Simultaneously solving the $H_0$ and $σ_8$ tensions with late dark energy

In a model independent approach, we derive generic conditions that any late time modification of the $Λ$CDM expansion history must satisfy in order to consistently solve both the $H_0$ and the $σ_8$ tensions. Our results are fully analytical and the method is merely based on the assumption that the late-time deviations from $Λ$CDM remain small. For the concrete case of a dark energy fluid with deviations encoded in the expansion history and the gravitational coupling constant, we present necessary conditions on its equation of state. Solving both the $H_0$ and $σ_8$ tensions requires that $w(z)$ must cross the phantom divide if $G_\text{eff}=G$. On the other hand, for $G_\text{eff}=G+δG(z)$ and $w(z)\leq -1$, it is required that $\displaystyle \frac{δG(z)}{G}<α(z)\frac{δH(z)}{H(z)}<0$ at some redshift $z$.

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