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Differential equations and recursive solutions for cosmological amplitudes
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Differential equations and recursive solutions for cosmological amplitudes
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Recently considerable efforts have been devoted to computing cosmological correlators and the corresponding wavefunction coefficients, as well as understanding their analytical structures. In this note, we revisit the computation of these ``cosmological amplitudes" associated with any tree or loop graph for conformal scalars with time-dependent interactions in the power-law FRW universe, directly in terms of iterated time integrals. We start by decomposing any such cosmological amplitude (for loop graph, the ``integrand" prior to loop integrations) as a linear combination of {\it basic time integrals}, one for each {\it directed graph}. We derive remarkably simple first-order differential equations involving such time integrals with edges ``contracted" one at a time, which can be solved recursively and the solution takes the form of Euler-Mellin integrals/generalized hypergeometric functions. By combining such equations, we then derive a complete system of differential equations for all time integrals needed for a given graph. Our method works for any graph: for a tree graph with $n$ nodes, this system can be transformed into the {\it canonical differential equations} of size $4^{n{-}1}$ quivalent to the graphic rules derived recently%so-called ``kinematic flow", and we also derive the system of differential equations for loop integrands {\it e.g.} of all-loop two-site graphs and one-loop $n$-gon graphs. Finally, we show how the differential equations truncate for the de Sitter (dS) case (in a way similar to differential equations for Feynman integrals truncate for integer dimensions), which immediately yields the complete symbol for the dS amplitude with interesting structures {\it e.g.} for $n$-site chains and $n$-gon cases.
Forward citations
Cited by 11 Pith papers
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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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A Graphical Coaction for FRW Integrals from Partial/Relative Twisted (Co)homology
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On-Shell Bootstrap of Loop Inflation Correlators with Spectral Dispersion
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.
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Cosmological Weight-Shifting Matrices
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Loop integrals in de Sitter spacetime: The parity-split IBP system and $\mathrm{d}\log$-form differential equations
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-l...
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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.
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From Cosmological Cuts to Yang--Mills Wavefunctions in de Sitter Space
Tree-level Yang-Mills de Sitter wavefunctions through six points are reconstructed from cosmological cuts into cut-detectable gluings plus a current-conservation completion, matching Feynman rules and suggesting an al...
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From Cosmological Cuts to Yang--Mills Wavefunctions in de Sitter Space
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de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs
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Kinematic Flow for Banana Loops and Unparticles
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.
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