nLab Starobinsky model of cosmic inflation




In phenomenology of cosmology, the Starobinsky model of cosmic inflation takes into account – and takes as the very source of the inflaton field – higher curvature corrections to the Einstein-Hilbert action of gravity, notably the term R 2R^2 (square of the Ricci curvature).

The Starobinsky model stands out among models of inflation as predicting a low value of the scalar-to-tensor ratio rr, specifically it predicts

r12N 2 r \sim \frac{12}{N^2}

where NN is the number of ee-foldings during inflation (see e.g. Kehagias-Dizgah-Riotto 13 (2.6)).

Observational support

Models of Starobinsky-type are favored by experimental results (PlanckCollaboration 13, BICEP2-Keck-Planck 15, PlanckCollaboration 15, BICEP3-Keck 18) which give a low upper bound on rr, well below 0.10.1 (whereas other models like chaotic inflation are disfavored by these values), see (PlanckCollaboration 13, page 12).

With respect to this data, the Starobinsky model (or “R 2R^2 inflation”) is the model with the highest Bayesian evidence (Rachen, Feb 15, PlanckCollaboration 15XX, table 6 on p. 18) as it is right in the center of the likelihood peak, shown in dark blue in the following plots (PlanckCollaboration 13, figure 1, also Linde 14, figure 5) and at the same time has the lowest number of free parameters :

This remains true with the data of (PlanckCollaboration 15), see (PlanckCollaboration 15 XIII, figure 22) and in the final analysis (PlanckCollaboration 18X, Fig 8), which gives the following (from here):

R 2R^2 inflation has the strongest evidence among the models considered here. However, care must be taken not to overinterpret small differences in likelihood lacking statistical significance. The models closest to R 2R^2 in terms of evidence are brane inflation and exponential inflation, which have one more parameter than R 2R^2 (PlanckCollaboration 15XX, p. 18)

This picture is further confirmed by observations of the BICEP/Keck collaboration reported in BICEP-Keck 2021, whose additional data singles out the dark blue area in the following (Fig. 5):

See also Ellis 13, Ketov 13, Efstathiou 2019, 50:49 for brief survey of Starobinsky inflation in relation to observation, and see Kehagias-Dizgah-Riotto 13 for more details. There it is argued that the other types of inflationary models which also reasonably fit the data are actually equivalent to the Starobinsky model during inflation.

Embedding into supergravity

Being concerned with pure gravity (the inflaton not being an extra matter field but part of the field of gravity) the Starobinsky model lends itself to embedding into supergravity (originally due to Cecotti 87, see e.g. Farakos-Kehagias-Riotto 13). Such embedding has been argued to improve the model further (highlighted e.g. in Ellis 13), for instance by

graphics grabbed from Dalianis 16, p. 8

More concretely, in Hiraga-Hyakutake 18 a simple model of 11-dimensional supergravity with its R 4R^4 higher curvature correction (see there) is considered and claimed to yield inflation with “graceful exit” and dynamical KK-compactification:

graphics from Hiraga-Hyakutake 18, p. 8



The model is due to

and the analysis of its predictions is due to:

  • Viatcheslav Mukhanov and G. V. Chibisov, JETP Lett. 33 (1981) 532 and: Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549 [spire:170051]

  • Aleksei Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett. 37 (1983) 66-69 [spire:187801]

  • Aleksei Starobinsky, The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy, Sov. Astron. Lett. 9 (1983) 302 [spire:199078]

The experimental data supporting the model is due to

See also

  • Debika Chowdhury, Jerome Martin, Christophe Ringeval, Vincent Vennin, Inflation after Planck: Judgment Day (arxiv:1902.03951)

  • Swagat S. Mishra, Varun Sahni, Canonical and Non-canonical Inflation in the light of the recent BICEP/Keck results [arXiv:2202.03467]

  • M. Chaichian, A. Ghal’e, M. Oksanen, An alternative approach to the Starobinsky model for inflation scenario [arXiv:2211.05212]

Review and exposition includes

  • Alex Kehagias, Azadeh Moradinezhad Dizgah, Antonio Riotto, Comments on the Starobinsky Model of Inflation and its Descendants, Phys. Rev. D 89, 043527 (2014) (arXiv:1312.1155)

  • Sergei Ketov, PLANCK mission, Starobinsky inflation and its realization in old-minimal supergravity, talk at Kavli IPMU Workshop: SUSY Model Building and Phenomenology, 2-4 December 2013 (pdf)

  • John Ellis, Planck-Compatible Inflationary Models, talk 2013 (pptx)

  • Andrei Linde, Inflationary Cosmology after Planck 2013 (arXiv:1402.0526)

  • Jörg Rachen, The Planck 2015 Results: Cosmology and Fundamental Physics from the Polarised CMB and Other Probes, IMAPP Special Seminar, Nijmegen, Feb.5, 2015

  • Ioannis Dalianis, Features and implications of the plateau inflationary potentials, Planck 2015 conference contribution (arXiv:1602.05026)

  • George P. Efstathiou on behalf of the PLANCK mission, The PLANCK legacy, inflation and the origin of structure in the universe, talk at University of Cambridge, January 28, 2019 (recording from 50:49)

Discussion with more general higher curvature corrections:

  • Gustavo Arciniega, Jose D. Edelstein, Luisa G. Jaime, Towards purely geometric inflation and late time acceleration (arXiv:1810.08166)

  • Gustavo Arciniega, Pablo Bueno, Pablo A. Cano, Jose D. Edelstein, Robie A. Hennigar, Luisa G. Jaimem, Geometric Inflation (arXiv:1812.11187)

Discussion of eternal inflation in Starobinsky-type models

  • Gabriela Barenboim, William Kinney, Wan-Il Park, Eternal Hilltop Inflation, Journal of Cosmology and Astroparticle Physics, Volume 2016, May 2016 (arXiv:1601.08140)

See also:

  • Dhong Yeon Cheong, Hyun Min Lee, Seong Chan Park, Beyond the Starobinsky model for inflation (arXiv:2002.07981)

  • Didam Duniya, Amare Abebe, Alvaro de la Cruz-Dombriz, Peter Dunsby, Imprint of f(R)f(R) gravity in the cosmic magnification [arXiv:2210.09303]

Embedding into supergravity

Discussion of embedding of Starobinsky inflation in supergravity originates in

and is further developed in the following articles:

Embedding into 11d supergravity

Discussion of Starobinsky inflation in 11-dimensional supergravity with its higher curvature corrections included (see there):

Embedding into superstring theory

Embedding of Starobinsky inflation into superstring theory is discussed in

Concerning further higher curvature corrections:

  • Sergei Ketov, Ekaterina O. Pozdeeva, Sergey Yu. Vernov, On the superstring-inspired quantum correction to the Starobinsky model of inflation [arXiv:2211.01546]

On quasi-realistic “flipped” SU ( 5 ) SU(5) -GUT, modeled in 4d heterotic string theory and subsuming realistic Starobinsky-type cosmic inflation:

Last revised on December 22, 2023 at 16:58:31. See the history of this page for a list of all contributions to it.