Abstract
In this paper we will study R2-like inflation in a non-local modification of gravity which contains quadratic in Ricci scalar and Weyl tensor terms with analytic infinite derivative form-factors in the action. It is known that the inflationary solution of the local R + R2 gravity remains a particular exact solution in this model. It was shown earlier that the power spectrum of scalar perturbations generated during inflation in the non-local setup remains the same as in the local R + R2 inflation, whereas the power spectrum of tensor perturbations gets modified due to the non-local Weyl tensor squared term. In the present paper we go beyond 2-point correlators and compute the non-Gaussian parameter fNL related to 3-point correlations generated during inflation, which we found to be different from those in the original local inflationary model and scenarios alike based on a local gravity. We evaluate non-local corrections to the scalar bi-spectrum which give non-zero contributions to squeezed, equilateral and orthogonal configurations. We show that fNL ∼ O(1) with an arbitrary sign is achievable in this model based on the choice of form-factors and the scale of non-locality. We present the predictions for the tensor-to-scalar ratio, r, and the tensor tilt, nt. In contrast to standard inflation in a local gravity, here the possibility nt > 0 is not excluded. Thus, future CMB data can probe non-local behaviour of gravity at high space-time curvatures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. 91B (1980) 99 [INSPIRE].
F.L. Bezrukov and M. Shaposhnikov, The Standard Model Higgs boson as the inflaton, Phys. Lett. B 659 (2008) 703 [arXiv:0710.3755] [INSPIRE].
A. Linde, Inflationary Cosmology after Planck 2013, in Proceedings, 100th Les Houches Summer School: Post-Planck Cosmology: Les Houches, France, July 8 – August 2, 2013, pp. 231–316, 2015, arXiv:1402.0526 [INSPIRE].
S.V. Ketov and A.A. Starobinsky, Embedding (R + R2 )-Inflation into Supergravity, Phys. Rev. D 83 (2011) 063512 [arXiv:1011.0240] [INSPIRE].
S.V. Ketov and A.A. Starobinsky, Inflation and non-minimal scalar-curvature coupling in gravity and supergravity, JCAP 08 (2012) 022 [arXiv:1203.0805] [INSPIRE].
T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 03 (2006) 009 [hep-th/0508194] [INSPIRE].
T. Biswas, A.S. Koshelev, A. Mazumdar and S.Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, JCAP 08 (2012) 024 [arXiv:1206.6374] [INSPIRE].
F. Briscese, A. Marcianò, L. Modesto and E.N. Saridakis, Inflation in (Super-)renormalizable Gravity, Phys. Rev. D 87 (2013) 083507 [arXiv:1212.3611] [INSPIRE].
B. Craps, T. De Jonckheere and A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, JCAP 11 (2014) 022 [arXiv:1407.4982] [INSPIRE].
A.S. Koshelev, L. Modesto, L. Rachwal and A.A. Starobinsky, Occurrence of exact R2 inflation in non-local UV-complete gravity, JHEP 11 (2016) 067 [arXiv:1604.03127] [INSPIRE].
A.S. Koshelev, K. Sravan Kumar and A.A. Starobinsky, R2 inflation to probe non-perturbative quantum gravity, JHEP 03 (2018) 071 [arXiv:1711.08864] [INSPIRE].
N.V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73 (1987) 1184 [Teor. Mat. Fiz. 73 (1987) 235] [INSPIRE].
Yu.V. Kuzmin, The convergent nonlocal gravitation (in Russian), Sov. J. Nucl. Phys. 50 (1989) 1011 [Yad. Fiz. 50 (1989) 1630] [INSPIRE].
E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
T. Biswas, A. Conroy, A.S. Koshelev and A. Mazumdar, Generalized ghost-free quadratic curvature gravity, Class. Quant. Grav. 31 (2014) 015022 [Erratum ibid. 31 (2014) 159501] [arXiv:1308.2319] [INSPIRE].
S. Talaganis, T. Biswas and A. Mazumdar, Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity, Class. Quant. Grav. 32 (2015) 215017 [arXiv:1412.3467] [INSPIRE].
L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].
E.T. Tomboulis, Renormalization and unitarity in higher derivative and nonlocal gravity theories, Mod. Phys. Lett. A 30 (2015) 1540005 [INSPIRE].
S. Talaganis and A. Mazumdar, High-Energy Scatterings in Infinite-Derivative Field Theory and Ghost-Free Gravity, Class. Quant. Grav. 33 (2016) 145005 [arXiv:1603.03440] [INSPIRE].
P. Donà, S. Giaccari, L. Modesto, L. Rachwal and Y. Zhu, Scattering amplitudes in super-renormalizable gravity, JHEP 08 (2015) 038 [arXiv:1506.04589] [INSPIRE].
