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REVIEW 3 major objections 5 minor 199 references

From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology

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Pith's one-line read The review argues that a Finslerian spacetime metric, depending on position and an internal direction, can generate an effective dark energy, with one variant fitting the data as well as the standard cosmological model.

desk verdict A useful self-review of the authors' Finslerian cosmology program with a new DESI DR2 comparison, but the 'viable alternative' claim is a parametric fit, not a geometric prediction. read the letter →

arxiv 2506.18422 v1 pith:SOGCZKCV submitted 2025-06-23 gr-qc hep-th

Amine Bouali , Himanshu Chaudhary , Lehel Csillag , Rattanasak Hama , Tiberiu Harko , Sorin V. Sabau , Shahab Shahidi This is my paper · ORCID
classification gr-qchep-th MSC 53B4083F0583D05 PACS 04.50.Kd98.80.-k
keywords FinslergeometryBarthelconnection(αβ)-metricsKropinametricRanderseffectivedarkenergycosmologicalconstantMCMCmodelselection
topics Dark Energy
open problems Dark Energy
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

Finsler geometry generalizes Riemannian geometry by letting the spacetime metric depend on an internal direction as well as on position. This review argues that for a specific class of Finsler metrics—the Randers and Kropina $(\alpha,\beta)$ metrics—the extra direction-dependence can be absorbed into a single time-dependent function, leaving generalized Friedmann equations that look like Einstein's equations plus correction terms. Those corrections behave like a dark energy component and can generate an effective cosmological constant without adding any new matter or energy. Confronting the models with cosmic chronometer, supernova, and baryon acoustic oscillation data, the paper reports that the Barthel–Kropina variant fits the late-time expansion history at least as well as the standard $\Lambda$CDM model. If right, the acceleration of the Universe may be a geometric residue of direction-dependent spacetime, not evidence for a cosmological constant or dark energy fluid.

What carries the argument

The machinery is the Barthel connection of an $(\alpha,\beta)$ metric, evaluated on the osculating vector field $Y=b$. For both Randers and Kropina metrics this connection coincides with the Levi–Civita connection of the Riemannian metric $\hat g_{IJ}(x)=g_{IJ}(x,b(x))$, so a direction-dependent Finsler geometry becomes an ordinary Riemannian geometry with extra structure. Imposing homogeneity and isotropy pins the one-form to $b=(a\eta,0,0,0)$, leaving one free function $\eta(t)$; every dark-energy-like term in the Friedmann equations is built from $\eta,\eta',\eta''$. The construction reduces an otherwise intractable Finsler gravity theory to standard Einstein equations plus corrections, at the cost that the function generating the dark energy is not fixed by any equation of motion.

What would settle it

Compute the gravitational field equations for $F=\alpha^2/\beta$ from an action principle on the unit tangent bundle instead of postulating Eq. (59); if the action's Friedmann equations do not reduce to Eqs. (71)–(72) with the same free $\eta(t)$, then the Barthel–Kropina fit is not the prediction of the geometry—the fitted $\eta(t)$ would need to be redetermined or the model abandoned.

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Extended reading notes

Core claim

On its own terms, the central claim is that the accelerating expansion of the Universe can emerge from Finslerian geometry rather than from a cosmological constant. In the Barthel–Kropina model, the Finsler function is $F=\alpha^2/\beta$, where $\alpha$ is the Riemannian FLRW metric and $\beta=b_I y^I$ is a one-form; after the osculating reduction the generalized Friedmann equations acquire terms built from $\eta(t)$, and these terms act as an effective dark energy. The statistical part of the review compares the Barthel–Randers variants and the Barthel–Kropina model with $\Lambda$CDM using cosmic chronometer, Type Ia supernova, and baryon acoustic oscillation data under an MCMC analysis. It finds $\chi^2_{\rm tot}=1762.37$ for Barthel–Kropina versus $1780.94$ for $\Lambda$CDM, with AIC lower by 14.6 and BIC lower by 3.7, while the Barthel–Randers variants are slightly less favored. The authors conclude that these Finslerian models are a statistically viable alternative to the standard cosmological model.

