of $N_{\mathrm{new}},~\mu,~v,~M$ and $T_{\mathrm{R}}$. Furthermore, in (\[eq:Nn\]), (\[eq:mu\]) and (\[eq:M\]), $N_{\mathrm{new}},~\mu,~M$ are written in terms of $\sigma_{i}$ and $n_s$. Here, we assume that the energy scale of the new inflation $v$ is approximately equal to that of the hybrid inflation $\mu$, and the reheating temperature $T_{\mathrm{R}}$ is $10^9\mathrm{GeV}$. For $\mu \simeq v$ the density of the gravitational waves is maximal so that our estimation gives the upper bound. With these assumptions, $f_{\mathrm{gw},0}$ and $\Omega_{\mathrm{gw},0}$ are written only by $\sigma_{i}$ and $n_s$ as $$\ln f_{\mathrm{gw},0}
=
-39.3-\frac{3}{2}\ln\sigma_{i} 
-\frac{1}{4}\ln\left[3\sigma_{i}^2-\left(n_s-1\right)\right]+N_{\mathrm{H}},$$ and $$\left\{
\begin{array}{ll}
\ln\Omega_{\mathrm{gw},0,\mathrm{no}}
&\hspace{-2ex}=
\ln\alpha -250 +3\ln\sigma_{i} 
+\frac{1}{2}\ln\left[3\sigma_{i}-\left(n_s-1\right)\right]
-2\ln\left[8\sigma_{i}^2-\left(n_s-1\right)\right]
+4N_{\mathrm{H}}\\[0.5\intextsep]
\ln\Omega_{\mathrm{gw},0,\mathrm{reh}}
&\hspace{-2ex}=
\ln\alpha +70.1 +12\ln\sigma_{i} 
+2\ln\left[3\sigma_{i}-\left(n_s-1\right)\right]
-2N_{\mathrm{H}}\\[0.5\intextsep]
\ln\Omega_{\mathrm{gw},0,\mathrm{ra}}
&\hspace{-2ex}=
\ln\alpha -15.12 +9\ln\sigma_{i}
+\frac{3}{2}\ln\left[3\sigma_{i}^2-\left(n_s-1\right)\right]
\\[0.5\intextsep]
\ln\Omega_{\mathrm{gw},0,\mathrm{ma}}
&\hspace{-2ex}=
\ln\alpha -12.5 +12\ln\sigma_{i}
+2\ln\left[3\sigma^2-\left(n_s-1\right)\right] -2N_{\mathrm{H}}
\end{array}
\right.$$ where domain of $\sigma_{i}$ is constrained as shown in Fig. \[fig:simami\]. Thus, for a given spectral index $n_s$ we can obtain $\Omega_{\mathrm{gw},0}$ as a function of the peak frequency $f_{\mathrm{gw},0}$. Fig. \[fig:Omega\_fg\] shows this relation. At low peak frequencies $f_{{\rm gw},0} \lesssim 40~\rm{Hz}$, the horizon entry occurs during the radiation dominated era. At high peak frequencies $40~{\rm Hz} \lesssim f_{{\rm gw},0} \lesssim 10^5~{\rm Hz}$, the horizon entry occurs during the reheating era. At higher peak frequencies $f_{{\rm gw},0} \gtrsim 