Dissecting the $H_0$ and $S_8$ tensions with Planck + BAO + supernova
  type Ia in multi-parameter cosmologies
  The mismatch between the locally measured expansion rate of the universe and
the one inferred from observations of the cosmic microwave background (CMB)
assuming the canonical $\Lambda$CDM model has become the new cornerstone of
modern cosmology, and many new-physics set ups are rising to the challenge.
Concomitant with the so-called $H_0$ problem, there is evidence of a growing
tension between the CMB-preferred value and the local determination of the
weighted amplitude of matter fluctuations $S_8$. It would be appealing and
compelling if both the $H_0$ and $S_8$ tensions were resolved at once, but as
yet none of the proposed new-physics models have done so to a satisfactory
degree. Herein, we adopt a systematic approach to investigate the possible
interconnection among the free parameters in several classes of models that
typify the main theoretical frameworks tackling the tensions on the universe
expansion rate and the clustering of matter. Our calculations are carried out
using the publicly available Boltzmann solver CAMB in combination with the
sampler CosmoMC. We show that even after combining the leading classes of
models sampling modifications of both the early and late time universe a
simultaneous solution to the $H_0$ and $S_8$ tensions remains elusive.
