First comprehensive constraints on the Finslerian models using cosmological observations

We perform the first comprehensive constraints on two Finslerian models, i.e., the simplest Finslerian $\Lambda$CDM (F$\Lambda$) and non-flat F$\Lambda$ models using the Type Ia supernovae, baryonic acoustic oscillations, cosmic microwave background and lensing observations. Using the most stringent constraints we can provide, we find that the constrained typical parameters of both Finslerian models are all consistent with zero at the $2\sigma$ confidence level (C.L.). This means that both Finslerian models just deviate very slightly from the $\Lambda$CDM one, which is also verified by using two geometrical diagnostics to distinguish three different models from each other. We also find that a spatially flat universe is still preferred in the framework of Finsler geometry, that our measured values of the spectral index $n_s$ of primordial power spectrum in both Fisnlerian models exclude the scale invariance at more than $8\sigma$ C.L., and that the current $H_0$ tension can be relieved from $3.4\sigma$ to $2.9\sigma$ and $2.8\sigma$ in the F$\Lambda$ and non-flat F$\Lambda$ models, respectively.