Load-bearing premise

The load-bearing premise is that fixing the internal Finsler direction to the one-form $b=(a\eta,0,0,0)$ and postulating the Einstein-like field equations (59) is the correct way to turn a Finsler spacetime into a gravity theory; if a variational principle or a different choice of internal direction changes those equations, the effective dark energy and all fitted results are not consequences of Finsler geometry.

Editorial extensions

If this is right

  • The late-time acceleration can be reproduced without a cosmological constant or dark energy fluid, so the dark-energy problem shifts from finding a substance to explaining why the spacetime metric carries a time-dependent internal direction.
  • The Barthel–Kropina model makes specific cosmographic predictions—transition redshift $z_{\rm tr}\approx 0.72$ and present jerk $j_0\approx 0.45$—that differ from $\Lambda$CDM and are testable with high-redshift surveys.
  • Because matter is not conserved in these theories, particle creation from geometry is a built-in consequence; its creation pressure and rate could be compared with thermodynamic bounds.
  • The same osculating Barthel construction applies to any $F=\alpha\,\phi(s)$, yielding a family of modified-gravity cosmologies that are no harder to compute than general relativity.

Reading between the lines

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

  • Editorial inference: the data comparison exercises only the background expansion; no perturbation equations are derived, so growth of structure, weak lensing, and CMB anisotropies remain uncalculated tests of the geometry.
  • Editorial inference: with $\eta(t)$ a free function possessing no dynamics, the good fit may partly reflect the model's flexibility; a version that derives $\eta$ from an action, or fixes it by symmetry, would show how much predictive power the framework truly has.
  • Editorial inference: deriving the same field equations from a Finsler–Einstein action for $F=\alpha^2/\beta$ would close the main gap between the postulated equations (59) and a fully variational theory, and would make the fitted $\eta(t)$ a solution rather than an input.
  • Editorial inference: the particle-creation channel ties the model to semiclassical gravity; entropy production requirements could constrain $\dot\phi$ and thereby select among the fitted expansion histories.
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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 / 5 minor

Summary. This manuscript reviews a line of work on Finslerian cosmological models based on the osculating Barthel connection for (α,β) metrics. It covers Barthel-Randers (F=α+β), Barthel-Kropina (F=α²/β), and conformally transformed Barthel-Kropina geometries: generalized Friedmann equations are derived, energy-balance equations are written, and a thermodynamic interpretation in terms of particle creation is developed. The paper then presents an MCMC analysis of three Barthel-Randers variants (linear, logarithmic, exponential φ(z)) and one Barthel-Kropina model, using Cosmic Chronometer, Pantheon+ SNe Ia, and DESI DR2 BAO data, with AIC/BIC model comparison against ΛCDM. The Abstract and Section 7 claim that these Finslerian models provide a satisfactory fit and represent a viable alternative to the standard cosmological model.

Significance. If the central claim were supported, the paper would establish a geometrically motivated alternative to dark energy, with Finslerian geometry generating an effective cosmological constant. The review usefully collects the mathematical machinery of osculating (α,β) geometries and shows that, for Y=b, the Barthel connection reduces to the Levi-Civita connection of an effective Riemannian metric; the derivation of the generalized Friedmann equations within that setup is self-contained and generally correct. The statistical pipeline is described in reasonable detail, and the paper makes explicit use of standard likelihood functions and convergence diagnostics. However, the central viability claim is not currently supported: the Finslerian terms are not predicted but are implemented through hand-chosen functions (φ(z) for Randers, initial conditions and EoS parameter for Kropina) that are fitted to the same data used for the comparison. The paper is therefore best read as a review of a specific phenomenological framework, and the observational comparison should be presented as a test of that parametrization rather than of Finsler gravity itself.

