Contents

Contents

Idea

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 2 R^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 r r , specifically it predicts

r ∼ 12 N 2 r \sim \frac{12}{N^2}

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

Models of this type are favored by experimental results (PlanckCollaboration 13, BICEP2-Keck-Planck 15, PlanckCollaboration 15) which give a low upper bound on r r around 0.1 0.1 (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 2 R^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 (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 2 R^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 2 R^2 in terms of evidence are brane inflation and exponential inflation, which have one more parameter than R 2 R^2 (PlanckCollaboration 15XX, p. 18)

See (Ellis 13, Ketov 13, Efstathiou 2019, 50:49) for brief survey and see (Kehagias-Dizgah-Riotto 13) for more details. There it is argued that the other types of models which also 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 4 R^4 higher curvature correction (see there) is considered and claimed to yield inflation with “graceful exit” and dynamical KK-compactification:

References

General

The model is due to

Aleksei Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91, 99 (1980); (doi:10.1016/0370-2693(80)90670-X)

and the analysis of its predictions is due to

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

Aleksei Starobinsky, Sov. Astron. Lett. 9, 302 (1983).

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)

Review and exposition includes

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)

Embedding into supergravity

Discussion of embedding of Starobinsky inflation in supergravity originates in

S. Cecotti, Higher derivative supergravity Is equivalent to standard supergravity coupled to matter, Phys. Lett. B 190, 86 (1987).

S. Cecotti, Sergio Ferrara, M. Porrati and S. Sabharwal, Nucl. Phys. B 306, 160 (1988).

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):

Katrin Becker, Melanie Becker, Supersymmetry Breaking, M-Theory and Fluxes, JHEP 0107:038,2001 (arXiv:hep-th/0107044)

Kazuho Hiraga, Yoshifumi Hyakutake, Inflationary Cosmology via Quantum Corrections in M-theory (arXiv:1809.04724)

Kazuho Hiraga, Yoshifumi Hyakutake, Scalar Cosmological Perturbations in M-theory with Higher Derivative Corrections (arxiv:1910.12483)

Embedding into superstring theory

Embedding of Starobinsky inflation into superstring theory is discussed in