On the Three Primordial Numbers

Abstract

Cosmological observations have provided us with the measurement of just three numbers that characterize the very early universe: $ 1-n_{s} $, $ N $ and $\ln\Delta_R^2$. Although each of the three numbers individually carries limited information about the physics of inflation, one may hope to extract non-trivial information from relations among them. Invoking minimality, namely the absence of ad hoc large numbers, we find two viable and mutually exclusive inflationary scenarios. The first is the well-known inverse relation between $1- n_{s} $ and $ N $. The second implies a new relation between $ 1-n_{s} $ and $\ln\Delta_R^2$, which might provide us with a handle on the beginning of inflation and predicts the intriguing $\textit{lower}$ bound on the tensor-to-scalar ratio $ r> 0.006 $ ($ 95\% $ CL).