Abstract
Primordial perturbations in our universe are believed to have a quantum origin, and can be described by the wavefunction of the universe (or equivalently, cosmological correlators). It follows that these observables must carry the imprint of the founding principle of quantum mechanics: unitary time evolution. Indeed, it was recently discovered that unitarity implies an infinite set of relations among tree-level wavefunction coefficients, dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a systematic set of “Cosmological Cutting Rules” which constrain wavefunction coefficients for any number of fields and to any loop order. These rules fix the discontinuity of an n-loop diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in terms of tree-level diagrams with fewer external fields. Our results apply with remarkable generality, namely for arbitrary interactions of fields of any mass and any spin with a Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application, we show how one-loop corrections in the Effective Field Theory of inflation are fixed by tree-level calculations and discuss related perturbative unitarity bounds. These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013) [DOI].
R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Diagrammar, NATO Sci. Ser. B 4 (1974) 177 [INSPIRE].
M.J.G. Veltman, Diagrammatica: The Path to Feynman rules, vol. 4, Cambridge University Press (2012) [DOI].
H. Goodhew, S. Jazayeri, M.H. Gordon Lee and E. Pajer, Cutting Cosmological Correlators, arXiv:2104.06587 [INSPIRE].
H. Goodhew, S. Jazayeri and E. Pajer, The Cosmological Optical Theorem, JCAP 04 (2021) 021 [arXiv:2009.02898] [INSPIRE].
S. Céspedes, A.-C. Davis and S. Melville, On the time evolution of cosmological correlators, JHEP 02 (2021) 012 [arXiv:2009.07874] [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].
D. Meltzer and A. Sivaramakrishnan, CFT unitarity and the AdS Cutkosky rules, JHEP 11 (2020) 073 [arXiv:2008.11730] [INSPIRE].
M.J.G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186 [INSPIRE].
P. Benincasa, New structures in scattering amplitudes: a review, Int. J. Mod. Phys. A 29 (2014) 1430005 [arXiv:1312.5583] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
C. Cheung, TASI Lectures on Scattering Amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics (TASI 2016), Boulder, CO, U.S.A., June 6–July 1, 2016, R. Essig and I. Low eds. (2018) [DOI] [arXiv:1708.03872] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
A. Hillman, Symbol Recursion for the dS Wave Function, arXiv:1912.09450 [INSPIRE].
N. Arkani-Hamed and P. Benincasa, On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes, arXiv:1811.01125 [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].
P. Benincasa, From the flat-space S-matrix to the Wavefunction of the Universe, arXiv:1811.02515 [INSPIRE].
P. Benincasa, Cosmological Polytopes and the Wavefuncton of the Universe for Light States, arXiv:1909.02517 [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
P. Benincasa and F. Cachazo, Consistency Conditions on the S-matrix of Massless Particles, arXiv:0705.4305 [INSPIRE].
D.A. McGady and L. Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev. D 90 (2014) 084048 [arXiv:1311.2938] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, Scattering Amplitudes For All Masses and Spins, arXiv:1709.04891 [INSPIRE].
E. Pajer, D. Stefanyszyn and J. Supeł, The Boostless Bootstrap: Amplitudes without Lorentz boosts, JHEP 12 (2020) 198 [arXiv:2007.00027] [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
P. Creminelli, Conformal invariance of scalar perturbations in inflation, Phys. Rev. D 85 (2012) 041302 [arXiv:1108.0874] [INSPIRE].
A. Kehagias and A. Riotto, Operator Product Expansion of Inflationary Correlators and Conformal Symmetry of de Sitter, Nucl. Phys. B 864 (2012) 492 [arXiv:1205.1523] [INSPIRE].
I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP 07 (2013) 015 [arXiv:1211.5482] [INSPIRE].
A. Ghosh, N. Kundu, S. Raju and S.P. Trivedi, Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation, JHEP 07 (2014) 011 [arXiv:1401.1426] [INSPIRE].
N. Kundu, A. Shukla and S.P. Trivedi, Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation, JHEP 04 (2015) 061 [arXiv:1410.2606] [INSPIRE].
N. Kundu, A. Shukla and S.P. Trivedi, Ward Identities for Scale and Special Conformal Transformations in Inflation, JHEP 01 (2016) 046 [arXiv:1507.06017] [INSPIRE].
E. Pajer, G.L. Pimentel and J.V.S. Van Wijck, The Conformal Limit of Inflation in the Era of CMB Polarimetry, JCAP 06 (2017) 009 [arXiv:1609.06993] [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The cosmological bootstrap: weight-shifting operators and scalar seeds, JHEP 12 (2020) 204 [arXiv:1910.14051] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization, arXiv:2005.04234 [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
C. Sleight and M. Taronna, From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing, arXiv:2007.09993 [INSPIRE].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
M. Baumgart and R. Sundrum, de Sitter Diagrammar and the Resummation of Time, JHEP 07 (2020) 119 [arXiv:1912.09502] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal n-point functions in momentum space, Phys. Rev. Lett. 124 (2020) 131602 [arXiv:1910.10162] [INSPIRE].
M. Baumgart and R. Sundrum, Manifestly Causal In-In Perturbation Theory about the Interacting Vacuum, JHEP 03 (2021) 080 [arXiv:2010.10785] [INSPIRE].
D. Green and E. Pajer, On the Symmetries of Cosmological Perturbations, JCAP 09 (2020) 032 [arXiv:2004.09587] [INSPIRE].
E. Pajer, Building a Boostless Bootstrap for the Bispectrum, JCAP 01 (2021) 023 [arXiv:2010.12818] [INSPIRE].
S. Jazayeri, E. Pajer and D. Stefanyszyn, From Locality and Unitarity to Cosmological Correlators, arXiv:2103.08649 [INSPIRE].
