amount of the gravitational waves $\Omega_{\mathrm{gw},0}$.

Peak frequency of gravitational waves
-------------------------------------

As a result of the lattice simulation, it is found that the spectrum of the gravitational waves has a sharp peak at $f_{\mathrm{gw},p}\simeq k_{\mathrm{peak}}/2\pi$, $$f_{\mathrm{gw},p}
=
\frac{0.35m_{\sigma}}{2\pi}.$$ This frequency decreases by the cosmic expansion and the present frequency $f_{\mathrm{gw},0}$ is given by $\left(a_{p}/a_0\right)f_{\mathrm{gw},p}$, where $a_{p}$ is the scale factor at the production of the gravitational waves. We assume that the scale factor $a_{p}$ is approximately equal to the scale factor at the end of the smooth hybrid inflation $a_{\mathrm{H}}$. Then, the ratio $a_0/a_{p}$ is given by $$\frac{a_0}{a_{p}}
\simeq
\frac{a_0}{a_{\mathrm{H}}}
=
8.3\times10^{33}~e^{N_{\mathrm{new}}}
\left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right)^{-1/3}
\left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{4/3}.$$ Thus, the present peak frequency of the gravitational waves $f_{\mathrm{gw},0}$ is given by $$\label{eq:4:fgw0}
f_{\mathrm{gw},0}
=
6.8\times10^7~e^{-N_{\mathrm{new}}}
\left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right)^{1/3}
\left(\frac{\mu}{M_{\mathrm{pl}}}\right)^{2/3}
\left(\frac{\mu M}{M_{\mathrm{pl}}^2}\right)^{-1/2}\mathrm{Hz}.$$

Relic density of gravitational waves
------------------------------------

In this subsection, we consider the effect of the cosmic expansion on the energy density of the gravitational waves. Just after the production, the energy density of the gravitational waves decreases as $a^{-4}$, like radiation. After the wavelength becomes larger than the horizon during the new inflation (horizon exit), the amplitude of the gravitational waves stops decreasing, and the energy density decreases as $a^{-2}$. After the end of 