Reconstructing inflationary potential from BICEP2 and running of tensor modes

Abstract

In this paper we will analyse the constraints on a sub-Planckian excursion of a single inflaton field, which would yield a large tensor to scalar ratio, while explaining the temperature anisotropy of the cosmic microwave background (CMB) radiation. In particular, our attempt will be to reconstruct the inflationary potential by constraining, $V(ϕ_0), V^{\prime}(ϕ_0), V^{\prime\prime}(ϕ_0), V^{\prime\prime\prime}(ϕ_0)$ and $V^{\prime\prime\prime\prime}(ϕ_0)$, in the vicinity of the field, $ϕ_0\ll M_p$, and the field displacement, $Δϕ\ll M_p$, where $M_p$ is the reduced Planck mass. We will provide, for the first time, a set of new {\it consistency} relationships for sub-Planckian excursion of the inflaton field, which would help us to differentiate sub-versus-super Planckian models of inflation. For a generic single field inflationary potential, we will be able to put a stringent bound on the potential energy density: $2.07\times10^{16} {\rm GeV}\leq\sqrt[4]{V_{\star}}\leq 2.40\times 10^{16} {\rm GeV}$, where inflation can occur on the flat potential within, $0.066 \leq\frac{\left |Δϕ\right|}{M_p}\,\leq 0.092$, for the following observational constraints: (Planck+WMAP-9+high L+BICEP2). We then provide a prediction for the spectral tilt ($n_{T}$), running ($α_{T}$) and running of running ($κ_{T}$) of the tensor modes within the window, $-0.019<n_{T}<-0.033$, $-2.97\times 10^{-4}<α_{T}(=dn_{T}/d\ln k)<2.86\times 10^{-5}$,and $-0.11\times 10^{-4}<κ_{T}(=d^{2}n_{T}/d\ln k^{2})<-3.58\times 10^{-4}$, in a model independent way. We also provide a simple example of an {\it inflection-point} model of inflation and reconstruct the potential in a model independent way to match the current observations.