major comments (3)
  1. [Section 4.1, Eq. (116)] The normalized Hubble function h(z) in Eq. (116) contains the denominator (1+z)φ'(z) − 2φ(z). With the stated forms φ(z)=1+βz, φ(z)=1+ln(1+βz), and φ(z)=e^{2βz}, and with the flatness constraint (117), evaluation at z=0 yields h(0) = (2−β)/(β−2) = −1 for the linear model, and similarly h(0)=−1 for the other two variants. This contradicts the requirement h(0)=1 for a normalized Hubble function and the fact that H(z)=H0 h(z) should be positive. Please correct the sign in the denominator (the likely form is 2φ(z) − (1+z)φ'(z)) and re-verify, and if necessary re-run, the Barthel-Randers MCMC results reported in Sections 5 and 6.
  2. [Section 3.4 and Section 4.1] The Barthel-Kropina field equations (71)–(72) contain four unknowns (a, η, ρ, p) and, as the text explicitly states, remain underdetermined even after imposing an equation of state; a supplementary relation is needed. Section 4.1 then introduces the integrated ODE system (118)–(120) for the Barthel-Kropina model without specifying the closure relation that connects f(z), u(z), and h(z) to the variables (a, η, ρ, p) of Section 3.4, and without defining the physical meaning of f and u. Consequently, the MCMC analysis of the Barthel-Kropina model is not reproducible from the manuscript, and the statistical comparison in Table 3 cannot be verified as a test of the Finsler-derived equations rather than of an unspecified phenomenological ansatz. Please provide the missing closure relation and definitions, or clearly declare (118)–(120) to be a separate phenomenological model.
  3. [Abstract, Sections 4 and 7, Table 3] The claim that the Finslerian models are a viable alternative to ΛCDM rests on fits in which the Finslerian input is a free function fitted to the data: the Barthel-Randers models use ad hoc forms φ(z)=1+βz, 1+ln(1+βz), and e^{2βz} in Eq. (116), with no equation of motion for φ (or η); and the Barthel-Kropina model fits the initial conditions f0, u0 and an equation-of-state parameter ω=1.045, which corresponds to a stiff fluid rather than baryonic matter. The ΔAIC=−14.6 preference for the Barthel-Kropina model in Table 3 therefore reflects the flexibility of a phenomenological parametrization (plus an unphysical matter component), not a successful prediction of Finsler geometry. The paper should explicitly state that η(t) has no dynamical law, that the fitted 'geometric dark energy' is a curve-fitting device, and should soften the 'viable alternative' conclusion accordingly.
minor comments (5)
  1. [Section 3.3, text after Eq. (70)] The sentence 'By substituting the expression of H² from the Friedmann equation (60), we recover equation (70)' appears circular; presumably the intended statement is that the first Friedmann equation reproduces the conservation equation (68) or a similar consistency check. Please clarify.
  2. [Section 5.4.3] The interpretation of the p-value is reversed: a p-value smaller than 0.05 indicates that the observed χ² is unlikely under the model, i.e., a poor fit, not evidence that the model is a good fit. The text should be corrected to avoid this statistical misunderstanding.
  3. [Section 6.1] There are typographical errors in the reported Gelman-Rubin statistics: the Barthel-Kropina line reads '1.0071.007' instead of a comma-separated list, and some braces in the Barthel-Randers lines are incorrectly placed. These should be fixed for clarity.
  4. [Section 4.1] The symbol h(z) is used for the normalized Hubble function in Eq. (116) and in the Barthel-Kropina system (118)–(120), but the paper never states the condition h(0)=1 for the Barthel-Kropina case. Please state this normalization explicitly, as it is used in the initial conditions for the ODE system.
  5. [Title page] The manuscript header includes 'Academic Editor', 'Received', 'Revised', 'Accepted', and a journal-style citation with a DOI placeholder. For an arXiv preprint this information is unusual and should be removed or updated to avoid confusion about the paper's status.

Circularity Check

3 steps flagged · score 6.0 of 10

The Finslerian 'dark energy' is a fitted free function: Randers uses hand-picked φ(z) with fitted β, Kropina closes an underdetermined system with fitted initial conditions and ω, so the statistical success is a fit rather than a geometric prediction.

