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
$\mathscr{L}_{\text{LBD}}=\frac{1}{16\pi}\left[\phi\left( R
+\frac{a}{\sqrt{-g}}{}^*RR +
b\mathcal{G}\right)-\frac{\omega_{\text{L}}}{\phi}\nabla_\alpha \phi
\nabla^\alpha\phi \right]$ 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
$\omega_{\text{L}}\to\infty$ 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.
