Simultaneously solving the H0 and σ8 tensions with late dark energy

In a model independent approach, we derive generic conditions that any late time modification of the ΛCDM expansion history must satisfy in order to consistently solve both the H0 and the σ8 tensions. Our results are fully analytical and the method is merely based on the assumption that the late-time deviations from ΛCDM 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 H0 and σ8 tensions requires that w(z) must cross the phantom divide if Geff=G. On the other hand, for Geff=G+δG(z) and w(z)≤−1, it is required that δG(z)G<α(z)δH(z)H(z)<0 at some redshift z.