The trouble beyond $H_0$ and the new cosmic triangles
  The distance ladder using supernovae yields higher values of the Hubble
constant $H_0$ than those inferred from measurements of the cosmic microwave
background (CMB) and galaxy surveys, a discrepancy that has come to be known as
the `Hubble tension'. This has motivated the exploration of extensions to the
standard cosmological model in which higher values of $H_0$ can be obtained
from CMB measurements and galaxy surveys. The trouble, however, goes beyond
$H_0$; such modifications affect other quantities, too. In particular, their
effects on cosmic times are usually neglected. We explore here the implications
that measurements of the age $t_{\rm U}$ of the Universe, such as a recent
inference from the age of the oldest globular clusters, can have for potential
solutions to the $H_0$ tension. The value of $H_0$ inferred from the CMB and
galaxy surveys is related to the sound horizon at CMB decoupling (or at
radiation drag), but it is also related to the matter density and to $t_{\rm
U}$. Given this observation, we show how model-independent measurements may
support or disfavor proposed new-physics solutions to the Hubble tension.
Finally, we argue that cosmological measurements today provide constraints
that, within a given cosmological model, represent an over-constrained system,
offering a powerful diagnostic tool of consistency. We propose the use of
ternary plots to simultaneously visualize independent constraints on key
quantities related to $H_0$ like $t_{\rm U}$, the sound horizon at radiation
drag, and the matter density parameter. We envision that this representation
will help find a solution to the trouble of and beyond $H_0$.
