perform the lattice simulations and calculate the evolution of the fluctuations of the scalar fields. Using the result of the lattice simulations, we estimate the energy density and the spectrum of the gravitational waves.

Order estimate
--------------

In this subsection, we roughly calculate the ratio $\Omega_{\mathrm{gw},p}$ of the energy density of the gravitational waves to the total energy density based on the naive estimation given by Felder and Kofman  [@Dufaux:2008dn; @Felder:2006cc]. Here and hereafter the subscript $p$ means the epoch, at which gravitational waves are produced.

Through the parametric resonance, the fluctuations of the scalar fields with the typical length scale $R_{\mathrm{peak}}\simeq k_{\mathrm{peak}}^{-1}$ are enhanced most. These fluctuations produce the gravitational waves. The enhanced fluctuations have self gravitational potential energy $E_{\mathrm{gw}}$. We can write down $E_{\mathrm{gw}}$ at $t_p$ as $$E_{\mathrm{gw}}
\sim
G\frac{
\left(\rho_{\mathrm{tot},p}R_{\mathrm{peak}}^3\right)^2
}{R_{\mathrm{peak}}}
\simeq
\rho_{\mathrm{tot},p}H_{p}^2R_{\mathrm{peak}}^5,$$ where $\rho_{\mathrm{tot},p}$ is the total density and $H_{p}$ is the Hubble parameter. We approximate that the order of the energy of the gravitational waves is comparable to the self gravitational potential energy of the enhanced scalar fields $E_{\mathrm{gw}}$. Under this approximation, the energy density of the gravitational waves $\rho_{\mathrm{gw},p}$ is estimated as $$\rho_{\mathrm{gw},p}
\sim
\frac{E_{\mathrm{gw}}}{R_{\mathrm{peak}}^3}
\sim
\rho_{\mathrm{tot},p}H_{p}^2R_{\mathrm{peak}}^2.$$ Thus, the ratio of the energy density 