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The way we organise perturbation theory is of fundamental importance both forcomputing the observables of relevance and for extracting fundamental physicsout of them. If on one hand the different ways in which the perturbativeobservables can be written make manifest different features (e.g. symmetries aswell as principles such as unitarity, causality and locality), on the otherhand precisely demanding that some concrete features are manifest lead todifferent ways of organising perturbation theory. In the context of flat-spacescattering amplitudes, a number of them are already known and exploited, whilemuch less is known for cosmological observables. In the present work, we showhow to systematically write down both the wavefunction of the universe and theflat-space scattering amplitudes, in such a way that they manifestly showphysical poles only. We make use of the invariant definition of suchobservables in terms of {\it cosmological polytopes} and their {\it scatteringfacet}. In particular, we show that such representations correspond totriangulations of such objects through hyperplanes identified by theintersection of their facets outside of them. All possible triangulations ofthis type generate the different representations. This allows us to provide ageneral proof for the conjectured all-loop causal representation of scatteringamplitudes. Importantly, all such representations can be viewed as makingexplicit a subset of compatible singularities, and our construction provides away to extend Steinmann relations to higher codimension singularities for boththe flat-space scattering amplitudes and the cosmological wavefunction.

Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe