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We review the status of the fourth-order (quartic in the spacetime curvature)terms induced by superstrings/M-theory (compactified on a warped torus) in theleading order with respect to the Regge slope parameter, and study their(non-perturbative) impact on the evolution of the Hubble scale in the contextof the four-dimensional FRW cosmology. After taking into account the quantumambiguities in the definition of the off-shell superstring effective action, wepropose the generalized Friedmann equations, find the existence of their (deSitter) exact inflationary solutions without a spacetime singularity, andconstrain the ambiguities by demanding stability and the scale factor dualityinvariance of our solutions. The most naive (Bel-Robinson tensor squared)quartic terms are ruled out, thus giving the evidence for the necessity ofextra quartic (Ricci tensor-dependent) terms in the off-shell gravitationaleffective action for superstrings. Our methods are generalizable to the higherorders in the spacetime curvature.

On the superstrings-induced four-dimensional gravity, and its applications to cosmology