Starobinsky inflation

Starobinsky inflation is a modification of general relativity used to explain cosmological inflation. It was the first model to describe how the universe could have gone through an extremely rapid period of exponential expansion. ^{[1] [2]}

History

In the Soviet Union, Alexei Starobinsky noted that quantum corrections to general relativity should be important for the early universe. These generically lead to curvature-squared corrections to the Einstein–Hilbert action and a form of f(R) modified gravity. The solution to Einstein's equations in the presence of curvature squared terms, when the curvatures are large, leads to an effective cosmological constant. Therefore, he proposed that the early universe went through an inflationary de Sitter era. ^[3] This resolved the cosmology problems and led to specific predictions for the corrections to the microwave background radiation, corrections that were then calculated in detail. Starobinsky originally used the semi-classical Einstein equations with free quantum matter fields. ^[4] However, it was soon realized that the late time inflation which is relevant for observable universe was essentially controlled by the contribution from a squared Ricci scalar in the effective action ^{[5] [6]}

${\displaystyle S={\frac {1}{2\kappa }}\int \left(R+{\frac {R^{2}}{6M^{2}}}\right){\sqrt {\vert g\vert }}\,\mathrm {d} ^{4}x,}$

where ${\displaystyle \kappa =8\pi G/c^{4}}$ and ${\displaystyle R}$ is the Ricci scalar. This action corresponds to the potential ^{[7] [8]}

${\displaystyle V(\phi )=\Lambda ^{4}\left(1-e^{-{\sqrt {2/3}}\phi /M_{p}}\right)^{2}}$

in the Einstein frame. As a result, the inflationary scenario associated to this potential or to an action including an ${\displaystyle R^{2}}$ term are referred to as Starobinsky inflation. To distinguish, models using the original, more complete, quantum effective action are then called (trace)-anomaly induced inflation. ^{[9] [10]}

Observables

Starobinsky inflation gives a prediction for primordial observables, e.g., the spectral tilt ${\displaystyle n_{s}}$ and the tensor-scalar ratio ${\displaystyle r}$: ${\displaystyle n_{s}\approx 1-{\frac {2}{N}},\quad r\approx {\frac {12}{N^{2}}},}$ ^[11] where ${\displaystyle N_{\ast }}$ is the number of e-foldings since the horizon crossing. As ${\displaystyle 50<N_{\ast }<60}$, these are compatible with experimental data, with 2018 CMB data from the Planck satellite giving a constraint of ${\displaystyle r<0.064}$ (95% confidence) and ${\displaystyle n_{s}=0.9649\pm 0.0042}$ (68% confidence). ^[11] The model also gives precise predictions to higher order observables, such as a negative running for the scalar spectral tilt: ${\displaystyle \alpha _{s}\approx -{\frac {1}{2}}(n_{s}-1)^{2}+{\frac {5}{48}}(n_{s}-1)^{3}}$, and a negative tensor tilt: ${\displaystyle n_{t}\approx -{\frac {3}{8}}(n_{s}-1)^{2}+{\frac {5}{16}}(n_{s}-1)^{3}}$. ^[12]

See also

• Alternatives to general relativity
• Inflation (cosmology)

References

1. Gribbin, John (1996). Companion to the cosmos. Boston: Little, Brown and Company. p. 221. ISBN   978-8-17-371245-6 – via Internet Archive.
2. Clery, Daniel (2014). "Pioneering the theory of cosmic inflation". Kavli Prize . Kavli Foundation. Archived from the original on 11 January 2023. Retrieved 13 January 2024.
3. Starobinsky, A. A. (December 1979). "Spectrum Of Relict Gravitational Radiation And The Early State Of The Universe". Journal of Experimental and Theoretical Physics Letters. 30: 682. Bibcode:1979JETPL..30..682S.; Starobinskii, A. A. (December 1979). "Spectrum of relict gravitational radiation and the early state of the universe". Pisma Zh. Eksp. Teor. Fiz. (Soviet Journal of Experimental and Theoretical Physics Letters). 30: 719. Bibcode:1979ZhPmR..30..719S.
4. Starobinsky, A.A (1980). "A new type of isotropic cosmological models without singularity". Physics Letters B. 91 (1): 99–102. Bibcode:1980PhLB...91...99S. doi:10.1016/0370-2693(80)90670-X.
5. Starobinsky, A.A. (1983). "The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy". Sov. Astron. Lett. 9 (9): 302.
6. Vilenkin, Alexander (1985). "Classical and quantum cosmology of the Starobinsky inflationary model". Physical Review D. 32 (10): 2511–2521. Bibcode:1985PhRvD..32.2511V. doi:10.1103/PhysRevD.32.2511. PMID   9956022.
7. Maeda, Ken-Ichi (1988). "Inflation as a Transient Attractor in R**2 Cosmology". Physical Review D. 37 (4): 858–882. doi:10.1103/PhysRevD.37.858. PMID   9958750.
8. Coule, David H.; Mijic, Milan B. (1988). "Quantum Fluctuations and Eternal Inflation in the R**2 Model". Int. J. Mod. Phys. A. 3 (3): 617–629. doi:10.1142/S0217751X88000266.
9. Hawking, Stephen; Hertog, Thomas; Reall, Harvey (2001). "Trace anomaly driven inflation". Physical Review D. 63 (8): 083504. arXiv: hep-th/0010232 . Bibcode:2001PhRvD..63h3504H. doi:10.1103/PhysRevD.63.083504. S2CID   8750172.
10. de Paula Netto, Tibério; Pelinson, Ana; Shapiro, Ilya; Starobinsky, Alexei (2016). "From stable to unstable anomaly-induced inflation". The European Physical Journal C. 76 (10): 1–14. arXiv: 1509.08882 . Bibcode:2016EPJC...76..544N. doi: 10.1140/epjc/s10052-016-4390-4 .
11. Akrami, Y.; et al. (2020). "Planck2018 results". Astronomy & Astrophysics. 641: A10. arXiv: 1807.06211 . doi:10.1051/0004-6361/201833887. S2CID   119089882.
12. Bianchi, Eugenio; Gamonal, Mauricio (14 November 2024). "Primordial power spectrum at N3LO in effective theories of inflation". Physical Review D. 110 (10). doi:10.1103/PhysRevD.110.104032. ISSN   2470-0010.