Starobinsky inflation

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.{{cite journal | author = Starobinsky, A. A. |title= Spectrum Of Relict Gravitational Radiation And The Early State Of The Universe |journal = Journal of Experimental and Theoretical Physics Letters |volume=30 |page= 682 |date=December 1979 |bibcode=1979JETPL..30..682S}}; {{cite journal |title=Spectrum of relict gravitational radiation and the early state of the universe |bibcode=1979ZhPmR..30..719S |journal = Pisma Zh. Eksp. Teor. Fiz. (Soviet Journal of Experimental and Theoretical Physics Letters) |page= 719 |date=December 1979 |volume=30 |author=Starobinskii, A. A. }} 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.{{cite journal |doi=10.1016/0370-2693(80)90670-X |title=A new type of isotropic cosmological models without singularity |journal=Physics Letters B |volume=91 |pages=99–102 |year=1980 |last1=Starobinsky |first1=A.A |issue=1 |bibcode=1980PhLB...91...99S }} 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{{cite journal |title=The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy |journal=Sov. Astron. Lett. |volume=9 |pages=302 |year=1983 |last1=Starobinsky |first1=A.A.|issue=9 |pmid= |bibcode= }}{{cite journal |doi=10.1103/PhysRevD.32.2511 |title=Classical and quantum cosmology of the Starobinsky inflationary model |journal=Physical Review D |volume=32 |pages=2511–2521 |year=1985 |last1=Vilenkin |first1=Alexander|issue=10 |pmid=9956022 |bibcode=1985PhRvD..32.2511V }}

:$$

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

where $\backslash kappa=8\backslash pi\; G/c^4$ and $R$ is the Ricci scalar. This action corresponds to the potential{{cite journal |doi=10.1103/PhysRevD.37.858 |title=Inflation as a Transient Attractor in R**2 Cosmology |journal=Physical Review D |volume=37 |pages=858–882 |year=1988 |last1=Maeda |first1=Ken-Ichi |issue= 4|pmid= 9958750|bibcode= }}{{cite journal |doi=10.1142/S0217751X88000266 |title=Quantum Fluctuations and Eternal Inflation in the R**2 Model |journal=Int. J. Mod. Phys. A |volume=3 |pages=617–629 |year=1988 |last1=Coule |first1=David H. |last2=Mijic |first2=Milan B.|issue= 3|pmid= |bibcode= }}

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 $R^2$ 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.{{cite journal |doi=10.1103/PhysRevD.63.083504 |title=Trace anomaly driven inflation |journal=Physical Review D |volume=63 |pages=083504 |year=2001 |last1=Hawking |first1=Stephen|last2=Hertog |first2=Thomas|last3=Reall |first3=Harvey|issue=8 |arxiv=hep-th/0010232 |bibcode=2001PhRvD..63h3504H |s2cid=8750172 }}{{cite journal |doi=10.1140/epjc/s10052-016-4390-4 |title=From stable to unstable anomaly-induced inflation |journal=The European Physical Journal C |volume=76 |pages=1–14 |year=2016 |last1=de Paula Netto |first1=Tibério|last2=Pelinson |first2=Ana|last3=Shapiro |first3=Ilya|last4=Starobinsky |first4=Alexei|issue=10 |arxiv=1509.08882 |bibcode=2016EPJC...76..544N |doi-access=free }}

Starobinsky inflation gives a prediction for primordial observables, e.g., the spectral tilt $n\_s$ and the tensor-scalar ratio $r$: $$

n_s \approx 1 - \frac{2}{N}, \quad r \approx \frac{12}{N^2},

{{cite journal |arxiv=1807.06211 |title=Planck2018 results |year=2020 |doi=10.1051/0004-6361/201833887 |last1=Akrami |first1=Y. |last2=Arroja |first2=F. |last3=Ashdown |first3=M. |last4=Aumont |first4=J. |last5=Baccigalupi |first5=C. |last6=Ballardini |first6=M. |last7=Banday |first7=A. J. |last8=Barreiro |first8=R. B. |last9=Bartolo |first9=N. |last10=Basak |first10=S. |last11=Benabed |first11=K. |last12=Bernard |first12=J.-P. |last13=Bersanelli |first13=M. |last14=Bielewicz |first14=P. |last15=Bock |first15=J. J. |last16=Bond |first16=J. R. |last17=Borrill |first17=J. |last18=Bouchet |first18=F. R. |last19=Boulanger |first19=F. |last20=Bucher |first20=M. |last21=Burigana |first21=C. |last22=Butler |first22=R. C. |last23=Calabrese |first23=E. |last24=Cardoso |first24=J.-F. |last25=Carron |first25=J. |last26=Challinor |first26=A. |last27=Chiang |first27=H. C. |last28=Colombo |first28=L. P. L. |last29=Combet |first29=C. |last30=Contreras |first30=D. |journal=Astronomy & Astrophysics |volume=641 |pages=A10 |s2cid=119089882 |display-authors=1 }}

where $N\_\backslash ast$ is the number of e-foldings since the horizon crossing. As

• Alternatives to general relativity
• Inflation (cosmology)

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Category:Inflation (cosmology)

Category:General relativity