Explicit curvature dependence of coupling constants

We consider a renormalizable quantum field theory, such as a grand unified theory, in a general curved spacetime. We use a new partially summed form of the proper-time or heat-kernel expansion of the propagator to obtain curvature-dependent logarithmic terms in the effective action. We demonstrate the connection between these terms and the renormalization group. The explicit form of the logarithmic curvature dependence permits us to obtain the modified gravitational field equations. The logarithmic terms are of importance for dynamical models of the early universe, including inflationary models. These terms are also of interest at the present time. We use them, together with the observed upper limit on the present value of the cosmological constant, to place an upper limit on the masses of the heaviest particles in renormalizable field theories. This upper limit is of the order of ${10}^{19}$ GeV for Higgs bosons in minimal SU(5) theory.