Simultaneously solving the $H_0$ and ${\sigma }_8$ tensions with late dark energy

Abstract

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 $w(z)\le -1$, it is required that $\frac{\delta G(z)}{G}<\alpha (z)\frac{\delta H(z)}{H(z)}<0$ at some redshift $z$.

4 pages, 2 figures, submission to PRL

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