Towards a Solution of the Negative Mode Problem in Quantum Tunnelling with Gravity

In the absence of gravity, one can prove that tunnelling instantons exhibit exactly one negative mode in their spectrum of fluctuations. It is precisely the existence of this tunnelling negative mode that warrants an interpretation of these solutions as mediating the decay of a metastable vacuum. In the presence of gravity the situation is much more subtle, not least because of diffeomorphism invariance. New complications arise here: in particular, the kinetic term of the fluctuations can change sign somewhere along the instanton. We show that in this case the mode functions remain non-singular, and the tunnelling negative mode continues to exist. Moreover, the eigenvalues vary continuously when the potential is varied such that the kinetic term of the fluctuations switches sign. However, the negative kinetic term implies the additional existence of an infinite tower of negative modes, whose significance and interpretation remain elusive.