Differential equations and recursive solutions for cosmological amplitudes
Abstract 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 o...
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| Format: | Article |
| Language: | English |
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2025-01-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP01(2025)001 |
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| author | Song He Xuhang Jiang Jiahao Liu Qinglin Yang Yao-Qi Zhang |
| author_facet | Song He Xuhang Jiang Jiahao Liu Qinglin Yang Yao-Qi Zhang |
| author_sort | Song He |
| collection | DOAJ |
| description | Abstract 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 basic time integrals, one for each 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 canonical differential equations of size 4 n−1 equivalent to the graphic rules derived recently , and we also derive the system of differential equations for loop integrands 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 e.g. for n-site chains and n-gon cases. |
| format | Article |
| id | doaj-art-8d8c35759aa3471e8f7d72f740bef406 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-8d8c35759aa3471e8f7d72f740bef4062025-01-19T12:07:34ZengSpringerOpenJournal of High Energy Physics1029-84792025-01-012025114310.1007/JHEP01(2025)001Differential equations and recursive solutions for cosmological amplitudesSong He0Xuhang Jiang1Jiahao Liu2Qinglin Yang3Yao-Qi Zhang4CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesAbstract 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 basic time integrals, one for each 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 canonical differential equations of size 4 n−1 equivalent to the graphic rules derived recently , and we also derive the system of differential equations for loop integrands 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 e.g. for n-site chains and n-gon cases.https://doi.org/10.1007/JHEP01(2025)001Cosmological modelsScattering Amplitudesde Sitter spaceEarly Universe Particle Physics |
| spellingShingle | Song He Xuhang Jiang Jiahao Liu Qinglin Yang Yao-Qi Zhang Differential equations and recursive solutions for cosmological amplitudes Journal of High Energy Physics Cosmological models Scattering Amplitudes de Sitter space Early Universe Particle Physics |
| title | Differential equations and recursive solutions for cosmological amplitudes |
| title_full | Differential equations and recursive solutions for cosmological amplitudes |
| title_fullStr | Differential equations and recursive solutions for cosmological amplitudes |
| title_full_unstemmed | Differential equations and recursive solutions for cosmological amplitudes |
| title_short | Differential equations and recursive solutions for cosmological amplitudes |
| title_sort | differential equations and recursive solutions for cosmological amplitudes |
| topic | Cosmological models Scattering Amplitudes de Sitter space Early Universe Particle Physics |
| url | https://doi.org/10.1007/JHEP01(2025)001 |
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