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We study the cosmological evolution of a scalar field in Horndeski gravity, assuming that the scalar field is subdominant with respect to the cosmic fluid. First, we analyze the most general shift-symmetric action that respects local Lorentz symmetry. We show that the bound on the speed of gravitational waves set by GW170817+GRB170817A imposes a constraint only on the linear coupling between the scalar and the Gauss-Bonnet invariant, and this constraint is rather mild. Then, we consider some interesting examples of theories that break shift symmetry, such as the Damour-Esposito-Far\`ese model of spontaneous scalarization and a theory with a quadratic coupling to the Gauss-Bonnet invariant that can lead to black hole scalarization. In both cases, tuning of cosmological initial conditions is necessary to keep the scalar field dormant during cosmic evolution.
Franchini et al. (Mon,) studied this question.
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