Two Kind of Dark Energy Models for Current Cosmology Tensions: Finslerian and Back-Reaction Effect Forms

In this presentation, we introduce two kind of dark energy models: 1. the Finslerian extension gravity framework for dark energy models. To reconcile the current tensions between high and low redshift observations, we perform the first constraints on the Finslerian cosmological models including the effective dark matter and dark energy components and find that all the four Finslerian models could alleviate effectively the Hubble constant ( $$H_0$$ H 0 ) tension as well as the amplitude of the root-mean-square density fluctuations ( $$\sigma _8$$ σ 8 ) tension between the Planck measurements and the local Universe observations at the 68 $$\%$$ % confidence level. The addition of a massless sterile neutrino and a varying total mass of active neutrinos to the base Finslerian (additionally) two-parameter model, respectively, reduces the $$H_0$$ H 0 tension from $$3.4\sigma $$ 3.4 σ to $$1.9\sigma $$ 1.9 σ and alleviates the $$\sigma _8$$ σ 8 tension better than the other three Finslerian models. Computing the Bayesian evidence, with respect to $$\Lambda $$ Λ CDM model, our analysis shows a weaker preference for the base Finslerian model and moderate preferences for its three one-parameter extensions. Based on the model-independent Gaussian Processes, we also propose a new linear relation which can describe the current redshift space distortions data, and by using the most stringent constraints provided so far, we have also obtained the limits on typical model parameters for three one-parameter extension models. 2. Back-reaction as an effective dark energy model for the $$H_o$$ H o tension problem. The Universe accelerated expansion phenomenon may be regarded as an averaged effect caused by small scale inhomogeneities of the universe matter-energy distribution with the use of the spatial averaged approach of Buchert, in which two models are considered, one of which assumes that the back-reaction term as $$\mathcal{Q}_\mathcal{D}$$ Q D and the averaged spatial Ricci scalar $$\left\langle \mathcal{R} \right\rangle _\mathcal{D}$$ R D obeys the scaling laws of the volume scale factor $$a_\mathcal{D}$$ a D at adequately late times, and another adopts the ansatz that the back-reaction term $$\mathcal{Q}_\mathcal{D}$$ Q D is simply a constant in the recent universe. Thanks to the effective geometry introduced by Larena et al. in their previous work, we confront these two back-reaction models with latest type Ia supernova and Hubble parameter observations, coming out with the results that the constant back-reaction model is slightly favored over the other model, and the best fitting back-reaction term in the scaling back-reaction model behaves almost like a constant. Also, the numerical results show that the constant back-reaction model predicts a smaller expansion rate and decelerated expansion rate than the other model does at redshifts higher than about $$(z >)1$$ ( z > ) 1 , and the both back-reaction terms begin to accelerate the universe expansions at a redshift z around 0.5.