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.[1] 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.[2] However, it was soon realized that the inflation was essentially controlled by the contribution from a squared Ricci scalar in the effective action[3]

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

where ${\displaystyle \kappa =8\pi G/c^{4}}$ and ${\displaystyle R}$ is the Ricci scalar. This action corresponds to the potential ${\displaystyle V(\phi )=\Lambda ^{4}\left(1-e^{-{\sqrt {2/3}}\phi /M_{p}}\right)^{2}}$ 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.[4][5]

Starobinsky inflation gives a prediction for the observables of the spectral tilt ${\displaystyle n_{s}}$ and the tensor-scalar ratio ${\displaystyle r}$: ${\displaystyle n_{s}=1-{\frac {2}{N}},\quad r={\frac {12}{N^{2}}},}$[6] where ${\displaystyle N}$ is the number of e-foldings since the horizon crossing. As ${\displaystyle 50<N<60}$, 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).[6]

• Alternatives to general relativity
• Inflation (cosmology)