The frozen vanilla model: Exploring dark sector interactions with delta effective theories

In this paper, we introduce a new interacting mechanism within the dark sector, encompassing both dark energy and dark matter, while grounding our analysis in the familiar framework of the $\mathrm{\Lambda CDM}$ model augmented by baryons and radiation components, including photons and neutrinos. The interaction between dark energy and dark matter is confined to the perturbative level. One significant advantage of this proposal is that all geometric probes yield constraints consistent with the so-called vanilla model or the extended vanilla model, where dark energy has a constant equation of state, $w_{x}$. However, the introduction of this new interacting mechanism affects several theoretical signatures, involving contrast dark matter and dark energy densities. We perform an exploratory analysis of those effects in the CMB power spectra, matter-power spectra, and redshift space distortions. For instance, it allows for a decrease/increment in the integrated Sachs-Wolfe (ISW) effect depending on the value taken by the interaction coupling. This effect could be observationally detected by looking for a cross-correlation between the ISW temperature fluctuations and the distribution of galaxies or quasars. At late times, the interaction in the dark sector becomes very effective, affecting the non-linear scale of structure formation. We discuss how the estimators $f\sigma_{8}(z)$ and $S_{8}(z)$ are affected by different interacting couplings; indicating that $f\sigma_{8}(z)$ can show a relative change of up to $15\%$ compared to the concordance model at low redshifts. Finally, we show how the various terms in the dark energy pressure perturbation (both adiabatic and non-adiabatic) are relevant for different scales, demonstrating the absence of large-scale instabilities.