The effective phase space and $e$-folding of the Starobinsky and extended Starobinsky model of inflation

Abstract: For zero spatial curvature, cosmological phase space of Starobinsky and extended Starobinsky inflationary model show three apparent attractors; the fixed angle attractor in the large field limit, the final attractor representing reheating phase in the small field region, and the apparent attractor corresponding to the slow-roll condition connecting between the large-field and small-field region. To consider the total $e$-folding likelihood of the model, Remmen-Carroll conserved measure is constructed and normalized. Using the measure, the total e-folding number $N$ and its expectation value $\left\langle N \right\rangle$ are calculated. Our results show that most classical slow-roll trajectories which intersect the Planck surface have $N<60$, and $\phi_{\rm UV}>5.5 M^{*}_{\rm Pl}$ is required for $N>60$. It is found that for $\phi_{\rm UV}\in [5.22,5.50]M^{*}_{\rm Pl}$ which satisfies the constraint on the spectral index, $n_s = 0.9658 \pm 0.0040\,\,\,(68\%\,\,{\rm CL})$, the expectation value $\langle N \rangle \simeq 3.5 - 4$ for trajectories intersecting the Planck surface in the Starobinsky model. For extended Starobinsky model with additional $R^3$ term parametrized by a coupling parameter $\alpha$, the expectation value when inflation starts from the top of the potential shifts to $\left\langle N \right\rangle = 4.025,4.336$ for $\alpha = 10^{-4},6.5\times 10^{-5}$ respectively. In the Starobinsky model even at very large inflaton cutoff $\phi_{\rm UV}$ where the field value is super-Planckian, the energy density from the (saturating) inflaton potential and the Hubble parameter are still sub-Planckian and therefore the inflation occurs within the semi-classical regime.