Abstract
The Hubble tension between direct measurements of the expansion rate and the prediction of the $\Lambda$CDM cosmological model calibrated on the Cosmic Microwave Background (CMB), is a strong motivation to explore alternative cosmological models. A popular class of such models includes an additional component of dark energy relevant in the early Universe, which solves the Hubble tension. These Early Dark Energy (EDE) models however typically worsen the $S_8$ tension between measurements of weak gravitational lensing and the $\Lambda$CDM expectation. Here, we show how combining EDE with scale-dependent primordial non-Gaussianities (sPNG) can in principle solve both tensions at once. Such sPNG are related to non-trivial inflationary physics, and observationally affect the non-linear regime of structure formation while leaving the linear regime mostly untouched. Depending on the amplitude of the sPNG, such models can either yield back the $\Lambda$CDM expectation for the PS in the non-linear regime, and hence an $S_8$ parameter compatible with $\Lambda$CDM, or can even reconcile the value of $S_8$ from weak-lensing measurements with the CMB, while solving the Hubble tension in all cases. In such models, both tensions would then be entirely related to pre-CMB physics of the early Universe.