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 .