numbers, $N$ $$\begin{aligned}
  \frac{d^2 \chi}{dN^2} + \left( \frac{3}{2} + \frac{\Gamma_{\chi}}{H\left(N\right)} \right) \frac{d \chi}{d N} + \frac{m_{\chi}^2}{H(N)^2} \chi &= 0 \,\, , \label{chifeqN} \\
\frac{\rho_{r}}{dN} + 4 \rho_{r} - \Gamma_{\chi} H(N) \left( \frac{d \chi}{d N}  \right)^{2}&= 0 \,\, , \label{Nrhor} \\
 \text{where} \,\, N \equiv \ln \left[ \frac{a}{a_{\osc}} \right] = \frac{2}{3} \ln \left[ 1 + \frac{\omega \left(t - \tosc \right)}{4} \right] \,\, , \,\,  H(N) &= \frac{\omega}{6} e^{-\frac{3}{2} N} \,\, .  \label{HN} \end{aligned}$$ As can be seen in Eq.(\[Nrhor\]), $\rho_{r}$ is created from the decaying of the $\chi$ field. At the end of the inflation, $\Gamma$ is typically much smaller than the Hubble parameter. Thus, at the beginning of the phase of the inflationary oscillations, the energy loss into particles is negligible compared to the energy loss due to the expansion of the space time. $\chi$ particle production becomes effective only when the Hubble expansion rate decreases to a value comparable to $\Gamma$. Thus, $\rho_{r}$ becomes the constant value of $\Gamma_{\chi} H(N_{\osc}) (d\chi/dN |_{N=N_{\osc}})^2$ at the not too long after the Universe has come into the oscillation phase. One can define the end of the reheating when the 90% of the energy density of the $\chi$ field is converted to that 