case “Matter time horizon entry". In each case, we can write down the $\Omega_{\mathrm{gw},0}$, as $$\label{eq:4:Omega0}
\left\{
\begin{array}{ll}
\hspace{-0.5\intextsep}\mathrm{No}~\mathrm{exit} &\hspace{-0.5\intextsep}
h^2\Omega_{\mathrm{gw},0,\mathrm{no}}
=
4.5\times10^{-17}~\mathrm{e}^{-4N_{\mathrm{new}}}
\left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right)^{4/3}
\left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{-4/3}
\Omega_{\mathrm{gw},\mathrm{p}},\\[1.0\intextsep]
&\hspace{-10ex}
    {\rm for}~~
    \mathrm{e}^{N_{\rm new}} <
    1.7 ~ \left(\frac{v}{\mu}\right)^{-2/3}
    \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2},
    \\[1.0\intextsep]
\hspace{-0.5\intextsep}\mathrm{Reh.}~\mathrm{entry} &\hspace{-0.5\intextsep}
h^2\Omega_{\mathrm{gw},0,\mathrm{reh}}
=
6.4\times10^{-19}~\mathrm{e}^{2N_{\mathrm{new}}}
\left(\frac{v}{\mu}\right)^{4}
\left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right)^{4/3}
\left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{-4/3}
\left(\frac{\mu M}{M_{\mathrm{pl}}^2}\right)^3
\Omega_{\mathrm{gw},\mathrm{p}},\\[1.0\intextsep]
&\hspace{-10ex}{\rm for}~~
    1.7 ~ \left(\frac{v}{\mu}\right)^{-2/3}
    \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2}
        <
        \mathrm{e}^{N_{\rm new}}
        <
        1.6\times 10^6 \left(
            \frac{T_{\mathrm{R}}}{10^{9}{\rm GeV}}
        \right)^{-2/3}
     \left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{2/3}
     \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2},
        \\[1.0\intextsep]
\hspace{-0.5\intextsep}\mathrm{Ra.}~\mathrm{entry} &\hspace{-0.5\intextsep}
h^2\Omega_{\mathrm{gw},0,\mathrm{ra}}
=
4.0\times10^{-6}
\left(\frac{v}{\mu}\right)^{4}
\left(\frac{\mu M}{M_{\mathrm{pl}}^2}\right)^2
\Omega_{\mathrm{gw},\mathrm{p}},\\[1.0\intextsep]
&\hspace{-10ex}{\rm for}~~
    1.6\times 10^6 \left(
            \frac{T_{\mathrm{R}}}{10^{9}{\rm GeV}}
        \right)^{-2/3}
     \left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{2/3}
     \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2}
        <
        \mathrm{e}^{N_{\rm new}}
        \\[.5\intextsep]
        &\hspace{10ex}
        <
            3.2\times 10^{24} ~
      \left(\frac{T_{\rm R}}{10^{9}{\rm GeV}}\right)^{1/3}
      \left(\frac{\mu}{M_{\rm pl}}\right)^{2/3}
      \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2},
        \\[1.0\intextsep]
\hspace{-0.5\intextsep}\mathrm{Ma.}~\mathrm{entry} &\hspace{-0.5\intextsep}
h^2\Omega_{\mathrm{gw},0,\mathrm{ma}}
=
8.7\times10^{-55}~\mathrm{e}^{2N_{\mathrm{new}}}
\left(\frac{v}{\mu}\right)^{4}
\left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right)^{-2/3}
\left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{-4/3}
\left(\frac{\mu M}{M_{\mathrm{pl}}^2}\right)^3
\Omega_{\mathrm{gw},\mathrm{p}},\\[1.0\intextsep]
&\hspace{-10ex}{\rm for}~~
    3.2\times 10^{24} ~
  \left(\frac{T_{\rm R}}{10^{9}{\rm GeV}}\right)^{1/3}
  \left(\frac{\mu}{M_{\rm pl}}\right)^{2/3}
  \left(\frac{\mu M}{M_{\rm pl}^2}\right)^{-1/2}
    <
    \mathrm{e}^{N_{\rm new}}.
\end{array}
\right.$$

$f_{\mathrm{gw},0}$ and $\Omega_{\mathrm{gw},0}$
------------------------------------------------

![ Relations between $\Omega_{\mathrm{gw},0}$ and $f_{\mathrm{gw},0}$ for given spectral indices $n_s= -0.04, -0.038, -0.03,  ~0$ and $0.1$. []{data-label="fig:Omega_fg"}](./pdf/Omega0.pdf){width="120mm"}

In this subsection, we calculate the relation between the amount of the gravitational waves today $\Omega_{\mathrm{gw},0}$ and the peak frequency of the gravitational wave spectrum $f_{\mathrm{gw},0}$. In (\[eq:4:fgw0\]) and (\[eq:4:Omega0\]), $f_{\mathrm{gw},0}$ and $\Omega_{\mathrm{gw},0}$ are written in terms 