Abstract
A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block — the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly streamlined derivation of these operators (and their generalizations) using tools of conformal field theory. Our results greatly simplify the theoretical foundations of the cosmological bootstrap program.
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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].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological polytopes and the wavefunction of the universe, arXiv:1709.02813 [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].
P. Benincasa, From the flat-space S-matrix to the wavefunction of the universe, arXiv:1811.02515 [INSPIRE].
C. Sleight, A Mellin space approach to cosmological correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping inflationary correlators in Mellin space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
P. Benincasa, Cosmological polytopes and the wavefuncton of the universe for light states, arXiv:1909.02517 [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
I. Antoniadis, P.O. Mazur and E. Mottola, Conformal invariance, dark energy, and CMB non-Gaussianity, JCAP 09 (2012) 024 [arXiv:1103.4164] [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].
N. Arkani-Hamed and J. Maldacena, Cosmological collider physics, arXiv:1503.08043 [INSPIRE].
S. Raju, New recursion relations and a flat space limit for AdS/CFT correlators, Phys. Rev. D 85 (2012) 126009 [arXiv:1201.6449] [INSPIRE].
S. Raju, Four point functions of the stress tensor and conserved currents in AdS4/CFT3, Phys. Rev. D 85 (2012) 126008 [arXiv:1201.6452] [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].
C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals, JHEP 07 (2013) 011 [arXiv:1304.6944] [INSPIRE].
C. Corianò and M.M. Maglio, Exact correlators from conformal Ward identities in momentum space and the perturbative T J J vertex, Nucl. Phys. B 938 (2019) 440 [arXiv:1802.07675] [INSPIRE].
C. Corianò and M.M. Maglio, On some hypergeometric solutions of the conformal Ward identities of scalar 4-point functions in momentum space, JHEP 09 (2019) 107 [arXiv:1903.05047] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
D. Anninos, T. Anous, D.Z. Freedman and G. Konstantinidis, Late-time structure of the Bunch-Davies de Sitter wavefunction, JCAP 11 (2015) 048 [arXiv:1406.5490] [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].
A. Bzowski, P. McFadden and K. Skenderis, Evaluation of conformal integrals, JHEP 02 (2016) 068 [arXiv:1511.02357] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Renormalised 3-point functions of stress tensors and conserved currents in CFT, JHEP 11 (2018) 153 [arXiv:1711.09105] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators, JHEP 07 (2018) 136 [arXiv:1805.11107] [INSPIRE].
H. Isono, T. Noumi and T. Takeuchi, Momentum space conformal three-point functions of conserved currents and a general spinning operator, JHEP 05 (2019) 057 [arXiv:1903.01110] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension, JHEP 10 (2019) 183 [arXiv:1908.04572] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes, JHEP 02 (2019) 040 [arXiv:1810.12459] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, New relation for Witten diagrams, JHEP 10 (2019) 274 [arXiv:1904.10043] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes. Part II. Gravitons, JHEP 12 (2019) 135 [arXiv:1908.01835] [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].
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].
J.A. Farrow, A.E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP 02 (2019) 130 [arXiv:1812.11129] [INSPIRE].
X. Chen and Y. Wang, Quasi-single field inflation and non-Gaussianities, JCAP 04 (2010) 027 [arXiv:0911.3380] [INSPIRE].
D. Baumann and D. Green, Signatures of supersymmetry from the early universe, Phys. Rev. D 85 (2012) 103520 [arXiv:1109.0292] [INSPIRE].
V. Assassi, D. Baumann and D. Green, On soft limits of inflationary correlation functions, JCAP 11 (2012) 047 [arXiv:1204.4207] [INSPIRE].
X. Chen and Y. Wang, Quasi-single field inflation with large mass, JCAP 09 (2012) 021 [arXiv:1205.0160] [INSPIRE].
S. Pi and M. Sasaki, Curvature perturbation spectrum in two-field inflation with a turning trajectory, JCAP 10 (2012) 051 [arXiv:1205.0161] [INSPIRE].
T. Noumi, M. Yamaguchi and D. Yokoyama, Effective field theory approach to quasi-single field inflation and effects of heavy fields, JHEP 06 (2013) 051 [arXiv:1211.1624] [INSPIRE].
D. Baumann, S. Ferraro, D. Green and K.M. Smith, Stochastic bias from non-Gaussian initial conditions, JCAP 05 (2013) 001 [arXiv:1209.2173] [INSPIRE].
V. Assassi, D. Baumann, D. Green and L. McAllister, Planck-suppressed operators, JCAP 01 (2014) 033 [arXiv:1304.5226] [INSPIRE].
J.-O. Gong, S. Pi and M. Sasaki, Equilateral non-Gaussianity from heavy fields, JCAP 11 (2013) 043 [arXiv:1306.3691] [INSPIRE].
H. Lee, D. Baumann and G.L. Pimentel, Non-Gaussianity as a particle detector, JHEP 12 (2016) 040 [arXiv:1607.03735] [INSPIRE].
A. Kehagias and A. Riotto, On the inflationary perturbations of massive higher-spin fields, JCAP 07 (2017) 046 [arXiv:1705.05834] [INSPIRE].
S. Kumar and R. Sundrum, Heavy-lifting of gauge theories by cosmic inflation, JHEP 05 (2018) 011 [arXiv:1711.03988] [INSPIRE].
H. An, M. McAneny, A.K. Ridgway and M.B. Wise, Quasi single field inflation in the non-perturbative regime, JHEP 06 (2018) 105 [arXiv:1706.09971] [INSPIRE].
