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We consider scalar-Gauss-Bonnet and modified Gauss-Bonnet gravities andreconstruct these theories from the universe expansion history. In particular,we are able to construct versions of those theories (with and without ordinarymatter), in which the matter dominated era makes a transition to the cosmicacceleration epoch. It is remarkable that, in several of the cases underconsideration, matter dominance and the deceleration-acceleration transitionoccur in the presence of matter only. The late-time acceleration epoch isdescribed asymptotically by de Sitter space but may also correspond to an exact$\Lambda$CDM cosmology, having in both cases an effective equation of stateparameter $w$ close to -1. The one-loop effective action of modifiedGauss-Bonnet gravity on the de Sitter background is evaluated and it is used toderive stability criteria for the ensuing de Sitter universe.Comment: LaTeX20 pages, 4 figures, version to apear in PR

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy