  Unstable heavy particles well above the TeV scale are unaccessible
experimentally. So far, Big-Bang Nucleosynthesis (BBN) provides the strongest
limits on their mass and lifetime, the latter being shorter than 0.1 second. We
show how these constraints could be potentially tremendously improved by the
next generation of Gravitational-Wave (GW) interferometers, extending to
lifetimes as short as $10^{-16}$ second. The key point is that these particles
may have dominated the energy density of the universe and have triggered a
period of matter domination at early times, until their decay before BBN. The
resulting modified cosmological history compared to the usually-assumed single
radiation era would imprint observable signatures in stochastic
gravitational-wave backgrounds of primordial origin. In particular, we show how
the detection of the GW spectrum produced by long-lasting sources such as
cosmic strings would provide a unique probe of particle physics parameters.
When applied to specific particle production mechanisms in the early universe,
these GW spectra could be used to derive new constraints on many UV extensions
of the Standard Model. We illustrate this on a few examples, such as
supersymmetric models where the mass scale of scalar moduli and gravitino can
be constrained up to $10^{10}$ GeV. Further bounds can be obtained on the
reheating temperature of models with only-gravitationally-interacting particles
as well as on the kinetic mixing of heavy dark photons at the level of
$10^{-18}$.
