We review the status of the fourth-order (quartic in the spacetime curvature) terms induced by superstrings/M-theory (compactified on a warpedtorus) in the leading order with respect to the Regge slope parameter, andstudy their (nonperturbative) impact on the evolution of the Hubble scalein the context of the four-dimensional FRW cosmology. After taking intoaccount the quantum ambiguities in the definition of the off-shell superstringeffective action, we propose the generalized Friedmann equations, find theexistence of their (de Sitter) exact inflationary solutions without a spacetimesingularity, and constrain the ambiguities by demanding stability andthe scale factor duality invariance of our solutions. The most naive (Bel-Robinson tensor squared) quartic terms are ruled out, thus giving the evidencefor the necessity of extra quartic (Ricci tensor-dependent) terms inthe off-shell gravitational effective action for superstrings. Our methods aregeneralizable to the higher orders in the spacetime curvature.
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