\rho_{\mathrm{gw}}(\tau)
&=
\frac{4\pi G}{a^4(\tau)V}
\int\frac{d^3k}{\left(2\pi\right)^3}
\sum_{ij}\\
&\hspace{1ex}\times
\left[
\left|
\int^{\tau_{f}}_{\tau_{i}}d\tau'
\cos\left(k\tau'\right)a\left(\tau'\right)T_{ij}^{TT}(\tau',\vec{k})
\right|^2
+
\left|
\int^{\tau_{f}}_{\tau_{i}}d\tau'
\sin\left(k\tau'\right)a\left(\tau'\right)T_{ij}^{TT}(\tau',\vec{k})
\right|^2
\right].
\end{split}$$ Thus, the spectrum of the gravitational waves just after the production is given by $$\label{eq:3:OmegaL}
\Omega_{\mathrm{gw}}\left(k\right)
=
\frac{1}{\rho_{\mathrm{tot}}}
\frac{d\rho_{\mathrm{gw}}(\tau_{f},k)}{d\ln k}
=
\frac{4G^2}{3\pi Va^4(\tau_{f})H^2(\tau_{f})}
S_{k}(\tau_{f}),$$ where $$\label{eq:3:sk}
\begin{split}
S_{k}(\tau_{f})
&=
k^3\int d\Omega_{k}
\sum_{ij}\\
&\times
\left[
\left|
\int^{\tau_{f}}_{\tau_{i}}d\tau'
\cos\left(k\tau'\right)a\left(\tau'\right)T_{ij}^{TT}(\tau',\vec{k})
\right|^2
+
\left|
\int^{\tau_{f}}_{\tau_{i}}d\tau'
\sin\left(k\tau'\right)a\left(\tau'\right)T_{ij}^{TT}(\tau',\vec{k})
\right|^2
\right],
\end{split}$$ and $d\Omega_{k}=d\cos\theta d\phi$.

Lattice simulation {#subsec:simulation}
------------------

The parametric resonance is a nonlinear phenomenon. Thus, in order to estimate the spectrum and energy density of produced gravitational waves precisely, we need to perform the lattice simulations to calculate the time evolution of the fluctuations of the scalar fields.

We perform the lattice simulations with three dimensional comoving box taking the cosmic expansion into account, where the scale factor is determined by the Friedmann equation. In order to resolve the oscillations of the inflaton, we set the time step $dt$ as $$dt = \frac{1}{10}\frac{1}{m_{\sigma}}<\frac{1}{m_{\sigma}}.$$ We set the step number of the time as $1500$, where the ratio of the scale factor at the end of the simulation to that at the beginning is $3.1$ [^1]. By the end of the simulation, the production of the gravitational waves 