Differential equations for cosmological correlators
-
1.
Institute for Advanced Study
-
2.
National Taiwan University
-
3.
University of Amsterdam
-
4.
California Institute of Technology
-
5.
Princeton University
-
6.
University of Chicago
-
7.
Scuola Normale Superiore di Pisa
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. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also 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 tr ϕ3 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.
Copyright and License
© The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3.
Acknowledgement
We are grateful to Ana Achucarro, Paolo Benincasa, Jan de Boer, Alessandra Caraceni, J. J. Carrasco, Xingang Chen, Claude Duhr, Alex Edison, Carolina Figueiredo, Dan Green, Thomas Grimm, Song He, Johannes Henn, Arno Hoefnagels, Yu-tin Huang, Michael Jones, Manki Kim, Barak Kol, Chia-Kai Kuo, Daniel Longenecker, Manuel Loparco, Scott Melville, Sebastian Mizera, Matteo Parisi, Julio Parra-Martinez, Nic Pavao, Andrzej Pokraka, Oliver Schlotterer, Leonardo Senatore, Chia-Hsien Shen, Melissa Sherman-Bennett, John Stout, Bernd Sturmfels, Simon Telen, Jaroslav Trnka, Kamran Salehi Vaziri, Dong-Gang Wang and Alexander Zhiboedov for insightful discussions. Thanks to Harry Goodhew and Kamran Salehi Vaziri for corrections to a previous version of the manuscript.
This project was initiated at the online workshop “Cosmological Correlators” [101], and we thank all participants for providing an exciting atmosphere during the workshop. GLP presented some of these results at various conferences and workshops — QCD meets Gravity 2021, Bethe Forum “Inflation,” DITP Holography Workshop at KU Leuven, Holography Workshop at Trinity College Dublin, Amplitudes 2022, MPI Group Retreat, Iberian Strings, “DarkCosmoGrav” in Pisa, “Scattering Amplitudes and Cosmology” at ICTP, “Quantum de Sitter Universe” at DAMTP, “Recent Trends in and out of the Swampland” at IBS, “Gravitational Waves in the Southern Hemisphere” at ICTP-SAIFR — and he is grateful for many interesting discussions and feedback from the participants. DB and AJ thank the participants of the Simons Symposium “Amplitudes Meet Cosmology” [102] for helpful discussions. GLP thanks the Universities of Leiden and Amsterdam for their hospitality and support. GLP also thanks the Weizmann Institute of Science — in particular, Ofer Aharony and Kfir Blum — for hosting an extended visit, and all the discussions with the groups at Weizmann, Hebrew University, Technion, and Tel Aviv University. NAH and DB thank the participants of the workshop “Current Topics in Fundamental Physics” (organized by Johannes Henn) for interesting discussions.
Funding
NAH is supported by the US Department of Energy (DOE) under contract DE–SC0009988. DB is supported by a Yushan Professorship at National Taiwan University funded by the Ministry of Education (MOE) NTU-112V2004-1. AH is supported by DOE (HEP) Award DE-SC0011632 and by the Walter Burke Institute for Theoretical Physics. AJ is supported in part by DOE (HEP) Award DE-SC0009924. HL is supported by the Kavli Institute for Cosmological Physics at the University of Chicago through an endowment from the Kavli Foundation and its founder Fred Kavli. GLP is supported by Scuola Normale, by a Rita-Levi Montalcini Fellowship from the Italian Ministry of Universities and Research (MUR), and by INFN (IS GSS-Pi).
The research of NAH and DB is funded by the European Union (ERC, UNIVERSE PLUS, 101118787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Files
| Name | Size | Download all |
|---|---|---|
|
md5:0e5e08e1dccd4b26d0f37e86effaa454
|
1.8 MB | Preview Download |
Additional details
- United States Department of Energy
- DE-SC0009988
- National Taiwan University
- Ministry of Education
- NTU-112V2004-1
- United States Department of Energy
- DE-SC0011632
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics -
- United States Department of Energy
- DE-SC0009924
- University of Chicago
- The Kavli Foundation
- Ministero dell'università e della ricerca
- Istituto Nazionale di Fisica Nucleare
- European Union
- 101118787
- SCOAP3
- Accepted
-
2025-07-27
- Caltech groups
- Walter Burke Institute for Theoretical Physics, Division of Physics, Mathematics and Astronomy (PMA)
- Publication Status
- Published