Abstract
We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sitter space and has no analogue in AdS space. This subtlety arises because in de Sitter space the metric at late times is a genuine degree of freedom, and hence to calculate correlation functions from the wave function of the Universe at late times, one must fix gauge completely. The resulting correlators are then invariant under a conformal transformation accompanied by a compensating coordinate transformation which restores the gauge.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XXII. Constraints on inflation, arXiv:1303.5082 [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity, arXiv:1303.5084 [INSPIRE].
D. Seery, M.S. Sloth and F. Vernizzi, Inflationary trispectrum from graviton exchange, JCAP 03 (2009) 018 [arXiv:0811.3934] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
K.N. Abazajian et al., Inflation Physics from the Cosmic Microwave Background and Large Scale Structure, arXiv:1309.5381 [INSPIRE].
E. Komatsu, Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background, Class. Quant. Grav. 27 (2010) 124010 [arXiv:1003.6097] [INSPIRE].
V. Desjacques and U. Seljak, Primordial non-Gaussianity from the large scale structure, Class. Quant. Grav. 27 (2010) 124011 [arXiv:1003.5020] [INSPIRE].
D. Seery, J.E. Lidsey and M.S. Sloth, The inflationary trispectrum, JCAP 01 (2007) 027 [astro-ph/0610210] [INSPIRE].
F. Arroja and K. Koyama, Non-gaussianity from the trispectrum in general single field inflation, Phys. Rev. D 77 (2008) 083517 [arXiv:0802.1167] [INSPIRE].
F. Arroja, S. Mizuno, K. Koyama and T. Tanaka, On the full trispectrum in single field DBI-inflation, Phys. Rev. D 80 (2009) 043527 [arXiv:0905.3641] [INSPIRE].
X. Chen, B. Hu, M. xin Huang, G. Shiu and Y. Wang, Large primordial trispectra in general single field inflation, JCAP 08 (2009) 008 [arXiv:0905.3494] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [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 and cosmic background radiation, Phys. Rev. Lett. 79 (1997) 14 [astro-ph/9611208] [INSPIRE].
F. Larsen, J.P. van der Schaar and R.G. Leigh, de Sitter holography and the cosmic microwave background, JHEP 04 (2002) 047 [hep-th/0202127] [INSPIRE].
F. Larsen and R. McNees, Inflation and de Sitter holography, JHEP 07 (2003) 051 [hep-th/0307026] [INSPIRE].
P. McFadden and K. Skenderis, Holographic Non-Gaussianity, JCAP 05 (2011) 013 [arXiv:1011.0452] [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. Bzowski, P. McFadden and K. Skenderis, Holographic predictions for cosmological 3-point functions, JHEP 03 (2012) 091 [arXiv:1112.1967] [INSPIRE].
P. McFadden and K. Skenderis, Cosmological 3-point correlators from holography, JCAP 06 (2011) 030 [arXiv:1104.3894] [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].
A. Kehagias and A. Riotto, The Four-point Correlator in Multifield Inflation, the Operator Product Expansion and the Symmetries of de Sitter, Nucl. Phys. B 868 (2013) 577 [arXiv:1210.1918] [INSPIRE].
K. Schalm, G. Shiu and T. van der Aalst, Consistency condition for inflation from (broken) conformal symmetry, JCAP 03 (2013) 005 [arXiv:1211.2157] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Holography for inflation using conformal perturbation theory, JHEP 04 (2013) 047 [arXiv:1211.4550] [INSPIRE].
I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP 07 (2013) 015 [arXiv:1211.5482] [INSPIRE].
J. Garriga and Y. Urakawa, Inflation and deformation of conformal field theory, JCAP 07 (2013) 033 [arXiv:1303.5997] [INSPIRE].
P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function, JCAP 10 (2004) 006 [astro-ph/0407059] [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 and M. Zaldarriaga, A Note on the Consistency Condition of Primordial Fluctuations, JCAP 08 (2012) 001 [arXiv:1203.6884] [INSPIRE].
P. Creminelli, C. Pitrou and F. Vernizzi, The CMB bispectrum in the squeezed limit, JCAP 11 (2011) 025 [arXiv:1109.1822] [INSPIRE].
N. Bartolo, S. Matarrese and A. Riotto, Non-Gaussianity in the Cosmic Microwave Background Anisotropies at Recombination in the Squeezed limit, JCAP 02 (2012) 017 [arXiv:1109.2043] [INSPIRE].
