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, $V({\varphi }_0),V^{\prime }({\varphi }_0),V^{\prime \prime }({\varphi }_0),V^{\prime \prime \prime }({\varphi }_0)$ and $V^{\prime \prime \prime \prime }({\varphi }_0)$, in the vicinity of the field, ${\varphi }_0\ll M_p$, and the field displacement, $\Delta \varphi \ll M_p$, 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: $2.07\times 10^{16}\mathrm{GeV}\le \sqrt[4]{V_{\star }}\le 2.40\times 10^{16}\mathrm{GeV}$, where inflation can occur on the flat potential within, $0.066\le \frac{|\Delta \varphi |}{M_p}\le 0.092$, for the following observational constraints: (Planck+WMAP-9+high L+BICEP2). We then provide a prediction for the spectral tilt ($n_T$), running (${\alpha }_T$) and running of running (${\kappa }_T$) of the tensor modes within the window, $-0.019<n_T<-0.033$, $-2.97\times 10^{-4}<{\alpha }_T(=d{n}_T/d\ln k)<2.86\times 10^{-5}$,and $-0.11\times 10^{-4}<{\kappa }_T(=d^2{n}_T/d\ln k^2)<-3.58\times 10^{-4}$, 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.

arXiv comment22 pages, 5 figures, A new set of consistency relationships have been added in the discussion, new citations added. arXiv admin note: text overlap with arXiv:1306.4496

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