Analytic spectrum of relic gravitational waves modified by neutrino free streaming and dark energy

Abstract

We include the effect of neutrino free streaming into the spectrum of relic gravitational waves (RGWs) in the currently accelerating universe. For the realistic case of a varying fractional neutrino energy density and a nonvanishing derivative of the mode function at the neutrino decoupling, the integro-differential equation of RGWs is solved by a perturbation method for the period from the neutrino decoupling to the matter-dominant stage. Incorporating it into the analytic solution of RGWs for the whole history of expansion of the universe, the analytic solution of RGWs is obtained, evolving from inflation up to the current acceleration. The resulting spectrum of RGWs covers the whole range of frequency $({10}^{\ensuremath{-}19}\char21{}{10}^{10})\text{ }\text{ }\mathrm{Hz}$ and improves the previous results. It is found that neutrino free streaming causes a reduction of the spectral amplitude by $\ensuremath{\sim}20%$ in the range $({10}^{\ensuremath{-}16}\char21{}{10}^{\ensuremath{-}10})\text{ }\text{ }\mathrm{Hz}$, and leaves the other portion of the spectrum almost unchanged. This agrees with the earlier numerical calculations. Examination is made on the difference between the accelerating and nonaccelerating models, and our analysis shows that the ratio of the spectral amplitude in the accelerating $\ensuremath{\Lambda}\mathrm{CDM}$ model over that in the CDM model is $\ensuremath{\sim}0.7$, and within the various accelerating models of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}g{\ensuremath{\Omega}}_{m}$ the spectral amplitude is proportional to ${\ensuremath{\Omega}}_{m}/{\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ for the whole range of frequency. Comparison with LIGO S5 run sensitivity shows that RGWs are not yet detectable by the present LIGO, and in the future LISA may be able to detect RGWs in some inflationary models.