Editing Cosmic inflation (section)

====Starobinsky inflation====
{{Main|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) gravity|{{math|''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.<ref>
{{cite journal
 |last=Starobinsky |first=A.A.
 |date=December 1979
 |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
 |bibcode=1979JETPL..30..682S
}}<br/>
{{cite journal
 |author=Starobinskii, A.A.
 |date=December 1979
 |title=Spectrum of relict gravitational radiation and the early state of the universe
 |journal=[[Pisma Zh. Eksp. Teor. Fiz.]]
 |volume=30  |page= 719
 |bibcode=1979ZhPmR..30..719S
}}
</ref> 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 used the action 
:<math>
S=\frac{1}{2} \int d^4 x \left(R + \frac{R^2}{6M^2} \right)
</math>
which corresponds to the potential
:<math>\quad V(\phi)=\Lambda^4 \left(1 - e^{-\sqrt{2/3} \phi/M^2_p} \right)^2 </math>
in the Einstein frame. This results in the observables: <math> n_s=1 - \frac{2}{N}, \qquad r=\frac{12}{N^2}.</math><ref>
{{cite journal
 |last1=Ade |first1=P.A.R.
 |display-authors=etal
 |year=2016
 |title=Planck 2015 results. XX. Constraints on inflation
 |journal=[[Astronomy & Astrophysics]]
 |volume=594 |page=17
 |arxiv=1502.02114 |doi=10.1051/0004-6361/201525898
 |bibcode=2016A&A...594A..20P |s2cid=119284788
}}
</ref>