From theoretical physicist M. B. Paranjape at Physics Today:
Another frequent concern expressed over the existence of negative mass is that it would cause an untenable instability of the universe. Stephen Hawking once told me that if negative mass existed, “the universe would be unstable and we would not be here to this day.” But negative mass exists only in an expanding universe, and because of energy conservation it can only be produced in positive–negative mass pairs. If there is a backreaction of the production of these pairs on the background cosmological energy, the production of negative mass should drive that energy density to zero, thus terminating the possibility of its production and quenching any instability. This mechanism could offer a means of resolving the long-standing problem of why the cosmological constant is so small. In the realm of speculation, the possibility of creating negative-mass bubbles in the laboratory could have incredible applications for energy production, warp-drive transportation, and armaments.
Rather than dismissing the idea of negative mass, researchers should try to use it to their advantage. Avenues for further study include looking for dynamical models of matter that would give rise to stable negative-mass configurations and exploring the consequences of negative mass in the inflationary phase of the early universe. A plasma of positive- and negative-mass particles during the inflationary epoch would have an important influence on the propagation of gravitational waves, an effect that might be observable in the cosmic microwave background. More.
Something to think about if you can’t sleep.