This follows from the fact that the coupling to matter displaces the dilaton field from the vicinity of the minimum of its potential, in the process increasing the Hubble rate during structure formation and therefore the friction term in the growth equation. This is a background effect which counterbalances the natural tendency of scalar-tensor models to increase the growth by the presence of an attractive scalar interaction between CDM particles, as observed in e.g.. This also could have interesting phenomenological consequences (though we here leave these for future work). We do not here try to match the hints for a time-dependent equation of state and its time drift described by the DESI collaboration, though we do observe that the dilaton-DM interactions these models have can easily appear to give a ‘phantom’ equation of state parameter, , if its dynamics are interpreted as being due to vanilla Dark Matter plus a single-field quintessence model.18 Trying to obtain the parameters and would require modifications to the dilaton potential without tampering with the axio-dilaton screening mechanism. If such a model could be built, some of the features of the model presented here such as the existence of early dark energy would remain. [...] If such a model could be built, some of the features of the model presented here such as the existence of early dark energy would remain. On the other hand, features such as a decrease in structure growth would depend on the dilaton potential which could for instance be taken of the thawing type instead of the frozen kind. Again, this is left for future work. The key constraint on the size of early dark energy fractions shown in Sect. 6 is not a physical constraint but calculational issue that limits us to only considering small axion-matter coupling potentials, . [...] Finally although the choices made for the functional forms of potentials and couplings are inspired by string constructions,19 we have not attempted to embed our model into a UV completion from first principles, such as from an extra-dimensional or string point of view. Of course this would be a step forward towards a better physical understanding of dark energy and its sensitivity to ultra-violet physics, and from this point of view its natural roots as the low-energy limit of models with supersymmetric large extra dimensions is suggestive, given the progress such models allow on understanding the UV side of the cosmological constant problem (see for a brief review). Much here is also left for future work. A The phase of the axion field Let us come back to the axion field and its variation in the presence of perturbations. At the background level we haveThis implies that at leading order we can writeup to an irrelevant constant .