Résumé
We perform a comprehensive analysis of the most common early-and late-universe solutions to the H-0, Ly-alpha, and S-8 discrepancies. When considered on their own, massive neutrinos provide a natural solution to the S-8 discrepancy at the expense of increasing the H-0 tension. If all extensions are considered simultaneously, the best-fit solution has a neutrino mass sum of similar to 0.4 eV, a dark energy equation of state close to that of a cosmological constant, and no additional relativistic degrees of freedom (d.o.f). However, the H-0 tension, while weakened, remains unresolved. Motivated by this result, we perform a nonparametric reconstruction of the evolution of the dark energy fluid density (allowing for negative energy densities), together with massive neutrinos. When all data sets are included, there exists a residual similar to 1.9 sigma tension with H-0. If this residual tension remains in the future, it will indicate that it is not possible to solve the H-0 tension solely with a modification of the late-universe dynamics within standard general relativity. However, we do find that it is possible to resolve the tension if either galaxy baryon acoustic oscillation (BAO) or joint light-curve analysis supernovae data are omitted. We find that negative dark energy densities are favored near redshift z similar to 2.35 when including the Ly-alpha BAO measurement (at similar to 2 sigma). This behavior may point to a negative curvature, but it is most likely indicative of systematics or at least an underestimated covariance matrix. Quite remarkably, we find that in the extended cosmologies considered in this work, the neutrino mass sum is always close to 0.4 eV regardless of the choice of external data sets, as long as the H-0 tension is solved or significantly decreased.