From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology
Abstract
1. Introduction—Early History and Modern Developments
- In the first step, the basic physical/empirical entities are postulated, such as the timelike worldlines for freely falling massive particles, the lightlike worldlines of light rays, and radar echoes between massive particle worldlines. These empirical elements provide enough structure to define a system of coordinates and allow for the construction of a differentiable structure on the spacetime set M, turning it into a smooth manifold.
- The conformal structure is established by requiring that, at each point in spacetime, the set of all possible directions (i.e., tangent vectors) splits into two components when the directions corresponding to massless (lightlike) trajectories are removed. This splitting reflects the causal distinction between future and past. Additionally, in a sufficiently small neighborhood V around the worldline of a massive particle, for any point not lying on the particle’s path, the function that maps p to the product of the radar emission time and reception time , i.e., , must be at least twice differentiable.
- Imposing that through each point in spacetime, and for each timelike direction, there exists one unique timelike (massive) trajectory passing through that point, which results in a projective structure. Each of these trajectories must admit a parametrization such that, in local coordinates near the point, the motion satisfies . This expresses the fact that particles move along straight lines in free fall.
- In the final step, compatibility between the conformal and projective structures is required. In particular, light rays must be special cases of particle geodesics in the limit of zero mass. This determines the metric up to a conformal factor, which leads to a Weyl structure. Through some technical steps, eliminating the second clock effect leads to a Lorentzian structure, i.e., a pseudo-Riemannian manifold.
2. Fundamentals of Finsler Geometry
- (i)
- is on .
- (ii)
- is positively 1-homogeneous: , for all and .
- (iii)
- For each , the Hessian matrix (29) is positive definite on .
3. Osculating -Type Cosmological Models
3.1. Mathematical Foundations of the Finslerian Cosmologies
3.1.1. Kropina and Geometries
3.1.2. The Barthel Connection
3.1.3. The Y-Osculating Riemann Geometry
- The case of the metrics.
- The curvature tensor.
3.2. Building Cosmological Models in Geometries
- The Universe is homogeneous and isotropic.
- The Riemannian metric is the FLRW metric.
- The Finsler metric depends on only.
- The 1-form b has vanishing space-like components.
- Matter moves along the Hubble flow.
- The matter content of the Universe is a perfect fluid.
- Geometric quantities.
- (iii)
- (iv)
- ;
- (v)
- ;
- (vi)
- Gravitational field equations.
- Flowchart of the algorithmic construction of osculating Barthel-type Finslerian gravitational theories.
3.3. Barthel–Randers Cosmology
- The energy conservation equation.
3.4. Barthel–Kropina Cosmology
- Energy balance equation.
- The general relativistic limit.
3.5. Conformal Barthel–Kropina Cosmology
- The generalized Friedmann equations.
3.6. Thermodynamic Interpretation of the Cosmologies
3.6.1. Irreversible Thermodynamics and Matter Creation
- Particle balance equations.
- The entropy flux.
- The creation pressure.
3.6.2. Application: Particle Creation in Barthel–Randers Cosmology
- Creation pressure in Barthel–Randers cosmology.
- The particle creation rate.
- The matter temperature.
3.6.3. Creation of Exotic Matter
4. Cosmological Implications of Barthel–Randers and Barthel–Kropina Models
4.1. Specific Cosmological Models
- Linear model: .
- Logarithmic model: .
- Exponential model: .
4.2. Methodology and Datasets
- Cosmic Chronometers: In this study, we use the Hubble measurements extracted based on the differential age approach. This technique leverages passively evolving massive galaxies, which formed at redshifts around , enabling a direct and model-independent determination of the Hubble parameter using the relationship . This method significantly reduces the reliance on astrophysical assumptions [163]. For our analysis, we use 15 Hubble measurements selected from the 31 Hubble measurements, which cover a redshift range from [164,165,166]. We use the likelihood function provided by Moresco in his GitLab repository3 which uses the full covariance matrix, accounting for both statistical and systematic uncertainties [167,168].
- Type Ia Supernova: We also use the Pantheon+ dataset without the SHOES calibration, which consists of 1701 light curves from 1550 Type Ia Supernovae (SNe Ia) covering a redshift range of [169]. To analyze this data, we adopt the likelihood function described in [170], which incorporates the total covariance matrix, , which includes both statistical () and systematic () uncertainties [171]. The likelihood function is given by where represents the residual vector, defined as the difference between the observed and theoretical distance moduli: , where . Here, is the inverse of the total covariance matrix. The model-predicted distance modulus is calculated as where the luminosity distance in a flat FLRW Universe is given by Here, c is the speed of light, and is the Hubble parameter. This formulation highlights the degeneracy between the nuisance parameter and the Hubble constant .
- Baryon Acoustic Oscillation: In our analysis, we also include the 13 recent Baryon Acoustic Oscillation (BAO) measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) [172]. These measurements cover a redshift range of . They were obtained from observations of the Bright Galaxy Sample (BGS), Luminous Red Galaxies (LRG1, LRG2, LRG3), Emission Line Galaxies (ELG1 and ELG2), Quasars (QSO), and Lyman- tracers4. The measurements are reported using three distance indicators: the Hubble distance , the comoving angular diameter distance , and the volume-averaged distance . To compare these with cosmological models, we compute the ratios , , and , where represents the sound horizon at the drag epoch, occurring around redshift . In a flat CDM model, Mpc [72]. However, in this study, we treat as a free parameter, allowing late-time observations to constrain the corresponding model parameters [173,174,175,176,177]. The chi-squared statistic for the BAO measurements is given by where and denote the vectors of theoretical predictions and observed measurements, respectively, and is the associated covariance matrix5.
