Inflationary field excursion in broad classes of scalar field models

In single-field slow-roll inflation models, the height and slope of the potential are made to satisfy certain conditions to match with observations. This in turn translates into bounds on the number of $e$-foldings and the excursion of the scalar field during inflation. In this work we consider broad classes of inflationary models to study how much the field excursion starting from the horizon exit to the end of inflation, $\mathrm{\Delta }\varphi$, can vary for the set of inflationary parameters given by Planck. We also derive an upper bound on the number of $e$-foldings between the horizon exit of a cosmologically interesting mode and the end of inflation. We comment on the possibility of having super-Planckian and sub-Planckian field excursions within the framework of single-field slow-roll inflation.

Figure 1

$N$ increases linearly for low values of $p$. Three values of ${\epsilon }_{1*}$ are used for three values of $\beta =2.15\times 10^3$, $1.00\times 10^4$, $4.64\times 10^5$.

Figure 2

$N$ approaches a maximum value for high values of $p$.

Figure 3

$\mathrm{\Delta }\varphi$ increases linearly for low values of $p$ and is sub-Planckian up to a certain value of $p$, beyond which it becomes super-Planckian. Three values of ${\epsilon }_{1*}$ are used to span the entire range of $\beta$ by choosing $\beta =2.15\times 10^3$, $1.00\times 10^4$, $4.64\times 10^5$.

Figure 4

$\mathrm{\Delta }\varphi$ becomes super-Planckian for $p>4$ with the given parameter choice and approaches a maximum value for high values of $p$. Here we have taken $\beta =2.15\times 10^3$, $1.00\times 10^4$, $4.64\times 10^5$, which lead to the values of ${\epsilon }_{1*}$ shown above.