Gravitational Stabilization of Scalar Potentials *

Using the formalism of Boucher we derive the conditions under which gravity can stabilize otherwise metastable or unstable scalar field configurations. For the case of a double-welled potential we rederive the stability criteria first obtained by Coleman and DeLuccia. Our derivation does not require the assumption of 0(4) symmetry or the thin-walled approximation used in their work. We also give the conditions for which gravity can stabilize the symmetric points of general quartic potentials and of a Coleman-Weinberg potential. Under certain circumstances gravitational effects can stabilize a scalar field configuration which would be metastable or unstable in the absence of gravity [14]. Consider for example, a potential with two local minima, one at ~ = ~+ and one at ~ = ~_, with an energy splitting V(¢+)-V(~_) = e. Normally, the state ~ = ~+ would be metastable. However, Coleman and DeLuccia [ 1 ] have shown that for sufficiently small e if V(q~+) ~< 0, the tunnelling rate between the states ~b = q~+ and ~b = ~b_ vanishes when gravity is included in the calculation. More recently, Boucher [4] has provided a general formalism for analysing the stability of various field theories based on Witten's proof of the positive energy theorem [5] applied to the energy as it is defined in the fiat [6] or anti-de Sitter [7] space cases. Here, we apply this formalism to the situation considered by Coleman and DeLuccia rederiving their stability conditions [ 1,8]. Our approach allows this to be done without requiring the assumption of 0(4) invariance and without using the thin-walled approximation. It also gives additional information which constrains the general form of the potential. In addition we derive the conditions for which gravity can stabilize the symmetric points of general quartic potentials and of a Coleman-Weinberg potential [9]. Boucher's formalism is remarkably simple to describe and implement. For any potential V(~) with an extremum V'(~+) = 0, a positive energy theorem for the state ~b = ¢+ can be proven using the approach of Witten provided that there exists a real function f(~) satisfying f(~b+) = [-V(~.)/3t¢] 1/2, (f)2 _ ~Kf2 ~<~ V(~b), (1) where K = 87rG. Clearly the first equation above requires that V(q~+) < O. Because of the resulting positive energy theorem, the existence of such a function is sufficient to prove that the state ~ = ~+ is both classically and semi-classically stable. We consider first …