P. Creminelli, J. Norena and M. Simonovic, Conformal consistency relations for single-field inflation, JCAP 07 (2012) 052 [arXiv:1203.4595] [INSPIRE].
P. Creminelli, A. Joyce, J. Khoury and M. Simonovic, Consistency Relations for the Conformal Mechanism, JCAP 04 (2013) 020 [arXiv:1212.3329] [INSPIRE].
K. Hinterbichler, L. Hui and J. Khoury, Conformal Symmetries of Adiabatic Modes in Cosmology, JCAP 08 (2012) 017 [arXiv:1203.6351] [INSPIRE].
V. Assassi, D. Baumann and D. Green, On Soft Limits of Inflationary Correlation Functions, JCAP 11 (2012) 047 [arXiv:1204.4207] [INSPIRE].
W.D. Goldberger, L. Hui and A. Nicolis, One-particle-irreducible consistency relations for cosmological perturbations, Phys. Rev. D 87 (2013) 103520 [arXiv:1303.1193] [INSPIRE].
K. Hinterbichler, L. Hui and J. Khoury, An Infinite Set of Ward Identities for Adiabatic Modes in Cosmology, JCAP 01 (2014) 039 [arXiv:1304.5527] [INSPIRE].
P. Creminelli, A. Perko, L. Senatore, M. Simonović and G. Trevisan, The Physical Squeezed Limit: Consistency Relations at Order q 2, JCAP 11 (2013) 015 [arXiv:1307.0503] [INSPIRE].
L. Berezhiani and J. Khoury, Slavnov-Taylor Identities for Primordial Perturbations, JCAP 02 (2014) 003 [arXiv:1309.4461] [INSPIRE].
T. Banks and W. Fischler, Holographic Theories of Inflation and Fluctuations, arXiv:1111.4948 [INSPIRE].
T. Banks, W. Fischler, T.J. Torres and C.L. Wainwright, Holographic Fluctuations from Unitary de Sitter Invariant Field Theory, arXiv:1306.3999 [INSPIRE].
J.M. Bardeen, Gauge Invariant Cosmological Perturbations, Phys. Rev. D 22 (1980) 1882 [INSPIRE].
J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Spontaneous Creation of Almost Scale — Free Density Perturbations in an Inflationary Universe, Phys. Rev. D 28 (1983) 679 [INSPIRE].
D.H. Lyth, Large Scale Energy Density Perturbations and Inflation, Phys. Rev. D 31 (1985) 1792 [INSPIRE].
D.S. Salopek and J.R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936 [INSPIRE].
S. Weinberg, Adiabatic modes in cosmology, Phys. Rev. D 67 (2003) 123504 [astro-ph/0302326] [INSPIRE].
S. Weinberg, Damping of tensor modes in cosmology, Phys. Rev. D 69 (2004) 023503 [astro-ph/0306304] [INSPIRE].
S. Weinberg, Cosmology, Oxford University Press, Oxford U.K. (2008).
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].
G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM(4) in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].
J.B. Hartle, High Energy Physics. Vol. 2: TASI Lectures On Quantum Cosmology, M.J. Bowick and F. Gursey eds., World Scientific, Singapore (1985).
S.M. Christensen and M.J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
M.J.G. Veltman, Lectures on Quantum Theory Of Gravitation in Les Houches 1975: Session XXVIII: Methods in Field Theory, R. Balian and J. Zinn-Justin eds., North-Holland/World Scientific, Amsterdam The Netherlands (1975).
S. Raju, Recursion Relations for AdS/CFT Correlators, Phys. Rev. D 83 (2011) 126002 [arXiv:1102.4724] [INSPIRE].
S. Raju, BCFW for Witten Diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [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 AdS 4 /CFT 3, Phys. Rev. D 85 (2012) 126008 [arXiv:1201.6452] [INSPIRE].
J. Smidt et al., A Measurement of Cubic-Order Primordial Non-Gaussianity (g NL and τ NL ) With WMAP 5-Year Data, arXiv:1001.5026 [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: 1401.1426
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Ghosh, A., Kundu, N., Raju, S. et al. Conformal invariance and the four point scalar correlator in slow-roll inflation. J. High Energ. Phys. 2014, 11 (2014). https://doi.org/10.1007/JHEP07(2014)011
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2014)011