Extensions to $\Lambda$CDM at Intermediate Redshifts to Solve the Tensions ?

Models of dark energy or modified gravity that tries to alleviate the tensions on the Hubble constant ($H_0$) and the matter fluctuation parameter ($\sigma_8$) are usually parameterized as function of either late or early time cosmic evolution. In this work we rather focus on one that could privilege extensions to $\Lambda$CDM on intermediate redshifts by mean of a Gaussian-like window function with a free moving centre $a_{Gwin}$ combined with a modified gravity parameter $\mu_{Gwin}$ and an extension of the equation of state parameter $\omega_{Gwin}$. Using different combinations of the latest available current datasets subject of the discrepancies, such as the cosmic microwave (CMB) background power spectrum, the baryonic acoustic scale (BAO) in galaxy distribution, Weak lensing (WL) shear and galaxy clustering cross correlations and local hubble constant measurements, we investigate whether such model could alleviate each or both $H_0$ and $\sigma_8$ tensions. We found when combining all probes that the $\sigma_8$ tension is alleviated while the $H_0$ is reduced with a small preference for a positive $\omega_{Gwin}$ without a particular preference for a redshift or a $\mu_{Gwin}$ different from its equivalent $\Lambda$CDM value. However, if we follow another approach and compare the two sets of the probes subject of discrepancy i.e. CMB+BAO vs WL+local $H_0$, we found that the model is able of solving the $\sigma_8$ discrepancy at the expense of a enlargement of the constraints, while the Hubble constant discrepancy is not that affected due to the fact that the two likelihood contours are stretched in parallel directions. We conclude that modifying $\Lambda$CDM cosmology at intermediate redshifts within our model, and the constraints from the datasets used in this study, are not likely a viable solution to solve both tensions.