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A higher order theory of dilaton gravity is motivated and presented. Although its Lagrangian density involves higher powers of the Riemann tensor and of the first two derivatives of the dilaton, the resulting equations of motion are quasi-linear in the second derivatives of both the metric tensor and of the dilaton. The higher order dilaton gravity with corrections up to arbitrary order in field derivatives is investigated in a co-dimension 1 brane scenario with general brane localized interactions. The derivation of the effective gravitational equations of motion at the brane is addressed in the covariant approach, with the results presented in detail for the first order dilaton gravity. The crucial influence of the bulk Z2 symmetry (with its fixed point at the brane position) on the form and the number of effective brane equations is shown and discussed. The generic influence of the bulk gravity solution on the effective Einstein-like equation at the brane is demonstrated. The resulting modified Friedmann equations at the brane are derived.

Higher Order Dilaton Gravity: Effective Equations at the Brane