  Instantons, which are non-perturbative solutions to Yang-Mills equations,
provide a signal for the occurrence of quantum tunneling between distinct
classes of vacua. They can give rise to decays of particles otherwise
forbidden. Using data collected at the Pierre Auger Observatory, we search for
signatures of such instanton-induced processes that would be suggestive of
super-heavy particles decaying in the Galactic halo. These particles could have
been produced during the post-inflationary epoch and match the relic abundance
of dark matter inferred today. The non-observation of the signatures searched
for allows us to derive a bound on the reduced coupling constant of gauge
interactions in the dark sector: $\alpha_X \lesssim 0.09$, for $10^{9} \lesssim
M_X/{\rm GeV} < 10^{19}$. Conversely, we obtain that, for instance, a reduced
coupling constant $\alpha_X = 0.09$ excludes masses $M_X \gtrsim 3\times
10^{13}~$GeV. In the context of dark matter production from gravitational
interactions alone, we illustrate how these bounds are complementary to those
obtained on the Hubble rate at the end of inflation from the non-observation of
tensor modes in the cosmological microwave background.
