A Laplace-space representation converts massive single-exchange cosmological correlators in de Sitter into a rapidly convergent series derived from flat-space integrals.
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Differential Equations for Cosmological Correlators
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abstract
Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and "twist factors" that depend on the cosmological evolution. These integrals are part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict -- by hand -- the equations for all tree graphs. While the rules of this "kinematic flow" are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in ${\rm tr}\,\phi^3$ theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.
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Derives integro-differential boundary equations from bulk locality for scale-breaking cosmological correlators with oscillating heavy-field masses and solves them analytically and numerically to reveal enhanced collider signals.
The double-exchange cosmological correlator is computed in KLF space, yielding a double series over hypergeometric functions that improves on prior four-layer representations.
Laplace transform converts cosmological correlator diagrams into flat-space integrals against kernels, yielding a closed-form rapidly convergent series for the massive single-exchange case valid across the full kinematic domain.
Constructs a graphical coaction for all-loop FRW integrals in conformally-coupled scalar theories via twisted (co)homology, with combinatorial description of kinematic flow and a public web app for computation.
Introduces spectral dispersion bootstrap combining dS spectral decomposition and dispersion relations to compute 3- and 4-point loop correlators with massive scalar and vector exchanges.
Introduces weight-shifting matrices for de Sitter diagrams, generalized with Kronecker products to arbitrary tree-level graphs, to derive massless wavefunction coefficients from conformally coupled seeds.
Four-point wavefunction coefficients for external conformally coupled scalars exchanging a particle of generic mass and spin are expressed in closed form as hypergeometric functions of Mandelstam invariants times Legendre polynomials in the cosmological Grassmannian.
De Sitter correlators in conformally coupled φ³ theory admit a time-integral representation built from flat-space correlators, revealing intrinsic simplifications including vanishing of odd conjugate-momentum graphs and a smaller symbol alphabet than the corresponding wavefunction coefficients.
A parity-split IBP system for n-propagator families in de Sitter space is identified, along with a conjecture that dlog-form differential equations extend to dS integrands with Hankel functions, verified for the one-loop bubble.
A graph-tubing combinatorial framework governs the first-order differential equations obeyed by master integrals for massive cosmological correlators in de Sitter space.
Computes inflationary bispectra and trispectra from tree-level unparticle exchanges using Mellin-Barnes methods and symmetry-based differential equations, revealing that full shapes are needed to distinguish unparticles from light particles.
The n-site chain graph contribution to the de Sitter cosmological wavefunction in conformally coupled φ³ theory is expressed explicitly in terms of Rudenko's quadrangular polylogarithms.
Banana loop cosmological correlators are captured by master integrals from tubings of marked graphs, with connection matrices derived from activation, merger, swap, and copy rules unique to unparticle exchanges.
Equal-time correlators are simpler than wavefunctions because they come from full-spacetime integrals; this implies fewer poles, cleaner factorization, and a systematic pole expansion whose first subleading term vanishes.
The all-loop two-site cosmological wavefunction coefficient admits an equivalent maximal-chain expansion on the Boolean lattice that unifies the shifted-tree decomposition and the tubing construction via finite-difference operators and cubical integrals.
In dynamical Chern-Simons inflation the parity-odd trispectrum is a double copy of the mixed bispectrum and parity-odd power spectrum via a prior factorization formula.
Reversing the direction of tubing evolution yields splitting rules that reproduce the kinematic flow differential equations at tree level and suggest time emerges from kinematic space in conformally coupled scalar models and tr phi^3 theory.
Review of so(1,D) representations for de Sitter space across all D, covering mixed symmetry and fermions, connected to propagating fields.
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Differential Equations for Massive Correlators
A graph-tubing combinatorial framework governs the first-order differential equations obeyed by master integrals for massive cosmological correlators in de Sitter space.