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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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Cited by 13 Pith papers
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Higher-Spin Correlators in dS
The five-point de Sitter higher-spin correlator is shown to be a spurious-singularity-free rational function organized by graph-theoretic orbits of the complete graph K5.
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On Cosmological Correlators at One Loop
The one-loop triangle in-in correlator of massless scalars in flat space is evaluated in closed form in terms of dilogarithms, with Landau analysis revealing its physical singularities and a partial-energy factorisati...
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Anatomy of Family Trees in Cosmological Correlators
A complete map of singularities and local series expansions for the hypergeometric building blocks of tree-level cosmological correlators.
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A physical basis for cosmological correlators from cuts
The physical subspace of FRW wavefunction integrals is characterized by shared cuts/residues, and the authors supply cut-tubing rules that enumerate its basis graphically.
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All Tree-Level Massive Cosmological Correlators via Spectral Gluing
Tree-level massive de Sitter correlators are constructed by gluing Lauricella-type vertex functions according to graph combinatorics, and the hypergeometric content collapses to rational functions once the dynamical p...
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Cosmological Correlators in KLF and the Double-Exchange
The double-exchange cosmological correlator is computed in KLF space, yielding a double series over hypergeometric functions that improves on prior four-layer representations.
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Bootstrapping the Cosmological Collider with Resonant Features
Oscillatory couplings with frequency above the heavy field mass remove the Boltzmann suppression of cosmological collider signals and produce new scale-dependent bispectrum shapes, with axion monodromy as a concrete r...
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Geometry of Kinematic Flow
The wavefunction coefficients of conformally coupled scalars in power-law cosmologies obey differential equations whose basis functions can be arranged on hypercubes and zonotopes, with a single merger rule generating...
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Massive Inflationary Amplitudes: Differential Equations and Complete Solutions for General Trees
Every tree-level massive inflation correlator in the signal region can be written as one massive family tree series plus its cuts, with 2I summation variables.
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Euler Discriminant of Complements of Hyperplanes
The Euler discriminant of families of hyperplane complements is the zero set of an explicit product of determinants indexed by connected square subgraphs.
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Correlators are simpler than wavefunctions
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.
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A non-perturbative construction of the de Sitter late-time boundary
The paper derives an inversion formula that defines de Sitter boundary operators as integrals of bulk fields against the bulk-to-boundary propagator, reproducing known two-point functions and perturbation theory.
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Cosmological Correlators at the Loop Level
Using the partial Mellin-Barnes method, the author derives analytic signals from one-loop bubble diagrams in inflation correlators, including the first analytic results for a de Sitter boost-breaking bubble.
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