Convergence Test
- Gelman–Rubin Statistic
- Convergence of Chains and Trace Plots
5. Comparing Barthel–Randers, Barthel–Kropina and CDM Models
5.1. Evolution of the Hubble Parameter, Hubble Residual, and BAO Distance Scales
5.2. Cosmographic Analysis of Barthel–Randers and Barthel–Kropina Models
Deceleration Parameter and Jerk Parameter
5.3. Statistical Assessment of Barthel–Randers and Barthel–Kropina Models
5.4. Goodness of Fit
5.4.1. Model Comparison Using AIC and BIC
5.4.2. Relative Comparison: AIC and BIC
- : Comparable support.
- : Considerably less support.
- : Strongly disfavored.
- : Weak evidence against the model.
- : Moderate evidence against the model.
- : Strong evidence against the model.
5.4.3. p-Value Statistics
6. Summary and Discussion of the Results
6.1. MCMC and Convergence Results
6.2. Hubble Parameter, Hubble Residual, and BAO Distance Scale Results
6.3. Cosmographic Results
6.4. Statistical Results
7. Discussion and Final Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
1 | https://emcee.readthedocs.io/en/stable/ (accessed on 14 June 2025) |
2 | https://getdist.readthedocs.io/en/latest/plot_gallery.html (accessed on 14 June 2025) |
3 | https://gitlab.com/mmoresco/CCcovariance (accessed on 14 June 2025) |
4 | https://github.com/CobayaSampler/bao_data (accessed on 14 June 2025) |
5 | https://github.com/CobayaSampler/bao_data/blob/master/desi_bao_dr2/desi_gaussian_bao_ALL_GCcomb_cov.txt (accessed on 14 June 2025) |
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Name | Finsler Function | Properties |
---|---|---|
Semi-Riemannian | Quadratic in y, reversible, i.e., . | |
Randers | Non-reversible, i.e., ; appears in EM analogs and Lorentz-violating physics. | |
Kropina | Singular on hypersurfaces; non-reversible. | |
Matsumoto | Non-reversible; used in irreversible mechanics; singular on hypersurfaces. | |
Bogoslovsky | Breaks full Lorentz invariance; ; used in very special relativity. | |
Funk | Defined on unit ball; forward complete; non-reversible. | |
Locally Minkowskian | General flat case; depends only on y, independent of x. | |
- metrics | Unifies and generalizes Randers, Kropina, and Matsumoto metrics via the scalar function . | |
General Lagrangian | , L 2-homogeneous in y | More general case; has some applications in mechanics. |
Optical/media | Includes anisotropic effects in optics through the direction-dependent refractive index . | |
Non-reversible Finsler | General F such that | Includes Funk, Randers, and non-reversible geometries. |
Cosmological Models | Parameter | JOINT |
---|---|---|
CDM Model | ||
BR (Linear Case) | ||
BR (Logarithmic Case) | ||
BR (Exponential Case) | ||
Barthel–Kropina Model | ||
Models | DoF | AIC | AIC | BIC | BIC | p-Value | ||
---|---|---|---|---|---|---|---|---|
CDM | 1780.94 | 1725 | 1.032 | 1788.94 | 0 | 1810.76 | 0.170 | 0.167 |
BR (Linear Case) | 1781.22 | 1724 | 1.033 | 1791.22 | 2.27 | 1818.49 | 7.74 | 0.164 |
BR (Logarithmic Case) | 1781.60 | 1724 | 1.033 | 1791.60 | 2.65 | 1818.87 | 8.11 | 0.163 |
BR (Exponential Case) | 1783.38 | 1724 | 1.034 | 1793.38 | 4.44 | 1820.65 | 9.89 | 0.155 |
Barthel–Kropina | 1762.37 | 1723 | 1.022 | 1774.37 | −14.57 | 1807.10 | −3.65 | 0.249 |
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Bouali, A.; Chaudhary, H.; Csillag, L.; Hama, R.; Harko, T.; Sabau, S.V.; Shahidi, S. From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology. Universe 2025, 11, 198. https://doi.org/10.3390/universe11070198
Bouali A, Chaudhary H, Csillag L, Hama R, Harko T, Sabau SV, Shahidi S. From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology. Universe. 2025; 11(7):198. https://doi.org/10.3390/universe11070198
Chicago/Turabian StyleBouali, Amine, Himanshu Chaudhary, Lehel Csillag, Rattanasak Hama, Tiberiu Harko, Sorin V. Sabau, and Shahab Shahidi. 2025. "From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology" Universe 11, no. 7: 198. https://doi.org/10.3390/universe11070198
APA StyleBouali, A., Chaudhary, H., Csillag, L., Hama, R., Harko, T., Sabau, S. V., & Shahidi, S. (2025). From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology. Universe, 11(7), 198. https://doi.org/10.3390/universe11070198