Quantum gravity effects on dark matter and gravitational waves

We explore how quantum gravity effects, manifested through the breaking of discrete symmetry responsible for both dark matter and domain walls, can have observational effects through cosmic microwave background observations and gravitational waves. To illustrate the idea we consider a simple model with two scalar fields and two $Z_2$ symmetries, one being responsible for dark matter stability, and the other spontaneously broken and responsible for domain walls, where both symmetries are assumed to be explicitly broken by quantum gravity effects. We show the recent gravitational wave spectrum observed by several pulsar timing array projects can help constrain such effects.