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 G_eff = G . On the other hand, for G_eff = 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.