  We build a mechanism of gravitational symmetry breaking (GSB) of a global
U(1) symmetry based on the relaxation of the equivalence principle due to the
mass variation of pseudo Nambu-Goldstone dark matter (DM) particles. This GSB
process is described by the modified cosmological convergence mechanism of the
Abnormally Weighting Energy (AWE) Hypothesis previously introduced by the
authors. Several remarkable constraints from the Hubble diagram of far-away
supernovae are derived, notably on the explicit and gravitational symmetry
breaking energy scales of the model. We then briefly present some consequences
on neutrino masses when this mechanism is applied to the particular case of the
breaking of lepton number symmetry.
