An exact analytic spectrum of relic gravitational waves in an accelerating universe

Abstract

An exact analytic calculation is presented for the spectrum of relic gravitational waves in the scenario of accelerating Universe $\Omega_{\Lambda}+\Omega_m = 1$. The spectrum formula contains explicitly the parameters of acceleration, inflation, reheating, and the (tensor/scalar) ratio, so that it can be employed for a variety of cosmological models. We find that the spectrum depends on the behavior of the present accelerating expansion. The amplitude of gravitational waves for the model $\Omega_{\Lambda}=0.65$ is about $\sim 50 %$ greater than that of the model $\Omega_{\Lambda}=0.7$, an effect accessible to the designed sensitivities of LIGO and LISA. The spectrum sensitively depends on the inflationary models with $a(\tau) \propto |\tau|^{1+\beta}$, and a larger $\beta$ yields a flatter spectrum, producing more power. The current LIGO results rule out the inflationary models of $\beta \geq -1.8$. The LIGO with its design sensitivity and the LISA will also be able to test the model of $\beta=-1.9$. We also examine the constraints on the spectral energy density of relic gravitational waves. Both the LIGO bound and the nucleosynthesis bound point out that the model $\beta=-1.8$ is ruled out, but the model $\beta=-2.0$ is still alive. The exact analytic results also confirm the approximate spectrum and the numerical one in our previous work.