Abstract
The R2 inflation which is an extension of general relativity (GR) by quadratic scalar curvature introduces a quasi-de Sitter expansion of the early Universe governed by Ricci scalar being an eigenmode of d’Alembertian operator. In this paper, we derive a most general theory of gravity admitting R2 inflationary solution which turned out to be higher curvature non-local extension of GR. We study in detail inflationary perturbations in this theory and analyse the structure of form-factors that leads to a massive scalar (scalaron) and massless tensor degrees of freedom. We argue that the theory contains only finite number of free parameters which can be fixed by cosmological observations. We derive predictions of our generalized non-local R2-like inflation and obtain the scalar spectral index ns ≈ 1 − \( \frac{2}{N} \) and any value of the tensor-to-scalar ratio r < 0.036. In this theory, tensor spectral index can be either positive or negative nt ≶ 0 and the well-known consistency relation r = –8nt is violated in a non-trivial way. We also compute running of the tensor spectral index and discuss observational implications to distinguish this model from several classes of scalar field models of inflation. These predictions allow us to probe the nature of quantum gravity in the scope of future CMB and gravitational wave observations. Finally we comment on how the features of generalized non-local R2-like inflation cannot be captured by established notions of the so-called effective field theory of single field inflation and how we must redefine the way we pursue inflationary cosmology.
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. B 91 (1980) 99 [INSPIRE].
A.A. Starobinsky, Nonsingular model of the universe with the quantum gravitational de sitter stage and its observational consequences, in the proceedings of the Second Seminar on Quantum Gravity, Moscow, October 1981, INR Press, Moscow (1982), p. 58–72 [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].
A. Vilenkin, Classical and Quantum Cosmology of the Starobinsky Inflationary Model, Phys. Rev. D 32 (1985) 2511 [INSPIRE].
M.B. Mijic, M.S. Morris and W.-M. Suen, The R2 Cosmology: Inflation Without a Phase Transition, Phys. Rev. D 34 (1986) 2934 [INSPIRE].
K.-I. Maeda, Inflation as a Transient Attractor in R2 Cosmology, Phys. Rev. D 37 (1988) 858 [INSPIRE].
Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10 [arXiv:1807.06211] [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [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].
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].
A.S. Koshelev, K. Sravan Kumar, A. Mazumdar and A.A. Starobinsky, Non-Gaussianities and tensor-to-scalar ratio in non-local R2-like inflation, JHEP 06 (2020) 152 [arXiv:2003.00629] [INSPIRE].
A.S. Koshelev, K.S. Kumar and A.A. Starobinsky, Analytic infinite derivative gravity, R2-like inflation, quantum gravity and CMB, Int. J. Mod. Phys. D 29 (2020) 2043018 [arXiv:2005.09550] [INSPIRE].
J. Ellis, D.V. Nanopoulos and K.A. Olive, Starobinsky-like Inflationary Models as Avatars of No-Scale Supergravity, JCAP 10 (2013) 009 [arXiv:1307.3537] [INSPIRE].
A. Kehagias, A. Moradinezhad Dizgah and A. Riotto, Remarks on the Starobinsky model of inflation and its descendants, Phys. Rev. D 89 (2014) 043527 [arXiv:1312.1155] [INSPIRE].
A. Linde, Inflationary Cosmology after Planck 2013, in the proceedings of the 100e Ecole d’Ete de Physique: Post-Planck Cosmology, France, 8 July – 2 August 2013, Oxford University Press (2015), p. 231–316 [https://doi.org/10.1093/acprof:oso/9780198728856.003.0006] [arXiv:1402.0526] [INSPIRE].
BICEP and Keck collaborations, Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett. 127 (2021) 151301 [arXiv:2110.00483] [INSPIRE].
F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite completion of the Starobinsky theory, Phys. Rev. D 89 (2014) 024029 [arXiv:1308.1413] [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].
L. Modesto, Nonlocal Spacetime-Matter, arXiv:2103.04936 [INSPIRE].
L. Modesto and G. Calcagni, Tree-level scattering amplitudes in nonlocal field theories, JHEP 10 (2021) 169 [arXiv:2107.04558] [INSPIRE].
L. Modesto and G. Calcagni, Early universe in quantum gravity, arXiv:2206.06384 [INSPIRE].
K. Sravan Kumar and L. Modesto, Non-local Starobinsky inflation in the light of future CMB, arXiv:1810.02345 [INSPIRE].
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].
A.O. Barvinsky, Y.V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, The One loop effective action and trace anomaly in four-dimensions, Nucl. Phys. B 439 (1995) 561 [hep-th/9404187] [INSPIRE].
A.O. Barvinsky, Y.V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, The Basis of nonlocal curvature invariants in quantum gravity theory. (Third order.), J. Math. Phys. 35 (1994) 3525 [gr-qc/9404061] [INSPIRE].