  1. fitted input called prediction [Section 4.1, Eq. (116); Section 3.3, Eqs. (60)-(62)]
    "Our starting point is the family of cosmological scenarios proposed in [120], where three distinct variants of the Barthel–Randers model were introduced. Each of these variants is characterized by a specific choice of the function φ(z). ... The form of φ(z) determines the behavior of each specific model and encodes the influence of the underlying Finslerian geometry."

    In the Randers derivation the entire Finslerian modification is carried by φ=1+aη (Eq. (62)), with η(x0) a free function that obeys no equation of motion. The normalized Hubble function (116) is then built from φ(z) plus fitted parameters Ωm0, β, H0, M, rd, while ΩΛ0 is defined by Eq. (117) in terms of φ′(0). The three 'variants' are three arbitrary ansätze for φ (linear, logarithmic, exponential). The abstract's 'effective dark energy component' is therefore not a prediction of the geometry: it is the chosen φ with its fitted β, and the reported χ2/AIC values test that parametrization, not Finsler gravity.

  2. fitted input called prediction [Section 3.4, Eqs. (71)-(73); Section 4.1, Eqs. (118)-(120); Table 2]
    "By considering an equation of state for the baryonic matter, p=p(ρ), the number of unknowns in the system of generalized Friedmann equations becomes three, and the system is still underdetermined. Therefore, to obtain solvable cosmological models, and to close the system, we must impose a supplementary independent relation."

    The text concedes that the Kropina Friedmann system (71)-(72) is underdetermined and requires an extra closure, but the review never states the relation that turns it into the integrated ODEs (118)-(120). The numerical solutions then treat f0, u0, and the baryonic equation-of-state parameter ω as free, with best fits f0=0.0420, u0=0.3641, ω=1.045 (Table 2). Since ω≈1.045 describes a stiff fluid, not standard baryonic matter, the ΔAIC=−14.6 advantage (Table 3) measures the flexibility of a six-parameter phenomenological H(z), not a successful prediction of Kropina geometry. The 'effective dark energy' appears because the free function and initial conditions are fitted to the same expansion data.

1 more flagged steps
  1. ansatz smuggled in via citation [Section 3.3-3.4 (Eqs. (60)-(61), (71)-(72)); Section 4.1 (after [120], [121])]
    "The generalized Friedmann equations in this geometry take the form [120]. ... In the case of the Barthel–Kropina geometry, we adopt the model proposed by [121]."

    The load-bearing evolution equations and the functional ansätze used for the fits are taken from the authors' own previous papers [120,121] rather than derived in this review. Those papers contain the same unconstrained η(t)/φ(z) input and, for Kropina, the same admitted 'supplementary independent relation' needed to close an underdetermined system (Section 3.4). Hence the statistical comparison tests a parametrization whose flexibility was fixed by the authors' earlier choices; the present paper adds MCMC fitting but not an independent derivation of the Finslerian 'dark energy' terms.

full rationale

Most of the mathematics in Sections 2-3.1 — the (α,β) metric formalism, the b-osculating Riemannian metric, and the theorem that the Barthel connection coincides with the Levi-Civita connection when Y=b — is genuine geometry, checkable independently of the cosmological claims, and not circular. The circularity enters when this geometry is converted into cosmology. The one-form β is fixed to (aη,0,0,0) by isotropy, but η(t) remains a completely free function with no equation of motion; the Friedmann equations therefore contain extra terms that are not determined by the Finsler structure. In the Barthel-Randers fits these terms become three arbitrary φ(z) ansätze with a fitted β, and ΩΛ0 is defined by Eq. (117) from φ′(0). In the Barthel-Kropina case the paper itself states that the system (71)-(72) is underdetermined and needs a supplementary relation; the integrated system (118)-(120) is then solved with fitted f0, u0, and ω≈1.045. The ΔAIC/BIC comparisons in Table 3 therefore measure how well a free function plus fitted initial conditions can match the expansion history, not whether Finsler geometry predicts dark energy. The abstract's claim that the geometric terms 'generate' an effective cosmological constant is a renaming of this fitted freedom. The score is 6 because the central statistical claim partially reduces by construction, even though the underlying Finsler geometry and the field-equation postulate are not themselves circular.

