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.
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