L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].
T. Biswas, T. Koivisto and A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity, JCAP 11 (2010) 008 [arXiv:1005.0590] [INSPIRE].
T. Biswas, T. Koivisto and A. Mazumdar, Nonlocal theories of gravity: the flat space propagator, in Proceedings, Barcelona Postgrad Encounters on Fundamental Physics, pp. 13–24, 2013, arXiv:1302.0532 [INSPIRE].
A.S. Koshelev, Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology, Class. Quant. Grav. 30 (2013) 155001 [arXiv:1302.2140] [INSPIRE].
T. Biswas, A.S. Koshelev and A. Mazumdar, Gravitational theories with stable (anti-)de Sitter backgrounds, Fundam. Theor. Phys. 183 (2016) 97, arXiv:1602.08475.
T. Biswas, A.S. Koshelev and A. Mazumdar, Consistent higher derivative gravitational theories with stable de Sitter and anti-de Sitter backgrounds, Phys. Rev. D 95 (2017) 043533 [arXiv:1606.01250] [INSPIRE].
S. Deser, M.J. Duff and C.J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys. B 111 (1976) 45 [INSPIRE].
A.A. Starobinsky, Evolution of small perturbations of isotropic cosmological models with one-loop quantum gravitational corrections, JETP Lett. 34 (1981) 438.
A.O. Barvinsky and G.A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B 333 (1990) 471 [INSPIRE].
S.W. Hawking, T. Hertog and H.S. Reall, Trace anomaly driven inflation, Phys. Rev. D 63 (2001) 083504 [hep-th/0010232] [INSPIRE].
S. Deser and R.P. Woodard, Nonlocal Cosmology, Phys. Rev. Lett. 99 (2007) 111301 [arXiv:0706.2151] [INSPIRE].
A.O. Barvinsky, Aspects of Nonlocality in Quantum Field Theory, Quantum Gravity and Cosmology, Mod. Phys. Lett. A 30 (2015) 1540003 [arXiv:1408.6112] [INSPIRE].
E. Witten, Noncommutative Geometry and String Field Theory, Nucl. Phys. B 268 (1986) 253 [INSPIRE].
E. Witten, Interacting Field Theory of Open Superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].
B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. 156B (1985) 315 [INSPIRE].
I.Ya. Aref’eva, A.S. Koshelev and S.Yu. Vernov, Exact solution in a string cosmological model, Theor. Math. Phys. 148 (2006) 895 [Teor. Mat. Fiz. 148 (2006) 23] [astro-ph/0412619] [INSPIRE].
K. Ohmori, A review on tachyon condensation in open string field theories, Ph.D. Thesis, Tokyo University, 2001, hep-th/0102085 [INSPIRE].
G. Calcagni, Cosmological tachyon from cubic string field theory, JHEP 05 (2006) 012 [hep-th/0512259] [INSPIRE].
G. Calcagni, M. Montobbio and G. Nardelli, A route to nonlocal cosmology, Phys. Rev. D 76 (2007) 126001 [arXiv:0705.3043] [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].
A. Conroy, A.S. Koshelev and A. Mazumdar, Geodesic completeness and homogeneity condition for cosmic inflation, Phys. Rev. D 90 (2014) 123525 [arXiv:1408.6205] [INSPIRE].
A. Conroy, A.S. Koshelev and A. Mazumdar, Defocusing of Null Rays in Infinite Derivative Gravity, JCAP 01 (2017) 017 [arXiv:1605.02080] [INSPIRE].
J. Edholm, A.S. Koshelev and A. Mazumdar, Behavior of the Newtonian potential for ghost-free gravity and singularity-free gravity, Phys. Rev. D 94 (2016) 104033 [arXiv:1604.01989] [INSPIRE].
A.S. Koshelev and A. Mazumdar, Do massive compact objects without event horizon exist in infinite derivative gravity?, Phys. Rev. D 96 (2017) 084069 [arXiv:1707.00273] [INSPIRE].
L. Buoninfante, A.S. Koshelev, G. Lambiase, J. Marto and A. Mazumdar, Conformally-flat, non-singular static metric in infinite derivative gravity, JCAP 06 (2018) 014 [arXiv:1804.08195] [INSPIRE].
L. Buoninfante, G. Harmsen, S. Maheshwari and A. Mazumdar, Nonsingular metric for an electrically charged point-source in ghost-free infinite derivative gravity, Phys. Rev. D 98 (2018) 084009 [arXiv:1804.09624] [INSPIRE].
