Scalar field theories in curved space

We investigate the behavior of scalar fields (φ) in curved space which have a potential V[φ]=$V_0$+(1/2)$m^2$ ${\phi }^2$+(1 /3!)η${\phi }^3$+(1/4!)λ${\phi }^4$ and a general coupling to gravity (1/2)ξR${\phi }^2$. The back-reaction of gravity strongly affects the stability of scalar fields. By examining the scalar field equations and the Einstein equations, we clarify conditions for the system to have an absolutely stable ground state in which φ is constant and a metric is either Minkowski, or de Sitter, or anti–de Sitter. We find that (i) cubic interactions cause instability, unless ξ=0, (ii) Higgs scalar fields in the standard model must have ξ<eq0 or ξ>eq(1/6), (iii) negative quartic interaction couplings (λ<0) can make sense, and (iv) a free scalar field with a tiny mass can reduce the bare large vacuum energy density $V_0$ to an extremely small value (∼$m^2$ $G^{\mathrm{-}1}$ ${\xi }^{\mathrm{-}1}$). Based on the last observation, the vanishing-cosmological-constant problem is viewed not as a problem of how to reduce the bare vacuum energy density, but as that of how to get a large gravitational constant (G≫‖$m^2$ $V_0$‖).