L. Senatore and M. Zaldarriaga, On Loops in Inflation, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
P. Adshead, C.P. Burgess, R. Holman and S. Shandera, Power-counting during single-field slow-roll inflation, JCAP 02 (2018) 016 [arXiv:1708.07443] [INSPIRE].
T. Grall and S. Melville, Inflation in motion: unitarity constraints in effective field theories with (spontaneously) broken Lorentz symmetry, JCAP 09 (2020) 017 [arXiv:2005.02366] [INSPIRE].
C. de Rham and S. Melville, Unitary null energy condition violation in P(X) cosmologies, Phys. Rev. D 95 (2017) 123523 [arXiv:1703.00025] [INSPIRE].
I. Babic, C.P. Burgess and G. Geshnizjani, Keeping an eye on DBI: power-counting for small-cs cosmology, JCAP 05 (2020) 023 [arXiv:1910.05277] [INSPIRE].
A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234 (1984) 189 [INSPIRE].
B.M. Gavela, E.E. Jenkins, A.V. Manohar and L. Merlo, Analysis of General Power Counting Rules in Effective Field Theory, Eur. Phys. J. C 76 (2016) 485 [arXiv:1601.07551] [INSPIRE].
C. Cheung, A.L. Fitzpatrick, J. Kaplan and L. Senatore, On the consistency relation of the 3-point function in single field inflation, JCAP 02 (2008) 021 [arXiv:0709.0295] [INSPIRE].
L. Senatore, K.M. Smith and M. Zaldarriaga, Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data, JCAP 01 (2010) 028 [arXiv:0905.3746] [INSPIRE].
Planck collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, Astron. Astrophys. 641 (2020) A9 [arXiv:1905.05697] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations. II. Can these corrections become large?, Phys. Rev. D 74 (2006) 023508 [hep-th/0605244] [INSPIRE].
P. Adshead, R. Easther and E.A. Lim, Cosmology With Many Light Scalar Fields: Stochastic Inflation and Loop Corrections, Phys. Rev. D 79 (2009) 063504 [arXiv:0809.4008] [INSPIRE].
L. Senatore and M. Zaldarriaga, On Loops in Inflation II: IR Effects in Single Clock Inflation, JHEP 01 (2013) 109 [arXiv:1203.6354] [INSPIRE].
G.L. Pimentel, L. Senatore and M. Zaldarriaga, On Loops in Inflation III: Time Independence of zeta in Single Clock Inflation, JHEP 07 (2012) 166 [arXiv:1203.6651] [INSPIRE].
D. Baumann and D. Green, Equilateral Non-Gaussianity and New Physics on the Horizon, JCAP 09 (2011) 014 [arXiv:1102.5343] [INSPIRE].
D. Baumann, D. Green and R.A. Porto, B-modes and the Nature of Inflation, JCAP 01 (2015) 016 [arXiv:1407.2621] [INSPIRE].
D. Baumann, D. Green, H. Lee and R.A. Porto, Signs of Analyticity in Single-Field Inflation, Phys. Rev. D 93 (2016) 023523 [arXiv:1502.07304] [INSPIRE].
P. Benincasa, A.J. McLeod and C. Vergu, Steinmann Relations and the Wavefunction of the Universe, Phys. Rev. D 102 (2020) 125004 [arXiv:2009.03047] [INSPIRE].
M. Celoria, P. Creminelli, G. Tambalo and V. Yingcharoenrat, Beyond Perturbation Theory in Inflation, arXiv:2103.09244 [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
A. Guerrieri, J. Penedones and P. Vieira, S-matrix Bootstrap for Effective Field Theories: Massless Pions, arXiv:2011.02802 [INSPIRE].
A. Guerrieri, J. Penedones and P. Vieira, Where is String Theory?, arXiv:2102.02847 [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
B. Bellazzini, Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034 [arXiv:1605.06111] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: Positivity Bounds for Particles with Spin, JHEP 03 (2018) 011 [arXiv:1706.02712] [INSPIRE].
G.N. Remmen and N.L. Rodd, Signs, Spin, SMEFT: Positivity at Dimension Six, arXiv:2010.04723 [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive Moments for Scattering Amplitudes, arXiv:2011.00037 [INSPIRE].
A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, arXiv:2011.02400 [INSPIRE].
S. Caron-Huot and V. Van Duong, Extremal Effective Field Theories, arXiv:2011.02957 [INSPIRE].
X. Li, C. Yang, H. Xu, C. Zhang and S.-Y. Zhou, Positivity in Multi-Field EFTs, arXiv:2101.01191 [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, arXiv:2012.15849 [INSPIRE].
T. Grall and S. Melville, Positivity Bounds without Boosts, arXiv:2102.05683 [INSPIRE].
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].
M. Mirbabayi, Infrared dynamics of a light scalar field in de Sitter, JCAP 12 (2020) 006 [arXiv:1911.00564] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Scalar 3-point functions in CFT: renormalisation, β-functions and anomalies, JHEP 03 (2016) 066 [arXiv:1510.08442] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2103.09832
Rights and permissions
Open Access . 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.
About this article
Cite this article
Melville, S., Pajer, E. Cosmological Cutting Rules. J. High Energ. Phys. 2021, 249 (2021). https://doi.org/10.1007/JHEP05(2021)249
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)249