Two dots on the surface of a balloon: that's all the reference points you have when trying to figure out how it inflates. That's roughly how cosmologists reconstruct the history of the universe's expansion. Only instead of dots, they use gravitational waves from mergers of ultra-dense objects: neutron stars and black holes. Such events are called "standard sirens"—by analogy with a sound whose loudness tells you exactly how far away the source is.

The authors of the new study took future catalogs from two giant detectors, the space-based LISA and the ground-based ET, and applied Gaussian processes to them. This mathematical tool doesn't impose a rigid shape on the expansion curve; instead, it lets the data speak for themselves. This approach helped reconstruct not only distances to distant galaxies but also key parameters: the Hubble parameter (expansion rate), the deceleration parameter, and the full "equation of state" of dark energy.

But the big surprise is that simply measuring distances isn't enough. If you only look at the "distance–redshift" plot, most competing cosmological models (including theories with interacting dark matter or early dark energy) merge into one indistinguishable band. To tell them apart, you need to take derivatives: that is, look at how fast the expansion rate changes. And that's where specific "transparency windows" appear: for ET, that's a redshift of about 1.7 (when the universe was roughly half its current age), and for LISA, about 2.7 (a third of its age). It's in these ranges that future observations will give the clearest answer as to which model is closer to the truth.

This approach is reminiscent of a doctor who doesn't just measure a patient's temperature but tracks its spikes at certain times. The authors have created a tool that precisely indicates at which "times" and for which symptoms to watch in order to diagnose the universe.