Assumptions & free parameters 9 free parameters · 6 assumptions · 1 invented entities

The central claim is carried by an unproven reduction: Finslerian effects are collapsed into one free time-dependent function, and the Einstein equations are then asserted on the osculating metric. The model's dark-energy-like terms come entirely from that function, whose form and parameters are fitted to the same data used to claim viability. The particle-creation and thermodynamic interpretation is an additional interpretive layer, not an independent constraint.

free parameters (9)
  • BR linear model parameter beta = 0.00213 +/- 0.00061
    Controls phi(z)=1+beta z in Eq. (116) and is fitted to CC + Pantheon+ + DESI DR2 data.
  • BR logarithmic model parameter beta = 0.00448 +/- 0.00055
    Controls phi(z)=1+ln(1+beta z) in Eq. (116) and is fitted to the same cosmological datasets.
  • BR exponential model parameter beta = 0.00820 +/- 0.00016
    Controls phi(z)=e^{2 beta z} in Eq. (116) and is fitted to the same cosmological datasets.
  • Functional form of phi(z) in BR models = Linear, logarithmic, or exponential, chosen by hand
    The Finslerian dark-energy source is not derived from the geometry; the functional form is selected ad hoc and then fitted.
  • BK initial condition f0 = 0.0420 +/- 0.0015
    Initial value f(0) for the Barthel-Kropina system (118)-(120); fitted to data.
  • BK initial condition u0 = 0.3641 +/- 0.0066
    Initial value u(0) for the Barthel-Kropina system; fitted to data.
  • BK equation-of-state parameter omega = 1.045 +/- 0.047
    Fitted matter equation-of-state parameter in the Barthel-Kropina system; the value is stiff and exotic for ordinary baryonic matter, which is not flagged in the text.
  • Standard joint fit parameters H0, Omega_m0, M, rd = H0 about 66-69, Omega_m0 about 0.288-0.306, M about -19.42 to -19.45, rd about 147.9-149.0
    Hubble constant, matter density, supernova absolute magnitude, and sound horizon are fitted jointly in all models, including Lambda-CDM.
  • Conformal factor phi(x0) in conformal Barthel-Kropina = Unconstrained
    The conformal factor enters Eqs. (83)-(85) but is not fixed by the theory or fitted in the present review.
assumptions (6)
  • standard math For (alpha, beta)-metrics evaluated at Y=b, the Cartan tensor vanishes and the Barthel connection equals the Levi-Civita connection of the osculating metric, Eqs. (51)-(52).
    Core mathematical fact used to reduce Finsler geometry to a Riemannian calculation; cited from references rather than proved in the review.
  • domain assumption The gravitational field equations are the Einstein equations for the osculating Barthel curvature, Eq. (59).
    Postulated as the natural extension of general relativity; no variational derivation from a Finsler action is given.
  • domain assumption The cosmological background is FLRW, the one-form has only a temporal component b0=a eta, and the matter four-velocity is comoving and described by a perfect fluid, Section 3.2.
    These assumptions enforce homogeneity, isotropy, and the standard matter content, excluding anisotropic or non-comoving Finslerian effects.
  • ad hoc to paper The Finslerian internal vector is identified with the one-form field, Y=b, Section 3.1.3.
    This particular osculating choice makes the Barthel connection the Levi-Civita connection; other choices would change the Friedmann equations.
  • domain assumption In Barthel-Randers cosmology the covariant divergence of the energy-momentum tensor with the Barthel connection vanishes, Eq. (68); in Barthel-Kropina it does not, and the non-conservation is interpreted as particle creation.
    Conservation or non-conservation is imposed from the connection rather than derived from an action, and the particle-creation interpretation is one possible reading.
  • ad hoc to paper The functional forms phi(z)=1+beta z, phi(z)=1+ln(1+beta z), and phi(z)=e^{2 beta z} are adopted to close the Barthel-Randers model.
    No geometric principle selects these forms; they are chosen for tractability and then fitted to data.
invented entities (1)
  • Effective time-dependent Finslerian dark-energy function phi(t)/eta(t)
    purpose: Adds extra terms to the Friedmann equations that mimic a cosmological constant and accelerate expansion.
    No unique prediction is provided: phi or eta is a free functional degree of freedom fitted to expansion data, with no independent handle in perturbations, CMB, or other observables.

