CMB spectrum in unified EFT of dark energy: scalar-tensor and vector-tensor theories

We study the cosmic microwave background (CMB) radiation in the unified description of the effective field theory (EFT) of dark energy that accommodates both scalar-tensor and vector-tensor theories. The boundaries of different classes of theories are universally parameterised by a new EFT parameter $\alpha_V$ characterising the vectorial nature of dark energy and a set of consistency relations associated with the global/local shift symmetry. After implementing the equations of motion in a Boltzmann code, as a demonstration, we compute the CMB power spectrum based on the $w$CDM background with the EFT parameterisation of perturbations and a concrete Horndeski/generalised Proca theory. We show that the vectorial nature generically prevents modifications of gravity in the CMB spectrum. On the other hand, while the shift symmetry is less significant in the perturbation equations unless the background is close to the $\Lambda$CDM, it requires that the effective equation of state of dark energy is in the phantom region $w_{\rm DE}<-1$. The latter is particularly interesting in light of the latest result of the DESI+CMB combination as the observational verification of $w_{\rm DE}>-1$ can rule out shift-symmetric theories including vector-tensor theories in one shot.