A. Bonanno and F. Saueressig, Asymptotically safe cosmology – A status report, Comptes Rendus Physique 18 (2017) 254 [arXiv:1702.04137] [INSPIRE].
A. Eichhorn, An asymptotically safe guide to quantum gravity and matter, Front. Astron. Space Sci. 5 (2019) 47 [arXiv:1810.07615] [INSPIRE].
L. Bosma, B. Knorr and F. Saueressig, Resolving Spacetime Singularities within Asymptotic Safety, Phys. Rev. Lett. 123 (2019) 101301 [arXiv:1904.04845] [INSPIRE].
J. Martin, C. Ringeval and V. Vennin, Encyclopædia Inflationaris, Phys. Dark Univ. 5-6 (2014) 75 [arXiv:1303.3787] [INSPIRE].
V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203 [INSPIRE].
LISA Cosmology Working Group collaboration, Measuring the propagation speed of gravitational waves with LISA, JCAP 08 (2022) 031 [arXiv:2203.00566] [INSPIRE].
A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682 [INSPIRE].
Y. Wang and W. Xue, Inflation and Alternatives with Blue Tensor Spectra, JCAP 10 (2014) 075 [arXiv:1403.5817] [INSPIRE].
A.S. Koshelev, K.S. Kumar and A.A. Starobinsky, Non-Gaussianities in generalized non-local R2-like inflation, JHEP 07 (2023) 094 [arXiv:2210.16459] [INSPIRE].
A.S. Koshelev, K. Sravan Kumar and A.A. Starobinsky, Tensor non-Gaussianities in generalized non-local R2-like inflation, in preparation.
C. Cheung et al., The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
G. Cabass et al., Snowmass white paper: Effective field theories in cosmology, Phys. Dark Univ. 40 (2023) 101193 [arXiv:2203.08232] [INSPIRE].
CMB-S4 collaboration, CMB-S4: Forecasting Constraints on Primordial Gravitational Waves, Astrophys. J. 926 (2022) 54 [arXiv:2008.12619] [INSPIRE].
LiteBIRD collaboration, Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey, PTEP 2023 (2023) 042F01 [arXiv:2202.02773] [INSPIRE].
A. Ricciardone, Primordial Gravitational Waves with LISA, J. Phys. Conf. Ser. 840 (2017) 012030 [arXiv:1612.06799] [INSPIRE].
CORE collaboration, Exploring cosmic origins with CORE: Inflation, JCAP 04 (2018) 016 [arXiv:1612.08270] [INSPIRE].
K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].
D. Anselmi, On the quantum field theory of the gravitational interactions, JHEP 06 (2017) 086 [arXiv:1704.07728] [INSPIRE].
P.D. Mannheim, Solution to the ghost problem in higher-derivative gravity, Nuovo Cim. C 45 (2022) 27 [arXiv:2109.12743] [INSPIRE].
C.M. Bender and P.D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100 (2008) 110402 [arXiv:0706.0207] [INSPIRE].
A. Salvio and A. Strumia, Agravity, JHEP 06 (2014) 080 [arXiv:1403.4226] [INSPIRE].
A. Salvio and A. Strumia, Agravity up to infinite energy, Eur. Phys. J. C 78 (2018) 124 [arXiv:1705.03896] [INSPIRE].
D. Anselmi, E. Bianchi and M. Piva, Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term, JHEP 07 (2020) 211 [arXiv:2005.10293] [INSPIRE].
A. Salvio and F. Sannino, From the Fermi Scale to Cosmology, Frontiers (2019) [https://doi.org/10.3389/978-2-88963-205-3] [INSPIRE].
A. Salvio, BICEP/Keck data and quadratic gravity, JCAP 09 (2022) 027 [arXiv:2202.00684] [INSPIRE].
J. Liu, L. Modesto and G. Calcagni, Quantum field theory with ghost pairs, JHEP 02 (2023) 140 [arXiv:2208.13536] [INSPIRE].
N.V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73 (1987) 1184 [INSPIRE].
E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
E.T. Tomboulis, Renormalization and unitarity in higher derivative and nonlocal gravity theories, Mod. Phys. Lett. A 30 (2015) 1540005 [INSPIRE].
E.T. Tomboulis, Nonlocal and quasilocal field theories, Phys. Rev. D 92 (2015) 125037 [arXiv:1507.00981] [INSPIRE].
L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [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.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].
A.S. Koshelev, K. Sravan Kumar, L. Modesto and L. Rachwał, Finite quantum gravity in dS and AdS spacetimes, Phys. Rev. D 98 (2018) 046007 [arXiv:1710.07759] [INSPIRE].
G. Calcagni and L. Modesto, Nonlocal quantum gravity and M-theory, Phys. Rev. D 91 (2015) 124059 [arXiv:1404.2137] [INSPIRE].
