  We provide a general analysis of the asymptotic behaviour of perturbative
contributions to observables in arbitrary power-law FRW cosmologies,
indistinctly the Bunch-Davies wavefunction and cosmological correlators. We
consider a large class of scalar toy models, including conformally-coupled and
massless scalars in arbitrary dimensions, that admits a first principle
definition in terms of (generalised/weighted) cosmological polytopes. The
perturbative contributions to an observable can be expressed as an integral of
the canonical function associated to such polytopes and to weighted graphs. We
show how the asymptotic behaviour of these integrals is governed by a special
class of nestohedra living in the graph-weight space, both at tree and loop
level. As the singularities of a cosmological process described by a graph can
be associated to its subgraphs, we provide a realisation of the nestohedra as a
sequential truncation of a top-dimensional simplex based on the underlying
graph. This allows us to determine all the possible directions -- both in the
infrared and in the ultraviolet --, where the integral can diverge as well as
their divergence degree. Both of them are associated to the facets of the
nestohedra, which are identified by overlapping tubings of the graph: the
specific tubing determines the divergent directions while the number of
overlapping tubings its degree of divergence. This combinatorial formulation
makes straightforward the application of sector decomposition for extracting
both leading and subleading divergences from the integral, as the sectors in
which the integration domain can be tiled are identified by the collection of
compatible facets of the nestohedra, with the latter that can be determined via
the graph tubings. Finally, the leading divergence can be interpreted as a
restriction of the canonical function of the relevant polytope onto a special
hyperplane.
