INFLATION FROM THE SUPERSTRING VACUUM
Quartic higher-derivative gravitational terms in the effective Lagrangian of the heterotic superstring theory renormalize the bare, four-dimensional gravitational coupling [Formula: see text], due to the reduction process [Formula: see text], according to the formula [Formula: see text], where A r and B r are the moduli for the physical space gij(xk) and internal space [Formula: see text], respectively. The Euler characteristic [Formula: see text] is negative for a three-generation Calabi–Yau manifold, and therefore both the additional terms, of tree-level and one-loop origin, produce a decrease in κ-2, which changes sign when κ-2 = 0. The corresponding tree-level critical point is [Formula: see text], if we set [Formula: see text] and λ = 15π2, for compactification onto a torus. Values [Formula: see text] yield the anti-gravity region κ-2 < 0, which is analytically accessible from the normal gravity region κ-2 > 0. The only non-singular, vacuum minimum of the potential [Formula: see text] is located at the point [Formula: see text], where [Formula: see text], the quadratic trace anomaly [Formula: see text] dominates over [Formula: see text], and a phase of de Sitter expansion may occur, as first envisaged by Starobinsky, in approximate agreement with the constraint due to the effect of gravitational waves upon the anisotropy of the cosmic microwave background radiation. There is no non-singular minimum of the potential [Formula: see text].