L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].
L. Modesto, M. Piva and L. Rachwal, Finite quantum gauge theories, Phys. Rev. D 94 (2016) 025021 [arXiv:1506.06227] [INSPIRE].
L. Modesto and L. Rachwał, Universally finite gravitational and gauge theories, Nucl. Phys. B 900 (2015) 147 [arXiv:1503.00261] [INSPIRE].
P. Donà et al., Scattering amplitudes in super-renormalizable gravity, JHEP 08 (2015) 038 [arXiv:1506.04589] [INSPIRE].
G. Calcagni, L. Modesto and G. Nardelli, Non-perturbative spectrum of non-local gravity, Phys. Lett. B 795 (2019) 391 [arXiv:1803.07848] [INSPIRE].
G. Calcagni, L. Modesto and G. Nardelli, Initial conditions and degrees of freedom of non-local gravity, JHEP 05 (2018) 087 [Erratum ibid. 05 (2019) 095] [arXiv:1803.00561] [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] [INSPIRE].
I. Dimitrijevic, B. Dragovich, Z. Rakic and J. Stankovic, New Cosmological Solutions of a Nonlocal Gravity Model, Symmetry 14 (2022) 3 [arXiv:2112.06312] [INSPIRE].
S.A. Appleby, R.A. Battye and A.A. Starobinsky, Curing singularities in cosmological evolution of F(R) gravity, JCAP 06 (2010) 005 [arXiv:0909.1737] [INSPIRE].
G. Calcagni and S. Kuroyanagi, Stochastic gravitational-wave background in quantum gravity, JCAP 03 (2021) 019 [arXiv:2012.00170] [INSPIRE].
A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
S. Weinberg, Effective Field Theory for Inflation, Phys. Rev. D 77 (2008) 123541 [arXiv:0804.4291] [INSPIRE].
P. Creminelli, J. Gleyzes, J. Noreña and F. Vernizzi, Resilience of the standard predictions for primordial tensor modes, Phys. Rev. Lett. 113 (2014) 231301 [arXiv:1407.8439] [INSPIRE].
P. Creminelli et al., Detecting Primordial B-Modes after Planck, JCAP 11 (2015) 031 [arXiv:1502.01983] [INSPIRE].
C.P. Burgess, Intro to Effective Field Theories and Inflation, arXiv:1711.10592 [INSPIRE].
D. Baumann, H. Lee and G.L. Pimentel, High-Scale Inflation and the Tensor Tilt, JHEP 01 (2016) 101 [arXiv:1507.07250] [INSPIRE].
D. Baumann, D. Green, H. Lee and R.A. Porto, Signs of Analyticity in Single-Field Inflation, Phys. Rev. D 93 (2016) 023523 [arXiv:1502.07304] [INSPIRE].
K. Aoki, S. Mukohyama and R. Namba, Positivity vs. Lorentz-violation: an explicit example, JCAP 10 (2021) 079 [arXiv:2107.01755] [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].
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].
C. Burrage, S. Cespedes and A.-C. Davis, Disformal transformations on the CMB, JCAP 08 (2016) 024 [arXiv:1604.08038] [INSPIRE].
X. Chen, M.H. Namjoo and Y. Wang, Quantum Primordial Standard Clocks, JCAP 02 (2016) 013 [arXiv:1509.03930] [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].
S. Ferrara, R. Kallosh, A. Linde and M. Porrati, Minimal Supergravity Models of Inflation, Phys. Rev. D 88 (2013) 085038 [arXiv:1307.7696] [INSPIRE].
T. Asaka et al., Reinterpretation of the Starobinsky model, PTEP 2016 (2016) 123E01 [arXiv:1507.04344] [INSPIRE].
Q.-G. Huang, A polynomial f(R) inflation model, JCAP 02 (2014) 035 [arXiv:1309.3514] [INSPIRE].
V.R. Ivanov, S.V. Ketov, E.O. Pozdeeva and S.Y. Vernov, Analytic extensions of Starobinsky model of inflation, JCAP 03 (2022) 058 [arXiv:2111.09058] [INSPIRE].
H. Motohashi, Consistency relation for Rp inflation, Phys. Rev. D 91 (2015) 064016 [arXiv:1411.2972] [INSPIRE].
K. Bamba and S.D. Odintsov, Inflationary cosmology in modified gravity theories, Symmetry 7 (2015) 220 [arXiv:1503.00442] [INSPIRE].
J.S. Martins, O.F. Piattella, I.L. Shapiro and A.A. Starobinsky, Inflation with Sterile Scalar Coupled to Massive Fermions and to Gravity, Grav. Cosmol. 28 (2022) 217 [arXiv:2010.14639] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, A proper fixed functional for four-dimensional Quantum Einstein Gravity, JHEP 08 (2015) 113 [arXiv:1504.07656] [INSPIRE].
