Simultaneously solving the $H_0$ and $\sigma_8$ tensions with late dark
  energy
  In a model independent approach, we derive generic conditions that any late
time modification of the $\Lambda$CDM expansion history must satisfy in order
to consistently solve both the $H_0$ and the $\sigma_8$ tensions. Our results
are fully analytical and the method is merely based on the assumption that the
late-time deviations from $\Lambda$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 $\sigma_8$ tensions requires that
$w(z)$ must cross the phantom divide if $G_\text{eff}=G$. On the other hand,
for $G_\text{eff}=G+\delta G(z)$ and $w(z)\leq -1$, it is required that
$\displaystyle \frac{\delta G(z)}{G}<\alpha(z)\frac{\delta H(z)}{H(z)}<0$ at
some redshift $z$.
