Starobinsky inflation

Starobinsky inflation is a modification of general relativity in order to explain cosmological inflation.

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. 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

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

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

Observables

Starobinsky inflation gives a prediction for the observables of the spectral tilt $n_{s}$ and the tensor-scalar ratio $r$: $n_{s}=1-{\frac {2}{N}},\quad r={\frac {12}{N^{2}}},$ where $N$ is the number of e-foldings since the horizon crossing. As $50<N<60$, these are compatible with experimental data, with 2018 CMB data from the Planck satellite giving a constraint of $r<0.064$ (95% confidence) and $n_{s}=0.9649\pm 0.0042$ (68% confidence).

See also

• Alternatives to general relativity
• Inflation (cosmology)