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On loops in inflation. (English) Zbl 1294.83099
Summary: We study loop corrections to correlation functions of inflationary perturbations. Previous calculations have found that in certain theories the two-point function can have a logarithmic running of the form $$\log(k/\mu)$$, where $$k$$ is the wavenumber of the perturbation, and $$\mu$$ is the renormalization scale. We highlight that this result would have profound consequences for both eternal inflation and the predictivity of standard inflation. We find a different result in the same cases. We consider two sets of theories: one where the inflaton has a large cubic self-interaction and one where the inflaton interacts gravitationally with $$N$$ massless spectator scalar fields. We find that there is a logarithmic running but of the form log($$H/\mu$$), where $$H$$ is the Hubble constant during inflation. We find this result in three independent ways: by performing the calculation with a sharp cutoff in frequency-momentum space, in dimensional regularization and by the simple procedure of making the loop integral dimensionless. For the simplest of our theories we explicitly renormalize the correlation function proving that the divergencies can be reabsorbed and that the correlation function for super-horizon modes does not depend on time (once the tadpole terms have been properly taken into account). We prove the time-independence of the super-horizon correlation function in several additional ways: by doing the calculation of the correlation function at finite time using both the regularizations and by developing a formalism which expresses loop corrections directly in terms of renormalized quantities at each time. We find this last formalism particularly helpful to develop intuition which we then use to generalize our results to higher loops and different interactions. In particular we argue correlation functions have no long-term time dependence even if the spectator fields have a potential.

##### MSC:
 83F05 Cosmology 81V22 Unified quantum theories 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T20 Quantum field theory on curved space or space-time backgrounds
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##### References:
 [1] Maldacena, JM, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP, 05, 013, (2003) [2] Weinberg, S., Quantum contributions to cosmological correlations, Phys. Rev., D 72, 043514, (2005) [3] Cheung, C.; Creminelli, P.; Fitzpatrick, AL; Kaplan, J.; Senatore, L., The effective field theory of inflation, JHEP, 03, 014, (2008) [4] Alishahiha, M.; Silverstein, E.; Tong, D., DBI in the sky, Phys. Rev., D 70, 123505, (2004) [5] Arkani-Hamed, N.; Creminelli, P.; Mukohyama, S.; Zaldarriaga, M., Ghost inflation, JCAP, 04, 001, (2004) [6] Senatore, L., Tilted ghost inflation, Phys. Rev., D 71, 043512, (2005) [7] Chen, X.; Huang, M-x; Kachru, S.; Shiu, G., Observational signatures and non-gaussianities of general single field inflation, JCAP, 01, 002, (2007) [8] Flauger, R.; McAllister, L.; Pajer, E.; Westphal, A.; Xu, G., Oscillations in the CMB from axion monodromy inflation, JCAP, 06, 009, (2010) [9] Zaldarriaga, M., Non-gaussianities in models with a varying inflaton decay rate, Phys. Rev., D 69, 043508, (2004) [10] Lyth, DH; Ungarelli, C.; Wands, D., The primordial density perturbation in the curvaton scenario, Phys. Rev., D 67, 023503, (2003) [11] Green, D.; Horn, B.; Senatore, L.; Silverstein, E., Trapped inflation, Phys. Rev., D 80, 063533, (2009) [12] Barnaby, N.; Huang, Z.; Kofman, L.; Pogosyan, D., Cosmological fluctuations from infra-red cascading during