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 $\left(N\right)$ are calculated. Our results show that most classical slow-roll trajectories which intersect the Planck surface have $N<60$, and ${\varphi }_{\mathrm{UV}}>5.5M_{\mathrm{Pl}}^{\ast }$ is required for $N>60$. It is found that for ${\varphi }_{\mathrm{UV}}\in [5.22,5.50]M_{\mathrm{Pl}}^{\ast }$ which satisfies the constraint on the spectral index, $n_s=0.9658\pm 0.0040\left(68\text{\%}\mathrm{CL}\right)$, the expectation value $〈N〉\simeq 3.5-4$ 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 $\left(N\right)=4.025,4.336$ for $\alpha =10^{-4},6.5\times 10^{-5}$ respectively. In the Starobinsky model even at very large inflaton cutoff ${\varphi }_{\mathrm{UV}}$ 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.