Cosmological Polytopes and the Wavefuncton of the Universe for Light States

TL;DR

The work investigates the late-time wavefunction of the universe for scalars with time-dependent masses and polynomial couplings in FRW cosmologies. It introduces a universal integrand for each Feynman diagram and derives recursion relations that map massive internal states to a massless conformally coupled seed via differential operators, framing these structures as a degenerate limit of an extended cosmological-polytope canonical form. The authors show that flat-space amplitudes arise as higher-codimension faces in this generalized polytope, with pole coefficients encoded by residues that can be expressed through flat-space processes; they further extend the polytope picture to $l=1$ states and perturbative mass insertions, offering a unified geometric framework and suggesting routes toward resummation and extension to spin. This work bridges the wavefunction of the universe with positive-geometry techniques, providing concrete recursion, polytope generalizations, and illustrative examples that illuminate how cosmological singularities relate to flat-space scattering data.

Abstract

We extend the investigation of the structure of the late-time wavefunction of the universe to a class of toy models of scalars with time-dependent masses and polynomial couplings, which contains general massive scalars in FRW cosmologies. We associate a universal integrand to each Feynman diagram contributing to the wavefunction of the universe. For certain (light) masses, such an integrand satisfies recursion relations involving differential operators, connecting states with different masses and having, as a seed, the massless scalar (which describes a conformally coupled scalar as a special case). We show that it is a degenerate limit of the canonical form of a generalisation of the cosmological polytopes describing the wavefunction for massless scalars. Intriguingly, the flat-space scattering amplitude appears as a higher codimension face: it is encoding the leading term in the Laurent expansion as the total energy is taken to zero, with the codimension of the face providing the order of the total energy pole. The same connection between the other faces and the Laurent expansion coefficients holds for the other singularities of the wavefunction of the universe, all of them connectable to flat-space processes. As the degenerate limit is taken, some of the singularities of the canonical form of the polytope collapse onto each other generating higher order poles. Finally, we consider the mass as a perturbative coupling, showing that the contribution to the wavefunction coming from graphs with mass two-point couplings can be identified with a degenerate limit of the canonical form of the cosmological polytope, if the perturbative expansion is done around the conformally coupled state; or as double degenerate limit of the canonical form of the extension of the cosmological polytopes introduced in the present paper, if the perturbative expansion is done around minimally coupled states.

Figures (9)

• Figure 1: Example of a Feynman graph contribution to the wavefunction of the universe (left) and its associated reduced graph (right), which is obtained from the former by suppressing the external lines.
• Figure 2: Cosmological polytopes constructed from the space of $n_e\,=\,2$ triangles. The (red) blue sides of the triangles (first line in the picture) are (non)-intersectable. Out of these two triangles, there are two ways of constructing more complicated objects: they can be intersected on the midpoint of one of the two (blue) intersectable sides ($\mathbf{x}'_i\,=\,\mathbf{x}'_j$), or in both ($\mathbf{x}_i\,=\,\mathbf{x}_j$; $\mathbf{x}'_i\,=\,\mathbf{x}'_j$), originating the polytopes on the bottom left and right respectively.
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