the vertical axis corresponds to $S_k$ defined in (\[eq:3:sk\]). []{data-label="fig:3:gw"}](./pdf/gw.pdf){width="12cm"}

With the lattice simulations, we can precisely calculate the energy density and the spectrum of the gravitational waves produced during the oscillatory phase. However, because of the cost of the simulation time, we cannot perform the lattice simulation for all parameters allowed in this model. In order to estimate the amount of the gravitational waves for various parameters, we use (\[eq:3:Omegap\]) determining the numerical constant $\alpha$ from the result of the lattice simulations. For the parameters given by (\[eq:3:parameter\]), (\[eq:3:Omegap\]) leads to $$\label{eq:3:Omegap_estimate1}
\Omega_{\mathrm{gw}}|_{\mathrm{estimate}}
\simeq
\alpha\times2.6\times10^{-4}.$$ Comparing (\[eq:3:Omegap\_Lattice1\]) with (\[eq:3:Omegap\_estimate1\]), we can determine the $\alpha$ as $$\alpha\simeq2.71\pm0.05~.$$ Thus, the amount of the gravitational waves just after the production is estimated as $$\Omega_{\mathrm{gw},p}
   \simeq \left(0.92 \pm 0.02\right)\frac{\mu M}{M_{\mathrm{pl}}^2}.$$

Present density of gravitational waves {#sec:present_density}
======================================

After the oscillatory phase, the universe is dominated by the false vacuum energy $v^4$ and the new inflation starts. After the new inflation, the universe experiences the reheating followed by the standard thermal history. Because of the cosmic expansion, the gravitational waves are red-shifted and their energy density decreases. Considering these effects, we calculate the present peak frequency of the gravitational wave spectrum $f_{\mathrm{gw},0}$, and the 