H. An, M. McAneny, A.K. Ridgway and M.B. Wise, Non-Gaussian enhancements of galactic halo correlations in quasi-single field inflation, Phys. Rev. D 97 (2018) 123528 [arXiv:1711.02667] [INSPIRE].
D. Baumann, G. Goon, H. Lee and G.L. Pimentel, Partially massless fields during inflation, JHEP 04 (2018) 140 [arXiv:1712.06624] [INSPIRE].
G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Shapes of gravity: tensor non-Gaussianity and massive spin-2 fields, JHEP 10 (2019) 182 [arXiv:1812.07571] [INSPIRE].
S. Kumar and R. Sundrum, Seeing higher-dimensional grand unification in primordial non-Gaussianities, JHEP 04 (2019) 120 [arXiv:1811.11200] [INSPIRE].
T. Liu, X. Tong, Y. Wang and Z.-Z. Xianyu, Probing P and CP-violations on the cosmological collider, JHEP 04 (2020) 189 [arXiv:1909.01819] [INSPIRE].
S. Kumar and R. Sundrum, Cosmological collider physics and the curvaton, JHEP 04 (2020) 077 [arXiv:1908.11378] [INSPIRE].
S. Alexander, S.J. Gates, L. Jenks, K. Koutrolikos and E. McDonough, Higher spin supersymmetry at the cosmological collider: sculpting SUSY rilles in the CMB, JHEP 10 (2019) 156 [arXiv:1907.05829] [INSPIRE].
S. Kim, T. Noumi, K. Takeuchi and S. Zhou, Heavy spinning particles from signs of primordial non-Gaussianities: beyond the positivity bounds, JHEP 12 (2019) 107 [arXiv:1906.11840] [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].
M. Gillioz, X. Lu and M.A. Luty, Scale anomalies, states, and rates in conformal field theory, JHEP 04 (2017) 171 [arXiv:1612.07800] [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Graviton scattering and a sum rule for the c anomaly in 4D CFT, JHEP 09 (2018) 025 [arXiv:1801.05807] [INSPIRE].
M. Gillioz, Momentum-space conformal blocks on the light cone, JHEP 10 (2018) 125 [arXiv:1807.07003] [INSPIRE].
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space, JHEP 01 (2020) 142 [arXiv:1908.04733] [INSPIRE].
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space, Commun. Math. Phys. 379 (2020) 227 [arXiv:1909.00878] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
M.S. Costa and T. Hansen, AdS weight shifting operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].
S. Weinberg, Six-dimensional methods for four-dimensional conformal field theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, SpringerBriefs Phys. (2016) [arXiv:1601.05000] [INSPIRE].
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 538] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
D. Anninos, F. Denef, R. Monten and Z. Sun, Higher spin de Sitter Hilbert space, JHEP 10 (2019) 071 [arXiv:1711.10037] [INSPIRE].
P. Creminelli, On non-Gaussianities in single-field inflation, JCAP 10 (2003) 003 [astro-ph/0306122] [INSPIRE].
S. Deser and R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Annals Phys. 154 (1984) 396 [INSPIRE].
L. Brink, R.R. Metsaev and M.A. Vasiliev, How massless are massless fields in AdSd, Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [INSPIRE].
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
S. Deser and A. Waldron, Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nucl. Phys. B 662 (2003) 379 [hep-th/0301068] [INSPIRE].
S. Deser, M. Sandora and A. Waldron, Nonlinear partially massless from massive gravity?, Phys. Rev. D 87 (2013) 101501 [arXiv:1301.5621] [INSPIRE].
C. de Rham, K. Hinterbichler, R.A. Rosen and A.J. Tolley, Evidence for and obstructions to nonlinear partially massless gravity, Phys. Rev. D 88 (2013) 024003 [arXiv:1302.0025] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, Conformal operators for partially massless states, JHEP 10 (2001) 016 [hep-th/0109096] [INSPIRE].
C. Brust and K. Hinterbichler, Free □k scalar conformal field theory, JHEP 02 (2017) 066 [arXiv:1607.07439] [INSPIRE].
C. Brust and K. Hinterbichler, Partially massless higher-spin theory, JHEP 02 (2017) 086 [arXiv:1610.08510] [INSPIRE].
E. Joung, K. Mkrtchyan and G. Poghosyan, Looking for partially-massless gravity, JHEP 07 (2019) 116 [arXiv:1904.05915] [INSPIRE].
N. Boulanger, C. Deffayet, S. Garcia-Saenz and L. Traina, Theory for multiple partially massless spin-2 fields, Phys. Rev. D 100 (2019) 101701 [arXiv:1906.03868] [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].
D. Simmons-Duffin, The conformal bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: new frontiers in fields and strings, World Scientific, Singapore (2017), pg. 1 [arXiv:1602.07982] [INSPIRE].
R.S. Erramilli, L.V. Iliesiu and P. Kravchuk, Recursion relation for general 3d blocks, JHEP 12 (2019) 116 [arXiv:1907.11247] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
V. Dobrev, G. Mack, V. Petkova, S. Petrova and I. Todorov, Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 63 (1977) 1.
Wolfram MathWorld webpage, http://mathworld.wolfram.com/.
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Baumann, D., Pueyo, C.D., Joyce, A. et al. The cosmological bootstrap: weight-shifting operators and scalar seeds. J. High Energ. Phys. 2020, 204 (2020). https://doi.org/10.1007/JHEP12(2020)204
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DOI: https://doi.org/10.1007/JHEP12(2020)204