Detecting relic gravitational waves in the CMB: Optimal parameters and their constraints

Abstract

The prospect of detecting relic gravitational waves, through their imprint in the cosmic microwave background radiation, provides an excellent opportunity to study the very early Universe. In the simplest viable theoretical models the relic gravitational wave background is characterized by two parameters, the tensor-to-scalar ratio $r$ and the tensor spectral index ${n}_{t}$. In this paper, we analyze the potential joint constraints on these two parameters, $r$ and ${n}_{t}$, using the data from the upcoming cosmic microwave background radiation experiments. Introducing the notion of the best-pivot multipole ${\ensuremath{\ell}}_{t}^{*}$, we find that at this pivot multipole the parameters $r$ and ${n}_{t}$ are uncorrelated, and have the smallest variances. We derive the analytical formulas for the best-pivot multipole number ${\ensuremath{\ell}}_{t}^{*}$, and the variances of the parameters $r$ and ${n}_{t}$. We verify these analytical calculations using numerical simulation methods, and find agreement to within 20%. The analytical results provide a simple way to estimate the detection ability for the relic gravitational waves by the future observations of the cosmic microwave background radiation.