Lyth bound, eternal inflation, and future cosmological missions

In this paper we provide a new expression for the variation of the inflaton field $\mathrm{\Delta }\varphi$ during the horizon crossing epoch in the context of single field slow roll inflationary models. Such an expression represents a generalization of the well-known Lyth bound. We also explore the consequences of a detection of permille order of the tensor-to-scalar ratio amplitude, $r$, as well as an improvement on the estimation of the scalar spectral index, $n_s$ and its running ${\alpha }_s$, by the upcoming cosmic microwave background (CMB) polarization experiments that will provide plausible constraints on the quantity $\mathrm{\Delta }\varphi$ during the horizon exit moment. In addition we discuss the relation between the local variation of the field and the possibilities of an eternal inflation. The results of the analysis are completely model independent.

Figure 1

Distribution functions for the minimum excursion of the scalar field during inflation, $\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}$, for the three different slow roll expansions. Here, we have supposed a tensor-to-scalar ratio amplitude of the order of ${\mu }_r\sim 3\times {10}^{-3}$ with $n_s=0.9680$, ${\alpha }_s=-0.0033$.

Figure 2

Distribution functions for the variable $\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}$ for the three different slow roll expansions. In this case, we have supposed a detection of primordial GW of the order of ${\mu }_r\sim 5\times {10}^{-3}$ with $n_s=0.9680$, ${\alpha }_s=-0.0033$.

Figure 3

Dispersion plot for the ratio $\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}({\mathrm{\Pi }}_1)/\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}({\mathrm{\Pi }}_0)$ in the case of a detection of $r\sim 3\times {10}^{-3}$ with $n_s=0.9680$, ${\alpha }_s=-0.0033$.

Figure 4

Dispersion plot for the ratio $\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}({\mathrm{\Pi }}_2)/\mathrm{\Delta }{\varphi }_{\mathrm{H}.\mathrm{c}.}({\mathrm{\Pi }}_1)$ in the case of a detection of $r\sim 3\times {10}^{-3}$ with $n_s=0.9680$, ${\alpha }_s=-0.0033$.

Figure 5

Third order Lyth bound as a function of the running ${\alpha }_s$. In this case we consider $n_s\sim 0.9680$ with a threshold value ${\alpha }_s\sim -1.2842\times {10}^{-5}$. Possible eternal inflation phases are associated with smaller variations of the field at horizon crossing of quantum fluctuations.