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Lovelock-Brans-Dicke gravity

According to Lovelock's theorem, the Hilbert-Einstein and the Lovelock actions are indistinguishable from their field equations. However, they have different scalar-tensor counterparts, which correspond to the Brans-Dicke and the \emph{Lovelock-Brans-Dicke} (LBD) gravities, respectively. In this paper the LBD model of alternative gravity with the Lagrangian density $\mathscr{L}_{\text{LBD}}=\frac{1}{16π}\left[ϕ\left( R +\frac{a}{\sqrt{-g}}{}^*RR + b\mathcal{G}\right)-\frac{ω_{\text{L}}}ϕ\nabla_αϕ\nabla^αϕ\right]$ is developed, where ${}^*RR$ and $\mathcal{G}$ respectively denote the topological Chern-Pontryagin and Gauss-Bonnet invariants. The field equation, the kinematical and dynamical wave equations, and the constraint from energy-momentum conservation are all derived. It is shown that, the LBD gravity reduces to general relativity in the limit $ω_{\text{L}}\to\infty$ unless the "topological balance condition" holds, and in vacuum it can be conformally transformed into the dynamical Chern-Simons gravity and the generalized Gauss-Bonnet dark energy with Horndeski-like or Galileon-like kinetics. Moreover, the LBD gravity allows for the late-time cosmic acceleration without dark energy. Finally, the LBD gravity is generalized into the Lovelock-scalar-tensor gravity, and its equivalence to fourth-order modified gravities is established. It is also emphasized that the standard expressions for the contributions of generalized Gauss-Bonnet dependence can be further simplified.