  An interesting test on the nature of the Universe is to measure the global
spatial curvature of the metric in a model independent way, at a level of
$|\Omega_k|<10^{-4}$, or, if possible, at the cosmic variance level of the
amplitude of the CMB fluctuations $|\Omega_k|\approx10^{-5}$. A limit of
$|\Omega_k|<10^{-4}$ would yield stringent tests on several models of
inflation. Further, improving the constraint by an order of magnitude would
help in reducing "model confusion" in standard parameter estimation. Moreover,
if the curvature is measured to be at the value of the amplitude of the CMB
fluctuations, it would offer a powerful test on the inflationary paradigm and
would indicate that our Universe must be significantly larger than the current
horizon. On the contrary, in the context of standard inflation, measuring a
value above CMB fluctuations will lead us to conclude that the Universe is not
much larger than the current observed horizon, this can also be interpreted as
the presence of large fluctuations outside the horizon. However, it has proven
difficult, so far, to find observables that can achieve such level of accuracy,
and, most of all, be model-independent. Here we propose a method that can in
principle achieve that, this is done by making minimal assumptions and using
distance probes that are cosmology-independent: gravitational waves, redshift
drift and cosmic chronometers. We discuss what kind of observations are needed
in principle to achieve the desired accuracy.
