the new inflation, the gravitational waves re-enter the horizon (horizon entry), and the energy density decreases as $a^{-4}$ again. Thus, the energy density of the gravitational waves today $\rho_{\mathrm{gw},0}$ is given by $$\rho_{\mathrm{gw},0} 
=
\left(\frac{a_0}{a_{\mathrm{H}}}\right)^{-4}
\left(\frac{a_{\mathrm{entry}}}{a_{\mathrm{exit}}}\right)^{2}
\rho_{\mathrm{gw},p}.$$ Here, $a_{\mathrm{exit}}$ is the scale factor at the horizon exit. $a_{\mathrm{entry}}$ is the scale factor at the horizon entry.

If $N_{\mathrm{new}}$ is sufficiently small, the wavelength never exceeds the horizon during the new inflation. We call this case “No horizon exit". As $\sigma_i$ becomes larger, $N_{\mathrm{new}}$ becomes larger and horizon exit occurs during the new inflation. After the new inflation, the horizon begins to grow, and the gravitational waves re-enter the horizon. If $N_{\mathrm{new}}$ is small, the horizon size at the end of the new inflation is not so large compared with the wavelength of the gravitational waves. Thus, the gravitational waves re-enter the horizon soon after the new inflation, i.e., before the reheating is completed. We call this case “Reheating time horizon entry". As $N_{\mathrm{new}}$ becomes larger, the horizon entry occurs later, i.e. during the radiation dominated era. We call this case “Radiation time horizon entry". When $N_{\mathrm{new}}$ becomes much larger, the horizon entry occurs during matter dominated era. We call this 