of the gravitational waves to the total energy density $\Omega_{\mathrm{gw},p}$ is given by $$\Omega_{\mathrm{gw},p}  
   \equiv \frac{\rho_{\mathrm{gw},p}}{\rho_{\mathrm{tot},p}}
   \sim \left(R_{\mathrm{{peak}}}H_{p}\right)^2.$$ In this paper we assume that the density of the produced gravitational waves is given by the above equation with introducing a numerical constant as $$\label{eq:3:Omegap0}
   \Omega_{\mathrm{gw},p}
   = \alpha\left(R_{\mathrm{{peak}}}H_{p}\right)^2,$$ where $\alpha$ will be determined by the numerical simulations presented in Sec. \[subsec:simulation\]. For the smooth hybrid new inflation model, from (\[eq:2:kpeak\]), $R_{\mathrm{peak}}$ is written as $$R_{\mathrm{peak}}
=
\frac{1}{0.35m_{\sigma}}=\frac{1}{0.35}\sqrt{\frac{M}{8\mu^3}}.$$ Using this relation, we can rewrite (\[eq:3:Omegap0\]) as $$\label{eq:3:Omegap}
   \Omega_{\mathrm{gw},p}
   =0.34~\alpha\left(\frac{\mu M}{M_{\mathrm{pl}}^2}\right).$$

Formalism for calculating gravitational wave spectrum
-----------------------------------------------------

Here, we review the formalism developed by Dufaux et al. [@Dufaux:2007pt] with which we calculate the spectrum of the gravitational waves produced from the fluctuations of the scalar fields.

We consider the Friedmann-Robertson-Walker metric including the linear perturbation in the spatial metric ${\cal H}_{ij}$. $$ds^2
=
a^2(\tau)
\left[
  -d\tau^2 + 
  \left(
  \delta_{ij} + {\cal H}_{ij}(\tau,\vec{x})
  \right)dx^{i}dx^{j}
\right],$$ where $\tau$ is the conformal time defined as $dt = ad\tau$. The amplitude of the gravitational waves $h_{ij}$ is defined as the transverse-traceless part of ${\cal H}_{ij}$ $$h_{ij}(\tau,\vec{x})
=
a(\tau){\cal H}_{ij}^{\mathrm{TT}}(\tau,\vec{x}).$$ The superscript TT means the transverse-traceless part. The evolution of $h_{ij}$ is described by the linearized Einstein 