  Using the data of the Pierre Auger Observatory, we report on a search for
signatures that would be suggestive of super-heavy particles decaying in the
Galactic halo. From the lack of signal, we present upper limits for different
energy thresholds above ${\gtrsim}10^8$\,GeV on the secondary by-product fluxes
expected from the decay of the particles. Assuming that the energy density of
these super-heavy particles matches that of dark matter observed today, we
translate the upper bounds on the particle fluxes into tight constraints on the
couplings governing the decay process as a function of the particle mass.
Instantons, which are non-perturbative solutions to Yang-Mills equations, can
give rise to decay channels otherwise forbidden and transform stable particles
into meta-stable ones. Assuming such instanton-induced decay processes, we
derive a bound on the reduced coupling constant of gauge interactions in the
dark sector: $\alpha_X \lesssim 0.09$, for $10^{9} \lesssim M_X/\text{GeV} <
10^{19}$. 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 during
the reheating epoch, we derive constraints on the parameter space that
involves, in addition to $M_X$ and $\alpha_X$, the Hubble rate at the end of
inflation, the reheating efficiency, and the non-minimal coupling of the Higgs
with curvature.
