Abstract
Light scalars in inflationary spacetimes suffer from logarithmic infrared divergences at every order in perturbation theory. This corresponds to the scalar field values in different Hubble patches undergoing a random walk of quantum fluctuations, leading to a simple toy “landscape” on superhorizon scales, in which we can explore questions relevant to eternal inflation. However, for a sufficiently long period of inflation, the infrared divergences appear to spoil computability. Some form of renormalization group approach is thus motivated to resum the log divergences of conformal time. Such a resummation may provide insight into De Sitter holography. We present here a novel diagrammatic analysis of these infrared divergences and their resummation. Basic graph theory observations and momen- tum power counting for the in-in propagators allow a simple and insightful determination of the leading-log contributions. One thus sees diagrammatically how the superhorizon sector consists of a semiclassical theory with quantum noise evolved by a first-order, interacting classical equation of motion. This rigorously leads to the “Stochastic Inflation” ansatz developed by Starobinsky to cure the scalar infrared pathology nonperturbatively. Our approach is a controlled approximation of the underlying quantum field theory and is systematically improvable.
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References
Y. Urakawa and T. Tanaka, Influence on Observation from IR Divergence during Inflation. I., Prog. Theor. Phys. 122 (2009) 779 [arXiv:0902.3209] [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].
Y. Urakawa and T. Tanaka, Natural selection of inflationary vacuum required by infra-red regularity and gauge-invariance, Prog. Theor. Phys. 125 (2011) 1067 [arXiv:1009.2947] [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].
L. Senatore and M. Zaldarriaga, The constancy of ζ in single-clock Inflation at all loops, JHEP 09 (2013) 148 [arXiv:1210.6048] [INSPIRE].
V. Assassi, D. Baumann and D. Green, Symmetries and Loops in Inflation, JHEP 02 (2013) 151 [arXiv:1210.7792] [INSPIRE].
T. Tanaka and Y. Urakawa, Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of secular growth in the Euclidean vacuum, PTEP 2013 (2013) 063E02 [arXiv:1301.3088] [INSPIRE].
T. Tanaka and Y. Urakawa, Large gauge transformation, Soft theorem, and Infrared divergence in inflationary spacetime, JHEP 10 (2017) 127 [arXiv:1707.05485] [INSPIRE].
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
A.A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the de Sitter background, Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016] [INSPIRE].
N.C. Tsamis and R.P. Woodard, Stochastic quantum gravitational inflation, Nucl. Phys. B 724 (2005) 295 [gr-qc/0505115] [INSPIRE].
B.-L. Hu, Infrared Behavior of Quantum Fields in Inflationary Cosmology — Issues and Approaches: an overview, arXiv:1812.11851 [INSPIRE].
M. Musso, A new diagrammatic representation for correlation functions in the in-in formalism, JHEP 11 (2013) 184 [hep-th/0611258] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
B. Garbrecht, G. Rigopoulos and Y. Zhu, Infrared correlations in de Sitter space: Field theoretic versus stochastic approach, Phys. Rev. D 89 (2014) 063506 [arXiv:1310.0367] [INSPIRE].
B. Garbrecht, F. Gautier, G. Rigopoulos and Y. Zhu, Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space, Phys. Rev. D 91 (2015) 063520 [arXiv:1412.4893] [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].
A.H. Guth, Eternal inflation and its implications, hep-th/0702178 [INSPIRE].
D. Seery, A parton picture of de Sitter space during slow-roll inflation, JCAP 05 (2009) 021 [arXiv:0903.2788] [INSPIRE].
V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].
A. Riotto and M.S. Sloth, The probability equation for the cosmological comoving curvature perturbation, JCAP 10 (2011) 003 [arXiv:1103.5876] [INSPIRE].
I. Moss and G. Rigopoulos, Effective long wavelength scalar dynamics in de Sitter, JCAP 05 (2017) 009 [arXiv:1611.07589] [INSPIRE].
J. Tokuda and T. Tanaka, Statistical nature of infrared dynamics on de Sitter background, JCAP 02 (2018) 014 [arXiv:1708.01734] [INSPIRE].
C.P. Burgess, R. Holman, L. Leblond and S. Shandera, Breakdown of Semiclassical Methods in de Sitter Space, JCAP 10 (2010) 017 [arXiv:1005.3551] [INSPIRE].
T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117.
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, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].
A. Kaya, On iϵ Prescription in Cosmology, JCAP 04 (2019) 002 [arXiv:1810.12324] [INSPIRE].
M. Baumgart and R. Sundrum, On iϵ Prescription in Cosmology, work in progress.
J. Kearney, H. Yoo and K.M. Zurek, Is a Higgs Vacuum Instability Fatal for High-Scale Inflation?, Phys. Rev. D 91 (2015) 123537 [arXiv:1503.05193] [INSPIRE].
R.D. Jordan, Effective Field Equations for Expectation Values, Phys. Rev. D 33 (1986) 444 [INSPIRE].
A.M. Polyakov, Infrared instability of the de Sitter space, arXiv:1209.4135 [INSPIRE].
M. Baumgart, C. Marcantonini and I.W. Stewart, Systematic Improvement of Parton Showers with Effective Theory, Phys. Rev. D 83 (2011) 034011 [arXiv:1007.0758] [INSPIRE].
Z. Nagy and D.E. Soper, Parton showers with quantum interference, JHEP 09 (2007) 114 [arXiv:0706.0017] [INSPIRE].
D. Neill and W.J. Waalewijn, Entropy of a Jet, Phys. Rev. Lett. 123 (2019) 142001 [arXiv:1811.01021] [INSPIRE].
H. Collins, R. Holman and T. Vardanyan, The quantum Fokker-Planck equation of stochastic inflation, JHEP 11 (2017) 065 [arXiv:1706.07805] [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. Rajaraman, de Sitter Space is Unstable in Quantum Gravity, Phys. Rev. D 94 (2016) 125025 [arXiv:1608.07237] [INSPIRE].
G. Geshnizjani and R. Brandenberger, Back reaction of perturbations in two scalar field inflationary models, JCAP 04 (2005) 006 [hep-th/0310265] [INSPIRE].
R. Brandenberger, L.L. Graef, G. Marozzi and G.P. Vacca, Backreaction of super-Hubble cosmological perturbations beyond perturbation theory, Phys. Rev. D 98 (2018) 103523 [arXiv:1807.07494] [INSPIRE].
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Baumgart, M., Sundrum, R. De Sitter diagrammar and the resummation of time. J. High Energ. Phys. 2020, 119 (2020). https://doi.org/10.1007/JHEP07(2020)119
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DOI: https://doi.org/10.1007/JHEP07(2020)119