Abstract
Sum rules connecting low-energy observables to high-energy physics are an interesting way to probe the mechanism of inflation and its ultraviolet origin. Unfortunately, such sum rules have proven difficult to study in a cosmological setting. Motivated by this problem, we investigate a precise analogue of inflation in anti-de Sitter spacetime, where it becomes dual to a slow renormalization group flow in the boundary quantum field theory. This dual description provides a firm footing for exploring the constraints of unitarity, analyticity, and causality on the bulk effective field theory. We derive a sum rule that constrains the bulk coupling constants in this theory. In the bulk, the sum rule is related to the speed of radial propagation, while on the boundary, it governs the spreading of nonlocal operators. When the spreading speed approaches the speed of light, the sum rule is saturated, suggesting that the theory becomes free in this limit. We also discuss whether similar results apply to inflation, where an analogous sum rule exists for the propagation speed of inflationary fluctuations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP10 (2006) 014 [hep-th/0602178] [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
J. Distler, B. Grinstein, R.A. Porto and I.Z. Rothstein, Falsifying Models of New Physics via WW Scattering, Phys. Rev. Lett.98 (2007) 041601 [hep-ph/0604255] [INSPIRE].
C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture, JHEP12 (2014) 087 [arXiv:1407.7865] [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].
B. Bellazzini, M. Lewandowski and J. Serra, Amplitudes’ Positivity, Weak Gravity Conjecture and Modified Gravity, arXiv:1902.03250 [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, Beyond Positivity Bounds and the Fate of Massive Gravity, Phys. Rev. Lett.120 (2018) 161101 [arXiv:1710.02539] [INSPIRE].
B. Bellazzini, C. Cheung and G.N. Remmen, Quantum Gravity Constraints from Unitarity and Analyticity, Phys. Rev.D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
C. Cheung and G.N. Remmen, Positivity of Curvature-Squared Corrections in Gravity, Phys. Rev. Lett.118 (2017) 051601 [arXiv:1608.02942] [INSPIRE].
C. Cheung and G.N. Remmen, Positive Signs in Massive Gravity, JHEP04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity Bounds for Massive Spin-1 and Spin-2 Fields, JHEP03 (2019) 182 [arXiv:1804.10624] [INSPIRE].
C. de Rham, S. Melville and A.J. Tolley, Improved Positivity Bounds and Massive Gravity, JHEP04 (2018) 083 [arXiv:1710.09611] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: Positivity Bounds for Particles with Spin, JHEP03 (2018) 011 [arXiv:1706.02712] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Massive Galileon Positivity Bounds, JHEP09 (2017) 072 [arXiv:1702.08577] [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].
Y. Hamada, T. Noumi and G. Shiu, Weak Gravity Conjecture from Unitarity and Causality, Phys. Rev. Lett.123 (2019) 051601 [arXiv:1810.03637] [INSPIRE].
W.-M. Chen, Y.-T. Huang, T. Noumi and C. Wen, Unitarity bounds on charged/neutral state mass ratios, Phys. Rev.D 100 (2019) 025016 [arXiv:1901.11480] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
C. Cordova, J. Maldacena and G.J. Turiaci, Bounds on OPE Coefficients from Interference Effects in the Conformal Collider, JHEP11 (2017) 032 [arXiv:1710.03199] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys.B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP10 (2001) 034 [hep-th/0106113] [INSPIRE].
A. Strominger, Inflation and the dS/CFT correspondence, JHEP11 (2001) 049 [hep-th/0110087] [INSPIRE].
P. McFadden and K. Skenderis, Holography for Cosmology, Phys. Rev.D 81 (2010) 021301 [arXiv:0907.5542] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett.43 (1986) 730 [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev.D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, renormalization group flows and the a-theorem in six-dimensional (1, 0) theories, JHEP10 (2016) 080 [arXiv:1506.03807] [INSPIRE].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys.A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett.118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
J. Kaplan and J. Wang, An Effective Theory for Holographic RG Flows, JHEP02 (2015) 056 [arXiv:1406.4152] [INSPIRE].
C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
P. Creminelli, M.A. Luty, A. Nicolis and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP12 (2006) 080 [hep-th/0606090] [INSPIRE].
O. Nachtmann, Positivity constraints for anomalous dimensions, Nucl. Phys.B 63 (1973) 237 [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP05 (2017) 065 [arXiv:1608.05101] [INSPIRE].
X. Chen, M.-x. Huang, S. Kachru and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation, JCAP01 (2007) 002 [hep-th/0605045] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in Strings 2001: International Conference, Mumbai, India, 5–10 January 2001 (2001) [hep-th/0106109] [INSPIRE].
