Initial Condition of Relic Gravitational Waves Constrained by LIGO S6 and Multiple Interferometers

The relic gravitational wave (RGW) generated during the inflation depends on the initial condition via the amplitude, the spectral index $n_t$ and the running index $alpha_t$. CMB observations so far have only constrained the tensor-scalar ratio $r$, but not $n_t$ nor $alpha_t$. Complementary to this, the ground-based interferometric detectors working at $sim 10^2$Hz are able to constrain the spectral indices that influence the spectrum sensitively at high frequencies. In this work we give a proper normalization of the analytical spectrum at the low frequency end, yielding a modification by a factor of $sim 1/50$ to the previous treatment. We calculate the signal-noise ratios (SNR) for various ($n_t,alpha_t$) at fixed $r=0.2$ by S6 of LIGO H-L, and obtain the observational upper limit on the running index $alpha_t<0.02093$ (i.e, at a detection rate $95%$ and a false alarm rate $5%$) at the default $(n_t=0,r=0.2)$. This is consistent with the constraint on the energy density obtained by LIGO-Virgo Collaboration. Extending to the four correlated detectors currently running, the calculated SNR improves slightly. When extending to the six correlated detectors of the second-generation in design, the calculated SNR is $sim 10^3$ times over the previous two cases, due to the high sensitivities. RGW can be directly detected by the six 2nd-generation detectors for models with $alpha_t>0.01364$.