Classical conformal invariance and superhorizon dynamics in de Sitter
Soft de Sitter Effective Theory is a well-motivated candidate for the correct effective late-time description of equal-time correlation functions in de Sitter space. In this work, we study its application to theories that enjoy classical conformal invariance, using the conformally-coupled $\phi^4$-theory as a toy model. While quantum effects generate non-trivial late-time dynamics in such models, we argue that it is not described by the standard construction of the effective theory as discussed thus far in the literature. We show that the tree-level matching of the trispectrum onto the effective theory does not fit into the expected power-counting scheme, and we contrast it with the matching of the tree-level bispectrum in the conformally-coupled $\phi^3$-theory, where it works consistently. We then propose a prescription to identify the leading superhorizon degrees of freedom in such theories, which should serve as the starting point for the construction of their late-time effective description. The interpretation of logarithms of the form $\log(-k\eta)$ in this context is briefly discussed.