This paper explains a new way to compute a wide class of cosmological correlators at tree level. Cosmological correlators are mathematical objects that describe how quantum fields in an expanding universe are related. For massive particles these correlators usually involve complicated multi-variable special functions. The authors show that this complexity hides a much simpler structure built from universal local pieces and elementary graph rules.

The main technical move is to use a spectral representation of the bulk-to-bulk propagator. A propagator is the function that describes how a particle moves between two points in spacetime, and the spectral representation rewrites that function as an integral over simpler components. With this representation the time integrals in any tree-level graph factorise into single-vertex objects the authors call vertex functions. Each vertex function depends only on the local data at one interaction point (its external energy, a parameter called twist, and the masses of legs attached there). The paper finds that these vertex functions are members of the Lauricella family of generalised hypergeometric functions — a multivariable generalisation of the familiar Gauss hypergeometric function.

Once the graphs are split into vertex functions, the remaining task is to glue them back together by evaluating the spectral integrals. The authors develop a “spectral gluing algorithm” that does this using simple graph combinatorics and the residue theorem from complex analysis. Which poles contribute to each integral is determined solely by combinatorial data encoded in the graph’s incidence matrix. A choice of a root vertex fixes the convergence region in kinematic space. The outcome are explicit multiple-series expressions that resum the dependence on internal energies away from soft limits and provide general solutions to the differential equations these correlators satisfy.