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As predicted by quantum chromodynamics(QCD), around $T\sim 190$ MeV in theearly universe, the QCD transition occurs during which the quarks are combinedinto the massive hadrons. This process reduces the effective relativisticdegree of freedom, and causes a change in the expansion behavior of theuniverse. Similarly, the $e^+e^-$ annihilation occurred around $T\sim 0.5$ Mevhas the same kind of effect. Besides, the dark energy also drives the presentstage accelerating expansion. We study these combined effects on the relicgravitational waves (RGWs). In our treatment, the QCD transition and the$e^+e^-$ annihilation, each is respectively represented by a short period ofexpansion inserted into the radiation era. Incorporating these effects, theequation of RGWs is analytically solved for a spatially flat universe, evolvingfrom the inflation up to the current acceleration, and the spectrum of RGWs isobtained, covering the whole range of frequency $>10^{-19}$ Hz. It is foundthat the QCD transition causes a reduction of the amplitude of RGWs by $\sim20%$ in the range $>10^{-9} $ Hz, and the $e^+e^-$ annihilation causes areduction $\sim 10%$ in the range $>10^{-12} $ Hz. In the presence of the darkenergy, the combination of the QCD transition and the $e^+e^-$ annihilation,causes a larger reduction of the amplitude by $\sim 30% $ for the range$>10^{-9} $ Hz, which covers the bands of operation of LIGO and LISA. Byanalysis, it is shown that RGWs will be difficult to detect by the presentLIGO, but can be tested by LISA for certain inflationary models.

Modifications by QCD transition and $e^+e^-$ annihilation on analytic spectrum of relic gravitational waves in accelerating universe