L. Buoninfante, A. Ghoshal, G. Lambiase and A. Mazumdar, Transmutation of nonlocal scale in infinite derivative field theories, Phys. Rev. D 99 (2019) 044032 [arXiv:1812.01441] [INSPIRE].
L. Buoninfante, G. Lambiase and A. Mazumdar, Ghost-free infinite derivative quantum field theory, Nucl. Phys. B 944 (2019) 114646 [arXiv:1805.03559].
L. Buoninfante et al., Towards nonsingular rotating compact object in ghost-free infinite derivative gravity, Phys. Rev. D 98 (2018) 084041 [arXiv:1807.08896] [INSPIRE].
L. Buoninfante, A.S. Koshelev, G. Lambiase and A. Mazumdar, Classical properties of non-local, ghost- and singularity-free gravity, JCAP 09 (2018) 034 [arXiv:1802.00399] [INSPIRE].
L. Buoninfante and A. Mazumdar, Nonlocal star as a blackhole mimicker, Phys. Rev. D 100 (2019) 024031 [arXiv:1903.01542] [INSPIRE].
L. Bosma, B. Knorr and F. Saueressig, Resolving Spacetime Singularities within Asymptotic Safety, Phys. Rev. Lett. 123 (2019) 101301 [arXiv:1904.04845] [INSPIRE].
L. Buoninfante, G. Lambiase, Y. Miyashita, W. Takebe and M. Yamaguchi, Generalized ghost-free propagators in nonlocal field theories, Phys. Rev. D 101 (2020) 084019 [arXiv:2001.07830] [INSPIRE].
A.S. Koshelev, K. Sravan Kumar, L. Modesto and L. Rachwa/l, Finite quantum gravity in dS and AdS spacetimes, Phys. Rev. D 98 (2018) 046007 [arXiv:1710.07759] [INSPIRE].
B. Knorr, C. Ripken and F. Saueressig, Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox, Class. Quant. Grav. 36 (2019) 234001 [arXiv:1907.02903] [INSPIRE].
Planck collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211 [INSPIRE].
Planck collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697 [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function, JCAP 10 (2004) 006 [astro-ph/0407059] [INSPIRE].
X. Chen, M.-x. Huang, S. Kachru and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation, JCAP 01 (2007) 002 [hep-th/0605045] [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].
A. De Felice and S. Tsujikawa, Primordial non-Gaussianities in general modified gravitational models of inflation, JCAP 04 (2011) 029 [arXiv:1103.1172] [INSPIRE].
A. De Felice and S. Tsujikawa, Shapes of primordial non-Gaussianities in the Horndeski’s most general scalar-tensor theories, JCAP 03 (2013) 030 [arXiv:1301.5721] [INSPIRE].
X. Gao, Primordial Non-Gaussianities of General Multiple Field Inflation, JCAP 06 (2008) 029 [arXiv:0804.1055] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Non-Gaussianity from inflation: Theory and observations, Phys. Rept. 402 (2004) 103 [astro-ph/0406398] [INSPIRE].
X. Chen and Y. Wang, Quasi-Single Field Inflation and Non-Gaussianities, JCAP 04 (2010) 027 [arXiv:0911.3380] [INSPIRE].
N. Barnaby, T. Biswas and J.M. Cline, p-adic Inflation, JHEP 04 (2007) 056 [hep-th/0612230] [INSPIRE].
N. Barnaby and J.M. Cline, Large NonGaussianity from Nonlocal Inflation, JCAP 07 (2007) 017 [arXiv:0704.3426] [INSPIRE].
A.A. Starobinsky, The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy, Sov. Astron. Lett. 9 (1983) 302 [INSPIRE].
Planck collaboration, Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20 [arXiv:1502.02114] [INSPIRE].
E.T. Tomboulis, Nonlocal and quasilocal field theories, Phys. Rev. D 92 (2015) 125037 [arXiv:1507.00981] [INSPIRE].
H. Hui et al., BICEP Array: a multi-frequency degree-scale CMB polarimeter, Proc. SPIE Int. Soc. Opt. Eng. 10708 (2018) 1070807 [arXiv:1808.00568] [INSPIRE].
BICEP2 and Keck Array collaborations, BICEP2/Keck Array x: Constraints on Primordial Gravitational Waves using Planck, WMAP and New BICEP2/Keck Observations through the 2015 Season, Phys. Rev. Lett. 121 (2018) 221301 [arXiv:1810.05216] [INSPIRE].
CMB-S4 collaboration, CMB-S4 Science Book, First Edition, arXiv:1610.02743 [INSPIRE].
K. Abazajian et al., CMB-S4 Science Case, Reference Design and Project Plan, arXiv:1907.04473 [INSPIRE].
Simons Observatory collaboration, The Simons Observatory: Science goals and forecasts, JCAP 02 (2019) 056 [arXiv:1808.07445] [INSPIRE].
