We investigate cosmological evolution in the scalar–tensor theory with the field derivative coupling to the double-dual of the Riemann tensor (the cubic-type theory). The theory can be seen as the straightforward extension of the scalar–tensor with the quadratic order field derivative coupling to the Einstein tensor (the quadratic-type theory). Both the field derivative couplings to the Einstein tensor and the double-dual of the Riemann tensor have been argued in terms of the successful realization of the self-tuning of the cosmological constant within the Horndeski theory. Assuming the constant potential given by the sum of the cosmological constant and the quantum vacuum energy, the shift symmetry for the scalar field and no matter fields, in the spatially-flat Friedmann–Lemaitre–Robertson–Walker spacetime, we can reduce the set of the field equations to the first-order ordinary nonlinear differential equation for the Hubble parameter, showing the existence of the self-tuned and runaway de Sitter solutions, in addition to the standard de Sitter solutions in general relativity and the finite Hubble singularities which can be reached within the finite time. We then argue the possible cosmological evolution in terms of the values of the effective cosmological constant, the kinetic coupling constants and the initial Hubble parameter. Although the behavior of the universe around each of the de Sitter solutions as well as the finite time singularities is very similar in both theories, we find that the crucial difference appears in terms of no bounce or turnaround behavior across the vanishing Hubble parameter as well as no limitation for the range of the Hubble parameter in the cubic-type theory.

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