P.O. Mazur and E. Mottola, Weyl cohomology and the effective action for conformal anomalies, Phys. Rev.D 64 (2001) 104022 [hep-th/0106151] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP05 (2003) 013 [astro-ph/0210603] [INSPIRE].
F. Piazza and F. Vernizzi, Effective Field Theory of Cosmological Perturbations, Class. Quant. Grav.30 (2013) 214007 [arXiv:1307.4350] [INSPIRE].
D. Baumann and L. McAllister, Inflation and String Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2015) [arXiv:1404.2601] [INSPIRE].
D. Baumann and D. Green, Equilateral Non-Gaussianity and New Physics on the Horizon, JCAP09 (2011) 014 [arXiv:1102.5343] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, arXiv:1811.00024 [INSPIRE].
I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP07 (2013) 015 [arXiv:1211.5482] [INSPIRE].
Planck collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697 [INSPIRE].
D. Harlow and H. Ooguri, Constraints on Symmetries from Holography, Phys. Rev. Lett.122 (2019) 191601 [arXiv:1810.05337] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the Asymptotics of 4D Quantum Field Theory, JHEP01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
A. Dymarsky, Z. Komargodski, A. Schwimmer and S. Theisen, On Scale and Conformal Invariance in Four Dimensions, JHEP10 (2015) 171 [arXiv:1309.2921] [INSPIRE].
A. Dymarsky, K. Farnsworth, Z. Komargodski, M.A. Luty and V. Prilepina, Scale Invariance, Conformality and Generalized Free Fields, JHEP02 (2016) 099 [arXiv:1402.6322] [INSPIRE].
Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept.569 (2015) 1 [arXiv:1302.0884] [INSPIRE].
X.-L. Qi, E.J. Davis, A. Periwal and M. Schleier-Smith, Measuring operator size growth in quantum quench experiments, arXiv:1906.00524 [INSPIRE].
D.A. Roberts and B. Swingle, Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories, Phys. Rev. Lett.117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the Operator Product Expansion, JHEP03 (2019) 201 [arXiv:1709.03597] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
M. Mezei, On entanglement spreading from holography, JHEP05 (2017) 064 [arXiv:1612.00082] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal Field Theories and Deep Inelastic Scattering, Phys. Rev.D 95 (2017) 065011 [arXiv:1601.05453] [INSPIRE].
A. Zhiboedov, On Conformal Field Theories With Extremal a/c Values, JHEP04 (2014) 038 [arXiv:1304.6075] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
D.M. Hofman, Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT, Nucl. Phys.B 823 (2009) 174 [arXiv:0907.1625] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP10 (2016) 141 [arXiv:1601.07904] [INSPIRE].
D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, JHEP06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
S.B. Giddings and R.A. Porto, The Gravitational S-matrix, Phys. Rev.D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].
J. Bonifacio, K. Hinterbichler, A. Joyce and R.A. Rosen, Massive and Massless Spin-2 Scattering and Asymptotic Superluminality, JHEP06 (2018) 075 [arXiv:1712.10020] [INSPIRE].
C. de Rham and A.J. Tolley, The Speed of Gravity, arXiv:1909.00881 [INSPIRE].
R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, Oscillations in the CMB from Axion Monodromy Inflation, JCAP06 (2010) 009 [arXiv:0907.2916] [INSPIRE].
A. Slosar et al., Scratches from the Past: Inflationary Archaeology through Features in the Power Spectrum of Primordial Fluctuations, arXiv:1903.09883 [INSPIRE].
E. Kiritsis, F. Nitti and L. Silva Pimenta, Exotic RG Flows from Holography, Fortsch. Phys.65 (2017) 1600120 [arXiv:1611.05493] [INSPIRE].
S.R. Behbahani, A. Dymarsky, M. Mirbabayi and L. Senatore, (Small) Resonant non-Gaussianities: Signatures of a Discrete Shift Symmetry in the Effective Field Theory of Inflation, JCAP12 (2012) 036 [arXiv:1111.3373] [INSPIRE].
M. Flory, Discrete scale invariance in holography revisited, Fortsch. Phys.66 (2018) 1700093 [arXiv:1711.03113] [INSPIRE].
D. Green, Disorder in the Early Universe, JCAP03 (2015) 020 [arXiv:1409.6698] [INSPIRE].
M.A. Amin and D. Baumann, From Wires to Cosmology, JCAP02 (2016) 045 [arXiv:1512.02637] [INSPIRE].
M.A.G. Garcia, M.A. Amin, S.G. Carlsten and D. Green, Stochastic Particle Production in a de Sitter Background, JCAP05 (2019) 012 [arXiv:1902.09598] [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1906.10226
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Baumann, D., Green, D. & Hartman, T. Dynamical constraints on RG flows and cosmology. J. High Energ. Phys. 2019, 134 (2019). https://doi.org/10.1007/JHEP12(2019)134
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)134