inflation, Phys. Rev., D 80, 043501, (2009) [13] Smith, KM; Senatore, L.; Zaldarriaga, M., Optimal limits on f_{NL}\^{local} from WMAP 5-year data, JCAP, 09, 006, (2009) [14] Senatore, L.; Smith, KM; Zaldarriaga, M., Non-gaussianities in single field inflation and their optimal limits from the WMAP 5-year data, JCAP, 01, 028, (2010) [15] Slosar, A.; Hirata, C.; Seljak, U.; Ho, S.; Padmanabhan, N., Constraints on local primordial non-gaussianity from large scale structure, JCAP, 08, 031, (2008) [16] Weinberg, S., Anthropic bound on the cosmological constant, Phys. Rev. Lett., 59, 2607, (1987) [17] Creminelli, P.; Dubovsky, S.; Nicolis, A.; Senatore, L.; Zaldarriaga, M., The phase transition to slow-roll eternal inflation, JHEP, 09, 036, (2008) [18] Dubovsky, S.; Senatore, L.; Villadoro, G., The volume of the universe after inflation and de Sitter entropy, JHEP, 04, 118, (2009) [19] Guth, AH, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev., D 23, 347, (1981) [20] A.D. Linde, Nonsingular Regenerating Inflationary Universe, Cambridge University Preprint, Print -82-0554, Cambridge U.K. (1982). [21] P.J. Steinhardt, Natural Inflation, in The Very Early Universe, G.W. Gibbons, S.W. Hawking and S. Siklos eds., Cambridge University Press, Cambridge U.K. (1983). [22] Vilenkin, A., The birth of inflationary universes, Phys. Rev., D 27, 2848, (1983) [23] Goncharov, AS; Linde, AD; Mukhanov, VF, The global structure of the inflationary universe, Int. J. Mod. Phys., A2, 561, (1987) [24] Arkani-Hamed, N.; Dubovsky, S.; Nicolis, A.; Trincherini, E.; Villadoro, G., A measure of de Sitter entropy and eternal inflation, JHEP, 05, 055, (2007) [25] Arkani-Hamed, N.; Dubovsky, S.; Senatore, L.; Villadoro, G., (no) eternal inflation and precision Higgs physics, JHEP, 03, 075, (2008) [26] Adshead, P.; Easther, R.; Lim, EA, Cosmology with many light scalar fields: stochastic inflation and loop corrections, Phys. Rev., D 79, 063504, (2009) [27] Miao, S-P; Woodard, RP, Leading log solution for inflationary Yukawa, Phys. Rev., D 74, 044019, (2006) [28] M. Musso, A new diagrammatic representation for correlation functions in the in-in formalism, [hep-th/0611258] [SPIRES]. [29] Bartolo, N.; Matarrese, S.; Pietroni, M.; Riotto, A.; Seery, D., On the physical significance of infra-red corrections to inflationary observables, JCAP, 01, 015, (2008) [30] Riotto, A.; Sloth, MS, On resumming inflationary perturbations beyond one-loop, JCAP, 04, 030, (2008) [31] Burgess, CP; Leblond, L.; Holman, R.; Shandera, S., Super-hubble de Sitter fluctuations and the dynamical RG, JCAP, 03, 033, (2010) [32] Sloth, MS, On the one loop corrections to inflation and the CMB anisotropies, Nucl. Phys., B 748, 149, (2006) [33] Sloth, MS, On the one loop corrections to inflation. II: the consistency relation, Nucl. Phys., B 775, 78, (2007) [34] Seery, D., One-loop corrections to the curvature perturbation from inflation, JCAP, 02, 006, (2008) [35] Seery, D., A parton picture of de Sitter space during slow-roll inflation, JCAP, 05, 021, (2009) [36] Urakawa, Y.; Maeda, K-i, One-loop corrections to scalar and tensor perturbations during inflation in stochastic gravity, Phys. Rev., D 78, 064004, (2008) [37] Strominger, A., The ds/CFT correspondence, JHEP, 10, 034, (2001) [38] E. Witten, Quantum gravity in de Sitter space, [hep-th/0106109] [SPIRES]. [39] McGreevy, J.; Silverstein, E.; Starr, D., New dimensions for wound strings: the modular transformation of geometry to topology, Phys. Rev., D 75, 044025, (2007) [40] Silverstein, E., Dimensional mutation and spacelike singularities, Phys. Rev., D 73, 086004, (2006)
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