Can late-time extensions solve the $H_0$ and $σ_8$ tensions?

We analyze the properties that any late-time modification of the $Λ$CDM expansion history must have in order to consistently solve both the $H_0$ and the $σ_8$ tensions. Taking a model-independent approach, we obtain a set of necessary conditions that can be applied to generic late-time extensions. Our results are fully analytical and merely based on the assumptions that the deviations from the $Λ$CDM background remain small. For the concrete case of a dark energy fluid with equation of state $w(z)$, we derive the following general requirements: (i) Solving the $H_0$ tension demands $w(z)<-1$ at some $z$ (ii) Solving both the $H_0$ and $σ_8$ tensions requires $w(z)$ to cross the phantom divide. Finally, we also allow for small deviations on the effective gravitational constant. In this case, our method is still able to constrain the functional form of these deviations.

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