a pleasure to thank Leonidas Christodoulou, Enrique Gaztañaga, Dragan Huterer, Eyal Kazin, Ofer Lahav and Shinji Tsujikawa for useful comments and conversations. We are grateful for the hospitality of the Centro de Sciencias de Benasque Pedro Pascual, where a part of this manuscript was prepared. The work of L.A. and I.S. is supported by the DFG through TRR33 “The Dark Universe”. M.K. acknowledges funding by the Swiss National Science Foundation. M.M. is supported by CNPq-Brazil. I.D.S. acknowledges STFC for financial support.

Details of Horndeski Properties {#sec:App}
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This appendix concerns the properties of the scalar-field theories described by the Horndeski Lagrangian given by the combination of terms presented in Eqs . The subscripts $,\phi$ and $,X$ denote derivation with respect to that variable. On a flat FRW background, the energy density and pressure are given by

$$\begin{aligned}
\rho_{x}= & 3H^{2}(1-w_{1})+2XK_{,X}-K-2XG_{3,\phi}+\\
 & +6\dot{\phi}H\left(XG_{3,X}-G_{4,\phi}-2XG_{4,\phi X}\right)+\nonumber \\
 & +12H^{2}\left(X\left(G_{4,X}+2XG_{4,XX}\right)-G_{5,\phi}-XG_{5,\phi X}\right)+\nonumber \\
 & +4\dot{\phi}XH^{3}\left(G_{5,X}+XG_{5,XX}\right)\,,\nonumber \\
P_{x}= & -\left(3H^{2}+2\dot{H}\right)(1-w_{1})+K-2XG_{3,\phi}+4XG_{4,\phi\phi}+\nonumber \\
 & +2\dot{\phi}Hw_{1,\phi}-4X^{2}H^{2}G_{5,\phi X}+2\dot{\phi}XH^{3}G_{5,X}+\frac{\ddot{\phi}}{\dot{\phi}}\left(w_{2}-2Hw_{1}\right)\,,\nonumber \end{aligned}$$

where, given a slight rearrangement of the results in Refs [@DeFelice:2011bh; @DeFelice:2011hq], we define four functions $w_i$ as

$$\begin{aligned}
w_{1}\equiv & 1+2\left(G_{4}-2XG_{4,X}+XG_{5,\phi}-\dot{\phi}XHG_{5,X}\right)\,,\label{eq:ws}\\
w_{2}\equiv & -2\dot{\phi}\left(XG_{3,X}-G_{4,\phi}-2XG_{4,\phi X}\right)+\nonumber \\
 & +2H\left(w_{1}-4X\left(G_{4,X}+2XG_{4,XX}-G_{5,\phi}-XG_{5,\phi X}\right)\right)-\nonumber \\
 & -2\dot{\phi}XH^{2}\left(3G_{5,X}+2XG_{5,XX}\right)\,,\nonumber \\
w_{3}\equiv & 3X\left(K_{,X}+2XK_{,XX}-2G_{3,\phi}-2XG_{3,\phi X}\right)+18\dot{\phi}XH\left(2G_{3,X}+XG_{3,XX}\right)-\nonumber \\
 & -18\dot{\phi}H\left(G_{4,\phi}+5XG_{4,\phi 