From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology
We present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also...
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| author | Amine Bouali Himanshu Chaudhary Lehel Csillag Rattanasak Hama Tiberiu Harko Sorin V. Sabau Shahab Shahidi |
| author_facet | Amine Bouali Himanshu Chaudhary Lehel Csillag Rattanasak Hama Tiberiu Harko Sorin V. Sabau Shahab Shahidi |
| author_sort | Amine Bouali |
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| description | We present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also on an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We examine in detail the possibility that Finsler-type geometries can describe the physical properties of the gravitational interaction, as well as the cosmological dynamics. In particular, we present and review the implications of a particular implementation of Finsler geometry, based on the Barthel connection, and of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula> geometries, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is a Riemannian metric, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is a one-form. For a specific construction of the deviation part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, in these classes of geometries, the Barthel connection coincides with the Levi–Civita connection of the associated Riemann metric. We review the properties of the gravitational field, and of the cosmological evolution in three types of geometries: the Barthel–Randers geometry, in which the Finsler metric function <i>F</i> is given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi></mrow></semantics></math></inline-formula>, in the Barthel–Kropina geometry, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>/</mo><mi>β</mi></mrow></semantics></math></inline-formula>, and in the conformally transformed Barthel–Kropina geometry, respectively. After a brief presentation of the mathematical foundations of the Finslerian-type modified gravity theories, the generalized Friedmann equations in these geometries are written down by considering that the background Riemannian metric in the Randers and Kropina line elements is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equations are also presented, and they are interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. We investigate the cosmological properties of the Barthel–Randers and Barthel–Kropina cosmological models in detail. In these scenarios, the additional geometric terms arising from the Finslerian structure can be interpreted as an effective geometric dark energy component, capable of generating an effective cosmological constant. Several cosmological solutions—both analytical and numerical—are obtained and compared against observational datasets, including Cosmic Chronometers, Type Ia Supernovae, and Baryon Acoustic Oscillations, using a Markov Chain Monte Carlo (MCMC) analysis. A direct comparison with the standard <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula>CDM model is also carried out. The results indicate that Finslerian cosmological models provide a satisfactory fit to the observational data, suggesting they represent a viable alternative to the standard cosmological model based on general relativity. |
| format | Article |
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| institution | Kabale University |
| issn | 2218-1997 |
| language | English |
| publishDate | 2025-06-01 |
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| spelling | doaj-art-bd487fd49f884fb4a10debb4dd29e27c2025-08-20T03:32:28ZengMDPI AGUniverse2218-19972025-06-0111719810.3390/universe11070198From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian CosmologyAmine Bouali0Himanshu Chaudhary1Lehel Csillag2Rattanasak Hama3Tiberiu Harko4Sorin V. Sabau5Shahab Shahidi6Laboratory of Physics of Matter and Radiation, Mohammed I University, Oujda BP 717, MoroccoDepartment of Physics, Babeș-Bolyai University, Kogălniceanu Street 1, 400084 Cluj-Napoca, RomaniaDepartment of Physics, Babeș-Bolyai University, Kogălniceanu Street 1, 400084 Cluj-Napoca, RomaniaFaculty of Science and Industrial Technology, Prince of Songkla University, Surat Thani Campus, Surat Thani 84000, ThailandDepartment of Physics, Babeș-Bolyai University, Kogălniceanu Street 1, 400084 Cluj-Napoca, RomaniaSchool of Biological Sciences, Department of Biology, Tokai University, Sapporo 005-8600, JapanSchool of Physics, Damghan University, Damghan 41167-36716, IranWe present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also on an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We examine in detail the possibility that Finsler-type geometries can describe the physical properties of the gravitational interaction, as well as the cosmological dynamics. In particular, we present and review the implications of a particular implementation of Finsler geometry, based on the Barthel connection, and of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula> geometries, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is a Riemannian metric, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is a one-form. For a specific construction of the deviation part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, in these classes of geometries, the Barthel connection coincides with the Levi–Civita connection of the associated Riemann metric. We review the properties of the gravitational field, and of the cosmological evolution in three types of geometries: the Barthel–Randers geometry, in which the Finsler metric function <i>F</i> is given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi></mrow></semantics></math></inline-formula>, in the Barthel–Kropina geometry, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>/</mo><mi>β</mi></mrow></semantics></math></inline-formula>, and in the conformally transformed Barthel–Kropina geometry, respectively. After a brief presentation of the mathematical foundations of the Finslerian-type modified gravity theories, the generalized Friedmann equations in these geometries are written down by considering that the background Riemannian metric in the Randers and Kropina line elements is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equations are also presented, and they are interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. We investigate the cosmological properties of the Barthel–Randers and Barthel–Kropina cosmological models in detail. In these scenarios, the additional geometric terms arising from the Finslerian structure can be interpreted as an effective geometric dark energy component, capable of generating an effective cosmological constant. Several cosmological solutions—both analytical and numerical—are obtained and compared against observational datasets, including Cosmic Chronometers, Type Ia Supernovae, and Baryon Acoustic Oscillations, using a Markov Chain Monte Carlo (MCMC) analysis. A direct comparison with the standard <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula>CDM model is also carried out. The results indicate that Finslerian cosmological models provide a satisfactory fit to the observational data, suggesting they represent a viable alternative to the standard cosmological model based on general relativity.https://www.mdpi.com/2218-1997/11/7/198Barthel connection(α,β) metricsFinslerian cosmologyposterior inferenceMCMC statistical analysis |
| spellingShingle | Amine Bouali Himanshu Chaudhary Lehel Csillag Rattanasak Hama Tiberiu Harko Sorin V. Sabau Shahab Shahidi From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology Universe Barthel connection (α,β) metrics Finslerian cosmology posterior inference MCMC statistical analysis |
| title | From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology |
| title_full | From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology |
| title_fullStr | From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology |
| title_full_unstemmed | From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology |
| title_short | From Barthel–Randers–Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology |
| title_sort | from barthel randers kropina geometries to the accelerating universe a brief review of recent advances in finslerian cosmology |
| topic | Barthel connection (α,β) metrics Finslerian cosmology posterior inference MCMC statistical analysis |
| url | https://www.mdpi.com/2218-1997/11/7/198 |
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