A Shift-Symmetric Goldstone Model of Early Dark Energy in Unimodular Gravity: Derivation Chain, Full-CMB Likelihood Test, and Structural Limits

AbstractWe construct an early dark energy (EDE) model in which a pseudo-Goldstone boson of aspontaneously broken global U(1) drives a transient pre-recombination expansion episode,and we embed the construction in unimodular gravity. The model is organised aroundthree symmetries: (i) the Goldstone shift symmetry, which forbids a direct f(θ)R couplingand therefore protects Newton's constant; (ii) the unimodular constraint, under whichvacuum energy does not gravitate and Λ enters as an integration constant; and (iii)diffeomorphism invariance, whose Bianchi identity, combined with (ii), rendersinter-sector energy exchange admissible without an ad hoc source term. From thisstructure we derive, rather than fit, several quantities that are ordinarily free parametersin axion-EDE constructions: the initial misalignment angle θ i , the fraction of dark matterparticipating in the transfer, the potential index n = 3, the multipole at which the EDEimprint is resonant, and a relation f/MPl = 1/√N between the Goldstone decay constant anda light-species count. We implement the model in CLASS_EDE and evaluate it against thePlanck 2018 plik_lite TT likelihood. At H0 = 72.0 km s-1 Mpc-1 the model reproduces Ωm =0.304, 100θs = 1.04189, sub-percent BAO distance ratios, S8 = 0.816 and a cosmic age of13.0 Gyr, so that every background observable is matched; the CMB temperaturelikelihood is nonetheless degraded relative to ΛCDM by Δχ2TT ≈ +100, against a gain ofΔχ2SH0ES ≈ −29. ΛCDM therefore remains preferred. We report this as a negative result.We state explicitly which elements of the construction are new and which are not: thesymmetry combination overlaps substantially with recent literature, the exponentialdark-matter coupling is that of the Early Dark Sector model, and the derived decayconstant f = 0.126 MPl exceeds the axion weak-gravity bound by more than an order ofmagnitude. The contribution we claim is correspondingly narrow, and we state it as amethodological message rather than a model proposal: when a construction of this class isconfronted with an uncompressed CMB temperature likelihood rather than withcompressed distance priors, the improvement in the local distance-ladder fit is outweighedseveral times over by the degradation of the temperature spectrum. Alongside this werecord an explicit chain of relations that fixes several quantities usually left free, togetherwith the two points at which that chain fails: the decay constant it selects is incompatiblewith the axion weak-gravity bound, and the improvement in S8 requires a component thatthe action does not supply