L.-H. Liu, T. Prokopec and A.A. Starobinsky, Inflation in an effective gravitational model and asymptotic safety, Phys. Rev. D 98 (2018) 043505 [arXiv:1806.05407] [INSPIRE].
A.R. Romero Castellanos, F. Sobreira, I.L. Shapiro and A.A. Starobinsky, On higher derivative corrections to the R + R2 inflationary model, JCAP 12 (2018) 007 [arXiv:1810.07787] [INSPIRE].
L. Sebastiani and R. Myrzakulov, F(R) gravity and inflation, Int. J. Geom. Meth. Mod. Phys. 12 (2015) 1530003 [arXiv:1506.05330] [INSPIRE].
A.S. Koshelev, Non-local SFT Tachyon and Cosmology, JHEP 04 (2007) 029 [hep-th/0701103] [INSPIRE].
A.S. Koshelev and S.Y. Vernov, Cosmological perturbations in SFT inspired non-local scalar field models, Eur. Phys. J. C 72 (2012) 2198 [arXiv:0903.5176] [INSPIRE].
A.S. Koshelev and S.Y. Vernov, Analysis of scalar perturbations in cosmological models with a non-local scalar field, Class. Quant. Grav. 28 (2011) 085019 [arXiv:1009.0746] [INSPIRE].
I.Y. Aref’eva and A.S. Koshelev, Cosmological Signature of Tachyon Condensation, JHEP 09 (2008) 068 [arXiv:0804.3570] [INSPIRE].
L. Buoninfante, G. Lambiase and M. Yamaguchi, Nonlocal generalization of Galilean theories and gravity, Phys. Rev. D 100 (2019) 026019 [arXiv:1812.10105] [INSPIRE].
L. Buoninfante et al., Generalized ghost-free propagators in nonlocal field theories, Phys. Rev. D 101 (2020) 084019 [arXiv:2001.07830] [INSPIRE].
L. Buoninfante, private communication.
A.S. Koshelev and A. Tokareva, Unitarity of Minkowski nonlocal theories made explicit, Phys. Rev. D 104 (2021) 025016 [arXiv:2103.01945] [INSPIRE].
L. Modesto and I.L. Shapiro, Superrenormalizable quantum gravity with complex ghosts, Phys. Lett. B 755 (2016) 279 [arXiv:1512.07600] [INSPIRE].
L. Modesto, Super-renormalizable or finite Lee–Wick quantum gravity, Nucl. Phys. B 909 (2016) 584 [arXiv:1602.02421] [INSPIRE].
D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum field theory, Phys. Rev. D 96 (2017) 045009 [arXiv:1703.05563] [INSPIRE].
D. Anselmi, Fakeons And Lee-Wick Models, JHEP 02 (2018) 141 [arXiv:1801.00915] [INSPIRE].
D. Anselmi, Diagrammar of physical and fake particles and spectral optical theorem, JHEP 11 (2021) 030 [arXiv:2109.06889] [INSPIRE].
D. Anselmi, Purely virtual particles versus Lee-Wick ghosts: Physical Pauli-Villars fields, finite QED, and quantum gravity, Phys. Rev. D 105 (2022) 125017 [arXiv:2202.10483] [INSPIRE].
M. Frasca, A. Ghoshal and A.S. Koshelev, Confining complex ghost degrees of freedom, Phys. Lett. B 841 (2023) 137924 [arXiv:2207.06394] [INSPIRE].
V. Muller, H.J. Schmidt and A.A. Starobinsky, The Stability of the De Sitter Space-time in Fourth Order Gravity, Phys. Lett. B 202 (1988) 198 [INSPIRE].
D. Müller, A. Ricciardone, A.A. Starobinsky and A. Toporensky, Anisotropic cosmological solutions in R + R2 gravity, Eur. Phys. J. C 78 (2018) 311 [arXiv:1710.08753] [INSPIRE].
Acknowledgments
AK is supported by FCT Portugal investigator project IF/01607/2015. This research work was supported by grants UID/MAT/00212/2019, COST Action CA15117 (CANTATA). KSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for Scientific Research No. JP20F20320 and No. JP21H00069. KSK would like to thank the Royal society for the Newton International Fellowship. AAS was supported by the RSF grant 21-12-00130. We thank L. Buoninfante, A. De Felice, T. Noumi, A. Tokareva and especially M. Yamaguchi for very useful discussions.
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: 2209.02515
Rights and permissions
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.
About this article
Cite this article
Koshelev, A.S., Kumar, K.S. & Starobinsky, A.A. Generalized non-local R2-like inflation. J. High Energ. Phys. 2023, 146 (2023). https://doi.org/10.1007/JHEP07(2023)146
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2023)146