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Abstract

In this paper we will list a few important goals that need to be addressed in
the next decade, also taking into account the current discordances between the
different cosmological probes, such as the disagreement in the value of the
Hubble constant $H_0$, the $\sigma_8$--$S_8$ tension, and other less
statistically significant anomalies. While these discordances can still be in
part the result of systematic errors, their persistence after several years of
accurate analysis strongly hints at cracks in the standard cosmological
scenario and the necessity for new physics or generalisations beyond the
standard model. In this paper, we focus on the $5.0\,\sigma$ tension between
the {\it Planck} CMB estimate of the Hubble constant $H_0$ and the SH0ES
collaboration measurements. After showing the $H_0$ evaluations made from
different teams using different methods and geometric calibrations, we list a
few interesting new physics models that could alleviate this tension and
discuss how the next decade's experiments will be crucial. Moreover, we focus
on the tension of the {\it Planck} CMB data with weak lensing measurements and
redshift surveys, about the value of the matter energy density $\Omega_m$, and
the amplitude or rate of the growth of structure ($\sigma_8,f\sigma_8$). We
list a few interesting models proposed for alleviating this tension, and we
discuss the importance of trying to fit a full array of data with a single
model and not just one parameter at a time. Additionally, we present a wide
range of other less discussed anomalies at a statistical significance level
lower than the $H_0$--$S_8$ tensions which may also constitute hints towards
new physics, and we discuss possible generic theoretical approaches that can
collectively explain the non-standard nature of these signals.[Abridged]