M. Hazumi et al., LiteBIRD: A Satellite for the Studies of B-Mode Polarization and Inflation from Cosmic Background Radiation Detection, J. Low. Temp. Phys. 194 (2019) 443 [INSPIRE].
S. Shandera et al., Probing the origin of our Universe through cosmic microwave background constraints on gravitational waves, Bull. Am. Astron. Soc. 51 (2019) 338 [arXiv:1903.04700] [INSPIRE].
NASA PICO collaboration, PICO: Probe of Inflation and Cosmic Origins, arXiv:1902.10541 [INSPIRE].
T. Kobayashi, N. Tanahashi and M. Yamaguchi, Multifield extension of G inflation, Phys. Rev. D 88 (2013) 083504 [arXiv:1308.4798] [INSPIRE].
L.C. Price, H.V. Peiris, J. Frazer and R. Easther, Gravitational wave consistency relations for multifield inflation, Phys. Rev. Lett. 114 (2015) 031301 [arXiv:1409.2498] [INSPIRE].
T. Kobayashi, M. Yamaguchi and J. Yokoyama, Generalized G-inflation: Inflation with the most general second-order field equations, Prog. Theor. Phys. 126 (2011) 511 [arXiv:1105.5723] [INSPIRE].
R.H. Brandenberger, String Gas Cosmology after Planck, Class. Quant. Grav. 32 (2015) 234002 [arXiv:1505.02381] [INSPIRE].
D. Seery and J.E. Lidsey, Primordial non-Gaussianities in single field inflation, JCAP 06 (2005) 003 [astro-ph/0503692] [INSPIRE].
J. Khoury, Fading gravity and self-inflation, Phys. Rev. D 76 (2007) 123513 [hep-th/0612052] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
D. Babich, P. Creminelli and M. Zaldarriaga, The shape of non-Gaussianities, JCAP 08 (2004) 009 [astro-ph/0405356] [INSPIRE].
E. Komatsu and D.N. Spergel, Acoustic signatures in the primary microwave background bispectrum, Phys. Rev. D 63 (2001) 063002 [astro-ph/0005036] [INSPIRE].
T. Takahashi, Primordial non-Gaussianity and the inflationary Universe, PTEP 2014 (2014) 06B105 [INSPIRE].
K. Sravan Kumar, S. Maheshwari and A. Mazumdar, Perturbations in higher derivative gravity beyond maximally symmetric spacetimes, Phys. Rev. D 100 (2019) 064022 [arXiv:1905.03227] [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].
X. Chen, Primordial Non-Gaussianities from Inflation Models, Adv. Astron. 2010 (2010) 638979 [arXiv:1002.1416] [INSPIRE].
C.T. Byrnes, Lecture notes on non-Gaussianity, Astrophys. Space Sci. Proc. 45 (2016) 135, [arXiv:1411.7002].
J. Ganc and E. Komatsu, A new method for calculating the primordial bispectrum in the squeezed limit, JCAP 12 (2010) 009 [arXiv:1006.5457] [INSPIRE].
M.H. Namjoo, H. Firouzjahi and M. Sasaki, Violation of non-Gaussianity consistency relation in a single field inflationary model, EPL 101 (2013) 39001 [arXiv:1210.3692] [INSPIRE].
J. Martin, H. Motohashi and T. Suyama, Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation, Phys. Rev. D 87 (2013) 023514 [arXiv:1211.0083] [INSPIRE].
A.A. Starobinsky, Inflaton field potential producing the exactly flat spectrum of adiabatic perturbations, JETP Lett. 82 (2005) 169 [astro-ph/0507193] [INSPIRE].
M. Alvarez et al., Testing Inflation with Large Scale Structure: Connecting Hopes with Reality, arXiv:1412.4671 [INSPIRE].
M. Liguori, A. Yadav, F.K. Hansen, E. Komatsu, S. Matarrese and B. Wandelt, Temperature and Polarization CMB Maps from Primordial non-Gaussianities of the Local Type, Phys. Rev. D 76 (2007) 105016 [Erratum ibid. D 77 (2008) 029902] [arXiv:0708.3786] [INSPIRE].
P.D. Meerburg et al., Primordial Non-Gaussianity, arXiv:1903.04409 [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
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2003.00629
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Koshelev, A.S., Kumar, K.S., Mazumdar, A. et al. Non-Gaussianities and tensor-to-scalar ratio in non-local R2-like inflation. J. High Energ. Phys. 2020, 152 (2020). https://doi.org/10.1007/JHEP06(2020)152
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)152