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.

17 pages, 10 figures, Accepted by PRD

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