  We investigate gravitational-wave backgrounds (GWBs) of primordial origin
that would manifest only at ultra-high frequencies, from kilohertz to 100
gigahertz, and leave no signal at either LIGO, Einstein Telescope, Cosmic
Explorer, LISA, or pulsar-timing arrays. We focus on GWBs produced by cosmic
strings and make predictions for the GW spectra scanning over high-energy scale
(beyond $10^{10}$ GeV) particle physics parameters. Signals from local string
networks can easily be as large as the Big Bang nucleosynthesis/cosmic
microwave background bounds, with a characteristic strain as high as $10^{-26}$
in the 10 kHz band, offering prospects to probe grand unification physics in
the $10^{14}-10^{17}$ GeV energy range. In comparison, GWB from axionic strings
is suppressed (with maximal characteristic strain $\sim 10^{-31}$) due to the
early matter era induced by the associated heavy axions. We estimate the needed
reach of hypothetical futuristic GW detectors to probe such GWB and, therefore,
the corresponding high-energy physics processes. Beyond the information of the
symmetry-breaking scale, the high-frequency spectrum encodes the microscopic
structure of the strings through the position of the UV cutoffs associated with
cusps and kinks, as well as potential information about friction forces on the
string. The IR slope, on the other hand, reflects the physics responsible for
the decay of the string network. We discuss possible strategies for
reconstructing the scalar potential, particularly the scalar self-coupling,
from the measurement of the UV cutoff of the GW spectrum.