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Cite this review

Pith. "Pith review of From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology." pith.science (2026). https://pith.science/paper/SOGCZKCV

@misc{pith2026250618422,
  author       = {Pith},
  title        = {Pith review of: From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SOGCZKCV}},
  note         = {Machine review of arXiv:2506.18422}
}
abstract

We review recent developments in cosmological models based on Finsler geometry and extensions of general relativity within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend on position and an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We explore whether Finsler-type geometries can describe gravitational interaction and cosmological dynamics. In particular, we examine the Barthel connection and $(\alpha, \beta)$ geometries, where $\alpha$ is a Riemannian metric and $\beta$ is a one-form. For a specific construction of $\beta$, the Barthel connection coincides with the Levi-Civita connection of the associated Riemann metric. We review gravitational field and cosmological evolution in three geometries: Barthel-Randers ($F = \alpha + \beta$), Barthel-Kropina ($F = \alpha^2 \beta$), and the conformally transformed Barthel-Kropina geometry. After presenting the mathematical foundations of Finslerian-type modified gravity theories, we derive generalized Friedmann equations in these geometries assuming a Friedmann-Lema\^itre-Robertson-Walker type metric. We also present the matter-energy balance equations, interpreting them from the perspective of thermodynamics with particle creation. The cosmological properties of Barthel-Randers and Barthel-Kropina models are explored in detail. The additional geometric terms in these models can be interpreted as an effective dark energy component, generating an effective cosmological constant. Several cosmological solutions are compared with observational data (Cosmic Chronometers, Type Ia Supernovae, Baryon Acoustic Oscillations) using MCMC analysis. A comparison with the $\Lambda$CDM model shows that Finslerian models fit observational data well, suggesting they offer a viable alternative to general relativity.

Figures

Figures reproduced from arXiv: 2506.18422 by the authors.

Figure 1
Figure 1. Flowchart of the algorithmic approach for the construction of the (α, β) gravitational models with the Barthel connection. Although specific examples are restricted to the cases of Barthel–Randers and Barthel– Kropina geometries, the formalism can be easily extended to any other choices of the Finslerian function F = αϕ(s), and the implementation of the geometrical model into a gravitational theoretical framework ca… view at source ↗
Figure 2
Figure 2. The figure shows the parameter constraints of the Barthel–Randers and Barthel–Kropina models using the CC + SNe Ia and BAO datasets together, and displaying both 1σ and 2σ confi￾dence intervals. The contours show the correlations between these parameters, with marginalized probability distributions along the diagonal. A similar trend is observed for rd , where the predicted central values across all models are close… view at source ↗
Figure 3
Figure 3. The figure illustrates the trace plots, which show the convergence behavior of the Markov chains for each parameter in the Barthel–Randers and Barthel–Kropina cosmological models [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: shows the evolution of the Hubble function and the Hubble residuals for the linear, logarithmic, and exponential cases of the Barthel–Randers model, as well as the Barthel–Kropina model. These are compared with the predictions from the ΛCDM model and CC measurements. T…
Figure 5
Figure 5. Figure 5: shows the evolution of different BAO distance scales. All three Barthel– Randers variants exhibit minimal deviation in both the volume-averaged distance DV/(rdz 2/3) and the transverse distance measure DM/(zDH), indicating strong align￾ment with cosmological observatio…
Figure 6
Figure 6. Figure 6: Evolution of the deceleration parameter q(z) and jerk parameter j(z) for the Barthel– Randers (linear, logarithmic, and exponential) and Barthel–Kropina models, and of the ΛCDM model. The left panel shows the evolution of q(z) as a function of redshift, while the right…

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