occupation number of the scalar fields in momentum space $n_{f}$ as $$n_{f}
=
\omega_{k}\left(
\frac{|\dot{f}_{k}|^2}{\omega_{k}^2} + |f_{k}|^2
\right),~
\omega_{k}^2 \equiv \frac{k^2}{a^2}+m_{f}^2:~f\in\{\sigma,~\psi,~\phi\},$$ where $f_{k}$ is the Fourier component of $f$. Fig. \[fig:3:n\_real\] shows the time evolution of $n_f$ for the scalar fields. We can see the enhancement of the scalar fields $\sigma$ and $\psi$ by the parametric resonance. [^3]

Using the result of the lattice simulations, we calculate the spectrum of the gravitational waves. Fig. \[fig:3:gw\] shows the spectrum of the gravitational waves produced from three real scalar fields. We can see that as the fluctuations of the scalar fields increase after the first inflation, gravitational waves are produced abundantly. This shows that the enhancement of the scalar fields by the parametric resonance works as the origin of the gravitational waves. Furthermore, we perform the simulations for 30 realizations with different initial random values of the scalar field fluctuations. Using the result of these 30 realizations and (\[eq:3:OmegaL\]), we estimate the amount of the gravitational waves $\Omega_{\mathrm{gw}}|_{\mathrm{Lattice}}$ as $$\label{eq:3:Omegap_Lattice1}
\Omega_{\mathrm{gw}}|_{\mathrm{Lattice}}
=
\left(6.1\pm0.1\right)\times10^{-4}.$$

![ Time evolution of the spectrum of the gravitational waves from $t = 30[1/m_{\sigma}]$ to $t=80[1/m_{\sigma}]$ for every $10[1/m_{\sigma}]$ time interval and at the final time $t=150[1/m_{\sigma}]$ . Here, 