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We consider dark energy models obtained from the general conformaltransformation of the Kropina metric, representing an $(\alpha, \beta)$ typeFinslerian geometry, constructed as the ratio of the square of a Riemannianmetric $\alpha$, and of the one-form $\beta$. Conformal symmetries do appear inmany fields of physics, and they may play a fundamental role in theunderstanding of the Universe. We investigate the possibility of obtainingconformal theories of gravity in the osculating Barthel-Kropina geometricframework, where gravitation is described by an extended Finslerian type model,with the metric tensor depending on both the base space coordinates, and on avector field. We show that it is possible to formulate a family of conformalBarthel-Kropina theories in an osculating geometry with second-order fieldequations, depending on the properties of the conformal factor, whose presenceleads to the appearance of an effective scalar field, of geometric origin, inthe gravitational field equations. The cosmological implications of the theoryare investigated in detail, by assuming a specific relation between thecomponent of the one-form of the Kropina metric, and the conformal factor. Thecosmological evolution is thus determined by the initial conditions of thescalar field, and a free parameter of the model. We analyze in detail threecosmological models, corresponding to different values of the theoryparameters. Our results show that the conformal Barthel-Kropina model couldgive an acceptable description of the observational data, and may represent atheoretically attractive alternative to the standard $\Lambda$CDM cosmology.

Conformal gravitational theories in the Barthel-Kropina type Finslerian geometry, and their cosmological implications