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

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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 $\alpha_t>0.01364$.