Relic Gravitational Waves with A Running Spectral Index and Its Constraints at High Frequencies

Abstract

We study the impact of a running index $\alpha_t$ on the spectrum of relic gravitational waves (RGWs) over the whole range of frequency $(10^{-18}\sim 10^{10})$ Hz and reveal its implications in RGWs detections and in cosmology. Analytical calculations show that, although the spectrum of RGWs on low frequencies is less affected by $\alpha_t\ne 0$, but, on high frequencies, the spectrum is modified substantially. Investigations are made toward potential detections of the $\alpha_t$-modified RGWs for several kinds of current and planned detectors. The Advanced LIGO will likely be able to detect RGWs with $\alpha_t\ge 0$ for inflationary models with the inflation index $\beta=-1.956$ and the tensor-scalar ratio $r= 0.55$. The future LISA can detect RGWs for a much broader range of ($\alpha_t$, $\beta$, $r$), and will have a better chance to break a degeneracy between them. Constraints on $\alpha_t$ are estimated from several detections and cosmological observations. Among them, the most stringent one is from the bound of the Big Bang nucleosynthesis (BBN), and requires $\alpha_t < 0.008$ rather conservatively for any reasonable ($\beta$, $r$), preferring a nearly power-law spectrum of RGWs. In light of this result, one would expect the scalar running index $\alpha_s$ to be of the same magnitude as $\alpha_t$, if both RGWs and scalar perturbations are generated by the same scalar inflation.