck evidence for a closed Universe and a possible crisis for cosmology,” Nature Astron.4(2019) no.2, 196 doi:10.1038/s41550- 019-0906-9 [arXiv:1911.02087 [astro-ph.CO]]

Pith/arXiv arXiv 2019

[13] [13]

Gaia Data Release 3 - The extragalactic content,

C. A. L. Bailer-Joneset al.[Gaia], “Gaia Data Release 3 - The extragalactic content,” Astron. Astrophys.674(2023) A41 doi:10.1051/0004-6361/202243232 [arXiv:2206.05681 [astro-ph.GA]]

Pith/arXiv arXiv 2023

[14] [14]

Mapping the Universe as a Bianchi I cosmology with Gaia data,

Valeri V. Makarov, A. Heinesen and T. Sch¨ ucker, “Mapping the Universe as a Bianchi I cosmology with Gaia data,” Class. Quantum Grav.43(2026) 125012 doi: 10.1088/1361- 6382/ae76b3, [arXiv:2605.30579 [astro-ph.CO]]. 15

Pith/arXiv arXiv 2026

[15] [15]

The Ellipsoidal Universe in the Planck Satellite Era,

P. Cea, “The Ellipsoidal Universe in the Planck Satellite Era,” Mon. Not. Roy. Astron. Soc.441(2014) no.2 1646 doi:10.1093/mnras/stu687 [arXiv:1401.5627 [astro-ph.CO]]

Pith/arXiv arXiv 2014

[16] [16]

Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant,

R. M. Wald, “ Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant,” Phys. Rev. D28(1983) 2118

1983

[17] [17]

Exact expressions for the properties of the zero-pressure Friedmann mod- els,

D. Edwards, “Exact expressions for the properties of the zero-pressure Friedmann mod- els,” Mon. R. Astron. Soc.159(1972) 51

1972

[18] [18]

Massive and massless Yang-Mills and gravitational fields,

H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys. B22(1970) 397. doi:10.1016/0550-3213(70)90416-5

work page doi:10.1016/0550-3213(70)90416-5 1970

[19] [19]

Linearized gravitation theory and the graviton mass,

V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett.12 (1970) 312

1970

[20] [20]

A posteriori correction of Gaia CRF proper motions using condi- tional probability and zero-parallax prior

Valeri V. Makarov, “A posteriori correction of Gaia CRF proper motions using condi- tional probability and zero-parallax prior”, to appear. A Exact solutions of Friedman’s equation with comoving dust in the limit of small curvature We use notations and results of reference [4], Ωm := 8π G ρF0 3H 2 F0 ,Ω Λ := Λ 3H 2 F0 >0,Ω κ := −κ a2 F0 H 2 F0 = 1−Ω m −Ω ...