Simple realization of inflaton potential on a Riemann surface

Abstract

The observation of the B-mode in the cosmic microwave background radiation combined with the so-called Lyth bound suggests the trans-Planckian variation of the inflaton field during inflation. Such a large variation generates concerns over inflation models in terms of the effective field theory below the Planck scale. If the inflaton resides in a Riemann surface and the inflaton potential is a multivalued function of the inflaton field when it is viewed as a function on a complex plane, the Lyth bound can be satisfied while keeping field values in the effective field theory within the Planck scale. We show that a multivalued inflaton potential can be realized starting from a single-valued Lagrangian of the effective field theory below the Planck scale.

Figures (2)

• Figure 1: A schematic picture of the inflaton potential on a Riemann surface. Left) A Riemann surface for the function $\phi^{1/4}$. We assume that the inflaton field takes a value on this surface. Right) Inflaton potential $(V \propto \phi^{1/4})$ on the Riemann surface.
• Figure 2: An inflaton trajectory on phases of $S$ and $\phi$ for $N=5$. The inflaton potential in Eq. (\ref{['eq:effective']}) is realized along the trajectory shown by the solid line. Arrowheads indicate the height of the potential along the trajectory, and the field values which maximize and minimize the potential in Eq. (\ref{['eq:effective']}) are denoted by points. Here, we have shifted the phases of $\phi$ and $S$ so that ${\rm arg} \phi = {\rm arg} S=0$ is the minimum of the potential. The shaded regions have large potentials by the last term in Eq. (\ref{['eq:scalar potential']}).