Abstract
Studying loop corrections to inflationary perturbations, with particular emphasis on infrared factors, is important to understand the consistency of the inflationary theory, its predictivity and to establish the existence of the slow-roll eternal inflation phenomena and its recently found volume bound. In this paper we prove that the ζ correlation function is time-independent at one-loop level in single clock inflation. While many of the one-loop diagrams lead to a time-dependence when considered individually, the time-dependence beautifully cancels out in the overall sum. We identify two subsets of diagrams that cancel separately due to different physical reasons. The first cancellation is related to the change of the background cosmology due to the renormalization of the stress tensor. It results in a cancellation between the non-1PI diagrams and some of the diagrams made with quartic vertices. The second subset of diagrams that cancel is made up of cubic operators, plus the remaining quartic ones. We are able to write the sum of these diagrams as the integral over a specific three-point function between two very short wavelengths and one very long one. We then apply the consistency condition for this three-point function in the squeezed limit to show that the sum of these diagrams cannot give rise to a time dependence. This second cancellation is thus a consequence of the fact that in single clock inflation the attractor nature of the solution implies that a long wavelength ζ perturbation is indistinguishable from a trivial rescaling of the background, and so results in no physical effect on short wavelength modes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
L. Senatore and M. Zaldarriaga, On loops in inflation, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].
S.B. Giddings and M.S. Sloth, Semiclassical relations and IR effects in de Sitter and slow-roll space-times, JCAP 01 (2011) 023 [arXiv:1005.1056] [INSPIRE].
C.T. Byrnes, M. Gerstenlauer, A. Hebecker, S. Nurmi and G. Tasinato, Inflationary infrared divergences: geometry of the reheating surface versus δN formalism, JCAP 08 (2010) 006 [arXiv:1005.3307] [INSPIRE].
L. Senatore and M. Zaldarriaga, On Loops in Single Field Inflation: Projection Effects, to appear.
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
D.H. Lyth, K.A. Malik and M. Sasaki, A general proof of the conservation of the curvature perturbation, JCAP 05 (2005) 004 [astro-ph/0411220] [INSPIRE].
D. Langlois and F. Vernizzi, Conserved non-linear quantities in cosmology, Phys. Rev. D 72 (2005) 103501 [astro-ph/0509078] [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].
E. Kahya, V. Onemli and R. Woodard, The Zeta-Zeta Correlator Is Time Dependent, Phys. Lett. B 694 (2010) 101 [arXiv:1006.3999] [INSPIRE].
P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga, The phase transition to slow-roll eternal inflation, JHEP 09 (2008) 036 [arXiv:0802.1067] [INSPIRE].
S. Dubovsky, L. Senatore and G. Villadoro, The volume of the universe after inflation and de Sitter entropy, JHEP 04 (2009) 118 [arXiv:0812.2246] [INSPIRE].
S. Dubovsky, L. Senatore and G. Villadoro, Universality of the volume bound in slow-roll eternal inflation, JHEP 05 (2012) 035 [arXiv:1111.1725] [INSPIRE].
M. Lewandowski, A. Perko, L. Senatore and G. Villadoro, The Volume of the Universe after Slow-Roll Corrections and its Universal Bound, in preparation.
N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, A measure of de Sitter entropy and eternal inflation, JHEP 05 (2007) 055 [arXiv:0704.1814] [INSPIRE].
P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function, JCAP 10 (2004) 006 [astro-ph/0407059] [INSPIRE].
M. Musso, A new diagrammatic representation for correlation functions in the in-in formalism, hep-th/0611258 [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].
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].
L. Senatore and M. Zaldarriaga, The effective field theory of multifield inflation, JHEP 04 (2012) 024 [arXiv:1009.2093] [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].
L. Senatore and M. Zaldarriaga, A naturally large four-point function in single field inflation, JCAP 01 (2011) 003 [arXiv:1004.1201] [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, JHEP 12 (2006) 080 [hep-th/0606090] [INSPIRE].
N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, Large non-Gaussianities in the effective field theory approach to single-field inflation: the bispectrum, JCAP 08 (2010) 008 [arXiv:1004.0893] [INSPIRE].
N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, Large non-Gaussianities in the effective field theory approach to single-field inflation: the trispectrum, JCAP 09 (2010) 035 [arXiv:1006.5411] [INSPIRE].
P. Creminelli, G. D’Amico, M. Musso, J. Norena and E. Trincherini, Galilean symmetry in the effective theory of inflation: new shapes of non-Gaussianity, JCAP 02 (2011) 006 [arXiv:1011.3004] [INSPIRE].
D. Baumann, L. Senatore and M. Zaldarriaga, Scale-invariance and the strong coupling problem, JCAP 05 (2011) 004 [arXiv:1101.3320] [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 and D. Green, Signatures of Supersymmetry from the Early Universe, Phys. Rev. D 85 (2012) 103520 [arXiv:1109.0292] [INSPIRE].
D. Lopez Nacir, R.A. Porto, L. Senatore and M. Zaldarriaga, Dissipative effects in the Effective Field Theory of Inflation, JHEP 01 (2012) 075 [arXiv:1109.4192] [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, arXiv:1111.3373 [INSPIRE].
D. Baumann and D. Green, A Field Range Bound for General Single-Field Inflation, arXiv:1111.3040 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Pimentel, G.L., Senatore, L. & Zaldarriaga, M. On loops in inflation III: time independence of ζ in single clock inflation. J. High Energ. Phys. 2012, 166 (2012). https://doi.org/10.1007/JHEP07(2012)166
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2012)166