Coming to the plots of the slow roll parameters $\epsilon_1, \epsilon_2, \epsilon_3$, respectively, in
Fig. 3 and in the left-hand side of
Fig. 4, we can see the fulfillment of the slow roll conditions, $\epsilon_1, \epsilon_2, \epsilon_3 \ll 1$ in the period, $10^4 \sim 10^7$ $t/t_{\rm Pl}$ (for exact values of the beginning and ending of the inflation see Table IIIB). Thus a slow roll phase of inflation is achieved. Next, our task is to calculate the number of e-folds of expansion generated during this slow roll phase of inflation for various values of the initial values of the scalar field $\phi$ and find out what value of $\phi_B$ gives rise to 60 e-folds of inflation. To this purpose we use $\ddot{a}=0$ to obtain the initial time and $\epsilon_1 = 1$ to obtain the end time of inflation. Since by definition $N_{\rm inf}= {\rm ln}({a_f}/{a_i})$, therefore, once we have the knowledge of starting and ending times of inflation, it is straintforward to carry out the calculation of $N_{\rm inf}$ from the right plot of
Fig. 5.
Table 1 shows $N_{\rm inf}$ as a function of $\phi_B$ for the Starobinsky potential in the Jordan frame for a kinetic energy dominated bounce with $\dot{\phi}_B<0$. Here, we begin with an initial value of $\phi_B=125 m_{\rm Pl}$ to achieve a small number of e-folds of expansion of the order of $\simeq 4$. Since this is not adequate to solve the puzzles of the Standard Model of cosmology, which requires $\sim 60$ e-folds of expansion, we increase in big steps the values of $\phi_B$ and see when $60$ is attained. It is found that $60.83$ number of e-folds are generated for initial condition $\phi_B=395m_{\rm Pl}$ and hence it is considered as the critical value of the initial value we are seeking for and the critical energy density at the bouncing to attain this is $r_w^c = 1.15075 \times 10^6$. Then, we continue this process to investigate the effects of $\phi_{B}$ on $N_{\rm inf}$ up to $\phi_B \simeq 1000 m_{\rm Pl}$ and plot it in the right-hand side of
Fig. 4. In
Fig. 6 we plot the effective equation of state $w_{\phi}$. Starting with a kinetic energy dominated bounce it shows the three phases of evolution, namely bouncing phase, transition from quantum bounce to classical universe and the slow roll inflation. During the bouncing phase as the kinetic energy dominates over potential, the effective equation of state $w_{\phi}\approx1$, while in the slow roll inflation phase $w_{\phi}$ becomes $-1$, which reflects the fact that the universe is dominated by the potential energy. We also note the universality of the solution in the bouncing phase. Finally in
Fig. 7 we plot the effective energy density due to scalar field $\rho_{\phi}$ on the right-hand side and the ratio $r_w$ on the left-hand side. The plot of $\rho_{\phi}$ in
Fig. 7 shows the fact that the energy density becomes maximal at the quantum bounce and then dilutes away as the universe expands to large volumes as expected. Whereas the right-hand side of
Fig. 7 demonstrates the dominance of kinetic over the potential energy throughout the evolution of the universe starting from the quantum bounce. As we initialize the universe with a kinetic energy dominated quantum bounce the ratio $r_w(t_B)\gg1$, and then it transits to $r_w\ll1$ as the universe becomes potential dominated in the slow roll inflation era.