We propose a viable dark energy scenario in the presence of cubic Horndeski interactions and a standard scalar-field kinetic term with two exponential potentials. We show the existence of new scaling solutions along which the cubic coupling G3 provides an important contribution to the field density that scales in the same way as the background fluid density. The solutions finally exit to the epoch of cosmic acceleration driven by a scalar-field dominated fixed point arising from the second exponential potential. We clarify the viable parameter space in which all the theoretically consistent conditions including those for the absence of ghost and Laplacian instabilities are satisfied on scaling and scalar-field dominated critical points. In comparison to quintessence with the same scalar potential, we find that the cubic coupling gives rise to some novel features: (i) the allowed model parameter space is wider in that a steeper potential can drive the cosmic acceleration; (ii) the dark energy equation of state wφ today can be closer to -1 relative to quintessence; (iii) even if the density associated with the cubic coupling dominates over the standard field density in the scaling era, the former contribution tends to be suppressed at low redshifts. We also compute quantities associated with the growth of matter perturbations and weak lensing potentials under the quasi-static approximation in the subhorizon limit and show that the cubic coupling leads to the modified evolution of perturbations which can be distinguished from quintessence.
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